Teacher's Guide: Workbook 3 - JUMPMath - Common Drive

Contents

Introduction: How to Use the JUMP Workbooks and the Teacher’s Guide, by John Mighton

Introduction Appendix 1: The Structure and Design of the Workbooks

Introduction Appendix 2: JUMP Math Instructional Approaches, by Dr. Melanie Tait

Introduction: Sample Problem Solving Lesson

Mental Math

Listing of Worksheet Titles

Patterns & Algebra Teacher’s Guide Workbook 3:1

BLACKLINE MASTERS Workbook 3 - Patterns & Algebra, Part 1

Number Sense Teacher’s Guide Workbook 3:1

BLACKLINE MASTERS Workbook 3 - Number Sense, Part 1

Measurement Teacher’s Guide Workbook 3:1

BLACKLINE MASTERS Workbook 3 - Measurement, Part 1

Probability & Data Management Teacher’s Guide Workbook 3:1

BLACKLINE MASTERS Workbook 3 - Probability & Data Management, Part 1

Geometry Teacher’s Guide Workbook 3:1

BLACKLINE MASTERS Workbook 3 - Geometry, Part 1

Patterns & Algebra Teacher’s Guide Workbook 3:2

BLACKLINE MASTERS Workbook 3 - Patterns & Algebra, Part 2

Number Sense Teacher’s Guide Workbook 3:2

BLACKLINE MASTERS Workbook 3 - Number Sense, Part 2

Measurement Teacher’s Guide Workbook 3:2

BLACKLINE MASTERS Workbook 3 - Measurement, Part 2

Probability & Data Management Teacher’s Guide Workbook 3:2

BLACKLINE MASTERS Workbook 3 - Probability & Data Management, Part 2

Geometry Teacher’s Guide Workbook 3:2

BLACKLINE MASTERS Workbook 3 - Geometry, Part 2

JUMPMath

Teacher’s Guide: Workbook 3

Copyright © 2008 JUMP Math

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by

any means, electronic or mechanical, including photocopying, recording, or any information storage

and retrieval system, without written permission from the publisher, or expressly indicated on the

page with the inclusion of a copyright notice.

JUMP Math

Toronto, Ontario

www.jumpmath.org

Writers: Dr. John Mighton, Dr. Sindi Sabourin, Dr. Anna Klebanov

Consultant: Jennifer Wyatt

Cover Design: Blakeley

Special thanks to the design and layout team.

Cover Photograph: © iStockphoto.com/Michael Kemter

ISBN: 978-1-897120-38-5

Printed and bound in Canada

Introduction 1WORKBOOK 3 Copyright © 2007, JUMP Math

Sample use only - not for sale

1. An Overview of JUMP Techniques and Principles

Over the past ten years our understanding of the brain has changed dramatically. Until recently,

neurologists believed that the brain was “structurally immutable by early childhood and that its

functions and abilities were programmed by genes” (see The Brain and the Mind, J. Schwartz).

Now, however, thanks to a series of groundbreaking experiments conducted over the last

decade, scientists believe that the human brain is much more “plastic” and malleable than

anyone had previously suspected, and that it can actually “rewire” itself to repair damage or to

develop new functions. Even older or impaired brains can develop new intellectual and creative

abilities, and can change their structure and their circuitry, through rigorous cognitive training.

As Philip Ross points out in his article “The Expert Mind,” which appeared in Scientifi c American

in 2006, this fact has profound implications for education:

The preponderance of psychological evidence indicates that experts are made not born.

What is more, the demonstrated ability to turn a child quickly into an expert—in chess,

music, and a host of other subjects—sets a clear challenge before the schools. Can

educators fi nd ways to encourage the kind of effortful study that will improve their reading

and math skills? Instead of perpetually pondering the question, “Why—can’t Johnny read?”

perhaps educators should ask, “Why should there be anything in the world that he can’t

learn to do?”

Over the past ten years the JUMP program itself has gathered a great deal of evidence, through

teacher testimonials and more rigorous large-scale pilots, that mathematical abilities can be

nurtured in all students, including those who have learning disabilities or who have traditionally

struggled at school. In a JUMP pilot that took place in over twenty schools in London England

between June 2006 and May 2007, a signifi cant number of elementary students who were

not expected to pass the National exams in mathematics did very well on the exams after

being taught the JUMP curriculum for a year (some students advanced as much as fi ve grade

levels in one year). In another JUMP pilot conducted in Toronto, teachers who had used the

JUMP materials for several months were asked to rate on a scale of 1 to 5 how much they

thought they had underestimated the weakest students in their class, where 5 meant “greatly

underestimated.” Ratings were given in ten categories that included enthusiasm for math,

ability to remember number facts, ability to concentrate, willingness to ask for harder work,

and ability to keep up with stronger students. In every category, all of the teachers circled

4 or 5. (If you would like to read about these and other results of the program, see the research

section of our website.)

If you are a teacher and you believe that some of the students in your class are not capable

of learning math,

I recommend that you read The End of Ignorance: Multiplying Our Human Potential, and consult

the JUMP website (at www.jumpmath.org) for testimonials from teachers who have tried the

program and for a report on current research on the program.

You are more likely to help all your students if you teach with the following principles in mind:

IntroductionHow to Use the JUMP Workbooks and the Teacher’s Guide by

John Mighton

Copyright © 2007, JUMP Math


2 TEACHER’S GUIDE Copyright © 2007, JUMP Math


(i) If a student doesn’t understand your explanation, assume there is something lacking in your explanation, not in your student.

When a teacher leaves students behind in math, it is often because they have not looked

closely enough at the way they teach. I often make mistakes in my lessons: sometimes I will

go too fast for a student or skip steps inadvertently. I don’t consider myself a natural teacher.

I know many teachers who are more charismatic or faster on their feet than I am. But I have had

enormous success with students who were thought to be unteachable because if I happen to

leave a student behind I always ask myself: What did I do wrong in that lesson? (And I usually

fi nd that my mistake lay in neglecting one of the principles listed below.)

I am aware that teachers work under diffi cult conditions, with over-sized classes and a

growing number of responsibilities outside the classroom. None of the suggestions in this

guide are intended as criticisms of teachers, who, in my opinion, are engaged in heroic work.

I developed JUMP because I saw so many teachers struggling to teach math in large and

diverse classrooms, with training and materials that were not designed to take account of the

diffi cult conditions in those classrooms. My hope is that JUMP will make the jobs of some

teachers easier and more enjoyable.

(ii) In mathematics, it is always possible to make a step easier.

A hundred years ago, researchers in logic discovered that virtually all of the concepts used

by working mathematicians could be reduced to one of two extremely basic operations,

namely, the operation of counting or the operation of grouping objects into sets. Most people

are able to perform both of these operations before they enter kindergarten. It is surprising,

therefore, that schools have managed to make mathematics a mystery to so many students.

A tutor once told me that one of her students, a girl in Grade 4, had refused to let her teach

her how to divide. The girl said that the concept of division was much too hard for her and

she would never consent to learn it. I suggested the tutor teach division as a kind of counting

game. In the next lesson, without telling the girl she was about to learn how to divide, the tutor

wrote in succession the numbers 15 and 5. Then she asked the child to count on her fi ngers by

multiples of the second number, until she’d reached the fi rst. After the child had repeated this

operation with several other pairs of numbers, the tutor asked her to write down, in each case,

the number of fi ngers she had raised when she stopped counting. For instance,

15 5 3

As soon as the student could fi nd the answer to any such question quickly, the tutor wrote, in

each example, a division sign between the fi rst and second number, and an equal sign between

the second and third.

15 ÷ 5 = 3

The student was surprised to fi nd she had learned to divide in 10 minutes. (Of course, the tutor

later explained to the student that 15 divided by fi ve is three because you can add 5 three times

to get 15: that’s what you see when you count on your fi ngers.)

In the exercises in the JUMP workbooks, we have made an effort to break concepts and skills

into steps that students will fi nd easy to master. Fitting the full curriculum into 350 pages was

not an easy task. The worksheets are intended as models for teachers to improve upon. We

have made a serious effort to introduce skillls and concepts in small steps and in a coherent

order, so that teachers can see where they need to create extra questons for practice (the lesson

plans in this guide provide many examples of extra questions) or where they need to fi ll in a

missing step in the development of an idea (this is usually outlined in the lesson plans as well).

Introduction



(iii) With a weaker student, the second piece of information almost always drives out the fi rst.

When a teacher introduces several pieces of information at the same time, students will often,

in trying to comprehend the fi nal item, lose all memory and understanding of the material that

came before (even though they may have appeared to understand this material completely

as it was being explained). With weaker students, it is always more effi cient to introduce one

piece of information at a time.

I once observed an intern from teachers college who was trying to teach a boy in a Grade 7

remedial class how to draw mixed fractions. The boy was getting very frustrated as the intern

kept asking him to carry out several steps at the same time.

I asked the boy to simply draw a picture showing the number of whole pies in the fraction

2 1__2 . He drew and shaded two whole pies. I then asked him to draw the number of whole pies

in 3 1__2 , 4

1__2 and 5

1__2 pies. He was very excited when he completed the work I had assigned him,

and I could see that he was making more of an effort to concentrate. I asked him to draw the

whole number of pies in 2 1__4 , 2

3__4 , 3

1__4 , 4

1__4 , then in 2

1__3 , 2

2__3 , 3

1__3 pies and so on. (I started

with quarters rather than thirds because they are easier to draw.) When the boy could draw

the whole number of pies in any mixed fraction, I showed him how to draw the fractional part.

Within a few minutes he was able to draw any mixed fraction. If I hadn’t broken the skill into two

steps (i.e. drawing the number of whole pies then drawing the fractional part) and allowed him

to practice each step separately, he might never have learned the concept.

As your weaker students learn to concentrate and approach their work with real excitement

(which generally happens after several months if the early JUMP units are taught properly), you

can begin to skip steps when teaching new material, or even challenge your students to fi gure

out the steps themselves. But if students ever begin to struggle with this approach, it is best to

go back to teaching in small steps.

(iv) Before you assign work, verify that all of your students have the skills they need to complete the work.

In our school system it is assumed that some students will always be left behind in

mathematics. If a teacher is careful to break skills and concepts into steps that every student

can understand, this needn’t happen. (JUMP has demonstrated this in scores of classrooms.)

Before you assign a question from one of the JUMP workbooks you should verify that all of your

students are prepared to answer the question without your help (or with minimal help). On most

worksheets, only one or two new concepts or skills are introduced, so you should fi nd it easy

to verify that all of your students can answer the question. The worksheets are intended as fi nal

tests that you can give when you are certain all of your students understand the material.

Always give a short diagnostic quiz before you allow students to work on a worksheet. In

general, a quiz should consist of four or fi ve questions similar to the ones on the worksheet.

Quizzes needn’t count for marks but students should complete quizzes by themselves, without

talking to their neighbours (otherwise you won’t be able to verify if they know how to do the

work independently). The quizzes will help you identify which students need an extra review

before you move on. If any of your students fi nish a quiz early, assign extra questions similar

to the ones on the quiz.

If tutors are assisting in your lesson, have them walk around the class and mark the quizzes

immediately. Otherwise check the work of students who might need extra help fi rst, then

take up the answers to the quiz at the board with the entire class (or use peer tutors to help

with the marking).

Introduction



Never allow students to work ahead in the workbook on material you haven’t covered with the

class. Students who fi nish a worksheet early should be assigned bonus questions similar to the

questions on the worksheet or extension questions from this guide. Write the bonus questions

on the board (or have extra worksheets prepared and ask students to answer the questions in

their notebooks. While students are working independently on the bonus questions, you can

spend extra time with anyone who needs help.

(v) Raise the bar incrementally.

Any successes I have had with weaker students are almost entirely due to a technique I use

which is, as a teacher once said about the JUMP method, “not exactly rocket science.” When

a student has mastered a skill or concept, I simply raise the bar slightly by challenging them to

answer a question that is only incrementally more diffi cult or complex than the questions I had

previously assigned. I always make sure, when the student succeeds in meeting my challenge,

that they know I am impressed. Students become very excited when they succeed in meeting

a series of graduated challenges. And their excitement allows them to focus their attention

enough to make the leaps I have described in The End of Ignorance. As I am not a psychologist

I can’t say exactly why the method of teaching used in JUMP has such a remarkable effect on

children who have trouble learning. But I am certain that the thrill of success and the intense

mental effort required to remember complex rules, and to carry out long chains of computation

and inference, helps open new pathways in their brains.

In designing the JUMP workbooks, we have made an effort to introduce only one or two skills

per page, so you should fi nd it easy to create bonus questions: just change the numbers in

an existing question or add an extra element to a problem on a worksheet. For instance, if you

have just taught students how to add a pair of three-digit numbers, you might ask students

who fi nish early to add a pair of four- or fi ve-digit numbers. This extra work is the key to the

JUMP program. If you become excited when you assign more challenging questions, you will

fi nd that even students who previously had trouble focusing will race to fi nish their work so

they can answer a bonus question too.

(vi) Repetition and practice are essential.

Even mathematicians need constant practice to consolidate and remember skills and concepts.

The new research in cognition, which I mentioned in Appendix 1, Section (ii), shows how

important it is to build component skills before students can understand the big picture.

(vii) Praise is essential.

We’ve found the JUMP program works best when teachers give their students a great deal of

encouragement. Because the lessons are laid out in steps that any student can master and,

because students having diffi culty can get extra help from our tutors, you’ll fi nd that you won’t

be giving false encouragement. If you proceed using these steps, your students should be

doing well on all their exercises. (This is one of the reasons kids love the program so much:

for many, it’s a thrill to be doing well at math.)

In this vein, we hope that you won’t use labels such as “mild intellectual defi cit” or “slow

learner” as reasons for expecting a poor performance in math from particular children. We

haven’t observed a student yet—even among scores of remedial students—who couldn’t learn

math. When math is taught in steps, children with attention defi cits and learning disabilities

can easily succeed, and thereby develop the confi dence and cognitive abilities they need to

do well in other subjects. Rather than being the hardest subject, math can be the engine of

learning for delayed students. This is one of JUMP’s cornerstone beliefs. If you disagree with

Introduction



this tenet, please reconsider your decision to use JUMP in your classroom. Our program will

only be fully effective if you embrace the philosophy.

(viii) Make math a priority.

I’ve occasionally met teachers who believe that because they survived school without knowing

much math or without ever developing a love of the subject, they needn’t devote too much

effort to teaching math in their own classes. There are two reasons why this attitude is harmful

to students.

REASON 1: It is easier to turn a good student into a bad student in mathematics than in any

other subject: mathematical knowledge is cumulative; when students miss a step or fall behind

they are often left behind permanently. Students who fall behind in mathematics tend to suffer

throughout their academic careers and end up being cut off from many jobs and opportunities.

REASON 2: JUMP has shown that mathematics is a subject where students who have reading

delays, attention defi cits and other learning diffi culties can experience immediate success

(and the enthusiasm, confi dence and sense of focus children gain from this success can

quickly spill over into other subjects): In neglecting mathematics, a teacher neglects a tool

that has the potential to transform the lives of weaker students.

2. What JUMP Looks Like in the Classroom

While the JUMP workbooks provide a good deal of guidance for students, these books are

not designed to be used without instruction, nor are they the whole program. Teachers should

use the workbooks and accompanying lesson plans in the Teacher’s Guide to design dynamic

lessons in which students are allowed to discover and explore ideas on their own. Students

benefi t when teachers are able to use a variety of instructional approaches and when they

are willing to experiment and try new methods in their classrooms. (See the essays “JUMP

Math Instructional Approaches” and “JUMP and the Process Standards for Mathematics”

in Appendix 2 by Dr. Melanie Tait in this Introduction for more information on different

approaches you can try in your classroom).

When I use JUMP workbooks in a lesson, it is usually only at the end of the lesson. I will build a

lesson around the material on a particular worksheet by creating questions or exercises that are

similar to the ones on the worksheet. I usually write the questions or instructions for the exercise

on the board and have students work in a separate notebook. I don’t generally spend too much

time at the board though—I will teach a skill or concept to the whole class at the same time,

giving lots of hints and guidance, asking each question in several different ways, and allowing

students time to think before I solicit an answer, so that every student can put their hand up

and so that students can discover the ideas for themselves. When I have presented a concept,

I will not go on until I have assessed whether all of the students are ready to move ahead.

I will give a mini-quiz (consisting of several questions) or task so I can see exactly what

students have understood or misunderstood. I normally allow students to try questions from

the workbooks only after I have gone through several cycles of explanations (or explorations)

followed by mini-quizzes.

Before a lesson, I prepare a stock of extra bonus questions which I write on the board from time

to time during the lesson for students who fi nish their quizzes or tasks early. Many examples

of bonus questions appear in the lessons. While faster students are occupied with these

questions, I circulate around the class doing spot checks on the work of the weaker students.

The bonus questions are usually simple extensions of the work on the quiz; for

Introduction



example, if students know how to add three-digit numbers I will assign four- and fi ve-digit

numbers; if they know how to fi nd the perimeter of simple shapes I will ask them to design

a shape of their own (perhaps a letter of their name) and fi nd its perimeter; if they can round

numbers to the nearest hundreds I will ask them to round numbers to the nearest thousands or

ten thousands. You should never underestimate how excited students will become at showing

off with these kinds of bonus questions when you become excited about their successes.

The bonus questions you create should generally be simple extensions of the material on

the worksheet: if you create questions that are too hard or that require too much background

knowledge, you may have to spend time helping stronger students who should be able to work

independently, and you won’t have time to help weaker students. At times, though, you will

want to assign more challenging questions that take the concepts taught in the lesson further:

that is why we have provided extension questions in the lesson plans. Six years of in-class

implementations of JUMP have shown that a teacher can always keep faster students engaged

with extra work, without leaving weaker students behind. But if, instead of assigning bonus

and extension questions, you allow some students to work ahead of others in the workbooks,

you will never be able to build the momentum and excitement that comes when an entire class

experiences success together.

The secret to bringing an entire class along at the same pace is to use “continuous assessment.”

When students are not able to keep up in a lesson it is usually because they are lacking one

or two basic skills that are needed for that lesson, or because they are being held back by a

simple misconception that is not diffi cult to correct. For instance, students who have trouble

continuing or seeing patterns in number sequences are usually held back by a poor grasp of

subtraction facts: they usually cannot determine quickly or accurately how much greater one

term in a sequence is than the preceding term. But I have found that these students are able to

take part in lessons on sequences when I have taken a few minutes before the lesson to teach

them how to fi nd the difference between numbers by counting up on their fi ngers. (Say the

smaller number with your fi st closed. Count up to the larger number raising one fi nger at a time

as you count—the number of fi ngers you have raised when you stop counting is the difference.)

To give another example, in fi nding the perimeter of a shape drawn on a grid, weaker students

will often overlook certain sides of the shape, particularly when a side is only one unit long.

When I start a lesson on perimeter, I insist that students write the length of each side of the

shape directly on the side, so I can see which students are overlooking sides. If you watch

“The Perimeter Lesson” video, you will see that several students in Grade 6 made mistakes

when they tried to calculate the perimeters of simple shapes, because they missed some of

the sides of the shapes. But because I caught this mistake early in the lesson, these students

were able to raise their hands to answer some challenging questions at the end of the lesson.

Paying attention to the small details can make all the difference in a lesson, and will determine

whether the teacher engages the entire class or only part of the class. If teachers learn to pay

attention to these details they will begin to anticipate when students will become confused or

make mistakes. Even the strongest students will make mistakes or misunderstand instructions,

but there is very little time for teachers to pay close attention to everyone. They must plan their

lessons so that they can assess quickly and consistently what students know. Teachers need

to make an effort to spot mistakes or misunderstandings right away, or they will have to spend

a good deal of time re-teaching material. Re-teaching material rather than teaching it properly

the fi rst time is always ineffi cient. If the teacher waits too long to correct an error, mistake can

be piled on mistake so that it becomes impossible to know exactly where a student went wrong.

And students can become confused and demoralized by making repeated mistakes, losing the

level of engagement and confi dence that they need to absorb the material effi ciently. This is

Introduction



why it is so important that teachers know how to break material into steps and introduce ideas

incrementally, so they can assess their students’ knowledge and provide help at each point

in the lesson.

Teachers may wonder how they could possibly make time to assess students’ work while

teaching the lesson. The key to keeping track during the lesson of what students know is

to present ideas in steps, to assign small sets of questions or problems that test students’

understanding of each step before the next one is introduced, and to insist that students

present their answers in a way that allows the teacher to spot mistakes easily. When I assign

mini-quizzes or tasks, I don’t mark every answer of every student. If, in teaching the material on

the quiz, I have avoided overwhelming students with too many steps or too much information,

and if, by seeing how many students raise their hands in the lesson, I have verifi ed that

students understand the material, I can be confi dent that students will be able to answer the

questions on the quiz. If I have taught material properly before I assign work based on the

material, it can take a matter of seconds to check if a weaker student needs extra help, and a

matter of minutes to give students more practice with the one concept or skill that they need

to master to do well on the quiz. (Of course, if I have included too many new skills or too much

new information on the quiz then this will not be the case.) Often I will only mark the answers

of students who I think may be struggling, and then either move on or take up the quiz with the

whole class. Sometimes I let faster students mark each others’ work or even give each other

challenge questions. I try to make sure that every student, even the faster ones, gets a check

mark or comment or special bonus question at some point in the lesson.

The point of constantly assigning tasks and quizzes is not to rank students or to encourage

them to work harder by making them feel inadequate. The quizzes are an opportunity for

students to show off what they know, to become more engaged in their work by meeting

incremental challenges, and to experience the collective excitement that can sweep through

a class when students experience success together. (I usually don’t even identify the work

I assign as a “quiz.”)

The research in cognition that shows the brain is plastic also shows that the brain can’t rewire

itself or register the effects of practice or training if it is not attentive. But a child’s brain can’t

be truly attentive unless the child is confi dent and excited and believes that there is a point in

being engaged in their work. When weaker students become convinced that they cannot keep

up with the rest of the class their brains begin to work less effi ciently, as they are never attentive

enough to consolidate new skills or develop new neural pathways. That is why it is so important

to constantly assess what weaker students know, to give them the skills they need to take

part in lessons and to give them opportunities to show off by answering questions in front

of their classmates.

Engaging the entire class in lessons is not simply a matter of fairness; it is also a matter of

effi ciency. While the idea may seem counterintuitive, teachers will enable faster students to

go further if they take care of slower students. Teachers can create a real sense of excitement

about math in the classroom simply by convincing the weaker students that they can do

well at the subject. The class will cover far more material in the year and stronger students

will no longer have to hide their love of math for fear of appearing strange or different.

I would not of course recommend that teachers teach all lessons in the style I have presented.

Such lessons require a great deal of energy and children will also benefi t from working

on projects individually or in small groups and circulating between activity centres in the

classroom. Teachers should sometimes teach more open ended lessons, in which students can

take the lesson in whatever direction they are inspired to explore. And teachers will sometimes

Introduction



want to allow their students to struggle more. I recently watched a wonderful lesson at the

Institute of Child Study at OISE/UT during which the teacher pretended to be confused about

fractions. The students were also confused at fi rst, but after working their way through their

misconceptions they became more and more excited about correcting their teacher. The most

effective mix of lessons for a class will likely vary depending on the level of the class and the

talents and tastes of the teacher. But if students are allowed to work on the same material and

to experience success together as a class in many of their lessons, and if teachers are careful

to assess and help their weaker students consistently so that differences are not created or

unnecessarily emphasized, I believe that students could go much further and have more fun

at school.

Whenever I work with a class for an extended period of time, I fi nd it easier to keep the weaker

students moving along with the class by spending ten to fi fteen minutes a week working with

them in small groups. Many schools have made guided reading lessons part of the school day

so that students get extra practice at reading. Schools could also make guided math lessons

standard. Because it is so easy to teach basic skills and concepts in math, these lessons

would take on average only a few extra minutes every day, and could be given in small groups

while other students worked independently. The students to whom I gave extra help did not

feel singled out or inadequate, because they always did well on my tests and were able to

participate in class, and I always made sure these students got to answer bonus questions

in front of the class.

On the following two pages, you will fi nd an annotated lesson plan from the Grade 5 Teacher’s

Guide that shows you how you can use the plans to prepare your lessons.

Introduction



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Grade Worksheet NameWorksheet Number

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Connections

between the

material and the

real world, other

strands and

other subjects.

Extra challenge

for students who

want them with

hints or scaffolding

if required.

Students are

sometimes asked

to consolidate

learning by writing

in a journal.

Blackline Masters

refer to the extra

sheets also

available with this

Teacher’s Guide.

Hands on activities

help students

consolidate their

knowledge.

Web pages provided

when appropriate.



3. Hints for Helping Students Who Have Fallen Behind

In response to questions asked by teachers using the JUMP program, I have compiled some

suggestions for helping students who are struggling with math. I hope you fi nd the suggestions

useful. (And I hope you don’t fi nd them impractical: I know, given the realities of the teaching

profession, that it is often hard to keep your head above water.)

(i) Teach Number Facts:

It is much easier to teach students their number facts than is generally believed. In the “Mental

Math” section of the JUMP Teacher’s Guide, you will fi nd a number of effective strategies to

help students learn their number facts (see, for instance, the section “How to Learn Your Times

Tables in a Week”). After you have taught these strategies, I would recommend giving students

who need extra practice daily two-minute drills and tests until they know their facts (you can

give a student the same sheet repeatedly until they have memorized the facts on it—that way

you don’t have to do a lot of extra work preparing materials). You might also send home extra

work or, whenever possible, ask parents to help their children memorize certain facts (don’t

overload the student—you might send home one times table or half a times table per night).

Students might also quiz each other using fl ash cards. JUMP has shown that students will

memorize material more quickly if their teacher is enthusiastic about their successes, no matter

how small those successes may seem. (You might even have some kind of reward system or

acknowledgment for facts learned.)

Trying to do mathematics without knowing basic number facts is like trying to play the piano

without knowing where the notes are: there are few things you could teach your students that

will have a greater impact on their academic career than a familiarity with numbers.

(ii) Give cumulative reviews.

Even mathematicians constantly forget new material, including material they once understood

completely. (I have forgotten things I discovered myself!) Children, like mathematicians, need a

good deal of practice and frequent review in order to remember new material.

Giving reviews needn’t create a lot of extra work for you. I would recommend that, once a

month, you simply copy a selection of questions from the workbook units you have already

covered onto a single sheet and Xerox the sheet for the class. Children rarely complain

about reviewing questions they already did a month or more ago (and quite often they won’t

even remember they did those particular questions). The most you should do is change a

few numbers or change the wording of the questions slightly. If you don’t have time to mark

the review sheets individually, you can take them up with the whole class (though I would

recommend looking at the sheets of any students you think might need extra help or practice).

(iii) Make mathematical terms part of your spelling lessons and post mathematical terms in the classroom.

In some areas of math, in geometry for instance, the greatest diffi culty that students face is in

learning the terminology. If you include mathematical terms in your spelling lessons, students

will fi nd it easier to remember the terms and to communicate about their work. You might

also create a bulletin board or math wall with pictures and mathematical terms, so students

can see the terms every day.

(iv) Find fi ve minutes, wherever possible, to help weaker students in small groups.

Whenever I have taught JUMP in a classroom for an extended time, I have found that I generally

Introduction



needed to set aside fi ve minutes every few days to give extra review and preparation to the

lowest four or fi ve students in the class. (I usually teach these students in a small group while the

other students are working on other activities.) Surprisingly, this is all it takes for the majority of

students to keep up (of course, in extreme cases, it may not be enough).

I know, given current class sizes and the amount of paperwork teachers are burdened with,

that it’s very hard for teachers to fi nd extra time to devote to weaker students but, if you can fi nd

the time, you will see that it makes an enormous difference to these students and to the class in

general. (By investing a little extra time in your weaker students, you may end up saving time as

you won’t have to deal so much with the extreme split in abilities that is common in most classes,

or with the disruptive behaviour that students who have fallen behind often engage in.)

(v) Teach denser pages in the workbooks in sections.

Even in this new edition of the workbooks, where we have made an effort to improve the layout,

several pages in our workbooks are more cramped than we would have liked, and some do

not provide enough practice or preparation. If you feel a worksheet is too dense or introduces

too many skills at once, assign only two or three questions from the worksheet at a time.

Give your students extra practice before they attempt the questions on the page: you can create

questions similar to the ones on the page by just changing the numbers or by changing the

wording slightly.

(vi) Change diffi cult behaviour using success and praise.

In my experience, diffi cult children respond much more quickly to praise and success than

to criticism and threats. Of course, a teacher must be fi rm with students, and must establish clear

rules and boundaries, but I’ve found it’s generally easier to get kids to adhere to rules

and to respect others if they feel admired and successful.

I have worked with hundreds of children with attention defi cits and behavioural problems

over the past 20 years (even in the correctional system), and I have had a great deal of

success changing behaviour using a simple technique: if I encounter a student who I think might

cause problems in a class I’ll say: “You’re very smart. I’d better give you something more chal-

lenging.” Then I give the student a question that is only incrementally harder—or that only looks

harder—than the one they are working on. For instance, if a student can add three fractions

with the same denominator, I give them a question with four fractions. (I never give a challenge

to a diffi cult student unless I’m certain they can do the question.) I always make sure, when the

student succeeds in meeting my challenge, that they know I am impressed. Sometimes I even

pretend to faint (students always laugh at this) or I will say: “You got that question but you’ll

never get the next one.” Students become very excited when they succeed in meeting a series

of graduated challenges. And their excitement allows them to focus their attention and make

the leaps I have described in The Myth of Ability. (Of course you don’t have to use my exact

techniques: teachers fi nd different ways to praise their students, but I think passion is essential.)

The technique of raising the bar is very simple but it seems to work universally: I have used it

in inner-city schools, in behavioural classes and even in the detention system and I have yet to

meet a student who didn’t respond to it. Children universally enjoy exercising their minds and

showing off to a caring adult.

Although JUMP covers the traditional curriculum, the program demands a radical change in the

way teachers deliver the curriculum: JUMP is based on the idea that success is not a by-product

of learning, it is the very foundation of learning. If you aren’t willing to give diffi cult students

graduated challenges that they can succeed at, and if you aren’t willing to be excited at their

successes, then you may leave those students behind unnecessarily.

Introduction



In every worksheet, we have tried to raise the bar incrementally and to break skills into minute

steps so that the teacher can gage precisely the size of the step and the student’s readiness

to attempt a new step.

I know that in a big class it’s extremely hard to give attention to diffi cult students, but sometimes

a few fi ve-minute sessions spent giving a student a series of graduated challenges (that you

know they can succeed at) can make all the difference to the student (and to your stress levels!).

NOTE: Once students develop a sense of confi dence in math and know how to work

independently, you can sometimes allow them to struggle more with challenges: students

need to eventually learn that it’s natural to fail on occasion and that solving problems sometimes

takes a great deal of trial and error.

(vii) Isolate the problem.

If your student is failing to perform an operation correctly, try to isolate the exact point or step at

which they’re faltering. Then, rather than making the student do an entire question right from the

beginning, give them a number of questions that have been worked out to the point where they

have trouble and have them practice doing just that one step until they master it. For instance,

when performing long division with a two-digit number, students sometimes guess

a quotient that is too small:

60 is larger then 46 so

the quotient 3 is too small

3 3 3

46 198 46 198 46 198

– 138 – 138

60

One of the JUMP students was struggling with this step—even after many explanations, the

student would forget what to do after performing the subtraction. Finally, the tutor wrote down

a number of examples that had been worked out up to the subtraction and simply asked the

student to check whether the remainder was larger than the divisor and, if so, to increase the

quotient by one. The student quickly mastered this step and was then able to move on to doing

the full question with ease.

NOTE: Students will also remember an operation better if they know why it works—the lesson

plans in this guide contain exercises that will help students understand various operations.

4. Hints for Helping Students Who Finish Work Early

(i) Assign students who fi nish work early bonus questions, or extension questions from this

guide. Avoid singling out students who work on extension questions as the class geniuses,

and, as much as possible, allow all of your students to try these questions (with hints

and guidance if necessary). Students won’t generally notice or care if some students

are working on harder problems, unless you make an issue of it. Your class will go much

further, and some of your students may eventually surprise you, if you make them all

feel like they are doing impressive work. There will always be differences in ability and

motivation between children, but those differences (particularly in speed) would probably

not have much bearing on long term success in mathematics if schools were not so intent

on making differences matter. Because a child’s level of confi dence and sense of self will

largely determine what they learn, teachers can easily create artifi cial differences in children

by singling out some as superior and others as inferior. I’ve learned not to judge students

too hastily: I’ve seen many slower students outpace faster ones as soon as they were given

a little help or encouragement.

Introduction



(ii) Even the most able students make mistakes, but sometimes it’s hard to convince a stronger

student to write out the steps of a solution or calculation so you can see where they went

wrong. If a student is reluctant to show their work, I will often say “I know you’re very clever,

and you can do the steps in your head, but I can’t always keep up with you, so I need you

to help me out and show me your steps occasionally.” I’ve also said. “Because you’re

so clever, you may want to help a friend or a brother or sister with math one day, so you’ll

need to know how to explain the steps.” I’ve found that students will generally show the

steps they took to solve a problem if they know there are good reasons for doing so (and

if they know I won’t always force them to write things out).

5. How to Get Through All of the Material in the Workbooks

The JUMP workbooks and Teacher’s Guide contain a great deal of material, not only because

they provide a substantial amount of review and practice, but also because they are complete

for the Atlantic, Ontario, and Western Curricula. We decided to make the workbooks so

comprehensive because we wanted to produce books that could be used across Canada,

but also because there are gaps in the curriculum in each part of the country. To give an

example of a serious gap: if children are taught to visualize and fi nd fractions of whole numbers

at an early age they understand percentages more readily. But these skills are not emphasized

in the curriculum of any part of Canada. In the JUMP books we introduce exercises on fi nding

fractions of whole numbers in Grade 3.

We recommend that you cover as much of the material in the books as you can. But if you

need to omit some sections, you should consult the Curriculum Guide on our website, which

will tell you which sections of the book are not required for your curriculum. Even if you cannot

cover every section of the book you might occasionally assign some of the material in the

sections you omitted to students who are able to work independently and who need extra work.

(But don’t let students work ahead of the class in sections you plan to cover.)

There are several things you can do to make sure you cover as much material as possible

during the year:

(i) Make math a priority and teach it as often as you can. Don’t feel that by spending time on

math you are taking time away from other subjects: by learning to do math your weaker

students will learn to focus, see patterns, generalize rules, synthesize information and make

deductions; all essential skills for other subjects. Many teachers have reported that the

gains students made in math quickly spilled over into other subjects.

(ii) Do everything you can to build the confi dence and capture the attention of weaker students:

they will learn more effi ciently and retain more of what they learn if their minds are engaged.

(iii) Give weaker students extra coaching and practice, and verify that they have the

background knowledge they need before and during lessons: you will waste less time

re-teaching material or dealing with the behaviour that comes with failure.

(iv) If you are certain that your students understand the material on a worksheet you can

sometimes assign it for homework or independent work. Try not to send home work that

students don’t understand: it’s not fair to expect parents to teach children math, as many

parents from disadvantaged families don’t have the time or expertise required.

(v) Make sure your lessons are clear and well scaffolded: students will learn the material more

quickly and will also develop an enduring belief that they are capable of succeeding in math.

(vi) Use the strategies in the Mental Math section of this Guide to build your students’ mental

Introduction



math skills. The less students have to struggle to remember math facts, the more mental

energy they can devote to learning and exploring new ideas.

6. The Fractions Unit

To prepare your students to use this book, you should set aside 40 to 50 minutes a day for

3 weeks to teach them the material in the JUMP Fractions Unit. You may print individual copies

of the unit from the JUMP website at no charge and you can order classroom sets (at cost)

from the University of Toronto Press. NOTE: For large numbers, this option is cheaper than

photocopying. The Fractions Unit has proven to be a remarkably effective tool for instilling a

sense of confi dence and enthusiasm about mathematics in students. The unit has helped many

teachers discover a potential in their students that they might not otherwise have seen. In a

recent survey, all of the teachers who used the Fractions Unit for the fi rst time acknowledged

afterwards that they had underestimated the abilities of some of their students. (For details

of this study, see the JUMP website at www.jumpmath.org.)

The Fractions Unit is very different from the units in JUMP’s grade-specifi c material. These

units follow the Ontario curriculum quite closely. The point of the Fractions Unit, however, is

largely psychological: students who complete the unit and do well on the Advanced Fractions

test show remarkable improvements in confi dence, concentration, and numerical ability. This

has been demonstrated, even with the lowest remedial students, in a number of classrooms.

For a detailed account of the purpose of the Fractions Unit, please see the Introduction of

the Teacher’s Manual for the Fractions Unit.

7. The Scope of the JUMP Program

Do not assume that JUMP is merely a remedial program or does not teach math conceptually

simply because it provides students with adequate practice and review. If you read both parts

of the workbook from cover to cover (along with all the accompanying activities and extensions

in the Teacher’s Guide) you will see that students are expected to advance to a very high

conceptual level, in some cases beyond grade level. We would recommend that you complete

Part 1 as soon as possible, so that you have time to cover the material in Part 2, which contains

a higher proportion of questions requiring problem solving and communication.

If you would like to know more about the rationale behind the design of the workbooks, and

about the research and psychology that support the JUMP methods, please see Appendix 1.

The essay in Appendix 2 is by Dr. Melanie Tait, currently an instructor at the Ontario Institute for

Studies in Education (OISE), gives a more in-depth account of the scope of the JUMP program.

As indicated in Dr. Tait’s essay, all of the Process Standards for the curriculum (Problem

Solving, Reasoning and Proving, Refl ecting, Selecting Tools and Computational Strategies,

Connecting, Representing and Communicating) are covered in the JUMP materials.

For an example of how to deliver a guided Problem Solving lesson, see the sample perimeter

lesson given in Section 9 below.

8. Feedback

The JUMP Math workbooks and Teacher’s Guides are still works in progress and are by no

means perfect. If any part of this program doesn’t lead you to the results envisioned, we would

welcome your feedback and ideas for improvements.

Introduction



2 TEACHER’S GUIDE

As much as possible, allow your students to extend and discover ideas on their own (without pushing them

so far that they become discouraged). It is not hard to develop problem solving lessons (where your students

can make discoveries in steps) using the material on the worksheets. Here is a sample problem solving lesson

you can try with your students.

1. Warm-up

Review the notion of perimeter from the worksheets. Draw the following diagram on a grid on the board

and ask your students how they would determine the perimeter. Tell students that each edge on the shape

represents 1 unit (each edge might, for instance, represent a centimeter).

Allow your students to demonstrate their method (EXAMPLE: counting the line segments, or adding

the lengths of each side).

2. Develop the Idea

Draw some additional shapes and ask your students to copy them onto grid paper and to determine

the perimeter of each.

Check Bonus Try Again?

The perimeters of the shapes

above are 10 cm, 10 cm and

12 cm respectively.

Have your students make a

picture of a letter from their

name on graph paper by

colouring in squares. Then

ask them to find the perimeter

and record their answer in words.

Ensure students only use

vertical and horizontal edges.

Students may need to use some

kind of system to keep them

from missing sides. Suggest that

your students write the length

of the sides on the shape.

3. Go Further

Draw a simple rectangle on the board and ask students to again find the perimeter.

Add a square to the shape and ask students how the perimeter changes.

Draw the following polygons on the board and ask students to copy the four polygons on their grid paper.

Sample Problem Solving Lesson

Isolate the

problem

Immediate

assessment

Introduce

one concept

at a time

Raise the bar

incrementally

Review and

test prior

knowledge

JUMP is trying to put together several problem-solving lessons that are separate from the

worksheets. We encourage grades 5 and 6 teachers to try this sample on perimeter in their

classrooms. All teachers can use the principles stated here in their regular JUMP lessons. NO

TE

The big picture is the end goal, not the starting point. Avoid overwhelming students with

too much information. TIP

1Introduction

9. Sample Problem Solving Lesson



Part 1 Number Sense 3WORKBOOK 4

Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to add an

additional square to each polygon and calculate the perimeter again.

4. Another Step

Draw the following shape on the board and ask your students, “How can you add a square to the following

shape so the perimeter decreases?”


Have your students

demonstrate where they

added the squares and how

they found the perimeter.

Ask your students to discuss

why they think the perimeter

remains constant when

the square is added in the

corner (as in the fourth

polygon above).

• Ask your students to calculate

the greatest amount the

perimeter can increase by

when you add a single square.

• Ask them to add 2 (or 3) squares

to the shape below and examine

how the perimeter changes.

• Ask them to create a T-table

where the two columns are

labelled “Number of Squares”

in the polygon and “Perimeter”

of the polygon (see the Patterns

section for an introduction to

T-tables). Have them add more

squares and record how the

perimeter continues to change.

Ask students to draw a single

square on their grid paper

and find the perimeter (4 cm).

Then have them add a square

and find the perimeter of the

resulting rectangle. Have them

repeat this exercise a few

times and then follow the same

procedure with the original

(or bonus) questions.


Discuss with your students

why perimeter decreases when

the square is added in the

middle of the second row.

You may want to ask them

what kinds of shapes have

larger perimeters and which

have smaller perimeters.

Ask your students to add two

squares to the polygons below

and see if they can reduce

the perimeter.

Have your students try the

exercise above again with six

square-shaped pattern blocks.

Have them create the polygon

as drawn above and find where

they need to place the sixth

square by guessing and

checking (placing the square

and finding the perimeter of

the resulting polygon).

Encourage

students to

communicate

their

understanding

Guide students

in small steps

to discover

ideas for

themselves

Scaffold when

necessary

Repetition that

is not tedious,

subtle variations

keep the task

interesting

Check: Do they

understand?

Bonus appears

harder but

requires no new

explanation;

allows teacher

to attend

to struggling

students

When teaching a skill or concept to the whole class, give lots of hints and guidance. Ask each

question in several different ways, and allow students time to think before soliciting an answer, so

that every student can put their hand up and so that students can discover ideas for themselves.TIP

2

NO

YES

Introduction



4 TEACHER’S GUIDE

5. Develop the Idea

Hold up a photograph that you’ve selected and ask your students how you would go about selecting a frame

for it. What kinds of measurements would you need to know about the photograph in order to get the right

sized frame? You might also want to show your students a CD case and ask them how they would measure

the paper to create an insert for a CD/CD-ROM.

Show your students how the perimeter of a rectangle can be solved with an addition statement (EXAMPLE:

Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of equal

lines and that, because of this, we only need two numbers to find the perimeter of a given rectangle.

3 cm ? Perimeter = cm

?

4 cm

Show your students that there are two ways to find this:

a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm

b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm

Ask your students to find the perimeters of the following rectangles (not drawn to scale).

1 cm ?

?

3 cm

3 cm ?

?

5 cm

4 cm ?

?

7 cm

6. Go Further

Demonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.

Sample Problem Solving Lesson (continued)


Take up the questions (the

perimeters of the rectangles

above, from left to right, are

8 cm, 16 cm and 22 cm).

Continue creating questions

in this format for your students

and gradually increase the size

of the numbers.

Have students draw a copy of

the rectangle in a notebook and

copy the measurements onto all

four sides. Have them create an

addition statement by copying

one number at a time and then

crossing out the measurement:

1 cm 1 cm

4 cm

4 cm

4 cm

4 cm

1 cm

+1 cm

Assign small

sets of

questions or

problems that

test students’

understanding

of each step

before the

next one is

introduced

In mathematics,

it is always

possible to

make a step

easier

Using larger

numbers makes

the problem

appear harder

and builds

excitement

After several months of building confi dence and excitement, try skipping steps when teaching new

material, or even challenge your students to fi gure out the steps themselves. But if students struggle,

go back to teaching in small steps. TIP

3

Use Extensions from the lesson plans for extra challenges. Avoid singling out students who work on

extensions as geniuses and allow all of your students to try these questions (with hints if necessary). TIP

4

Make connec-

tions explicit

(Example:

between math

and the real

world, between

strands or

between math

and other

subjects)

Introduction



Part 1 Number Sense 5WORKBOOK 4

Ask your students to draw all the rectangles they can with a perimeter of 12 cm.

7. Raise the Bar

Draw the following rectangle and measurements on paper:

1 cm ? Perimeter = 6 cm

?

?

Ask students how they would calculate the length of the missing sides. After they have given some

input, explain to them how the side opposite the one measured will always have the same measurement.

Demonstrate how the given length can be subtracted twice (or multiplied by two and then subtracted)

from the perimeter. The remainder, divided by two, will be the length of each of the two remaining sides.

Draw a second rectangle and ask students to find the lengths of the missing sides using the methods

just discussed.

2 cm ? Perimeter = 14 cm

?

?



After your students have

finished, ask them whether they

were able to find one rectangle,

then two rectangles, then three

rectangles.

Ask students to find (and draw)

all the rectangles with a

perimeter of 18 cm. After they

have completed this, they can

repeat the same exercise for

rectangles of 24 cm or 36 cm.

If students find only one (or zero)

rectangles, they should be

shown a systematic method

of finding the answer and then

given the chance to practise

the original question.

On grid paper, have students

draw a pair of lines with lengths

of 1 and 2 cm each.

Ask them to draw the other three

sides of each rectangle so that

the final perimeter will be 12 cm

for each rectangle, guessing and

checking the lengths of the other

sides. Let them try this method

on one of the bonus questions

once they accomplish this.

1 cm 2 cm

Keeping your

students

excited will

help them

focus on harder

problems

Bonus

questions do

not require

much

background

knowledge

or extra

explanations,

so the stronger

students can

work independ-

ently while you

help weaker

students

Continuously

allow students

to show

off what

they learned

Scaffold

problem-solving

strategies

(Example:

systematic

search) when

necessary

Only assign questions from the workbook after going through several cycles of explanations

followed by mini-quizzes. Do not allow any student to work ahead of others in the workbooks. TIP

5

Insist that students present their answers in a way that allows you to spot mistakes easily, for

example: “I can’t always keep up with you, so I need you to show me your steps occasionally”

or “Because you’re so clever, you may want to help a friend with math one day, so you’ll need

to know how to explain the steps.”

TIP

6

Introduction



6 TEACHER’S GUIDE

8. Assessment

Draw the following diagrams of rectangles and perimeter statements, and ask students to complete the

missing measurements on each rectangle.

a) b) c)

2 cm ?

4 cm

?

Perimeter = 12 cm

? 4 cm

?

?

Perimeter = 18 cm

? ?

6 cm

?

Perimeter = 18 cm


Check that students can

calculate the length of the

sides (2 cm, 2 cm, 5 cm

and 5 cm).

Give students more problems

like above. For example:

Side = 5 cm; Perimeter = 20 cm




Be sure to raise the numbers

incrementally on bonus questions.

Give students a simple

problem to try (similar to the

first demonstration question).

1 cm

Perimeter = 8 cm

Provide them with eight

toothpicks (or a similar object)

and have them create the

rectangle and then measure

the length of each side.

Have them repeat this with

more questions.

Check Bonus

Answers for the above questions (going

clockwise from the sides given):

a) 2 cm, 4 cm

b) 3 cm, 6 cm, 3 cm

c) 5 cm, 4 cm, 5 cm

Draw a square and inform your students that

the perimeter is 20 cm. What is the length of

each side? (Answer: 5 cm.) Repeat with other

multiples of four for the perimeter.

Observe the

excitement

that can sweep

through a class

when students

experience

success

together

Continuous

assessment

is the secret

to bringing

an entire class

along at the

same pace

Use hands-on

activities with

concrete

materials to

consolidate

learning

A full size copy of this lesson can be downloaded from our website.

NO

TE

If a student is failing to perform an operation correctly, isolate the problematic step. Give them a

number of questions that have been worked out to the point where they have trouble and have them

practice doing just that one step until they master it. TIP

7Introduction



The scientifi c evidence now suggests that children are born with roughly equal potential, and that

what becomes of them is largely a matter of nurture, not nature. So how does so much potential

ultimately disappear in so many children? Why do we observe such extreme differences in

mathematical ability in students even by Grade 3? And why are the majority of adults convinced that

they are not capable of learning math? I will discuss these questions in some detail, as the answers

I will propose should help you understand the structure and design of the JUMP workbooks.

The theories of education on which current textbooks and math programs are based do not take

proper account of the conditions teachers face in real classrooms, nor do they take account of the

new psychological models that have been created in the last ten years to explain how the brain

works and how children learn. The JUMP workbooks were initially developed after thousands of

hours of observations of children (both by myself and by the teachers who helped us develop the

books) and have been refi ned repeatedly over the past six years through testing and feedback from

teachers and educational experts and also as a result of what we have learned from research in

education and psychology.

Some Features of the Workbooks and the Rationale for Those Features

(i) The workbooks provide a review of the curriculum that goes back at least two years.

In a typical elementary class, even among children who are only eight years old, an enormous

difference exists between the weakest and the strongest students. The most knowledgeable

will be able to recite their multiplication tables to twelve, while the most delayed have trouble

counting by twos. The fastest will fi nish a page of work before the slowest have found their pencils.

And the most eager will wave their hands to answer questions while the most distracted stare

vacantly into space.

This gap in knowledge, ability and motivation—which is already pronounced in Grade 3 and which

grows steadily until, by Grade 9, students must be separated into streams—makes it very diffi cult

for teachers to help all of their students.

Some educational authorities seem to believe that if children manage to discover or understand

mathematical operations or concepts they will always fi nd it easy to apply the concepts to

new situations and will be able to recall the concepts immediately, even if they haven’t had an

opportunity to think about them for a year. This assumption certainly does not refl ect my experience

as a mathematician. I discovered original (and rather elementary) algorithms in knot theory that

I mastered only after months of practice, but if you were to ask me how one of these algorithms

works now, I would have to devote several weeks of hard work to remembering the answer.

At the beginning of every school year, teachers are faced with many new students who have

either forgotten much of the previous years work or who did not succeed in learning it properly in

the fi rst place. Most textbooks will include only material that is mandated by the curriculum for a

particular grade level. They will often have a single page at the beginning of a chapter with material

the student should already have learned, perhaps titled “Do You Remember?”. The answer for

most students is “No, I don’t” or “I didn’t learn it properly the fi rst time.” The JUMP workbooks

are designed to take account of the fact that forgetting (and occasionally struggling to master

concepts) is a natural part of learning. In the fi rst workbook for each year we provide extensive

scaffolding and review that goes back at least two years.

Appendix 1

The Structure and Design of the Workbooks



Research in education suggests that if teachers begin their year by assessing the level of their

students and providing whatever review and remediation is necessary, rather than sticking rigidly

to the curriculum, they will ultimately cover far more curriculum by the end of the school year

(see for instance the papers of Dylan Wiliam, including Beyond the Black Box: Raising Standards

Through Classroom Assessment).

The review exercises in Workbook 1 are designed to give your students a solid foundation for each

of the strands in the curriculum. If you have a particularly strong class, you might assign the more

basic review exercises only to small groups of students as extra work, so you can fi nish Workbook

1 more quickly. (You should aim to fi nish Workbook 1 by January so you have time to cover the

material in Workbook 2 which, together with the material in Workbook 1 and the Teacher’s Guide,

covers the full curriculum at grade level.) Many teachers have told us that they were able to use the

review exercises in Workbook 1 (and the confi dence building exercises in our Fractions Unit, which

you can download for free from our website, and which I will describe later) to quickly close the

gap between their stronger and weaker students, leaving enough time to cover the curriculum over

the course of the year. For suggestions on how to provide enough review for your students, and

still cover all of the material in the workbooks, please see the section “How to Get Through all of

the Material in the Workbooks” (Section 5) above.

(ii) The workbooks introduce one or two concepts at a time, and provide clear explanations and rigorous guidance for students.

In “The Expert Mind,” Philip Ross argues that logical and creative abilities, intuition, and expertise

can be fostered in children through practice and rigorous instruction. But, as Ross points out, the

research in cognition shows that to become an expert in a game like chess it is not enough for

the student to play the game without guidance or instruction. The kind of training in which chess

experts engage, which includes playing small sets of moves over and over, memorizing positions,

and studying the techniques of the masters, appears to play a greater role in the development

of ability than the actual playing out of a game. That is why, according to Ross,

it is possible for enthusiasts to spend thousands of hours playing chess or golf or a musical

instrument without ever advancing beyond the amateur level and why a properly trained student

can overtake them in a relatively short time. It is interesting to note that time spent playing chess,

even in tournaments, appears to contribute less than such study to a player’s progress; the

main training value of such games is to point up weaknesses for future study.

The idea that children who spend a great deal of time playing a game or exploring a subject,

either on their own or with a little guidance, will not necessarily become good at a subject, whereas

a person who is rigorously trained can become an expert in a relatively short time, runs counter to

current educational practice.

Over the past twenty years many educational theorists have claimed that if children are allowed to

play with concrete materials, such as blocks and Cuisinaire rods and fraction strips, and to explore

ideas with a little guidance from a teacher, they can turn themselves into experts in mathematics.

According to this view, effective teachers can create conditions in their classrooms that will allow

their students to construct knowledge and make discoveries, whether on their own or with the

help of their peers. Teachers shouldn’t put too much emphasis on specifi c knowledge or skills

in a subject, nor should they fi ll up their students’ heads with facts. The content of a subject is not

as important as the way students learn the subject. A famous saying in education, which I have

heard several math consultants repeat, captures this philosophy: “The teacher should be the guide

on the side and not the sage on the stage.” This approach to teaching, which usually goes under

the name of discovery-based learning or inquiry-based learning, is now mandated in one form

or the other in curricula across Canada and the United States.

Introduction: Appendix 1



None of the ideas behind discovery-based learning are unreasonable in themselves. I believe

very strongly, for instance, that teachers should allow students to discover things independently

whenever they can. The teaching methods in JUMP have been characterized by some educators

as rote learning, perhaps because I recommend that teachers guide weaker students in small

steps until they have developed the skills and confi dence to do more independent work. But,

as I will explain later, I always encourage students to take the step themselves, and as much

as possible help them understand why they took the step. This method of teaching, which I call

guided discovery, is very different from rote learning. Even when students are capable of taking

only the smallest steps, they are still actively engaged in making discoveries and constructing their

knowledge of mathematics, and are therefore lead to understand math at a deeper conceptual

level. As students gain confi dence through their successes and as they become more engaged

in their work, I encourage them to work more independently.

In “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of

Constructivist, Discovery, Problem-Based, Experiential and Inquiry Based Teaching,” Paul Kirshner,

John Sweller and Richard Clark argue…

After half a century of advocacy associated with instruction using minimal guidance, it appears

that there is no body of research supporting the technique. In so far as there is any evidence

from controlled studies, it almost uniformly supports direct, strong instructional guidance rather

than constructivist-based minimal guidance during the instruction of novice to intermediate

learners. Even for students with considerable prior knowledge, strong guidance while learning

is most often found to be equally effective as unguided approaches. Not only is unguided

instruction normally less effective: there is also evidence that it may have negative results when

students acquire misconceptions or incomplete or disorganized knowledge.

The authors present several reasons why, based on the architecture of the brain, instruction with

minimal guidance is not likely to be effective. They argue, for instance, that unguided instruction

does not take account of the limitations of a student’s working memory: the mind can only retain

so much new information or so many component steps at one time.

The JUMP materials were developed on the assumption, now borne out by research in psychology

and education, that younger students usually need a good deal of guidance and practice to learn

mathematics. In the workbooks and Teacher’s Guides we have made an effort to break mathematical

concepts and operations into the most basic elements of perception and understanding and

to introduce ideas in a sequence of clear and logical steps. For any given topic, we present a

variety of different ways of looking at or representing the topic: for instance, in the Grades 3 to 6

workbooks, the concept of division is presented in a variety of ways: through skip counting and

repeated addition or subtraction, as the operation of forming a particular number of sets or a

particular size of set, as the inverse of multiplication, through student investigations of various

ideas (such as the meaning of the remainder), and through more formal presentations of standard

algorithms. Students are allowed to master and understand each representation fully, at the

same time as they are introduced to applications of the particular representation and connections

between the representation and other strands in the curriculum. All of the review and preparation

that students will need to understand a particular topic are presented on the worksheets or in

the Teacher’s Guide.

Although students will often fi nd it easy to work through material in the workbooks independently

(because of the clear presentation of ideas) the workbooks were not designed to be used without

instruction. We encourage teachers to create dynamic lessons based on the worksheets, in which

students discover the ideas on the worksheets on their own (guided by questions posed by the

teacher, by hints, and by their own investigations) and in which students are also allowed to show




off their knowledge (by answering questions or explaining their ideas in front of their peers, by

completing mini-quizzes and assignments which are marked or taken up immediately with the

class, and by attempting bonus and extension questions that take ideas further). The workbooks

were designed as tools for assessment or practice rather than as replacements for the lesson.

I will say more about how you can use the JUMP materials to create lessons that will engage

your students later.

(iii) The workbooks do not overwhelm students with information.

In the article “Applications and Misapplications of Cognitive Psychology to Mathematics Education”

the Nobel prize winning cognitive scientist Herb Simon makes a case against a number of ideas

that are popular in educational philosophy now, including the notion that children only really

understand concepts that they discover by themselves (rather than those they are taught by a

teacher), the notion that knowledge cannot be represented or taught symbolically, and the notion

that knowledge can only be communicated in complex learning situations. To the claim that

knowledge can always be communicated best in complex learning situations, Simon provides

contrary evidence that...

...a learner who is having diffi culty with components can easily be overwhelmed by the

processing demands of a complex task. Further, to the extent that many components are

well mastered, the student wastes much less time repeating these mastered operations

to get an opportunity to practice the few components that need additional effort.

The JUMP workbooks take account of the fact that a student can easily be overwhelmed by too

much information. Children in Canada now are required to work from math textbooks in which

a great many new terms and concepts are introduced on each page and in which the pages are

crammed with pictures and information. While these books contain some very good exercises and

explanations, all the extra bells and whistles, as well as the amount of new information on each

page can actually distract or confuse children. A study that appeared several years ago in Scientifi c

American found that the more complex a pop-up book is, the less effective it is for teaching reading.

The authors of the study speculate that this is probably true of materials for older children as well.

In Asia, where students do much better in math, the texts are extremely spare.

When designing the JUMP workbooks the JUMP staff and I deliberately made the layout very

simple. We also tried to reduce the number of words on each page and introduce only one or two

new concepts per page. There is a great deal of consistency in the design and in the way topics

are introduced at each grade level, with similar formatting in many of the questions, so that the

material seems familiar to children who have been in the program for some time. Teachers in a very

successful JUMP pilot in London were initially afraid their students wouldn’t like the workbooks

because they didn’t have fl ashy pictures or cartoons. When the pilot ended, teachers reported that

students loved the workbooks, not only because they had so much success with the material, but

also because the books didn’t make them feel like they were doing “baby-math.”

On some of the pages in the workbooks (usually entitled “Concepts in...,” “Topics in...,” or

“Problems and Puzzles”) the text is denser. We recommend that you only assign a few questions

from these pages at a time and that you read the questions to students if necessary.

(iv) The workbooks provide enough repetition and practice.

Adults think that repetition is tedious so they seldom give children the practice they need to

consolidate their understanding of skills and concepts. Anyone who has read a story to a child or

watched a TV show like Blues Clues (where the same episode is played fi ve times a week) knows

how much children love repetition. In The First Idea, a book on how intelligence evolved in humans,




Stanley Greenspan and Stuart Shanker argue that infants develop cognitive abilities by learning

to decipher subtle patterns in the voices, facial expressions and emotions of their caregivers.

Even older children love to observe and create subtle variations on a pattern endlessly. According

to Greenspan and Shanker these kinds of activities, especially when performed together with

a loving and responsive caregiver or instructor, help develop and consolidate neural pathways

in the child’s brain.

Practice doesn’t have to be painful for children, and repetition doesn’t have to involve what teachers

call “drill and kill.” If teachers are careful to introduce subtle variations into the work they assign,

if they constantly raise the bar without raising it too far, if they make learning into a game with

different twists and turns, and if they allow all students to succeed rather than creating unnecessary

hierarchies, then kids will practice with real enjoyment.

The workbooks offer your students many opportunities to practice the steps of important operations

and procedures, and to build the component skills and concepts they will need to understand

math deeply.

(v) The workbooks and Teacher’s Guides help students develop a sense of numbers and a mastery of basic operations.

It is a serious mistake to think that students who don’t know number facts can get by in mathematics

using a calculator or other aids. Students can certainly perform operations and produce numbers

on a calculator, but if they don’t have a sense of numbers, they will not be able to tell if their

answers are correct, nor can they develop a talent for solving mathematical problems. To solve

problems, students must be able to see patterns in numbers and make estimates and predictions

about numbers. A calculator cannot provide those abilities. Trying to do mathematics without

knowing basic number facts is like trying to play the piano without knowing where the notes are.

Our workbooks, lesson plans, and The Mental Math section of our Teacher’s Guide provide

strategies students can use to quickly learn their number facts and basic operations.




JUMP Math is based on the belief that with support and encouragement, all children will succeed

at math. When teachers believe that all students can succeed, they will strive to establish a

classroom environment where all students feel comfortable participating and taking risks.

In JUMP classrooms, if students don’t understand, the teacher must assume responsibility and

fi nd another way to explain the material. Three essential characteristics for a JUMP teacher are the

ability to diagnose where students “are at”, customize instruction to suit individual children, and

improvise to meet their needs.

As Glickman (1991) writes:

Effective teaching is not a set of generic practices, but instead is a set of context-driven

decisions about teaching. Effective teachers do not use the same set of practices for every

lesson… Instead, what effective teachers do is constantly refl ect about their work, observe

whether students are learning or not, and, then adjust their practice accordingly. (p. 6)

JUMP teachers, like all excellent teachers, know their students well and use a variety of creative

instructional strategies to meet their needs. JUMP teachers are constantly checking in with

students to make sure everyone is moving forward. A JUMP class is a busy and interactive

learning environment.

JUMP recognizes that teachers are skilled professionals with unique strengths and teaching

preferences. Accordingly, the JUMP Math program is designed to accommodate a number

of instructional approaches and strategies. Teachers are encouraged to vary instructional

approaches and strategies to suit the class and the needs of individual students.

JUMP’s approach is built on the belief that all children can learn when provided with the appropriate

learning conditions in the classroom. Learning is supported through explicit instruction, interaction

with the teacher and classmates, and independent learning and practice. The program is

composed of well-defi ned learning objectives organized into smaller, sequentially organized units.

Generally, units consist of discrete topics which all students begin together. Students who do not

satisfactorily complete a topic are given additional instruction until they succeed. Students who

master the topic early engage in enrichment activities until the entire class can progress together.

In a JUMP classroom, the teacher employs a variety of instructional techniques, with frequent and

specifi c feedback using diagnostic and formative assessment. Students require numerous feedback

loops, based on small units of well-defi ned, appropriately-sequenced outcomes. Teachers assess

student progress in a variety of ways and adjust their programs accordingly.

JUMP lessons are often dynamic: as soon as the teacher has explained or demonstrated a concept

or operation, students are allowed to ‘show off’ their understanding through scaffolded tasks and

quizzes that can be checked individually or taken up with the whole class. Students enjoy being

able to apply their knowledge and they benefi t from the immediate feedback. This way of teaching

allows the teacher to assess what students know before moving on.

Appendix 2

JUMP Math Instructional Approaches by

Dr. Melanie Tait



Fundamentals of JUMP Math Instruction

There are several fundamental principles that guide effective JUMP instruction:

ESTABLISHING A SAFE ENVIRONMENT: Teachers must create a safe environment where students

can feel challenged without feeling threatened. This is perhaps especially important in math

classrooms. Math anxiety can have a negative effect on children’s (and teachers’) self-confi dence,

enjoyment of math and motivation to learn math. (Tobias, 1980). JUMP is founded on the notion

that given the right kind and amount of support and encouragement, all children, except perhaps

those with severe brain damage, are able to learn mathematics.

The Fractions Unit, for example, which is a wonderful way to begin the JUMP program, is designed

to improve confi dence, concentration and numerical ability while making math fun and interesting.

JUMP teachers demonstrate, through their attitude to mathematics and their students, that it

is important to persevere to solve problems and that people only learn through making errors.

Patience, praise, encouragement and positive feedback are essential parts of the program.

ELICITING AND ENCOURAGING PARTICIPATION: Teachers need to elicit participation in order

to assess learning and to ensure that every student feels their contributions are valid and valued.

Some possible ways to elicit and encourage participation include:

• Share the expectation that all students will participate.

• Remind everyone that all questions, answers and suggestions will be respected.

• Ask if students understand before proceeding with the lesson.

• Ask for volunteers at times and call on specifi c students at other times.

• Ask for a contribution from someone who has not yet spoken.

• Encourage quiet students (EXAMPLE: direct questions, pre-arrange questions).

• Offer challenging and thought-provoking ideas for discussion.

• Use open-ended questions. (What do you think about…? Why do you think it is important to…?)

• Plan interactive activities (EXAMPLE: small-group discussions, solving problems in groups).

• Give students time to think before they answer.

• Rephrase your question. Use different wording give an example, or different examples.

• Provide hints.

• Show approval for student ideas (EXAMPLE: positive comments, praise for trying).

• Answer questions in a meaningful way.

• Incorporate student ideas into lessons.

MAKING CONNECTIONS EXPLICIT: Connecting new concepts to real life, other subject areas

or other mathematical ideas may help students relate to the content and engage in the lesson.

Seeing relationships helps students to understand mathematical concepts on a deeper level

and to appreciate that mathematics is more than a set of isolated skills and concepts but rather

something relevant and useful. JUMP explicitly make links between math skills and concepts

from the different strands and ensures that all prerequisite knowledge is reviewed or retaught

before going on to new material.

There are many ways to involve children in meaningful and relevant mathematics activities. Baking

for a class fundraiser will require application of measurement skills and money concepts. Keeping

track of the statistics in the NHL or World Cup calls for data management skills. To introduce

a lesson on perimeter, students could be asked what they would need to know if they were

responsible for installing a fence to go around the school playground and then actually measure

the perimeter of the yard before working with standard algorithms. Links between math and other

subject areas, such as the arts, social studies and science, can easily be made. Children’s literature




can also provide a lifelike context for math learning. Books provide children with experiences that

they might otherwise not have, and many lend themselves beautifully to mathematical activities.

JUMP materials also highlight the relevance of mathematics in careers and the media.

DIAGNOSING AND RESPONDING: Once students are interested and paying attention,

teachers need to determine where students “are at” in their comprehension and application

of mathematics concepts. Core JUMP lessons begin with diagnostic checks of understanding.

The teacher uses questions, discussion, student demonstrations on the board or diagnostic

pencil and paper quizzes to verify understanding and determine the entry point for the lesson.

Nothing should be assumed—for instance, before learning to fi nd the perimeter of a rectangle,

students must be able to add a sequence of numbers.

Gaps in understanding can only be addressed if they are identifi ed. The teacher can continue to

work with the others individually or in small groups by “buying time” through assigning interesting

“bonus” questions which “raise the bar” incrementally. For example, if the teacher is verifying that

students can add two-digit numbers with regrouping, those students who demonstrate competence

with this skill can solve three-digit addition questions. This allows for one-on-one or small group

tutorial time during the course of whole class lessons and is an important feature of a JUMP lesson.

Two characteristics of a good JUMP teacher are the abilities to improvise and customize to meet

the needs of students. The students and their learning are more important than the lesson plan.

Responding to the results of diagnostic questioning in an appropriate and timely way is crucial to

the success of subsequent lessons. On occasion, this will mean that the teacher will need to go

back to a previously presented concept for review or additional practice before moving forward.

BREAKING CONCEPTS DOWN INTO DISCRETE SEGMENTS: To successfully use the JUMP

method, teachers must break down new concepts and skills into small, sequential parts. The

JUMP teacher demonstrates the concept or skill, explaining it as she goes and inviting student

participation as appropriate. (“What should go here?” “What does this mean?” “Do you think

this is right?” “How else could you do this?”) As the lesson progresses, she continues to

constantly assesses whether concepts or procedures have been understood and then revisits

the instruction using different words, examples or questions to help those who do not understand

while providing “bonus” questions which extend the ideas in a manageable way for those who

are ready to tackle them.

GUIDING DISCOVERY: Guided discovery is another important element of the JUMP program.

By using a variety of questioning styles, examples and activities, teachers lead students to

understanding. Perhaps more importantly, though, teachers need to be able to simplify processes

and procedures so that students are able to move forward from their current level towards

discovering the pattern, rule or generalization. Guided discovery works best in a safe classroom

environment because once students begin to trust the teacher and feel confi dent in their ability

to progress, their attention and behaviour improve, they are more likely to take risks and their

perseverance increases. The guided discovery strategy can be used with an entire class or with

a small group or individual.

According to Mayer (2003), who believes that guided discovery tends to result in better long term

retention and transfer of understanding and skills, guided discovery both encourages learners

to search actively for how to apply rules and makes sure that the learner comes into contact with

the rule to be learned. Teachers must give students appropriate and timely guidance, but must

also have enough patience to let the learning process develop. Flexible thinking, a variety of readily

available strategies and approaches, a willingness and ability to simplify problems and patience

are valuable attributes for JUMP teachers.




ASSIGNING INDEPENDENT PRACTICE: Most JUMP lessons end with independent practice in

the form of assigned questions in the workbooks, problems to complete in the student’s math

notebook or homework. This follow-up practice is an important component of the JUMP program.

Independent practice helps students to retain and reinforce newly learned material as well as giving

the teacher another way to assess learning and retention.

Instructional Strategies and Approaches

JUMP lessons lend themselves well to a variety of instructional strategies and approaches.

Teachers are encouraged to develop and use these and other strategies as needed to respond

to the needs of their students.

EXPLICIT TEACHING is a core teaching strategy in a JUMP classroom. Topics and content are

broken down into small parts and taught individually in a logical order. The teacher directs the

learning, providing explanations, demonstration and modeling the skills and behaviours needed for

success. Student listening and attention are important. Explicit teaching involves setting the stage

for the lesson, telling students what they will be doing, showing them how to do it and guiding their

application of the new learning through multiple opportunities for practice until independence is

attained. This approach is modeled on our professional development videos.

EXPLANATIONS AND DEMONSTRATIONS are an important part of explicit teaching and key

in JUMP instruction. The teachers explains the rule, procedure or process and then demonstrates

how it is applied through examples and modeling. The demonstration provides the link between

knowing about the rule to being able to use the rule. Research has shown that demonstrations

are most effective when learners are able to follow them clearly and when brief explanations and

discussion occur during the demonstration (Arends, 1998).

CONCEPT FORMATION enables students to develop and refi ne their ability to recall and

discriminate between key ideas, to see commonalities and identify relationships, to formulate

concepts and generalizations, to explain how they have organized data, and to present evidence

to support their organization of the data involved.

CONCEPT ATTAINMENT is an indirect instruction strategy that used a structured inquiry

procedure. Students fi gure out the attributes of a group or category that has been given to them

by the teacher. To do so, they compare or contrast examples that contain the attributes of the

concept with those that do not.

INTERACTIVE INSTRUCTION relies heavily on discussion and sharing among participants.

As is illustrated in videos of JUMP classrooms, they are highly interactive learning environments

and rely on various groupings, including whole class, small groups or pairs.

COOPERATIVE LEARNING is a useful interactive instructional strategy. Students work in

groups which are carefully structured and monitored by the teacher. Specifi c work goals, time

allotments, roles and sharing techniques are set by the teacher in order to ensure that all students

are included and engaged. Think-pair-share, for example, is a strategy designed to increase

classroom participation and “think” time while helping students clarify their thinking by explaining

it to a partner. It is easy to use on the spur of the moment and in large class settings. Cooperative

learning strategies can be used in a variety of ways in the JUMP classroom. Some examples of

cooperative learning activities that work well in the JUMP classroom:

1. Have students turn to their partner and compare their answers to a problem.

2. Ask students to write a math problem, solve it, and then exchange problems with a partner.

They check each other’s work and talk about it.




3. As an introduction to graphing, show the class pictures of different kinds of graphs from

newspapers, magazines and other sources on the overhead projector. Before beginning

to work in groups, the teacher reviews role expectations and what brainstorming means

(giving as many ideas as possible with all ideas being acceptable).

Students are assigned to groups of four. Each person in the group has a role. The recorder

writes down the thoughts of the group, the reporter shares the group’s ideas with the class,

the timekeeper makes sure the group is on task and completes their work on time, and the

encourages compliments group members on their participation and contributions. Groups

are given three minutes to do their work and paper on which to record it. Students brainstorm

different places in their live that they see graphs. The recorder writes down the ideas. The

timekeeper makes sure the task is completed within three minutes. The encourager thanks

group members for participating and compliments them on their contributions. The reporter

shares the group’s list with the whole class.

The teacher creates a master list and facilitates a discussion about graphs and their purpose.

4. Have students play the 2-D sorting games in the Teacher’s Guide individually (SEE: page 243),

using their own sorting rules. Next, each student shares his sorting rule with a partner. The

partners discuss their rules and come to some agreement about their criteria. They then share

their ideas with another pair of partners. Work is done when everyone in the group has agreed

on the sorting rule and can explain it to the teacher.

When using a cooperative learning approach, the teacher should verify that all students have

understood the material before moving on. The JUMP worksheets can be used to assess what

students have learned after a cooperative lesson.

Tips about cooperative learning strategies can be found at:

http://olc.spsd.sk.ca/DE/PD/ instr/ strats/coop/index.html

INDEPENDENT PRACTICE is an important part of the JUMP program. Following instruction,

students are assigned practice questions from the student workbooks to consolidate their skills.

Teachers can check the assigned work to make sure students have understood the lesson and

are ready to move forward. Alternatively, if they have not understood the lesson, the teacher

can decide how to reteach the content in a different way. It is suggested that JUMP questions

be assigned to be completed at the end of the lesson or for homework.

Specifi c Instructional Techniques

WAIT TIME: Wait time is a key element of JUMP instruction. It is the time between asking the

question and soliciting a response. Wait time gives students a chance to think about their

answer and leads to longer and clearer explanations. It is particularly helpful for more timid

students, those who are slower to process information and students who are learning English

as a second language.

Studies about the benefi ts of increasing wait time to three seconds or longer confi rm that

there are increases in student participation, better quality of responses, better overall classroom

performance, more questions asked by students and more frequent, unsolicited contributions.

Teachers who increase their wait time tend to ask a greater variety of questions, are more likely

to modify their instruction to accommodate students’ comments and questions and demonstrate

higher expectations for their students’ success.




Increasing wait time has two apparent benefi ts for student learning (Tobin, 1987). First, it allows

students more time to process and be actively engaged with the subject matter. Second, it appears

to change the nature of teacher-student discussions and interactions.

QUESTIONING: As in any classroom, questioning is an important instructional skill in a JUMP

classroom. Strategic questioning helps teachers assess student learning, improves involvement

and can help students deepen their understanding. In a JUMP lesson, questions are initially used

to diagnosis levels of understanding. As the lesson progresses, the teacher uses questions to

break down concepts and skills into smaller steps as needed, guiding student understanding

incrementally. Questions can be used to elicit specifi c pieces of information or to stimulate thought

and creativity. Socratic questions are useful to extend thinking and promote communication.

They might begin with phrases like “Why do you think that?”, “How do you know that is true?”,

“Is there another way of explaining that?”, “Could you give me an example?” (From http://set.lanl.

gov/programs/CIF/Resource/Handouts/SocSampl.htm)

When questioning is used well:

• a high degree of student participation occurs as questions are widely distributed;

• an appropriate mix of low and high level cognitive questions is used;

• student understanding is increased;

• student thinking is stimulated, directed, and extended;

• feedback and appropriate reinforcement occur;

• students’ critical thinking abilities are honed; and,

• student creativity is fostered.

The teacher should begin by obtaining the attention of the students before the question is asked.

The question should be addressed to the entire class before a specifi c student is asked to respond.

Volunteers and non-volunteers should be called upon to answer, and the teacher should encourage

students to speak to the whole class when responding. However, the teacher must be sensitive to

each student’s willingness to speak publicly and never put a student on the spot.

Good questions should be carefully planned, clearly stated, and to the point in order to achieve

specifi c objectives. Teacher understanding of questioning technique, wait time, and levels of

questions is essential. Teachers should also understand that asking and responding to questions

is viewed differently by different cultures. The teacher must be sensitive to the cultural needs of the

students and aware of the effects of his or her own cultural perspective in questioning. In addition,

teachers should realize that direct questioning might not be an appropriate technique for all

students. (From http://olc.spsd.sk.ca/DE/PD/instr/questioning.html)

SCAFFOLDING is the guidance, support and assistance a teacher or more competent learner

provides to students that allows students to gain skill and understanding. It extends the range

of what students would be able to do independently and is only used when needed. Scaffolding

is a basic JUMP instructional skill that is ingrained both in the materials and in the lesson format.

Scaffolding involves several steps:

1. Task defi nition—what is the specifi c objective?

2. Establish a reasonable sequence

3. Model performance—demonstrate the learning strategy or skill while thinking aloud,

explaining, answering one’s own questions

4. Provide prompts, cures, hints, links, partial solutions, guides and structures; ask

leading questions; make connections obvious




5. Withdraw when the student is able to work independently

Demonstrating, explaining and questioning are all examples of scaffolding. As with questioning and

other instructional strategies, scaffolding must be customized to suit individual students or groups

of students. Pairing students and using cooperative learning strategies are two ways to provide

scaffolding for students. Scaffolding needs to be tailored to meet the needs of specifi c students—

it is intended to help the student move closer to being able to complete the task independently.

Scaffolding is demonstrated in the JUMP Guided Lessons and on the professional development

video clips.

DRILL AND PRACTICE is an instructional technique that helps students learn the building blocks

for more meaningful learning. Students are given opportunities to drill and practice their basic math

facts throughout the program. Teachers are encouraged to assign homework to help students

consolidate their skills and to provide information on student understanding for program planning.

There are many web sites designed to help students learn and practice math facts (for a partial list,

see Technology Links later in this essay).

Groupings in the JUMP Math Classroom

WHOLE CLASS INSTRUCTION: JUMP lessons usually begin and end with a whole class grouping.

The teacher sets the scene for the lesson with examples, questions or connections to previous

lessons, other subject areas or real life experiences. At the end of the lesson, there is an opportunity

to summarize what was covered, assign independent practice or set up the next lesson. As

reported in Education for All: Report of the Expert Panel on Literacy and Numeracy Instruction

(2005), research has shown that whole class instruction in mathematics is effective when both

procedural skill and conceptual knowledge are explicitly targeted for instruction, and this type of

instruction improves outcomes for children across ability and grade levels (Fuchs et al., 2002).

INDIVIDUAL WORK: There are several ways in which students may work as individuals during a

JUMP lesson. For diagnostic and formative purposes, students are often asked to solve a problem

or demonstrate a skill in their notebooks. If the need arises, students who need extra help are

briefl y supported individually by the teacher. Another routine part of JUMP lessons is the completion

of independent practice based on the content of the lesson, either as a continuation of the lesson

or as homework. There is an acknowledgement in the JUMP program that in order to properly

determine student levels, some individual work and assessment is necessary.

PAIRS ACTIVITIES: Students are often encouraged to work with a partner using the Think-Pair-

Share cooperative learning strategy. This enables them to share their thinking and discuss the

concept or skill they are learning with a fellow learner. Peer support is also encouraged when

appropriate. Students consolidate their skills when they are required to explain their reasoning

to someone else.

SMALL GROUP WORK: Students work in small groups to solve problems, play games and discuss

their work. They may also work in small tutorial groups on occasion when several students have

similar questions or diffi culties. Structured cooperative learning groups promote the participation

of all students and encourage mutual support. Cooperative group responsibility and structural

guidelines are very important to the success of this strategy. Students must understand that they

are responsible for their own work and the work of the group as a whole. The group is only suc-

cessful if everyone understands. Students must be willing to help if a group member asks for it or

needs it. Another strategy is to allow small groups of students to ask the teacher questions only

when everyone in the group has the same question. This encourages those who understand some




aspect of the problem or skill to teach the others in the group. A useful place to learn more about

cooperative learning and its benefi ts for students, including a self-guided tutorial and links to

other sites, is:

http://olc.spsd.sk.ca/DE/PD/instr/strats/coop/index.html

Both pairs activities and small group work promote communication about mathematics. Through

communication with their classmates, students are able to refl ect upon and clarify their ideas,

consolidate skills and deepen their understanding.

TUTORIAL GROUPS are structured by the teacher to support students who have similar needs or

interests. When the teacher identifi es a group of students who all need more practice or enrichment

on a particular skill, she can work with that group separately from the rest of the class in a focused

way to support their learning. This can often be accomplished in just a few minutes while others

are working independently or at another time, such as during the lunch period or after school if the

teacher’s time permits.

Classroom Management

Classroom management is an important skill for all teachers. Teaching new material requires

attentiveness. The teacher’s responsibility is to make sure everyone is following the lesson

and respectful of others’ contributions. Eye contact, paying attention, taking turns, listening,

participating and celebrating effort and success are all important aspects of a well-managed

JUMP classroom. In classrooms where JUMP has been implemented, teachers often report that

students are engaged and focused on the lessons and practice materials.

A useful and interesting reference about classroom management is Classroom Teacher’s Survival

Guide: Practical Strategies, Management Techniques, and Reproducibles for New and Experienced

Teacher (Partin, R. L., 2005).

Technology Links

WEB SITES: There are a number of excellent web sites that support mathematical learning and

problem-solving, as well as sites to help teachers plan interesting and creative lessons. An excellent

user-friendly source of information and ideas for using the internet in teaching mathematics is

Mathematics on the Internet: A Resource for K–12 Teachers by J. A. Ameis. This guide includes help

locating resources, planning lessons, engaging students in problem-solving and communication,

as well as links to professional development in the areas of assessment, collaboration, and gender,

multi-cultural, and special needs concerns.

VIRTUAL MANIPULATIVES: An excellent article about using virtual manipulatives to support

student learning can be found at http://my.nctm.org/eresources/view_media.asp?article_id=1902.

This article explains the difference between different kinds of manipulative sites available on the

internet, the advantages of using virtual manipulatives in the classroom, and questions to help

teachers assess different sites.

MATH CENTRAL: http://mathcentral.uregina.ca/

This bilingual site, provided by the University of Regina, offers a number components for teachers,

including resource sharing, lesson planning, teaching ideas, Teacher Talk, Quandaries and

Queries, and Monthly Problems.




NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS: http://illuminations.nctm.org/

This very rich site contains resources for teachers and students, including activities organized

by grade and links to other excellent sites.

THE MATH FORUM: http://www.mathforum.org/

This site provides resources for both teachers and students, including lessons, puzzles,

problems and links to other valuable sites.

THE NATIONAL LIBRARY OF VIRTUAL MANIPULATIVES: http://nlvm.usu.edu/en/nav/vlibrary.html

This site provides numerous engaging and useful activities for various grade levels.

MATH FACTS: http://home.indy.rr.com/lrobinson/mathfacts/mathfacts.html

This site provides practice on math facts as well as links to other mathematics sites for

students and teachers.

A+ MATH: http://www.aplusmath.com/

This site was developed to help students practice their math skills interactively. There are a

variety of games and activities.

CENSUS AT SCHOOL: http://www19.statcan.ca/r000_e.htm

Census at School is an international online project that engages students from Grades 4 to 12 in

statistical enquiry. Students discover how to use and interpret data about themselves as part of their

classroom learning in math, social sciences or information technology. They also learn about the

importance of the national census in providing essential information for planning education, health,

transportation and many other services.

ENVIRONMENT CANADA: www.ec.gc.ca

Bilingual; weather and environmental information.

STOCK MARKET: www.globeinvestor.com

Up-to-the-minute Canadian stock market research and information.

STATISTICS CANADA: http://www.statscan.ca

Canadian statistical data on a variety of topics, with useful information for teachers.

JUMP and the Process Standards for Mathematics

PROBLEM SOLVING: Like many math programs, problem solving is the basis for JUMP Math.

What distinguishes JUMP Math’s approach is the way that problem solving is taught and practiced.

Prerequisite skills are identifi ed and reviewed or retaught before students begin new sorts of

problems. They are led through the stages of solving problems step by step in a logical and

sequential way by the teacher who models each step. Students are given lots of practice before

they tackle problems independently. Problem-solving strategies and activities are found throughout

both the JUMP workbooks and Teacher’s Guide.

REASONING AND PROVING: When students explain their reasoning to others, they are clarifying

their own thinking as well as helping others to clarify theirs. As an integral part of the problem

solving process, students must be able to explain their reasoning in a variety of ways, orally or on

paper, using words, a picture, chart or model. In JUMP lessons, the teacher might ask students

to explain to the whole class, to their small group or to a partner. In the workbooks, students

frequently explain their work using words, diagrams, or pictures.




REFLECTING: Students in JUMP classrooms are frequently asked to share how they solved a

problem and to consider how others have solved it. They do this in the whole class setting, in small

groups and pairs and individually, orally and in writing.

SELECTING TOOLS AND COMPUTATIONAL STRATEGIES: Students in JUMP classrooms are

encouraged to use a variety of methods to work with numbers and solve problems, including

pencil and paper calculations, mental computations, estimation, calculators, models, drawings and

manipulative materials. There is a specifi c section on Mental Math in the Teacher’s Guide.

To complete JUMP activities and solve problems, students must use tools (ruler, protractor,

calculator), concrete materials and models (2- & 3-D shapes, base-ten materials for example),

charts (hundreds chart, times tables), pictures, and other tools.

CONNECTING: Connecting new concepts to real life, other subject areas or other mathematical

ideas may help students relate to the content and engage in the lesson. Seeing relationships helps

students to understand mathematical concepts on a deeper level and to appreciate that mathematics

is more than a set of isolated skills and concepts but rather something relevant and useful.

Whenever new skills or concepts are introduced, JUMP teachers review or reteach the prerequisite

skills necessary to move forward. This often entails explicitly making the links between and among

the fi ve strands in the curriculum. (Please see the JUMP Math: Teacher’s Manual for the Fraction

Unit—Second Edition).

There are many ways to involve children in meaningful and relevant mathematics activities. Baking

for a class fundraiser will require application of measurement skills and money concepts. Keeping

track of statistics in the NHL or World Cup calls for data management skills. To introduce a lesson

on perimeter, students could be asked what they would need to know if they were responsible

for installing a fence to go around the school playground and then actually measure the perimeter

of the yard before working with standard algorithms.

REPRESENTING: The JUMP program includes numerous opportunities to represent mathematical

ideas and relationships in a variety of ways. JUMP teachers explicitly teach and model mathematical

notation, conventions and representations. As children learn and practice new concepts and skills,

they are asked to represent their thinking and their work in different ways.

COMMUNICATING: Students in JUMP classrooms are encouraged to communicate frequently

with the teacher and each other. Oral participation is a key component of the program and JUMP

classrooms are typically highly interactive. Using a variety of questioning techniques, cooperative

learning strategies and wait time, teachers ensure that all students are participating. Teachers

model strategies while explaining their thinking out loud, teach appropriate symbols and vocabulary

to facilitate written communication, encourage talk about the problem-solving process and

encourage students to seek clarifi cation or ask for help when they are unsure or do not understand.

Throughout the materials, students are asked to communicate their answers to problems using

words, mathematical symbols, pictures, concrete materials or abstract models. Mathematics itself is

a kind of language, with its own rules and grammatical structures, and math teaching and activities

should help students become fl uent in the language of mathematics.




Works Cited

1. Arends, R. Learning to Teach. New York: McGraw-Hill, 1998.

2. Fuchs, L. S., D. Fuchs, L. Yazdian and S.R. Powell. “Enhancing First-Grade Children’s

Mathematical Development with Peer-Assisted Learning Strategies.” School Psychology

Review 31.4 (2002): 569-583.

3. Glickman, C. “Pretending Not to Know What We Know.” Educational Leadership 48.8

(1991): 4-10.

4. Mayer, R. E. Learning and Instruction. Upper Saddle River: Prentice Hall, 2003.

5. Tobias, Sheila. Overcoming Math Anxiety. Boston: Houghton Miffl in Company, 1980.

6. Tobin, Kenneth. “The Role of Wait Time in Higher Cognitive Level Learning.”

Review of Educational Research 57 Spring 1987: 69-95.


Sample Problem Solving Lesson

page 1

Teacher’s Guide - Introduction

As much as possible, allow your students to extend and discover ideas on their own (without pushing

them so far that they become discouraged). It is not hard to develop problem solving lessons (where your

students can make discoveries in steps) using the material on the worksheets. Here is a sample problem

solving lesson you can try with your students.

1. Warm-up

Review the notion of perimeter from the worksheets. Draw the following diagram on a grid on the

board and ask your students how they would determine the perimeter. Tell students that each edge

on the shape represents 1 unit (each edge might, for instance, represent a centimeter).

Allow your students to demonstrate their method (e.g., counting the line segments, or adding the

lengths of each side).

2. Develop the Idea

Draw some additional shapes and ask your students to copy them onto grid paper and to determine

the perimeter of each.


The perimeters of the shapes above

are 10 cm, 10 cm and 12 cm

respectively.

Have your students make a picture of a

letter from their name on graph paper

by colouring in squares. Then ask them

to find the perimeter and record their

answer in words. Ensure students only

use vertical and horizontal edges.

Students may need to use some kind

of system to keep them from missing

sides. Suggest that your students write

the length of the sides on the shape.

3. Go Further

Draw a simple rectangle on the board and ask students to again find the perimeter.

Add a square to the shape and ask students how the perimeter changes.

Draw the following polygons on the board and ask students to copy the four polygons on their

grid paper.


page 2


Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to

add an additional square to each polygon and calculate the perimeter again.


Have your students demonstrate where

they added the squares and how they

found the perimeter.

Ask your students to discuss why they

think the perimeter remains constant

when the square is added in the corner

(as in the fourth polygon above).

� Ask your students to calculate the

greatest amount the perimeter can

increase by when you add a single

square.

� Ask them to add 2 (or 3) squares to

the shape below and examine how

the perimeter changes.

� Ask them to create a T-table where

the two columns are labelled

“Number of Squares” in the polygon

and “Perimeter” of the polygon (see

the Patterns section for an

introduction to T-tables). Have them

add more squares and record how

the perimeter continues to change.

Ask students to draw a single square

on their grid paper and find the

perimeter (4 cm). Then have them add

a square and find the perimeter of the

resulting rectangle. Have them repeat

this exercise a few times and then

follow the same procedure with the

original (or bonus) questions.

4. Another Step

Draw the following shape on the board and ask your students, “How can you add a square to the

following shape so the perimeter decreases?”


Discuss with your students why

perimeter decreases when the square

is added in the middle of the second

row. You may want to ask them what

kinds of shapes have larger perimeters

and which have smaller perimeters.

Ask your students to add two squares

to the polygons below and see if they

can reduce the perimeter.

Have your students try the exercise

above again with six square-shaped

pattern blocks. Have them create the

polygon as drawn above and find

where they need to place the sixth

square by guessing and checking

(placing the square and finding the

perimeter of the resulting polygon).


page 3


5. Develop the Idea

Hold up a photograph that you’ve selected and ask your students how you would go about selecting

a frame for it. What kinds of measurements would you need to know about the photograph in order to

get the right sized frame? You might also want to show your students a CD case and ask them how

they would measure the paper to create an insert for a CD/CD-ROM.

Show your students how the perimeter of a rectangle can be solved with an addition statement (e.g.,

Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of

equal lines and that, because of this, we only need two numbers to find the perimeter of a given

rectangle.

Perimeter = ___ cm

Show your students that there are two ways to find this:

a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm

b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm

Ask your students to find the perimeters of the following rectangles (not drawn to scale).


Take up the questions (the perimeters

of the rectangles above, from left to

right, are 8 cm, 16 cm and 22 cm).

Continue creating questions in this

format for your students and gradually

increase the size of the numbers.

Have students draw a copy of the

rectangle in a notebook and copy the

measurements onto all four sides.

Have them create an addition

statement by copying one number at a

time and then crossing out the

measurement:

6. Go Further

Demonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.

?

3 cm ?

4 cm

?

1 cm ?

3 cm

?

3 cm ?

5 cm

?

4 cm ?

7 cm

4 cm

4 cm

1 cm 1 cm

4 cm

4 cm

1 cm

+ 1 cm


page 4


Ask your students to draw all the rectangles they can with a perimeter of 12 cm.


After your students have finished, ask

them whether they were able to find

one rectangle, then two rectangles,

then three rectangles.

Ask students to find (and draw) all the

rectangles with a perimeter of 18 cm.

After they have completed this, they

can repeat the same exercise for

rectangles of 24 cm or 36 cm.

If students find only one (or zero)

rectangles, they should be shown a

systematic method of finding the

answer and then given the chance to

practise the original question.

On grid paper, have students draw a

pair of lines with lengths of 1 and 2 cm

each.

Ask them to draw the other three sides

of each rectangle so that the final

perimeter will be 12 cm for each

rectangle, guessing and checking the

lengths of the other sides. Let them try

this method on one of the bonus

questions once they accomplish this.

7. Raise the Bar

Draw the following rectangle and measurements on paper:

Perimeter = 6 cm

Ask students how they would calculate the length of the missing sides. After they have given some

input, explain to them how the side opposite the one measured will always have the same

measurement. Demonstrate how the given length can be subtracted twice (or multiplied by two and

then subtracted) from the perimeter. The remainder, divided by two, will be the length of each of the

two remaining sides.

Draw a second rectangle and ask students to find the lengths of the missing sides using the methods

just discussed.

Perimeter = 14 cm

?

1 cm ?

?

?

2 cm ?

?

1 cm 2 cm


page 5



Check that students can calculate the

length of the sides (2 cm, 2 cm, 5 cm

and 5 cm).

Give students more problems like

above.

For example:

� Side = 5 cm; Perimeter = 20 cm




Be sure to raise the numbers

incrementally on bonus questions.

Give students a simple problem to try

(similar to the first demonstration

question).

1 cm Perimeter = 8 cm

Provide them with eight toothpicks

(or a similar object) and have them

create the rectangle and then measure

the length of each side. Have them

repeat this with more questions.

8. Assessment

Draw the following diagrams of rectangles and perimeter statements, and ask students to complete

the missing measurements on each rectangle.

a)

Perimeter = 12 cm

b)

Perimeter = 18 cm

c)

Perimeter = 18 cm

Check Bonus

Answers for the above questions (going clockwise from the

sides given):

a) 2 cm, 4 cm

b) 3 cm, 6 cm, 3 cm

c) 5 cm, 4 cm, 5 cm

Draw a square and inform your students that the perimeter

is 20 cm. What is the length of each side? (Answer: 5 cm.)

Repeat with other multiples of four for the perimeter.

4 cm

2 cm ?

?

6 cm

? ?

?

?

? 4 cm

?

Mental Math 1WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math


Teacher

If any of your students don’t know their addition and subtraction facts, teach them to add and

subtract using their fi ngers by the methods taught below. You should also reinforce basic facts

using drills, games and fl ash cards. There are mental math strategies that make addition and

subtraction easier: some effective strategies are taught in the next section. (Until your students

know all their facts, allow them to add and subtract on their fi ngers when necessary.)

1. Add:

a) 5 + 2 = b) 3 + 2 = c) 6 + 2 = d) 9 + 2 =

e) 2 + 4 = f) 2 + 7 = g) 5 + 3 = h) 6 + 3 =

i) 11 + 4 = j) 3 + 9 = k) 7 + 3 = l) 14 + 4 =

m) 21 + 5 = n) 32 + 3 = o) 4 + 56 = p) 39 + 4 =

2. Subtract:

a) 7 – 5 = b) 8 – 6 = c) 5 – 3 = d) 5 – 2 =

e) 9 – 6 = f) 10 – 5 = g) 11 – 7 = h) 17 – 14 =

i) 33 – 31 = j) 27 – 24 = k) 43 – 39 = l) 62 – 58 =

Teacher

To prepare for the next section, teach your students to add 1 to any number mentally (by counting

forward by 1 in their head) and to subtract 1 from any number (by counting backward by 1).

Mental MathAddition and Subtraction

To ADD 4 + 8, Grace says the greater number (8) with her fi st closed. She counts up from 8, raising one

fi nger at a time. She stops when she has raised the number of fi ngers equal to the lesser number (4):

She said “12” when she raised her 4th fi nger, so: 4 + 8 = 12

To SUBTRACT 9 – 5, Grace says the lesser number (5) with her fi st closed. She counts up from 5 raising

one fi nger at a time. She stops when she says the greater number (9):

She has raised 4 fi ngers when she stopped, so: 9 – 5 = 4



Teacher

Students who don’t know how to add, subtract or estimate readily are at a great disadvantage

in mathematics. Students who have trouble memorizing addition and subtraction facts can still

learn to mentally add and subtract numbers in a short time if they are given daily practice in

a few basic skills.

SKILL 1: Adding 2 to an Even Number

This skill has been broken down into a number of sub-skills. After teaching each sub-skill, you

should give your students a short diagnostic quiz to verify that they have learned the skill. I have

included sample quizzes for Skills 1 to 4.

i) Naming the next one-digit even number:

Numbers that have ones digit 0, 2, 4, 6 or 8 are called the even numbers. Using drills or

games, teach your students to say the sequence of one-digit even numbers without hesitation.

Ask students to imagine the sequence going on in a circle so that the next number after 8 is 0

(0, 2, 4, 6, 8, 0, 2, 4, 6, 8… ) Then play the following game: name a number in the sequence

and ask your students to give the next number. Don’t move on until all of your students have

mastered the game.

ii) Naming the next greatest two-digit even number:

CASE 1: Numbers that end in 0, 2, 4 or 6

Write an even two-digit number that ends in 0, 2, 4 or 6 on the board. Ask your students to name

the next greatest even number. Students should recognize that if a number ends in 0, then the

next even number ends in 2; if it ends in 4 then the next even number ends in 6, etc. For instance,

the number 54 has ones digit 4: so the next greatest even number will have ones digit 6.

CASE 2: Numbers that end in 8

Write the number 58 on the board. Ask students to name the next greatest even number.

Remind your students that even numbers must end in 0, 2, 4, 6, or 8. But 50, 52, 54 and 56

are all less than 58 so the next greatest even number is 60. Your students should see that an

even number ending in 8 is always followed by an even number ending in 0 (with a tens digit

that is one higher).

iii) Adding 2 to an even number:

Point out to your students that adding 2 to any even number is equivalent to fi nding the

next even number: EXAMPLE: 46 + 2 = 48, 48 + 2 = 50, etc. Knowing this, your students

can easily add 2 to any even number.

QU

IZ

Name the next greatest even number:

a) 52 : b) 64 : c) 36 : d) 22 : e) 80 :

QU

IZ

Name the next greatest even number:

a) 58 : b) 68 : c) 38 : d) 48 : e) 78 :

Addition and Subtraction



SKILL 2: Subtracting 2 from an Even Number

i) Finding the preceding one-digit even number:

Name a one-digit even number and ask your students to give the preceding number in the

sequence. For instance, the number that comes before 4 is 2 and the number that comes

before 0 is 8. (REMEMBER: the sequence is circular.)

ii) Finding the preceding two-digit number:


Write a two-digit number that ends in 2, 4, 6 or 8 on the board. Ask students to name the

preceding even number. Students should recognize that if a number ends in 2, then the

preceding even number ends in 0; if it ends in 4 then the preceding even number ends

in 2, etc. For instance, the number 78 has ones digit 8 so the preceding even number has

ones digit 6.


Write the number 80 on the board and ask your students to name the preceding even number.

Students should recognize that if an even number ends in 0 then the preceding even number

ends in 8 (but the ones digit is one less). So the even number that comes before 80 is 78.

ii) Subtracting 2 from an even number:

Point out to your students that subtracting 2 from an even number is equivalent to fi nding the

preceding even number: EXAMPLE: 48 – 2 = 46, 46 – 2 = 44, etc.

QU

IZ

Subtract:

a) 58 – 2 = b) 24 – 2 = c) 36 – 2 = d) 42 – 2 = e) 60 – 2 =

QU

IZ

Name the preceding even number:

a) 40 : b) 60 : c) 80 : d) 50 : e) 30 :

QU

IZ

Add:

a) 26 + 2 = b) 82 + 2 = c) 40 + 2 = d) 58 + 2 = e) 34 + 2 =

QU

IZ

Name the preceding even number:

a) 48 : b) 26 : c) 34 : d) 62 : e) 78 :




SKILL 3: Adding 2 to an Odd Number

i) Naming the next one-digit odd number:

Numbers that have ones digit 1, 3, 5, 7, and 9 are called the odd numbers. Using drills or

games, teach your students to say the sequence of one-digit odd numbers without hesitation.

Ask students to imagine the sequence going on in a circle so that the next number after 9 is 1

(1, 3, 5, 7, 9, 1, 3, 5, 7, 9…). Then play the following game: name a number in the sequence

and ask you students to give the next number. Don’t move on until all of your students have

mastered the game.

ii) Naming the next greatest two-digit odd number:


Write an odd two-digit number that ends in 1, 3, 5, or 7 on the board. Ask you students to

name the next greatest odd number. Students should recognize that if a number ends in 1,

then the next even number ends in 3; if it ends in 3 then the next even number ends in 5, etc.

For instance, the number 35 has ones digit 5: so the next greatest even number will have

ones digit 7.


Write the number 59 on the board. Ask students to name the next greatest number. Remind

your students that odd numbers must end in 1, 3, 5, 7, or 9. But 51, 53, 55, and 57 are all

less than 59. The next greatest odd number is 61. Your students should see that an odd

number ending in 9 is always followed by an odd number ending in 1 (with a tens digit that

is one higher).

iii) Adding 2 to an odd number:

Point out to your students that adding 2 to any odd number is equivalent to fi nding the next odd

number: EXAMPLE: 47 + 2 = 49, 49 + 2 = 51, etc. Knowing this, your students can easily add

2 to any odd number.

QU

IZ

Name the next greatest odd number:

a) 59 : b) 69 : c) 39 : d) 49 : e) 79 :

QU

IZ

Add:

a) 27 + 2 = b) 83 + 2 = c) 41 + 2 = d) 59 + 2 = e) 35 + 2 =

QU

IZ

Name the next greatest odd number:

a) 51 : b) 65 : c) 37 : d) 23 : e) 87 :




SKILL 4: Subtracting 2 from an Odd Number

i) Finding the preceding one-digit odd number:

Name a one-digit even number and ask your students to give the preceding number in the

sequence. For instance, the number that comes before 3 is 1 and the number that comes

before 1 is 9. (REMEMBER: the sequence is circular.)

ii) Finding the preceding odd two-digit number:


Write a two-digit number that ends in 3, 5, 7 or 9 on the board. Ask students to name the

preceding even number. Students should recognize that if a number ends in 3, then the

preceding odd number ends in 1; if it ends in 5 then the preceding odd number ends in 3,

etc. For instance, the number 79 has ones digit 9, so the preceding even number has

ones digit 7.


Write the number 81 on the board and ask your students to name the preceding odd number.

Students should recognize that if an odd number ends in 1 then the preceding odd number

ends in 9 (but the ones digit is one less). So the odd number that comes before 81 is 79.

iii) Subtracting 2 from an odd number:

Point out to your students that subtracting 2 from an odd number is equivalent to fi nding the

preceding even number: EXAMPLE: 49 – 2 = 47, 47 – 2 = 45, etc.

SKILLS 5 and 6

Once your students can add and subtract the numbers 1 and 2, then they can easily

add and subtract the number 3: Add 3 to a number by fi rst adding 2, then 1 (EXAMPLE:

35 + 3 = 35 + 2 + 1). Subtract 3 from a number by subtracting 2, then subtracting 1

(EXAMPLE: 35 – 3 = 35 – 2 – 1).

QU

IZ

Subtract:

a) 59 – 2 = b) 25 – 2 = c) 37 – 2 = d) 43 – 2 = e) 61 – 2 =

QU

IZ

Name the preceding odd number:

a) 41 : b) 61 : c) 81 : d) 51 : e) 31 :

QU

IZ

Name the preceding odd number:

a) 49 : b) 27 : c) 35 : d) 63 : e) 79 :





NOTE: All of the addition and subtraction tricks you teach your students should be reinforced

with drills, fl ashcards and tests. Eventually students should memorize their addition and subtraction

facts and shouldn’t have to rely on the mental math tricks. One of the greatest gifts you can give

your students is to teach them their number facts.

SKILLS 7 and 8

Add 4 to a number by adding 2 twice (EXAMPLE: 51 + 4 = 51 + 2 + 2). Subtract 4 from a number

by subtracting 2 twice (EXAMPLE: 51 – 4 = 51 – 2 – 2).

SKILLS 9 and 10

Add 5 to a number by adding 4 then 1. Subtract 5 by subtracting 4 then 1.

SKILL 11

Students can add pairs of identical numbers by doubling (EXAMPLE: 6 + 6 = 2 × 6). Students

should either memorize the 2 times table or they should double numbers by counting on

their fi ngers by 2s.

Add a pair of numbers that differ by 1 by rewriting the larger number as 1 plus the smaller

number (then use doubling to fi nd the sum): EXAMPLE: 6 + 7 = 6 + 6 + 1 = 12 + 1 = 13;

7 + 8 = 7 + 7 + 1 = 14 + 1 = 15, etc.

SKILLS 12, 13 and 14

Add a one-digit number to 10 by simply replacing the zero in 10 by the one-digit number:

EXAMPLE: 10 + 7 = 17.

Add 10 to any two-digit number by simply increasing the tens digit of the two-digit number by 1:

EXAMPLE: 53 + 10 = 63.

Add a pair of two-digit numbers (with no carrying) by adding the ones digits of the numbers

and then the tens digits: EXAMPLE: 23 + 64 = 87.

SKILLS 15 and 16

To add 9 to a one-digit number, subtract 1 from the number and then add 10: EXAMPLE:

9 + 6 = 10 + 5 = 15; 9 + 7 = 10 + 6 = 16, etc. (Essentially, the student simply has to subtract 1

from the number and then stick a 1 in front of the result.)

To add 8 to a one-digit number, subtract 2 from the number and add 10: EXAMPLE:

8 + 6 = 10 + 4 = 14; 8 + 7 = 10 + 5 = 15, etc.

SKILLS 17 and 18

To subtract a pair of multiples of ten, simply subtract the tens digits and add a zero for the

ones digit: EXAMPLE: 70 – 50 = 20.

To subtract a pair of two-digit numbers (without carrying or regrouping), subtract the ones digit

from the ones digit and the tens digit from the tens digit: EXAMPLE: 57 – 34 = 23.



Mental MathFurther Strategies

Further Mental Math Strategies

1. Your students should be able to explain how to use the strategies of “rounding the subtrahend

(EXAMPLE: the number you are subtracting) up to the nearest multiple of ten.”

EXAMPLES:

a) 37 – 19 = 37 – 20 + 1

b) 64 – 28 = 64 – 30 + 2

c) 65 – 46 = 65 – 50 + 4

PRACTICE QUESTIONS:

a) 27 – 17 = 27 – + d) 84 – 57 = 84 – +

b) 52 – 36 = 52 – + e) 61 – 29 = 61 – +

c) 76 – 49 = 76 – + f) 42 – 18 = 42 – +

NOTE: This strategy works well with numbers that end in 6, 7, 8 or 9.

2. Your students should be able to explain how to subtract by thinking of adding.

EXAMPLES:

a) 62 – 45 = 5 + 12 = 17

b) 46 – 23 = 3 + 20 = 23

c) 73 – 17 = 6 + 50 = 56

PRACTICE QUESTIONS:

a) 88 – 36 = + = d) 74 – 28 = + =

b) 58 – 21 = + = e) 93 – 64 = + =

c) 43 – 17 = + = f) 82 – 71 = + =

3. Your students should be able to explain how to “use doubles.”

EXAMPLES:

a) 12 – 6 = 6 6 + 6 = 12

b) 8 – 4 = 4

PRACTICE QUESTIONS:

a) 6 – 3 = d) 18 – 9 =

b) 10 – 5 = e) 16 – 8 =

c) 14 – 7 = f) 20 – 10 =

Subtrahend

Subtrahend rounded to the nearest tens

You must add 1 because 20 is 1 greater than 19

You must add 2 because 30 is 2 greater than 28

Count by ones from 45 to the nearest tens (50)

Count from 50 until you reach the fi rst number (62)

The sum of counting up to the nearest ten

and the original number is the difference

What method did we use here?

Minuend

If you add the subtrahend to itself and

the sum is equal to the minuend then the

subtrahend is the same as the difference

Same value as

minuend

Minuend plus itself



NOTE TO TEACHER: Teaching the material on these worksheets may take several lessons.

Students will need more practice than is provided on these pages. These pages are intended

as a test to be given when you are certain your students have learned the materials fully.

Teacher

Teach SKILLS 1, 2, 3 AND 4 as outlined on pages 2 through 5 before you allow your students to

answer Questions 1 through 12:

1. Name the even number that comes after the number. Answer in the blank provided:

a) 32 b) 46 c) 14 d) 92 e) 56

f) 30 g) 84 h) 60 i) 72 j) 24

2. Name the even number that comes after the number:

a) 28 b) 18 c) 78 d) 38 e) 68

3. Add. REMEMBER: adding 2 to an even number is the same as fi nding the next even number:

a) 42 + 2 = b) 76 + 2 = c) 28 + 2 = d) 16 + 2 =

e) 68 + 2 = f) 12 + 2 = g) 36 + 2 = h) 90 + 2 =

i) 70 + 2 = j) 24 + 2 = k) 66 + 2 = l) 52 + 2 =

4. Name the even number that comes before the number:

a) 38 b) 42 c) 56 d) 72 e) 98

f) 48 g) 16 h) 22 i) 66 j) 14

5. Name the even number that comes before the number:

a) 30 b) 70 c) 60 d) 10 e) 80

6. Subtract. REMEMBER: subtracting 2 from an even number is the same as fi nding the preceding

even number:

a) 46 – 2 = b) 86 – 2 = c) 90 – 2 = d) 14 – 2 =

e) 54 – 2 = f) 72 – 2 = g) 12 – 2 = h) 56 – 2 =

i) 32 – 2 = j) 40 – 2 = k) 60 – 2 = l) 26 – 2 =

7. Name the odd number that comes after the number:

a) 37 b) 51 c) 63 d) 75 e) 17

f) 61 g) 43 h) 81 i) 23 j) 95

8. Name the odd number that comes after the number:

a) 69 b) 29 c) 9 d) 79 e) 59

Mental MathExercises



9. Add. REMEMBER: Adding 2 to an odd number is the same as fi nding the next odd number:

a) 25 + 2 = b) 31 + 2 = c) 47 + 2 = d) 33 + 2 =

e) 39 + 2 = f) 91 + 2 = g) 5 + 2 = h) 89 + 2 =

i) 11 + 2 = j) 65 + 2 = k) 29 + 2 = l) 17 + 2 =

10. Name the odd number that comes before the number:

a) 39 b) 43 c) 57 d) 17 e) 99

f) 13 g) 85 h) 79 i) 65 j) 77

11. Name the odd number that comes before the number:

a) 21 b) 41 c) 11 d) 91 e) 51

12. Subtract. REMEMBER: Subtracting 2 from an odd number is the same as fi nding the preceding

odd number.

a) 47 – 2 = b) 85 – 2 = c) 91 – 2 = d) 15 – 2 =

e) 51 – 2 = f) 73 – 2 = g) 11 – 2 = h) 59 – 2 =

i) 31 – 2 = j) 43 – 2 = k) 7 – 2 = l) 25 – 2 =

Teacher

Teach SKILLS 5 AND 6 as outlined on pages 5 and 6 before you allow your students to answer

Questions 13 and 14:

13. Add 3 to the number by adding 2, then adding 1 (EXAMPLE: 35 + 3 = 35 + 2 + 1):

a) 23 + 3 = b) 36 + 3 = c) 29 + 3 = d) 16 + 3 =

e) 67 + 3 = f) 12 + 3 = g) 35 + 3 = h) 90 + 3 =

i) 78 + 3 = j) 24 + 3 = k) 6 + 3 = l) 59 + 3 =

14. Subtract 3 from the number by subtracting 2, then subtracting 1 (EXAMPLE: 35 – 3 = 35 – 2 – 1):

a) 46 – 3 = b) 87 – 3 = c) 99 – 3 = d) 14 – 3 =

e) 8 – 3 = f) 72 – 3 = g) 12 – 3 = h) 57 – 3 =

i) 32 – 3 = j) 40 – 3 = k) 60 – 3 = l) 28 – 3 =

15. Fred has 49 stamps. He gives 2 stamps away. How many stamps does he have left?

16. There are 25 minnows in a tank. Alice adds 3 more to the tank. How many minnows are

now in the tank?

Exercises



Exercises

Teacher

Teach SKILLS 7 AND 8 as outlined on page 6.

17. Add 4 to the number by adding 2 twice (EXAMPLE: 51 + 4 = 51 + 2 + 2):

a) 42 + 4 = b) 76 + 4 = c) 27 + 4 = d) 17 + 4 =

e) 68 + 4 = f) 11 + 4 = g) 35 + 4 = h) 8 + 4 =

i) 72 + 4 = j) 23 + 4 = k) 60 + 4 = l) 59 + 4 =

18. Subtract 4 from the number by subtracting 2 twice (EXAMPLE: 26 – 4 = 26 – 2 – 2):

a) 46 – 4 = b) 86 – 4 = c) 91 – 4 = d) 15 – 4 =

e) 53 – 4 = f) 9 – 4 = g) 13 – 4 = h) 57 – 4 =

i) 40 – 4 = j) 88 – 4 = k) 69 – 4 = l) 31 – 4 =

Teacher


19. Add 5 to the number by adding 4, then adding 1 (or add 2 twice, then add 1):

a) 84 + 5 = b) 27 + 5 = c) 31 + 5 = d) 44 + 5 =

e) 63 + 5 = f) 92 + 5 = g) 14 + 5 = h) 16 + 5 =

i) 9 + 5 = j) 81 + 5 = k) 51 + 5 = l) 28 + 5 =

20. Subtract 5 from the number by subtracting 4, then subtracting 1 (or subtract 2 twice,

then subtract 1):

a) 48 – 5 = b) 86 – 5 = c) 55 – 5 = d) 69 – 5 =

e) 30 – 5 = f) 13 – 5 = g) 92 – 5 = h) 77 – 5 =

i) 45 – 5 = j) 24 – 5 = k) 91 – 5 = l) 8 – 5 =

Teacher

Teach SKILLS 11 as outlined on page 6.

21. Add:

a) 6 + 6 = b) 7 + 7 = c) 8 + 8 =

d) 5 + 5 = e) 4 + 4 = f) 9 + 9 =

22. Add by thinking of the larger number as a sum of two smaller numbers:

a) 6 + 7 = 6 + 6 + 1 b) 7 + 8 = c) 6 + 8 =

d) 4 + 5 = e) 5 + 7 = f) 8 + 9 =



Teacher

Teach SKILLS 12, 13 AND 14 as outlined on page 6.

23. a) 10 + 3 = b) 10 + 7 = c) 5 + 10 = d) 10 + 1 =

e) 9 + 10 = f) 10 + 4 = g) 10 + 8 = h) 10 + 2 =

24. a) 10 + 20 = b) 40 + 10 = c) 10 + 80 = d) 10 + 50 =

e) 30 + 10 = f) 10 + 60 = g) 10 + 10 = h) 70 + 10 =

25. a) 10 + 25 = b) 10 + 67 = c) 10 + 31 = d) 10 + 82 =

e) 10 + 43 = f) 10 + 51 = g) 10 + 68 = h) 10 + 21 =

i) 10 + 11 = j) 10 + 19 = k) 10 + 44 = l) 10 + 88 =

26. a) 20 + 30 = b) 40 + 20 = c) 30 + 30 = d) 50 + 30 =

e) 20 + 50 = f) 40 + 40 = g) 50 + 40 = h) 40 + 30 =

i) 60 + 30 = j) 20 + 60 = k) 20 + 70 = l) 60 + 40 =

27. a) 20 + 23 = b) 32 + 24 = c) 51 + 12 = d) 12 + 67 =

e) 83 + 14 = f) 65 + 24 = g) 41 + 43 = h) 70 + 27 =

i) 31 + 61 = j) 54 + 33 = k) 28 + 31 = l) 42 + 55 =

Teacher


28. a) 9 + 3 = b) 9 + 7 = c) 6 + 9 = d) 4 + 9 =

e) 9 + 9 = f) 5 + 9 = g) 9 + 2 = h) 9 + 8 =

29. a) 8 + 2 = b) 8 + 6 = c) 8 + 7 = d) 4 + 8 =

e) 5 + 8 = f) 8 + 3 = g) 9 + 8 = h) 8 + 8 =

Teacher


30. a) 40 – 10 = b) 50 – 10 = c) 70 – 10 = d) 20 – 10 =

e) 40 – 20 = f) 60 – 30 = g) 40 – 30 = h) 60 – 50 =

31. a) 57 – 34 = b) 43 – 12 = c) 62 – 21 = d) 59 – 36 =

e) 87 – 63 = f) 95 – 62 = g) 35 – 10 = h) 17 – 8 =

Mental MathPractice Sheet



Multiples of Ten

STUDENT: In the exercises below, you will learn several ways to use multiples of ten in mental

addition or subtraction.

1. Warm up:

a) 536 + 100 = b) 816 + 10 = c) 124 + 5 = d) 540 + 200 =

e) 234 + 30 = f) 345 + 300 = g) 236 – 30 = h) 442 – 20 =

i) 970 – 70 = j) 542 – 400 = k) 160 + 50 = l) 756 + 40 =

2. Write the second number in expanded form and add or subtract one digit at a time.

The fi rst one is done for you:

a) 564 + 215 = =

b) 445 + 343 = =

c) 234 + 214 = =

3. Add or subtract mentally (one digit at a time):

a) 547 + 312 = b) 578 – 314 = c) 845 – 454 =

4. Use the tricks you’ve just learned:

a) 845 + 91 = b) 456 + 298 = c) 100 – 84 = d) 1000 – 846 =

Mental MathAdvanced

EX

AM

PL

E 3

Sometimes in subtraction, it helps to think of a multiple of ten as a sum of 1 and a number consisting

entirely of 9s (EXAMPLE: 100 = 1 + 99; 1000 = 1 + 999). You never have to borrow or exchange

when you are subtracting from a number consisting entirely of 9s.

100 – 43 = 1 + 99 – 43 = 1 + 56 = 57

1000 – 543 = 1 + 999 – 543 = 1 + 456 = 457

Do the subtraction, using 99 instead of 100,

and then add 1 to your answer.

EX

AM

PL

E 2 If one of the numbers you are adding or subtracting is close to a number with a multiple of ten, add

the multiple of ten and then add or subtract an adjustment factor:

645 + 99 = 645 + 100 – 1 = 745 – 1 = 744

856 + 42 = 856 + 40 + 2 = 896 + 2 = 898

EX

AM

PL

E 1 542 + 214 = 542 + 200 + 10 + 4 = 742 + 10 + 4 = 752 + 4 = 756

827 – 314 = 827 – 300 – 10 – 4 = 527 – 10 – 4 = 517 – 4 = 713

Sometimes you will need to carry:

545 + 172 = 545 + 100 + 70 + 2 = 645 + 70 + 2 = 715 + 2 = 717

564 + 200 + 10 + 5 779



Purpose

If students know the pairs of one-digit numbers that add up to particular target numbers,

they will be able to mentally break sums into easier sums.

EXAMPLE: As it is easy to add any one-digit number to 10, you can add a sum more readily

if you can decompose numbers in the sum into pairs that add to ten.

7 + 5 = 7 + 3 + 2 = 10 + 2 = 12

To help students remember pairs of numbers that add up to a given target number I developed

a variation of “Go Fish” that I have found very effective.

The Game

Pick any target number and remove all the cards with value greater than or equal to the target

number out of the deck. In what follows, I will assume that the target number is 10, so you would

take all the tens and face cards out of the deck (Aces count as one).

The dealer gives each player 6 cards. If a player has any pairs of cards that add to 10 they are

allowed to place these pairs on the table before play begins.

Player 1 selects one of the cards in his or her hand and asks the Player 2 for a card that adds to

10 with the chosen card. For instance, if Player 1’s card is a 3, they may ask the Player 2 for a 7.

If Player 2 has the requested card, the fi rst player takes it and lays it down along with the card from

their hand. The fi rst player may then ask for another card. If the Player 2 doesn’t have the requested

card they say: “Go fi sh,” and the Player 1 must pick up a card from the top of the deck. (If this card

adds to 10 with a card in the player’s hand they may lay down the pair right away). It is then Player

2’s turn to ask for a card.

Play ends when one player lays down all of their cards. Players receive 4 points for laying down all

of their cards fi rst and 1 point for each pair they have laid down.

NOTE: With weaker students I would recommend you start with pairs of numbers that add to 5.

Take all cards with value greater than 4 out of the deck. Each player should be dealt only 4 cards

to start with.

I have worked with several students who have had a great deal of trouble sorting their cards and

fi nding pairs that add to a target number. I’ve found the following exercise helps:

Give your student only three cards; two of which add to the target number. Ask the student

to fi nd the pair that add to the target number. After the student has mastered this step with

3 cards repeat the exercise with 4 cards, then 5 cards, and so on.

NOTE: You can also give your student a list of pairs that add to the target number. As the

student gets used to the game, gradually remove pairs from the list so that the student learns

the pairs by memory.

Mental MathGame: Modifi ed Go Fish

These numbers add to 10.



Student Name

Can Add

1 to

Any Number

Can Subtract

1 from

Any Number

Can Add

2 to

Any Number

Can Subtract

2 from

Any Number

Knows

All Pairs that

Add to 5

Can Double

1-Digit

Numbers

Mental MathChecklist #1




Student Name

Can Add

Near Doubles.

EXAMPLE:

6 + 7 =

6 + 6 + 1

Can Add a

1-Digit Number

to Any

Multiple of 10.

EXAMPLE:

30 + 6 = 36

Can Add Any

1-Digit Number

to a Number

Ending in 9.

EXAMPLE:

29 + 7 =

30 + 6 = 36

Can Add 1-Digit

Numbers by

“Breaking” them

Apart into Pairs

that Add to 10.

EXAMPLE:

7 + 5 =

7 + 3 + 2 =

10 + 2

Can Subtract

Any Multiple of

10 from 100.

EXAMPLE:

100 – 40 = 60



Student Name

Can Mentally

Make Change

from a Dollar.

SEE: Workbook

Sheets on Money.

Can Mentally

Add Any

Pair of 1-Digit

Numbers.

Can Mentally

Subtract Any

Pair of 1-Digit

Numbers.

Student Can Multiply

and Count by:


2 3 4 5 6 7 8 9



Teacher

Trying to do math without knowing your times tables is like trying to play the piano without knowing

the location of the notes on the keyboard. Your students will have diffi culty seeing patterns in

sequences and charts, solving proportions, fi nding equivalent fractions, decimals and percents,

solving problems etc. if they don’t know their tables.

Using the method below, you can teach your students their tables in a week or so. (If you set aside

fi ve or ten minutes a day to work with students who need extra help, the pay-off will be enormous.)

There is really no reason for your students not to know their tables!

DAY 1: Counting by 2s, 3s, 4s and 5s

If you have completed the Fractions Unit you should already know how to count and multiply by

2s, 3s, 4s and 5s. If you don’t know how to count by these numbers you should memorize the

hands below:

If you know how to count by 2s, 3s, 4s and 5s, then you can multiply by any combination of these

numbers. For instance, to fi nd the product 3 × 2, count by 2s until you have raised 3 fi ngers.

2 4 6

DAY 2: The Nine Times Table

The numbers you say when you count by 9s are called the MULTIPLES of 9 (zero is also a multiple

of 9). The fi rst ten multiples of 9 (after zero) are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. What happens

when you add the digits of any of these multiples of 9 (EXAMPLE: 1 + 8 or 6 + 3)? The sum

is always 9!

Here is another useful fact about the nine times table: Multiply 9 by any number between 1 and

10 and look at the tens digit of the product. The tens digit is always one less than the number

you multiplied by:

9 × 4 = 36 9 × 8 = 72 9 × 2 = 18

You can fi nd the product of 9 and any number by using the two facts given above. For instance, to

fi nd 9 × 7, follow these steps:

STEP 1: 9 × 7 = 9 × 7 =

Mental MathHow to Learn Your Times Tables in a Week

3 is one less than 4 7 is one less than 8 1 is one less than 2

Subtract 1 from the number

you are multiplying by 9: 7 – 1 = 6

Now you know the

tens digit of the product.

3 × 2 = 6

5

15 2010255

4

12 16820

43

9 12615

32

6 8410

2



Teacher

1. Make sure your students know how to subtract (by counting on their fi ngers if necessary)

before you teach them the trick for the nine times table.

2. Give a test on STEP 1 (above) before you move on.

STEP 2: 9 × 7 = 9 × 7 =

Practise these two steps for all of the products of 9: 9 × 2, 9 × 3, 9 × 4, etc.

DAY 3: The Eight Times Table

There are two patterns in the digits of the 8 times table. Knowing these patterns will help you

remember how to count by 8s.

STEP 1: You can fi nd the ones digit of the fi rst fi ve multiples of 8, by starting at 8 and counting

backwards by 2s.

8

6

4

2

0

STEP 2: You can fi nd the tens digit of the fi rst fi ve multiples of 8, by starting at 0 and count up by 1s.

08

16

24

32

40

(Of course you don’t need to write the 0 in front of the 8 for the product 1 × 8.)

STEP 3: You can fi nd the ones digit of the next fi ve multiples of 8 by repeating step 1:

8

6

4

2

0

STEP 4: You can fi nd the remaining tens digits by starting at 4 and count up by 1s.

48

56

64

72

80

Practise writing the multiples of 8 (up to 80) until you have memorized the complete list. Knowing

the patterns in the digits of the multiples of 8 will help you memorize the list very quickly. Then you

will know how to multiply by 8:

8 × 6 = 48

How to Learn Your Times Tables in a Week

Count by eight until you have 6 fi ngers up: 8, 16, 24, 32, 40, 48.

So the missing digit is 9 – 6 = 3

(You can do the subtraction on your fi ngers if necessary).

These two digits add to 9.



DAY 4: The Six Times Table

If you have learned the eight and nine times tables, then you already know 6 × 9 and 6 × 8.

And if you know how to multiply by 5 up to 5 × 5, then you also know how to multiply by 6

up to 6 × 5! That’s because you can always calculate 6 times a number by calculating 5 times

the number and then adding the number itself to the result. The pictures below show why this

works for 6 × 4:

6 × 4 = 4 + 4 + 4 + 4 + 4 + 4

6 × 4 = 5 × 4 + 4 = 20 + 4 = 24

Similarly:

6 × 2 = 5 × 2 + 2; 6 × 3 = 5 × 3 + 3; 6 × 5 = 5 × 5 + 5.

Knowing this, you only need to memorize 2 facts:

ONE: 6 × 6 = 36 TWO: 6 × 7 = 42

Or, if you know 6 × 5, you can fi nd 6 × 6 by calculating 6 × 5 + 6.

DAY 5: The Seven Times Table

If you have learned the six, eight and nine times tables, then you already know:

6 × 7, 8 × 7 and 9 × 7.

And since you also already know 1 × 7 = 7, you only need to memorize 5 facts:

1. 2 × 7 = 14 2. 3 × 7 = 21 3. 4 × 7 = 28 4. 5 × 7 = 35 5. 7 × 7 = 49

If you are able to memorize your own phone number, then you can easily memorize these 5 facts!

NOTE: You can use doubling to help you learn the facts above. 4 is double 2, so 4 × 7 (= 28)

is double 2 × 7 (= 14). 6 is double 3, so 6 × 7 (= 42) is double 3 × 7 (= 21).

Try this test every day until you have learned your times tables:

1. 3 × 5 = 2. 8 × 4 = 3. 9 × 3 = 4. 4 × 5 =

5. 2 × 3 = 6. 4 × 2 = 7. 8 × 1 = 8. 6 × 6 =

9. 9 × 7 = 10. 7 × 7 = 11. 5 × 8 = 12. 2 × 6 =

13. 6 × 4 = 14. 7 × 3 = 15. 4 × 9 = 16. 2 × 9 =

17. 9 × 9 = 18. 3 × 4 = 19. 6 × 8 = 20. 7 × 5 =

21. 9 × 5 = 22. 5 × 6 = 23. 6 × 3 = 24. 7 × 1 =

25. 8 × 3 = 26. 9 × 6 = 27. 4 × 7 = 28. 3 × 3 =

29. 8 × 7 = 30. 1 × 5 = 31. 7 × 6 = 32. 2 × 8 =

How to Learn Your Times Tables in a Week

Plus one more 4.

Plus one more 4.

JUMP Math – Workbook 3 (3rd Edition)

page 1

1 Listing of Worksheet Titles

PART 1 Patterns & Algebra

PA3-1 Counting .................................................................................................................1

PA3-2 Preparation for Increasing Sequences...................................................................4

PA3-3 Increasing Sequences............................................................................................5

PA3-4 Counting Backwards ..............................................................................................7

PA3-5 Preparation for Decreasing Sequences ...............................................................10

PA3-6 Decreasing Sequences ........................................................................................11

PA3-7 Increasing and Decreasing Sequences ...............................................................12

PA3-8 Attributes ..............................................................................................................14

PA3-9 Patterns Where Two Attributes Change ..............................................................16

PA3-10 Repeating Patterns...............................................................................................17

PA3-11 Extending Repeating Patterns .............................................................................20

PA3-12 Finding Cores in Patterns.....................................................................................21

PA3-13 Making Patterns with Squares .............................................................................22

PA3-14 Making Patterns with Squares (Advanced)..........................................................24

PA3-15 Extending a Pattern Using a Rule ........................................................................25

PA3-16 Identifying Patterns Rules ....................................................................................27

PA3-17 Introduction to T-tables ........................................................................................28

PA3-18 T-tables ................................................................................................................31

PA3-19 Problems and Puzzles .........................................................................................32

Number Sense

NS3-1 Place Value – Ones, Tens, and Hundreds...........................................................33

NS3-2 Place Value ..........................................................................................................34

NS3-3 Writing and Reading Number Words ...................................................................35

NS3-4 Writing Numbers...................................................................................................36

NS3-5 Representation with Base Ten Materials .............................................................38

NS3-6 Representation in Expanded Form ......................................................................40

NS3-7 Representing Numbers - Review .........................................................................42

NS3-8 Comparing Numbers ............................................................................................43

NS3-9 Comparing and Ordering Numbers......................................................................45

NS3-10 Differences of 10 and 100 ....................................................................................46

NS3-11 Comparing Numbers (Advanced).........................................................................48

NS3-12 Counting by 2s .....................................................................................................49

NS3-13 Counting by 5s and 25s .......................................................................................50

NS3-14 Counting by 2s, 3s and 5s....................................................................................51

NS3-15 Counting Backwards by 2s and 5s.......................................................................52

NS3-16 Counting by 10s ...................................................................................................53

NS3-17 Counting by 2s, 3s, 4s, 5s and 10s ......................................................................54

NS3-18 Counting by 100s .................................................................................................55

NS3-19 Regrouping...........................................................................................................56

NS3-20 Regrouping (Advanced) .......................................................................................58

NS3-21 Adding 2-Digit Numbers .......................................................................................59

NS3-22 Adding with Regrouping (or Carrying)..................................................................60

NS3-23 Adding with Money...............................................................................................62

NS3-24 Adding 3-Digit Numbers .......................................................................................63

NS3-25 Subtracting 2- and 3-Digit Numbers.....................................................................65


page 2


NS3-26 Subtracting by Regrouping...................................................................................67

NS3-27 Subtracting by Regrouping Hundreds..................................................................69

NS3-28 Mental Math..........................................................................................................70

NS3-29 Parts and Totals ...................................................................................................72

NS3-30 Parts and Totals (Advanced)................................................................................74

NS3-31 Sums and Differences..........................................................................................76

NS3-32 Larger Numbers ...................................................................................................77

NS3-33 Concepts in Number Sense .................................................................................78

NS3-34 Arrays ...................................................................................................................79

NS3-35 Adding Sequences of Numbers ...........................................................................81

NS3-36 Multiplication and Repeated Addition...................................................................82

NS3-37 Multiplying by Skip Counting ................................................................................83

NS3-38 Multiplying by Adding On .....................................................................................85

NS3-39 Doubles ................................................................................................................87

NS3-40 Topics in Multiplication .........................................................................................88

NS3-41 Concepts in Multiplication ....................................................................................90

NS3-42 Pennies, Nickels and Dimes ................................................................................91

NS3-43 Quarters ...............................................................................................................93

NS3-44 Counting by Two or More Coin Values ................................................................94

NS3-45 Counting by Different Denominations ..................................................................96

NS3-46 Least Number of Coins ........................................................................................98

NS3-47 Dimes and Pennies ............................................................................................100

NS3-48 Making Change Using Mental Math ...................................................................101

NS3-49 Lists ....................................................................................................................103

NS3-50 Organized Lists ..................................................................................................105

Measurement

ME3-1 Estimating Lengths in Centimetres ....................................................................107

ME3-2 Measuring in Centimetres ..................................................................................108

ME3-3 Rulers .................................................................................................................109

ME3-4 Measuring Centimetres with Rulers ...................................................................110

ME3-5 Drawing to Centimetre Measurements...............................................................111

ME3-6 Estimating in Centimetres ..................................................................................112

ME3-7 Estimating in Metres...........................................................................................113

ME3-8 Estimating in Metres (Advanced) .......................................................................114

ME3-9 Kilometres ..........................................................................................................116

ME3-10 Ordering and Assigning Appropriate Units.........................................................118

ME3-11 Measuring Perimeter ..........................................................................................121

ME3-12 Perimeter............................................................................................................123

ME3-13 Exploring Perimeter............................................................................................124

ME3-14 Investigations .....................................................................................................125

ME3-15 Measuring Mass .................................................................................................127

ME3-16 Measuring Capacity ...........................................................................................128

ME3-17 Measuring Temperature.....................................................................................129

Probability & Data Management

PDM3-1 Introduction to Classifying Data .........................................................................130

PDM3-2 Venn Diagrams...................................................................................................132


page 3


PDM3-3 Introduction to Tallying Data ..............................................................................135

PDM3-4 Reading Data from a Tally Chart........................................................................136

PDM3-5 Introduction to Pictographs ................................................................................137

PDM3-6 Pictograph Scale ................................................................................................138

PDM3-7 Displaying Data on a Pictograph........................................................................140

PDM3-8 Introduction to Bar Graphs .................................................................................141

PDM3-9 Bar Graphs .........................................................................................................143

PDM3-10 Bar Graphs (Advanced) .....................................................................................145

PDM3-11 Collecting Data ...................................................................................................146

PDM3-12 Practice with Surveys.........................................................................................147

PDM3-13 Blank Tally Chart and Bar Graph .......................................................................149

PDM3-14 Collecting and Interpreting Data.........................................................................150

Geometry

G3-1 Sides and Vertices .............................................................................................151

G3-2 Introduction to Angles ........................................................................................153

G3-3 Equilateral Shapes .............................................................................................155

G3-4 Quadrilaterals and Other Polygons ....................................................................156

G3-5 Tangrams ...........................................................................................................158

G3-6 Congruency ........................................................................................................160

G3-7 Congruency (Advanced) ....................................................................................161

G3-8 Recognizing and Drawing Congruent Shapes ...................................................162

G3-9 Exploring Congruency with Geoboards..............................................................163

G3-10 Exploring Congruency with Grids.......................................................................164

G3-11 Symmetry ...........................................................................................................165

G3-12 Lines of Symmetry..............................................................................................166

G3-13 Completing Symmetric Shapes..........................................................................167

G3-14 Comparing Shapes.............................................................................................168

G3-15 Sorting Shapes by Property ...............................................................................170

G3-16 Finding Polygons................................................................................................172

G3-17 Puzzles and Problems .......................................................................................173

PART 2 Patterns & Algebra

PA3-20 Patterns Involving Time......................................................................................176

PA3-21 Calendars ...........................................................................................................178

PA3-22 Number Lines .....................................................................................................179

PA3-23 Mixed Patterns ...................................................................................................180

PA3-24 Describing and Creating Patterns ......................................................................185

PA3-25 2-Dimensional Patterns......................................................................................188

PA3-26 Patterns in Two Times Tables............................................................................191

PA3-27 Patterns in the Five Times Tables......................................................................192

PA3-28 Patterns in the Eight and Nine Times Tables.....................................................193

PA3-29 Patterns in the Times Tables (Advanced) ..........................................................195

PA3-30 Patterns with Increasing Gaps ...........................................................................196

PA3-31 Patterns with Larger Numbers ...........................................................................197

PA3-32 Extending and Predicting Positions....................................................................198


page 4


PA3-33 Equations ...........................................................................................................199

PA3-34 Adding and Subtracting Machines .....................................................................200

PA3-35 Equations (Advanced) ........................................................................................201

PA3-36 Problems and Puzzles .......................................................................................202

Number Sense

NS3-51 Ordinal Numbers ................................................................................................204

NS3-52 Round to the Nearest Ten..................................................................................206

NS3-53 Round to the Nearest Hundreds ........................................................................208

NS3-54 Rounding ............................................................................................................210

NS3-55 Estimating Sums and Differences ......................................................................211

NS3-56 Estimating...........................................................................................................212

NS3-57 Mental Math and Estimation...............................................................................214

NS3-58 Multiplication and Division (Review)...................................................................216

NS3-59 Sharing – Knowing the Number of Sets.............................................................218

NS3-60 Sets ....................................................................................................................220

NS3-61 Two Ways of Sharing .........................................................................................222

NS3-62 Division...............................................................................................................225

NS3-63 Dividing by Skip Counting ..................................................................................226

NS3-64 Division and Multiplication..................................................................................230

NS3-65 Knowing When to Multiply or Divide...................................................................231

NS3-66 Remainders ........................................................................................................233

NS3-67 Multiplication and Division..................................................................................235

NS3-68 Multiplication and Division (Review)...................................................................236

NS3-69 Patterns Made with Repeated Addition..............................................................238

NS3-70 Counting by Dollars and Coins...........................................................................239

NS3-71 Dollar and Cent Notation....................................................................................240

NS3-72 Counting and Changing Units ............................................................................242

NS3-73 Converting Between Dollar and Cent Notation ..................................................244

NS3-74 Canadian Bills and Coins ...................................................................................245

NS3-75 Adding Money ....................................................................................................246

NS3-76 Subtracting Money .............................................................................................248

NS3-77 Estimating...........................................................................................................249

NS3-78 Equal Parts.........................................................................................................251

NS3-79 Models of Fractions............................................................................................252

NS3-80 Fractions of a Region or a Length......................................................................254

NS3-81 Equal Parts of a Set ...........................................................................................255

NS3-82 Parts and Wholes...............................................................................................257

NS3-83 Sharing Fractions ...............................................................................................260

NS3-84 Comparing Fractions..........................................................................................263

NS3-85 Fractions Greater than One ...............................................................................265

NS3-86 Puzzles and Problems .......................................................................................266

NS3-87 Decimal Tenths ..................................................................................................267

NS3-88 Word Problems (Warm Up)................................................................................269

NS3-89 Word Problems...................................................................................................270

NS3-90 Planning Party ....................................................................................................271

NS3-91 Additional Problems ...........................................................................................272

NS3-92 Charts.................................................................................................................273


page 5


NS3-93 Arrangements and Combinations.......................................................................274

NS3-94 Arrangements and Combinations (Advanced) ...................................................275

NS3-95 Guess and Check...............................................................................................276

NS3-96 Puzzles...............................................................................................................277

Measurement

ME3-18 Analog Clock Faces ...........................................................................................278

ME3-19 Hands on an Analog Clock.................................................................................279

ME3-20 Telling Time → The Hour Hand .........................................................................280

ME3-21 Telling Time → Five−Minute Intervals................................................................282

ME3-22 Telling Time → Putting It Together! ...................................................................284

ME3-23 Digital Clock Faces.............................................................................................286

ME3-24 Timelines ............................................................................................................287

ME3-25 Intervals of Time.................................................................................................290

ME3-26 Estimating Time Intervals...................................................................................292

ME3-27 Cumulative Reviews...........................................................................................293

ME3-28 Area....................................................................................................................294

ME3-29 Area in Square Centimetres...............................................................................297

ME3-30 Half Squares.......................................................................................................298

ME3-31 Puzzles and Problems .......................................................................................300

ME3-32 Investigating Units of Area .................................................................................302

Probability & Data Management

PDM3-15 Outcomes ...........................................................................................................303

PDM3-16 Even Chances ....................................................................................................304

PDM3-17 Even, likely and Unlikely ....................................................................................307

PDM3-18 Outcomes ...........................................................................................................308

PDM3-19 Describing Probabilities (Advanced) ..................................................................309

PDM3-20 Fair Games.........................................................................................................310

PDM3-21 Experiments and Expectation ............................................................................311

PDM3-22 Cumulative Review.............................................................................................312

Geometry

G3-18 Introduction to Coordinate Systems...................................................................314

G3-19 Coordinate Systems...........................................................................................316

G3-20 Introduction to Slides (or Translations) ..............................................................318

G3-21 Slides..................................................................................................................319

G3-22 Slides (Advanced) ..............................................................................................320

G3-23 Slides on a Grid..................................................................................................321

G3-24 Coordinate Systems and Maps ..........................................................................322

G3-25 Mapping Exercise...............................................................................................325

G3-26 Flips ....................................................................................................................326

G3-27 Rotations ............................................................................................................327

G3-28 Flips and Slides ..................................................................................................328

G3-29 Turns ..................................................................................................................329

G3-30 Rotations ............................................................................................................330

G3-31 Rotations (Advanced).........................................................................................331

G3-32 Flips, Slides and Turns.......................................................................................332


page 6


G3-33 Building Pyramids...............................................................................................334

G3-34 Building Prisms...................................................................................................335

G3-35 Edges, Vertices and Faces ................................................................................336

G3-36 Pyramid Nets......................................................................................................339

G3-37 Prism Nets..........................................................................................................340

G3-39 Drawing Pyramids and Prisms ...........................................................................342

G3-40 Properties of Pyramids and Prisms....................................................................343

G3-41 Sorting 3-D Shapes............................................................................................345

G3-42 Classifying Shapes and Making Patterns ..........................................................346

G3-43 Geometry in the World .......................................................................................348

G3-44 Problems and Puzzles .......................................................................................349

Patterns & Algebra Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

A Note for Patterns Before students can create or recognize a pattern in a sequence of numbers, they must be able to tell how

far apart the successive terms in a particular sequence are. There is no point in introducing students to

sequences if they don’t know how to find the gap between a given pair of numbers, either by applying their

knowledge of basic addition and subtraction, or by counting on their fingers as described below.

For weaker students, use the following method for recognizing gaps:

How far apart are 8 and 11?

STEP 1:

Say the lower number (“8”) with your fist closed.

8

STEP 2: Count up by ones, raising your thumb first, then one finger at a time until you have reached

the higher number (11).

9 10 11

STEP 3: The number of fingers you have up when you reach the final number is the answer

(in this case you have three fingers up, so three is the difference between 8 and 11.)

Using the method above, you can teach even the weakest student to find the difference between two

numbers in one lesson. (You may have to initially hold your student’s fist closed when they say the first

number—some students will want to put their thumb up to start—but otherwise students find this method

easy.)

Eventually, you should wean your student off using their fingers to find the gap between a pair of

numbers. The exercises in the MENTAL MATH section of this manual will help with this.


PA3-1 Counting

Goal: Students will find differences between numbers mentally, by using fingers or by using a number line.

Prior Knowledge Required: Count to 100

Vocabulary: difference, number line.

To introduce the topic of patterns, tell your students that they will often encounter patterns in sequences of

numbers in their daily lives. For instance, the sequence 25, 20, 15, 10... might represent the amount of

money in someone’s savings account (each week) or the amount of water remaining in a leaky fish tank

(each hour).

Ask your students to give other interpretations of the pattern. You might also ask them to say what the

numbers in the pattern mean for a particular interpretation. For instance, if the pattern represents the

money in a savings account, the person started with $25 and withdrew $5 each week.

Tell your students that you would like to prepare them to work with patterns and you will start by making

sure they are able to find the difference between two numbers. There is nothing wrong with counting on

their fingers until they have learned their number facts—they are our built-in calculator. If there are students

who know their subtraction facts, tell them they are in perfect shape and can do the calculations mentally.

Show students how to find the difference between two numbers using the method described in the note at

the beginning of this unit. (Say the smaller number with your fist closed. Count up to the larger number

raising one finger at a time. The number of fingers you have raised when you say the larger number is the

difference).

NOTE: Before you allow students to try any of the questions on a particular worksheet you should assign

sample questions (like the ones on the worksheet) for the whole class to discuss and solve. You should

also give

a mini quiz or assessment consisting of several questions that students can work on independently (either

in a notebook or individual pieces of paper). The quiz or assessment will tell you whether students are

ready to do the work on the worksheet.

The bonus questions provided in a lesson plan may either be assigned during the lesson, (to build

excitement, by allowing students to show off with harder looking questions) or during the assessments (for

students who have finished their work early, so you have time to help slower students).

Sample Questions:

Find the difference:

a) 3 7 b) 23 25 c) 39 42 d) 87 92

Bonus:

a) 273 277 b) 1768 1777

c) 1997 2003 d) 32 496 32 502


Assessment:

Find the difference:

a) 4 7 b) 21 25 c) 37 42 d) 70 77

To introduce the topic on the second worksheet for this unit draw a number line on the board.

Show your students how to find the difference between 6 and 9 as shown below:

What number added to 6 gives 9? 6 + ? = 9

Count 3 spaces between 6 and 9 on a number line, checking with each hop:

Add 1, get 7, 7 is 1 more than 6.

Add 2, get 8, 8 is 2 more than 6.

Add 3, get 9, 9 is 3 more than 6,

SO: 6 + 3 = 9 AND: 3 is called the difference between 9 and 6

Ask several volunteers to come to the board and find the difference between 4 and 10, 2 and 7, 0 and 8.

Draw another number line and ask students to find the difference (or gap) between the numbers below.

a) 10 12 b) 15 25 c) 15 20 d) 14 19 e) 12 18 f) 18 24

Tell your students that the expression 33 36 (where they are expected to fill in the gap) could also be

written 33 + ____ = 36.

Ask a couple of volunteers to write the old exercises in the new way and to fill in the missing number.

4 10 4 + ____ = 10

Assessment:

0 1 2 3 4 5 6 7 8 9 10

1 2 3

15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 29

10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 24

?


Find the missing number by counting up from the smaller to the larger number.

a) 20 + ____ = 23 b) 23 + ____ = 28 c) 18 + ____ = 25

d) 24 + ____ = 29 e) 22 + ____ = 31

Point at a pair of numbers on the number line, say 16 and 13. Ask your students to find the difference

between them. Which number is larger? How much larger than 13 is 16? Give an example—Jack has 13

stickers, Jill has 16. Who has more? How many more does Jill have?

Sample Questions:

Fill in the missing numbers:

a) 28 is ____ more than 22 b) 29 is ____ more than 25 c) 36 is ____ more than 25

Bonus:


a) 67 is ____ more than 63 b) 78 is ____ more than 71 c) 83 is ____ more than 78

d) 107 + ____ = 111 e) 456 + ____ = 459 f) 987 + ____ = 992.

NOTE: It is extremely important that your students learn their number facts and eventually move beyond

using their fingers. We recommend that you use the exercises in the MENTAL MATH section of this Guide

to help students learn their number facts.

Activity: Let your students play the Modified Go Fish Game in pairs (see the MENTAL MATH section for

instructions). Start with the number 5 as the target number and proceed gradually to 10. This will help them

find larger differences between numbers.


PA3-2 Preparation for Increasing Sequences

Goals: Students will find a number that is more than another number using a given difference.

Prior Knowledge Required: Count to 100. Find difference between two numbers.

Vocabulary: difference

This worksheet is an essential step before starting increasing sequences.

Warm-up:

Add the number in the circle to the number beside it. Write your answer in the blank:

a) 6 _____ b) 7 _____ c) 9 _____

d) 19 _____

e) 17 _____ f) 26 _____ g) 14 _____ h) 23 _____

Then show an example: What is 4 more than 11? Use finger counting. Practice questions like:

a) _____ is 5 more than 7 b) _____ is 8 more than 3 c) _____ is 6 more than 9

Point out to your students that questions like the ones they just solved are the reverse of those they solved

in the last lesson—they now know the difference, but do not know one of the numbers.

Assessment:


a) _____ is 3 more than 13 b) _____ is 6 more than 15 c) _____ is 10 more than

26

d) _____ is 4 more than 32 e) _____ is 7 more than 21 f) _____ is 8 more than 17

Bonus:


a) _____ is 4 more than 59 b) _____ is 6 more than 75

2 3 2 4

5 3 6 8


PA3-3 Increasing Sequences

Goal: Students will extend increasing sequences and solve simple problems using increasing sequences.


Vocabulary: difference, increasing sequence, pattern, number line.

In this lesson, weaker students can use their fingers to extend a pattern as follows:

EXAMPLE: Extend the pattern 3, 6, 9… up to six terms.

STEP 1: Identify the gap between successive pairs of numbers in the sequence (You may count on

your fingers, if necessary). The gap, in this example, is three. Check that the gap between

successive terms in the sequence is always the same, otherwise you cannot continue the pattern

by adding a fixed number. (Your student should write the gap between each pair of successive

terms above the pairs.)

3 , 6 , 9 , _____ , _____,

STEP 2: Say the last number in the sequence with your fist closed. Count by ones until you have

raised three fingers (the gap between the numbers). The number you say when you have raised

your third finger is the next number in the sequence.

3 , 6 , 9 _____ , _____ ,

STEP 3: Continue adding terms to extend the sequence.

3 , 6 , 9 , _____ , _____ , _____

To start the lesson write the sequence: 1, 2, 3, 4, 5, … on the board. Ask your students what the next

number will be. Then write the sequence 2, 4, 6, 8, … Ask what the next number is this time. Ask students

how they guessed that the next number is 10 (skip counting by 2). Ask what they did to the previous

number to get the next one. Ask what the difference between the numbers is.

Tell your students that the next question is going to be harder, but that you still expect them to find the

answer. Write the sequence 2, 5, 8, 11… with room to write circles between the numbers:

2 , 5 , 8 , 11 ,

Ask students if they can see any pattern in the numbers. If your students need help, fill in the difference

between the first two terms in the circle.

2 , 5 , 8 , 11

Add another circle and ask students to find the difference between 5 and 8.

2 , 5 , 8 , 11 .

3

3

3 3

3 3

12

12 18 15


After students have found the difference between 8 and 11, ask them if they can predict the next term in the

sequence.

Explain that sequences like the one you just showed them are common in daily life. Suppose George wants

to buy a skateboard. He saves $7 during the first month and, in each of the next months, he saves $4 more.

How much money does he have after two months? Write 7 _____ , and ask a volunteer to fill in the

next number.

Ask volunteers to continue George’s savings’ sequence. Mention that patterns of this kind are called

“increasing sequences” and ask if the students can explain why they are called increasing. Point out that in

all of the exercises so far the difference between terms in sequence has been the same. Later your

students will study more advanced sequences in which the difference changes.

Extend the patterns:

a) 6 , 9 , 12 , 15 , _____ , _____ b) 5 , 11 , 17 , 23 , _____

c) 2 , 10 , 18 , 26 , _____

Do not progress to word problems before all your students can extend a given pattern.

Bonus:

a) 99, 101, 103, … b) 654, 657, 660, …

Now ask your students to solve the following problem:

A young wizard learned three new incantations every day during his studies at the wizards’ school. One

sunny Monday morning he told his friend that he already knows 12 incantations. How many incantations

will he know by Wednesday?

Draw a table on the board:

Number of

incantations 12

Days Monday Tuesday Wednesday

Students should see that to solve the problem they need to extend an increasing sequence (the “gap” here

is 3, as the wizard learns 3 spells per day).

Bonus:

A baby elephant grows 2 cm a day. Today it is 140 cm tall. What will its height be tomorrow?

When will it reach a height of 150 cm?

Sample Word Problems:

1. A newborn Saltwater Crocodile is about 25 cm long. It grows 5 cm in a month during the

first 4 months. How long is a 2-month-old Saltie? 4-mth old one? If its length is 40 cm,

how old is it?

2. An apple tree sapling grows 3 cm in a month. On May 1, it is 10 cm tall. What will its height be on

August 31?

4


Extensions:

1. For each sequence, create a word problem that goes with the sequence.

a) 7 , 9 , 11 , 13 , ___ , ___ b) 5 , 9 , 13 , 17 , ___ , ___

2. ANOTHER GAME: Each group needs two dice (say blue and red) and a token for each player. Use the

“Number Line Chasing Game” from the BLM. The blue die shows the difference in the pattern. The red

die shows the number of terms in the sequence to calculate. The players start at 1. If he throws 3 with

the blue die and 2 with the red die, he will have to calculate the next

2 terms of the sequence: 1 , _____ , _____ (4 and 7). 7 is a “smiley face”. When the player lands

on a “smiley face”, he jumps to the next “smiley face”, in this case 17. His next sequence will start at 17.

The first player to reach 100 is the winner.

Activities:

1. A Game for Pairs

One student writes a number, the other writes the difference in the increasing sequence. Then the

players take turns writing sequence terms. They have to continue the sequence up to 6 numbers.

2. Advanced

The first player writes a number and the second player writes the second number in the sequence. The

first player has to find the difference and then continue the sequence.

3 3


0 1 2 3 4 5 6 7 8 9 10

3 2 1

PA3-4 Counting Backwards

Goal: Students will find the difference between two numbers counting backwards using fingers and

a number line.

Prior Knowledge Required: Count backwards from 100 to 1.


ASK A RIDDLE: I am a number between 1 and 10. Subtract me from 8, and get 5. Who am I?

Explain that you can use a number line to find the difference. Draw the number line on the board and show how to find the difference by counting backwards. WRITE: “5 is 3 less than 8.” Show a couple more examples on the board, this time using volunteers.

Sample problems:

I am a number between 1 and 10. Who am I?

a) If you subtract me from 26, you get 21.

b) If you subtract me from 32, you get 27.

c) If you subtract me from 34, you get 26.

Ask your students how they could solve the problem when they do not have a number line. Remind them

how to find the difference counting backwards on their fingers.

You might need to spend more time with students who need more practice finding the gap between pairs of

numbers by subtracting or counting backwards. Your students will not be able to extend or describe

patterns if they cannot find the gap between pairs of numbers. You also might use flashcards similar to

those you used with the worksheet PA3-1.

Draw two number lines on the board:

20 21 22 23 24 25 26 27 28 29 30

20 21 22 23 24 25 26 27 28 29 30

20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 34


What number added to 24 is 27? Solve by counting forward on the upper line.

What number is subtracted from 27 to get 24? Solve by counting backward on the lower line. (Draw arrows

to indicate the “hops” between the numbers on the number line).

What are the similarities between the methods? What are the differences?

Students should notice that both methods give the difference between 24 and 27 and are merely different

ways to subtract. ASK: How many times did you hop? What did you measure? (distance, difference, gap)

What is this called in mathematics? What other word for the gap did we use? (difference).

When you measured the distance (or difference) between two points on the number line, the first time you

went from left to right, and the second time you went from right to left.

What operation are you performing when you go from the smaller number to the larger? (adding) From the

larger number to the smaller one? (subtracting)

You might even draw arrows: from right to left and write “–”, and from left to right and write “+” below and

above the respective number lines.

Congratulate your students on discovering an important mathematical fact—addition and subtraction are

opposite movements on a number line, and both counting backwards and forwards are used to find the

difference.

Assessment:

MORE RIDDLES: I am a number between 1 and 10. Who am I?

a) If you subtract me from 27, you get 21.

b) If you subtract me from 32, you get 24.

c) If you subtract me from 34, you get 27.

Bonus: Find the gap between the numbers:

a) 108 105 b) 279 274 c) 241 238 d) 764 759

20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 34


PA3-5 Preparation for Decreasing Sequences

Goal: Students will find the number that is less than the other number using a given difference.

Prior Knowledge Required: Count to 100. Find the difference between two numbers.


Repeat the lesson PA3-2: Preparation for Increasing Sequences, using decreasing sequences and

counting backwards.


PA3-6 Decreasing Sequences

Goal: Students will find the differences between two numbers by subtraction, and extend decreasing

sequences.


Vocabulary: difference, decreasing sequence, pattern

Remind your students what an increasing sequence is. Ask them where they encounter increasing

sequences in life. POSSIBLE EXAMPLES: height, length, weight of plants and animals, growing buildings,

etc. Are all sequences that they will encounter in life increasing? Ask students to discuss this point and give

examples of sequences that are not increasing.

Give an example, such as: Jenny was given $20 as a monthly allowance. After the first week she still had

16 dollars, after the second week she had 12 dollars, after the third week she had 8 dollars. If this pattern

continues, how much money will she have after the fourth week? The amount of money that Jenny has

decreases: 20, 16, 12, 8, so this type of sequences is called a “decreasing sequence”. Write the term on the

board and later include it in spelling tests together with “increasing sequence”.

Have your students do some warm-up exercises, such as:

1. Subtract the number in the circle from the number beside it. The minus reminds you that you are

subtracting. Write your answer in the blank:

a) 13 – 2

_____ b) 12 – 6

_____ c) 11 – 8

_____ d) 9 – 5

_____

2. Fill in the missing numbers:

a) _____ is 4 less than 37 b) _____ is 5 less than 32

Then write two numbers on the board:

14 , 11

Ask what the difference between the numbers is. Write the difference in the circle with a minus sign then

ask students to extend the sequence.

14 , 11 , 8 , _____ , _____

Assessment:

Extend the decreasing sequences:

a) 21 , 19 , 17 , 15 , _____ , _____ c) 48 , 43 , 38 , 33 , _____ , _____

b) 34 , 31 , 28 , _____ , _____ , _____


Bonus:

Extend the decreasing sequences:

a) 141 , 139 , 137 , 135 , _____

b) 548 , 541 , 534 , 527 , _____

c) 234 , 224 , 214 , _____ , _____

Activity:

A Game for Two

The students will need two dice and a hundreds chart (to help with counting backwards). The first player

throws both dice (blue and red). The number on the red die determines how many numbers in the sequence

the player has to calculate and the number on the red die indicates the difference of the decreasing

sequence. The first number in this sequence is 100. The next player starts at the number the first player

finished at. The player who ends at 0 or below is the winner.

EXAMPLE: The first throw yields red 4, blue 3. The first player has to calculate four numbers in the

sequence whose difference is 3 and whose first number is 100. The player calculates: 97, 94, 91, 88. The

next player starts with 88 and rolls the dice.

ADVANCED VERSION: Give 1 point for each correct term in the sequence and 5 points to the first player

to reach 0. The player with the most points is the winner.

ALTERNATIVE: Use the “Number Line Chasing Game” from the BLM. Each player uses a moving token,

similar to the game in section PA3-3.

Extension: If you used the “Number Line Chasing Game” from the BLM or the Activity above, your

students might explore the following question:

Two players start at the same place. John throws the dice and receives red 3 and blue 4.

Mike gets red 4 and blue 3. Do they end at the same place? Does this always happen? Why?


PA3-7 Increasing and Decreasing Sequences

Goal: Students will distinguish between increasing and decreasing sequences, and extend sequences.

Prior Knowledge Required: Count to 100. Find difference between two numbers

Increasing sequences Decreasing sequences

Vocabulary: difference, increasing sequence, decreasing sequence, pattern

Remind your students about how they can extend a sequence using the gap provided. Give examples for

both increasing and decreasing sequences. For increasing sequences, you might draw an arrow pointing

upwards beside the sequence, and for decreasing sequences, an arrow pointing downwards.

Write several sequences on the board and ASK: Does the sequence increase or decrease? What sign

should you use for the gap? Plus or minus? What is the difference between the numbers?

Let your students practice finding the next term in a sequence using the gap provided:

a) 4 , 9 , … b) 25 , 22 , … c) 23 , 26 , … d) 49 , 44 , …

Ask students what they would do if they have the beginning of the sequence but they do not know the gap:

4 , 7 , 10 , …

First ask if the sequence goes up or down. Is the sign ‘+’ or ‘–‘? Then find the difference. When the gaps

are filled in, extend the sequence.

Assessment:

Extend the sequences. First find the difference (gap). (GRADING TIP: Check to see if the students have

the correct “gap”):

a) 21 , 18 , 15 , ____ , ____ , ____ d) 51 , 49 , 47 , ____ , ____ , ____

b) 34 , 36 , 38 , ____ , ____ , ____ e) 84 , 74 , 64 , ____ , ____ , ____

c) 47 , 50 , 53 , ____ , ____ , ____ f) 60 , 65 , 70 , ____ , ____ , ____

+5 –3 +3 –5


Bonus:

a) 128 , 134 , 140 , ____ , ____ , ____

b) 435 , 438 , 441 , ____ , ____ , ____

c) 961 , 861 , 761 , ____ , ____ , ____

Activity:

A Game

Calculating three terms of a sequence each time, one point for each correct term. Students will need two

dice of differentcolours (say red and blue) and a cardboard coin with the signs “+” and “–” written on its

sides. The students throw the dice and the coin.

The number on the die is the difference between terms in sequence. The coin determines whether the

sequence is increasing (“+”) or decreasing (“-“). For example, if the die shows 4 and the coin shows “-“,

the student has to write: 30, 26, 22, 18. Give a point for each correct term.

Students could also play the game using the Number Line Chasing Game from the BLM.

Extension:

1. You could make up some fanciful word problems for your students, possibly based on things they are

reading:

A newborn dragon (called Herbert by its proud owner) grows 50 cm every day. At hatching, it is 30 cm

long. How long will Herbert be in 3 days? In a week?

2. A GAME FOR PAIRS: The first player throws a die so that the second player does not see the result.

The die indicates the difference between terms in a sequence. The first player chooses the first term of

the sequence. The first player then gives the second player the first and the third terms in the

sequence.

For instance, if the die gives 3, the first player might select 13 as the first term and write: 13, _____ , 19.

The second player then has to guess the second term and the difference. In this case, the second term

is 16.

(The trick is to find the number that is exactly in the middle of the two given terms) After students have

tried the game, you might ask them to make the T-table of differences:

Students will be soon able to see the pattern.

The die The difference between the first

and the second terms

The difference between the first

and the third terms

1

2

…


PA3-8 Attributes

Goal: Students will distinguish the attributes that change in a sequence, and make sequences with the

given number of changing attributes.

Prior Knowledge Required: Geometric shapes: triangle, square, circle, pentagon

Vocabulary: attribute,colour, shape, size

Draw two similar triangles of different sizes on the board. Ask your students to describe the shapes. At this

stage, the description can be simple: large triangle, small triangle. Start by making a table, adding columns

and rows when you need them.

Shape triangle triangle

Size large small

Add another triangle of a different colour. Add the “colour” row. Then add a circle.

Shape triangle triangle triangle circle

Size large small small small

Colour white white red white

Add more shapes and ask volunteers to fill in the table.

You might also introduce attributes like orientation, visual pattern (dots, stripes, etc) and others.

FOR EXAMPLE:

1. Large red triangle, new attribute—orientation.

2. Large circle, new attribute—pattern.

Explain that the properties are called “attributes”, and write that word in the top-left cell of your table.

Ask students to practice in pairs—the first player says a shape and lists two attributes. The second player

then draws two versions of that shape: both shapes should have the first attribute, but only one shape

should have the second attribute. For example, the first player says “triangle, colour, size” and the second

player then draws two orange triangles, one large and one small.

For practice, give your students several patterns and ask them to write the attribute that changes. If

students have problems deciding which attribute changes you might give them a limited list to choose from.

W W

W W R W


Sample Patterns:

a) b)

c)

Assessment:

Write the attribute that changes in each pattern:

a)

b)

Activity:

A Game for Pairs or Groups

Give your students a set of pattern blocks or beads. The beads may differ not only in size, colour and

shape, but also in texture (rough, smooth), visual pattern (striped, solid), etc. One player lays out three

shapes that share one attribute. The other has to add a shape that shares the same attribute.

If the second player succeeds, they receive a point. If not, the first player has to add another object with the

same attribute to help his peer. For a more advanced version ask students to say which attributes remain

the same and which ones change in their sequence of shapes. How many attributes change, how many

remain the same? It is a good exercise to ask students to write down all the possible attributes for their set

of shapes before they start playing. The second player only has two chances to guess the common attribute

before the game starts again, with the players roles reversed.

Extensions:

1. When your students have learned some terms from geometry, you can play the following game: Set out

three shapes that all share a common attribute and a fourth shape that lacks that attribute. Ask your

students to tell you which shape doesn’t belong in the set (and why).

EXAMPLES:

Three of the shapes have exactly one right angle. Three of the shapes are 4-sided. The parallelogram doesn’t belong. The triangle doesn’t belong.


2. Circle the word that tells you which attribute of a figure or figures changes in the pattern:

a) b)

shape position size size orientation number

c) d)

number size shape shape position size

e) f)

size orientation number number size shape


PA3-9 Patterns Where Two Attributes Change

Goal: Students will distinguish the attributes that changed in a sequence, and make sequences where two

attributes change.

Prior Knowledge Required: Geometric shapes: triangle, square, circle,

pentagon, cube, cylinder, cone, ball

Vocabulary: attribute, shape, colour, size.

Remind your students of what an attribute is are and ask them to tell you which attribute changed in the

following pattern:

Ask your students to name some attributes in a collection of objects, and to draw a sequence where at least

one of the attributes changes. If any of the following attributes are missing from the student’s list you should

add them to the list: shape, size,colour, visual pattern, number and position.

Give your students several patterns where two attributes change and ask them to name the attributes that

change.

a) b) c)

For an activity students could play the advanced version of the game in the Activity from the previous

section.

Assessment:

Write the attributes that changed in each pattern. How many attributes changed?

Activity:

SET® is an excellent game for developing pattern recognition skills. For an easy introduction, use the

SET® game with only two attributes—for example, take only red filled shapes.

If you do not have the game, go to http://www.setgame.com/ for rules. You may also find the online

version there.


Extensions:

1. Show your students a necklace or bracelet and ask them to describe how the pattern changes:

Students might mention colour, size, shape, texture (rough, smooth), visual pattern (striped, solid), etc.

2. Give each student a set of pattern blocks or beads and ask them to pick out a pair of shapes that differ

by one attribute, two attributes etc. Ask them to find a pair of shapes that differ by the greatest number

of attributes. (For instance, they might pick a pair of beads that differ in shape, size, colour, texture

(rough, smooth), visual pattern (striped, solid), etc.)

3. Circle the two words that tell you which attributes of a figure or figures changes in the pattern:

a)

shape orientation size number colour

b)

orientation size number shape colour

c)

orientation size number shape colour


PA3-10 Repeating Patterns

Goal: Students will find the core of the pattern and continue the repeating pattern.

Prior Knowledge Required: Ability to count. Attributes.

Vocabulary: attribute, length of core, core

Explain that a pattern is repeating if it consists of a “core” of terms or figures repeated over and over.

Show an EXAMPLE:

core

The core of the pattern is the part that repeats. Circle it. Write several more examples, such as:

a) b)

c) R A T R A T R A d) S T O P S T O P S e) u 2 u 2 u 2 u 2 u

Show the pattern M O M M O M and extend it as follows: M O M M O M O M M O M. Ask students if they

agree with the way you extended the sequence. You may pretend not to see your mistake or even to ask

the students to grade your extension. Give more difficult examples of sequences where the core starts and

ends with the same symbol, such as:

a) b) A Y A A Y A A Y c) 1 2 3 1 1 2 3 1 1

Warn your students that these patterns are trickier.

Explain that the “length” of the core is the number of terms in the core.

For instance, the pattern 3, 2, 7, 3, 2, 7… has a core of length 3. It’s core is “3, 2, 7”. Ask volunteers to find

the length of the cores for the examples you’ve drawn.

Assessment:

Circle the core of the pattern, and then continue the pattern:

a) C A T D O G C A T D O G C A T ___________________________________ Core length: _____

b) A M A N A A M A N A A M A ______________________________________ Core length: _____


Circle the core of the pattern, and then continue the pattern.

Activity: Give your students a set of beads or pattern blocks and ask them to make a repeating pattern in

which one or more attributes change. Ask them to describe precisely how their pattern changes: EXAMPLES:

a)

In this example only the size of the shape changes.

b)

In this example both the size and the colour of the shape changes.

Extensions:

1. Can you always find the core of a pattern that repeats in some way? Do the following sequences have a

core? No. Can you still continue the pattern?

a) b)

2. What are the similarities and the differences of the two patterns of coloured counters below?

Pattern 1

Pattern 2

Explain to your students that these patterns were made by two different students according to the same

rule. Both students had counters of two colours only, and the rule described the way the attributes

changed: change colour, change colour, stay the same, change colour, change colour, stay the same,

repeat. Ask your students to describe the following patterns in terms of the change of attributes:

a)

b)

R

R

R

R

B R R B B B

B R R B B B

R

R

R

R

R W R R R W

B Y B B B Y


3. Make a pattern of your own with blocks or beads, changing at least two attributes. Explain which

attributes you used in making your pattern, and how the attributes changed. Try to make both

sequences that have cores and sequences that do not. To make a pattern, you can change the colour,

shape, size, position or orientation of a figure or you can change the number of times a figure

occurs. Try to describe the change in the attribute as a rule.

EXAMPLES:

The shape changes, the size, colour and position stay the same.

The core consists of two squares and two circles.

The rule for the change of attributes is: Start with a small square. Stay the same, change shape, stay the

same, change shape, repeat.

The size changes, the shape, colour and position stay the same.

The core consists of one large square and two small squares.

The rule for the change of attributes is: Start with a large square. Change size, stay the same, change

size, change size, stay the same, change size, repeat.

The size and position change, the shape and the colour stay the same.

The core consists of one large square and two small squares.

The rule for the change of attributes is: Start with a large square. Change size and turn the square, turn

the square, change size, change size and turn the square, turn the square, change size, repeat.

Draw your pattern.

3. Ask your students to name some attributes of articles of clothing. For instance, shirts may have:

long sleeves short sleeves collars no collars stripes checks

• Make a list of the attributes on the board.

• Think of a simple repeating pattern that involves those attributes.

• Arrange students who are wearing articles of clothing that fit your pattern in a row.

• The rest of the class should try to guess the rule for your pattern.

Make your students aware that patterns result from repeating an action (say in music), operation,

transformation or in making some other change (colour, shape orientation, etc).


PA3-11 Extending Repeating Patterns

Goal: Students will continue a repeating pattern given its core.

Prior Knowledge Required: Ability to count. Attributes.

Vocabulary: attribute, core, length of core, repeating pattern

Draw several groups of figures that are the cores of repeating patterns. Ask your students to continue the

patterns. You might also use the activity below for practice.

Draw several cores and extend some of the patterns in a wrong way. Ask the students to grade your work,

and then to correct your mistakes. Sample sequences:

R W W R W W R R W W R W W R R W W R W W R W R W R W R

Assessment:

Check if the following sequences were extended correctly and if not, write the correct extensions.

R W W R W W R R W W R W R R R W R R R W R W R W R W R

A A B B A A B B A C A C A A C A C A G Y G G Y G G Y

Activity: A game in pairs: The students will need a pair of dice and a set of coloured beads of different

shapes, patterns and sizes. If beads are not available, students might use pattern blocks or draw the

patterns. Make a list of at least 6 attributes of the shapes. The first player rolls the dice. He has to build the

core of the pattern, where the larger number on the dice is the length of the core, the other number is the

number of attributes that should change in the core. The second player has to continue the pattern and to

name the attributes that changed in the core.

For example, if the roll is 5 and 3; the attributes that change might be pattern, size and orientation:


PA3-12 Finding Cores in Patterns

Goal: Students will find cores of repeating patterns.

Prior Knowledge Required: Ability to count. Attributes. Ordinal numbers.


Show your students how to recognize the core of a pattern. Draw a sequence of blocks with a simple

repeating pattern like the shown one below (where B stands for blue and Y for yellow).

Then draw a rectangle around a set of blocks in the sequence and follow the steps below:

STEP 1:

Ask students to say how many blocks you have enclosed in the rectangle (in this case, three).

STEP 2:

Ask students to check if the pattern (BYY) inside the rectangle recurs exactly in the next three blocks of the

sequence, and in each subsequent block of three, until they have reached the end of the sequence (if you

had enclosed four blocks in the rectangle, then students would check sets of four, and so on). If the pattern

in the rectangle recurs exactly in each set of three boxes, as in the diagram above, then it is the core of the

sequence. Otherwise students should erase your rectangle, guess another core and repeat steps 1 and 2.

(Students should start by looking for a shorter core than the one you selected.)

You could also have your students do this exercise with coloured blocks, separating the blocks into sets,

rather than drawing rectangles around them. Draw several sequences on the board and circle the cores in

some of them in the wrong way. Ask your students to grade your work. If they think you have not got it right,

ask them to correct the mistake.

EXAMPLES:

A A L L A A L L A A A S A S A A S A S A G E G G E G G E

Find the core and the length of the core:

R E D D R E D D D A D D A D D A G R E G G R E G

B Y Y B Y Y B Y Y

B Y Y B Y Y B Y Y


Draw a pattern on the board:

Ask volunteers to circle the core of the pattern, find its length and continue the pattern. Ask which figures are

circles. (1st, 3

rd, 5

th, etc) What will the 20

th figure be – a circle or a diamond? Write down the sequence of the

places for circles (1, 3, 5, 7, …) and diamonds (2, 4, 6, 8, …), and ask students to say which sequence the

number 20 belongs to. Challenge them to predict the shapes of 36th, 55

th, 99

th figures.


PA3-13 Making Patterns with Squares and PA3-14 Making Patterns with Squares (Advanced) 14

Goal: Students will make patterns with blocks and squares.

Prior Knowledge Required: Pattern extension


Lay out a couple of blocks and ask volunteers to add blocks in the positions you indicate. After that draw

several simple shapes on the board and ask more volunteers to draw an extra block in each of the places

marked by an arrow.

Sample problems:

After that draw several sequences of blocks, first adding only one block at a time, then two and or more

blocks, and ask your students to identify which blocks were added each time.

Assessment:

1. Shade two squares that were added to the figure.

2. Shade the squares added to each previous figure to get the next one in the pattern. Then draw the next

figure in the pattern on grid paper.

Activities:

1. Ask your students to build the sequence of QUESTION 5 a) of the worksheets using blocks. Then ask

them to build the next figure in the sequence. Repeat for QUESTIONS 5 b), c), and d).

2. Teach your students the terms “horizontal”, “vertical”, “column”, “row”, “right” and “left”. Ask them to

describe as precisely as they can (using mathematical terms) how the changes in QUESTION 5 a)

occur. For instance, they might say “the horizontal arm grows by one block each time” or even more

precisely: “You start with a column of 2 blocks. Place one block beside the top block on the right hand

side. Then place another block beside this one…” and so on.


3. Have two students sit across from each other at a table. Place a barrier between the students to that

they can see each other but neither student can see the part of the table directly in front of the other

student. The first student builds a sequence of figures that grows in a regular way (as in QUESTION 5 a)

and describes their sequence as they build it. The second student has to try to build the same sequence.

The students then raise the barrier to see if their sequences match.

Extension: Give each student a set of 40 blocks of the same shape (either squares or triangles) and ask

them to build a sequence of 3 shapes in which the number of blocks in each shape grows by a fixed amount

(as in QUESTION 1). Have them count the number of blocks they have left after making 3 shapes and

predict using a T-table or calculator whether they have enough blocks to build the next step (or the next two

steps) in the pattern.


PA3-15 Extending a Pattern Using a Rule

Goal: Students will extend patterns using a verbal rule.

Prior Knowledge Required: Addition. Subtraction. Extension of patterns. Sequence.

Vocabulary: pattern rule

This section provides a basic introduction to pattern rules. For more advanced work including word

problems, applications and communication, see sections PA3-20 TO 32.

Write a number sequence on the board: 10, 13, 16, 19, …

Draw the circles for differences.

10 , 13 , 16 , 19 , _____

Then ask them if they could describe the sequence without simply listing the individual terms. For a

challenge you might ask students to work in pairs. One student writes an increasing sequence without

showing their partner. The student must get their partner to reproduce their sequence but they are not

allowed to give more than one term of the sequence. Students should see that the best way to describe a

sequence is to give the first term and the difference between terms. The sequence above could be

described by the rule “Start at 10 and add 3 to get the next term.”

For practice, give your students questions such as:

Continue the patterns:

a) (add 3) 14, 17, ____ , ____ , ____ , ____ b) (subtract 2) 35, 33, ____ , ____ , ____ , ____

c) (add 5) 12, 17, ____ , ____ , ____ , ____ d) (subtract 10) 77, 67, ____ , ____ , ____ , ____

e) (add 4) 15, 19, ____ , ____ , ____ , ____ f) (subtract 5) 97, 92, ____ , ____ , ____ , ____

Create a pattern of your own: ____ , ____ , ____ , ____

My rule: ___________________________________________________________________________

For practice you might assign the game in the activity below.

Tell your students the following story and ask them to help the friends:

Bonnie and Leonie are two Grade 3 friends who are doing their homework together. The sequence in

their homework is: 43, 36, 29, 22, … Bonnie said: “If you count backwards from 43 to 36 you say eight

numbers:

1 2 3 4 5 6 7 8

43, 42, 41, 40, 39, 38, 37, 36


So the difference between 43 and 36 is 8.”

Leonie said “If you count the number of “hops” on a number line between 43 and 36, you see there are

7 hops so the difference is 7.”

Who is right?

Assessment:

Continue the patterns:

a) (add 4) 32 , 36 , ____ , ____ , ____ , ____ b) (subtract 6) 77 , 71 , ____ , ____ , ____ , ____

Create a pattern of your own: ____ , ____ , ____ , ____

My rule: ___________________________________________________________________________

Activity:

A Game

The students need a die and a coin. The coin has the signs “+” and “–” on the sides. The player throws the

coin and the die and uses the results to write a sequence. The die gives the difference. The coin indicates

whether the difference is added or subtracted (according to the sign that faces up). The player chooses the

first number of the sequence. The game might also be played in pairs, where one of the players checks the

results of the other and supplies the initial term of the sequence. Give one point for each correct term.

Extension: Extend the sequences according to the rules:

a) (add 15) 12 , 27 , ____ , ____ , ____ , ____

b) (subtract 12) 177 , 165 , ____ , ____ , ____ , ____

c) (add 102) 12 , 114 , ____ , ____ , ____ , ____

d) (subtract 21) 276 , 255 , ____ , ____ , ____ , ____

e) (multiply by 2) 3 , 6 , ____ , ____ , ____ , ____ , ____

f) (add two previous numbers) 1, 2 , 3 , 5 , 8 , ____ , ____ , ____ , ____

36 37 38 39 40 41 42 43


PA3-16 Identifying Pattern Rules

Goal: Students will identify simple pattern rules.

Prior Knowledge Required: Addition. Subtraction. Ordinal Numbers

Vocabulary: increasing sequence, decreasing sequence, term

Tell your students that today you will make the task from the previous lesson harder—you will give them a

sequence, but you will not tell them what the rule is; they have to find it themselves. Give them an

example—write a sequence on the board: 2, 6, 10, 14… Is it an increasing or a decreasing sequence?

What was added? Let students practice with questions like:

Write the rule for each pattern:

a) 69, 64, 59, 54 subtract ____

b) 98, 96, 94 subtract ____

c) 25, 28, 31, 34 add ____

d) 35, 39, 43, 47 add ____

e) 43, 54, 65, 76 add ____

f) 119, 116, 113 subtract ____

Tell another story about pattern fans Bonnie and Leonie: The teacher asked them to write a sequence that

is given by the rule “add 2”. Bonnie has written the sequence: “3, 5, 7, 9…”. Leonie has written the

sequence: “2, 4, 6, 8…”. They quarrel—whose sequence is the right one?

Ask your students: how can you ensure that the sequence given by a rule is the one that you want? What

do you have to add to the rule? Students should see that you need a single number as a starting point.

Write several sequences on the board and ask your students to make rules for them.

Explain to your students that mathematicians need a word for the members of the sequences. If your

sequence is made of numbers, you might say “number”. But what if the members of the sequence are

figures? Show an example of a pattern made of blocks or other figures.

Tell the students that the general word for a member of a sequence is “term”.

Form a line of volunteers. Ask them to be a sequence. Ask each student in the sequence to say in order:

“I am the first/second/third… term.” Ask each term in the sequence to do some task, such as: “Term 5, hop

3 times”, or “All even terms, hold up your right arm!” Ask your sequence to decide what simple task each of

them will do, and ask the rest of the class to identify which term has done what. Thank and release your

sequence.

MORE PRACTICE:

a) What is the third term of the sequence 2, 4, 6, 8?

b) What is the fourth term of the sequence 17, 14, 11, 8?

c) Extend each sequence and find the sixth term:

(i) 5, 10, 15, 20 (ii) 8, 12, 16, 20 (iii) 131, 126, 121, 116


Bonus:

What operation was performed on each term in the sequence to make the next term:

2, 4, 8, 16… (HINT: neither “add”, nor “subtract”. ANSWER: Multiply the term by 2.)

Activity: Divide your students into groups or pairs and ask them to play a game: One player or group

writes a sequence, the other has to guess what the rule is. Each correctly solved problem gives 5 points.

Warn them that the rules have to be simple—otherwise they will have to give clues. Each clue lowers the

score by 1 point for both players or groups.

Extensions:

1. Find the mistake in each pattern and correct it.

a) 2, 5, 7, 11 add 3

b) 7, 12, 17, 21 add 5

c) 6, 8, 14, 18 add 4

d) 29, 27, 26, 23 subtract 2

e) 40, 34, 30, 22 subtract 6

2. Divide the students into groups or pairs. One player writes a rule, the other has to write a sequence

according to the rule, but that sequence must have one mistake in it. The first player has to find the

mistake in the sequence. Warn the students that making mistakes on purpose is harder than simply

writing a correct answer, because you have to solve the problem correctly first! The easiest version is to

make the last term wrong. More challenging is to create a mistake in any other term.

3. Find the missing number in each pattern. Explain the strategy you used to find the number.

a) 2, 4, ____, 8

b) 9, 7, ____, 3

c) 7, 10, ____, 16

d) 16, ____, 8, 4

e) 3, ____, 11, 15

f) 15, 18, ____, 24, ____, 30

g) 14, ____, ____, 20

h) 57, ____, ____, 45

4. One of these sequences was not made by a rule. Find the sequence and state the rules for the other

two sequences. (Identify the starting number and the number added or subtracted.)

a) 25, 20, 15, 10 b) 6, 8, 10, 11 c) 9, 12, 15, 18


PA3-17 Introduction to T-tables

Goal: The students will be able to create a T-table for growing block patterns and to identify rules of

number patterns using T-tables.

Prior Knowledge Required: Addition

Skip counting

Subtraction. Number patterns.

Vocabulary: T-table, growing pattern, chart, core, term

Draw the following sequence of figures on the board and tell your students that the pictures show several

stages in the construction of a castle made of blocks:

Ask your students to imagine that they want to keep track of the number of blocks used in each stage of the

construction of the castle (perhaps because they will soon run out of blocks and will have to buy some

more). A simple way to to keep track of how many blocks are needed for each stage of the construction is

to make a T-table (the central part of the chart resembles a T- hence the name). Draw the following table on

the board and ask students to help you fill in the number of blocks used in each stage of construction.

Figure Number of

Blocks

1 4

2 6

3 8

Ask students to describe how the numbers in the table change—they should notice that the number of

blocks in each successive figure increases by 2, or that the difference between successive terms in the right

hand column is 2. Write the number 2 (the “gap” between terms) in a circle between each pair of terms.

Figure Number of

Blocks

1 4

2 6

3 8

2

2


Ask students if they can state a rule for the pattern in the table (Start at 4 and add 2) and predict how many

blocks will be used for the fifth figure. Students should see that they can continue the pattern in the chart

(by adding the gap to each new term) to find the answer. Point out that the T-table allows them to calculate

the number of blocks needed for a particular structure even before they have built it.

Draw the following sequence of figures on the board and ask students to help you make a T-table and to

continue the table up to five terms.

Before you fill in any numbers, ask your students if they can predict the gap between terms in the T-table.

They should see—even without subtracting terms in the table—that the gap between terms is 3 because

the castle has three towers and you add one block to each tower at every stage of the construction.

Draw the following T-table on the board:

Figure Number of

Blocks

1 13

2 18

3 23

Tell students that the T-table gives the number of blocks used at each stage in the construction of a castle.

The castle was built in the same way as the others (towers are separated by a gate with a triangular roof,

and one block is added to each of the towers at each stage), but this particular castle has more towers. Ask

students if they can guess, from the pattern in the number of blocks, how many towers the castle has. From

the fact that the gap is 5, students should see that the castle must have five towers. Ask a student to come

to the board to draw a picture of the first stage in the building of the castle. Then ask students to help you

extend the T-table to five terms by adding the gap to successive terms.

After completing these exercises, give students a quiz with several questions like QUESTIONS 1 a) and

2 a) on the worksheet, or have them work through the actual questions on the worksheet.

Activities:

1. Ask your students to construct a sequence of shapes (for instance, castles or letters of the alphabet)

that grow in a fixed way. You might also use pattern blocks for this activity.


Ask students to describe how their pattern grows and to predict how many blocks they would need to

make the 6th figure. (If students have trouble finding the answer in a systematic way, suggest that they

use a T-table to organize their calculation.) This activity is also good for assessment.

2. Give each student a set of blocks and ask them to build a sequence of figures that grows in a regular

way (according to some pattern rule) and that could be a model for a given T-table. Here are some

sample T-tables you can use for this exercise.

Extensions:

1. A castle was made by adding one block at a time to each of four towers. Towers are separated by a

gate with a triangular roof. Altogether 22 blocks were used. How high are the towers? How many

blocks are not in the towers?

2. Claude used one kind of block to build a structure. He added the same number of blocks to his structure

at each stage of its construction. He made a mistake though in copying down the number of blocks at

each stage. Can you find his error and correct it?

3. You want to construct a block castle following the steps shown below. You would like each tower to be

5 blocks high. Each block costs five cents and you have 80 cents altogether. Do you have enough

money to buy all the blocks you need? (HINT: Make a T-table with three columns: Figure, Number of

Blocks and Cost).

STEP 1 STEP 2 STEP 3

Figure Number of

Blocks

1 4

2 6

3 8

Figure Number of

Blocks

1 3

2 7

3 11

Figure Number of

Blocks

1 1

2 5

3 9

Figure Number of

Blocks

1 5

2 7

3 11

4 14


PA3-18 T-tables

Goal: Students will extend patterns using T-tables.

Prior Knowledge Required: Addition. Subtraction. Skip counting. Number patterns.

Ordinal numbers.

Vocabulary: T-table, chart, term

Draw several figure patterns on the board (or build them using one type of pattern block for each pattern).

Ask your students to predict how many blocks you will need for the next figure in each pattern. Ask them how

they made their prediction.

Invite volunteers to draw T-tables for the patterns you built. Use more volunteers to extend the tables and to

check the results by building the next figure in the pattern. For one of the simpler figures you might say that

each block costs two cents and ask the students to find the cost of each figure in the pattern. Ask students

to make a new table for the cost of the figures. For the figures in the picture, you may also ask: Each block

costs 3 cents. I have 40 cents; will that be enough to build the sixth figure?

Assessment:

Shade the blocks added to each figure to make the next one.

How many blocks will the 6th figure in the pattern have?

Extension: In many sequences that students will encounter in life—for instance in a recipe—two

quantities will vary in a regular way.

If a recipe for muffins calls for 3 cups of flour for every 2 cups of blueberries, you can keep track of how

many cups of each ingredient you need by making a double chart. Give students several questions about

recipies that they can solve with a double chart.

Number of

Muffins Trays

Number of Cups

of Flowers

Number of Cups

of Blueberries

1 3 2

2 6 4

3 9 6


PA3-19 Problems and Puzzles

This worksheet is the final review and may be used for practice.

Workbook 3 - Patterns & Algebra, Part 1 1BLACKLINE MASTERS

Hundreds Charts _______________________________________________________2

Number Line Chasing Game ______________________________________________3

PA3 Part 1: BLM List



Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100


Number Line Chasing GameS

tart

Fin

ish

10

80

50

40

30

70

1

10

0

2 8

116 9

14

12

16

1922

24

26

29

32

34

36

39

44

38

42

46

41

59

48

49

51

52

54

56

61

58

62

69

64

66

68

71

74

79

76

72

78

81

84

86

89

82

96

92

91 90

18

= M

ISS

A T

UR

N

= T

AK

E A

NO

TH

ER

TU

RN

= G

O T

O T

HE

NE

XT

= G

O T

O T

HE

PR

EV

IOU

S10

20

4

21

312

8

60

88

94

98 99

Number Sense Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

NS3-1 Place Value – Ones, Tens and Hundreds

Goals: Students will identify the place value of digits in 2- and 3-digit numbers.

Prior Knowledge Required: Number Words — one, ten, hundred— and their corresponding

numerals

Vocabulary: the numbers from 1–10, both the sounds and the numerals

Photocopy the BLM “Place Value Cards” and cut out the three cards. Write the number 321 on the board,

leaving extra space between all the digits, and hold the “ones” card under the 3.

ASK: Did I put the card in the right place? Is 3 the ones digit? Have a volunteer put the card below the

correct digit. Invite volunteers to position the other cards correctly. Cards can be affixed to the board

temporarily using tape or sticky tack.

Now erase the 3 and take away the hundreds card. ASK: Are these cards still in the right place?

Write the 3 back in, put the hundreds card back beneath the 3, erase the 1, and remove the ones card.

ASK: Are these cards still in the right place? Have a volunteer reposition the cards correctly. Repeat this

process with 3 1 (erase the 2).

Write 989 on the board and ask students to identify the place value of the underlined digit. (NOTE: If you

give each student a copy of the BLM “Place Value Cards,” individuals can hold up their answers. Have

students cut out the cards before you begin.) Repeat with several 2- and 3-digit numbers that have an

underlined digit.

Vary the question slightly by asking students to find the place value of a particular digit without underlining it.

(EXAMPLE: Find the place value of the digit 4 in the numbers: 401, 124, 847.) Continue until students can

identify place value correctly and confidently. Include examples where you ask for the place value of the digit 0.

Then introduce the place value chart and have students write the digits from the number 231 in the correct

column:

Do more examples together. Include numbers with 1, 2, and 3 digits and have volunteers come to the board

to write the numbers in the correct columns.

Extensions:

1. Teach students the Egyptian system for writing numerals, to help them appreciate the utility of

place value.

1 = (stroke) 10 = (arch) 100 = (coiled rope)

Hundreds Tens Ones

a) 231 2 3 1


Write the following numbers using both the Egyptian and our Arabic systems:

234

848

423

Invite students to study the numbers for a moment, then ASK: What is different about the Egyptian system

for writing numbers? (It uses symbols instead of digits. You have to show the number of ones, tens, and so

on individually—if you have 7 ones, you have to draw 7 strokes. In our system, a single digit (7) tells you how

many ones there are.) Review the ancient Egyptian symbols for 1, 10, and 100, and ask students to write a

few numbers the Egyptian way and to translate those Egyptian numbers into regular numbers (using Arabic

numerals). Emphasize that the order in which you write the symbols doesn’t matter:

234 = =

ASK: Does the order in which you write regular digits matter? Is 234 the same as 423? In the Egyptian way,

does the value of a symbol depend on its place? In our way, does the value of a digit depend on its place?

Are the ones, tens and so on always in the same place in our system? In the Egyptian system? Why is our

way called a place value system?

Have students write a number that is really long to write the Egyptian way (EXAMPLE: 798). ASK: How is our

system more convenient? Why is it helpful to have a place value system (i.e. the ones, tens, and so on are

always in the same place)? Having a place value system allows you to use the same symbol to mean many

different values. The digit 7, for example, can mean 7 ones, 7 tens or 7 hundreds depending on where it is

in the number.

Students might want to invent their own number system using the Egyptian system as a model.

2. Have students identify and write numbers given specific criteria and constraints.

a) Write a number between 30 and 40.

b) Write an even number with a 6 in the tens place.

c) Write a number that ends with a zero.

d) Write a 2-digit number.

e) Write an odd number greater than 70.

f) Write a number with a tens digit one more than its ones digit.

Harder

g) Which number has both digits the same: 34, 47, 88, 90?

h) Write a number between 50 and 60 with both digits the same.

i) Find the sum of the digits in each of these numbers: 37, 48, 531, 225, 444, 372.

j) Write a 2-digit number where the sum of the digits is 11.

k) Write a 2-digit number where the digits are the same and the sum of the digits is 14.

l) Write a 3-digit number where the digits are the same and the sum of the digits is 15.


Bonus:

Is there a 2-digit number satisfying the same conditions?

m) Which number has a tens digit one less than its ones digit: 34, 47, 88, 90?

n) Write a 2-digit number with a tens digit eight less than its ones digit.

o) Write a 3-digit number where all three digits are odd.

p) Write a 3-digit number where the ones digit is equal to the sum of the hundreds digit

and the tens digit.

Make up more such questions, or have students make up their own.


NS3-2 Place Value

Goals: Students will understand the value of digit in 2-, and 3-digit numbers.

Prior Knowledge Required: Place Value: Ones, Tens, Hundreds

Vocabulary: ones, tens, and hundreds digit, value

Write 836 on the board. SAY: The number 836 is a 3-digit number. What is the place value of the digit 8?

(If necessary, point to each digit as you count aloud from the right: ones, tens, hundreds). SAY: The 8 is in

the hundreds place, so it stands for 800. What does the digit 3 stand for? (30) The 6? (6)

Explain that 836 is just a short way of writing 800 + 30 + 6. The 8 actually has a value of 800, the 3 has a

value of 30, and the 6 has a value of 6. Another way to say this is that the 8 stands for 800, and so on.

ASK: What is 537 short for? 480? 35? 601? Write out the corresponding addition statements for each

number (also known as the expanded form).

ASK: What is the value of the 6 in 608? In 306? In 762? In 506?

ASK: In the number 831, what does the digit 3 stand for? The 1? The 8?

ASK: What is the value of the 0 in 340? In 403? In 809? Emphasize that 0 always has a value of 0, no matter

what position it is in.

ASK: In the number 856, what is the tens digit? Ones? Hundreds? Repeat for 350, 503, 455, 770, 820.

Write the following numbers on the board: 350, 503, 435, 537, 325, 753. Ask students to identify which digit,

the 5 or the 3, is worth more in each number. Students should be using the phrases introduced in the

lesson—stands for, has a value of, is short for. (EXAMPLE: In 350, the 5 stands for 50 and the 3 stands for

300, so the digit 3 is worth more.)

Extension: If your students are familiar with the concept “how many times more”, ASK: What is the value

of the first 1 in the number 11? What is the value of the second 1? How many times more is the first 1 worth

than the second 1? Repeat with more numbers in which the digit 1 is repeated (EXAMPLES: 131, 110, 101,

211, 171).


NS3-3 Writing and Reading Number Words

Goals: Students will read and write number words to twenty and multiples of ten up to ninety.

Prior Knowledge Required: Reading and writing number words to ten

Place value (ones and tens)

Saying the alphabet

Vocabulary: numeral, number word, ones and tens digits

Write the following words on the board, all in a row:

eighteen thirteen seventeen sixteen nineteen fifteen

Ask the class to read the words out loud together and then ask volunteers to write the corresponding

numerals under the words.

ASK: What number does the word “teen” remind you of? Guide them by asking them to look at the letters—is

it spelled almost the same as a number they know? Tell them that eighteen is 8 + 10 = 18. ASK: Where can

you see “eight” in eighteen? Where can you see a word that looks like “ten” in the

word eighteen?

Have volunteers fill in the blanks with the correct number words:

a) fourteen = ________ + ten b) seventeen = ________ + seven

c) eighteen = ________ + _______ d) nineteen = ________ + __________

e) thirteen = ________ + _______ f) fifteen = ________ + __________

g) _______ = six + ten h) twelve = ________ + __________

i) eleven = ________ + _______

Have individual students write the missing words in their notebooks:

a) sixteen = ________ + ten b) seventeen = ________ + ten

c) nineteen = nine + _______ d) thirteen = ________ + ten

e) fourteen = ________ + four f) fifteen = ________ + ten

Have student volunteers circle the beginning letters that are the same.

a) six sixteen b) five fifteen c) nine nineteen

d) four fourteen e) three thirteen f) two twelve

Then, for each pair above, have students write the correct numerals in their notebooks and to circle the digits

that are in common.

Repeat the above exercise with ending letters instead of beginning letters for the following pairs.

a) thirteen fourteen b) seventeen eighteen c) nineteen fifteen

Then write on the board: twenty = 20 two = 2

ASK: What two beginning letters do those words have in common? (tw) What digit is in both numbers? (2)


Write on the board: thirty. ASK: Can anyone think of a word for a 1-digit number that starts with the same two

letters? (three) Then write: thirty = 0 three = 3

Have a volunteer fill in the blank.

Write: forty = 0 fifty = 0 thirty = sixty =

Have volunteers fill in the blanks by looking carefully at the beginning letters and asking themselves what

one-digit number those letters remind them of.

ASK: What ones digit do these numbers all have? What letter do the words all end with? Tell them that any

number word ending with “y” will always mean a number having ones digit 0.

Ask volunteers to guess how the following number words are written as numbers:

eighty ninety seventy

Challenge them to find a 2-digit number having ones digit 0 whose number word doesn’t end with “y”. (10)

Have students write the numerals for the following number words individually:

a) thirty thirteen three b) twenty two twelve

c) four fourteen forty d) eighteen eighty eight

e) seven ninety thirteen eighty nine fourteen

f) nineteen sixty forty fifteen twelve eight

Have students write individually the number word ending for these words:

a) 30 = thir____ b) 20 = twen ______ c) 13 = thir_____

d) 17 = seven____ e) 40 = for____ f) 80 = eigh___

g) 18 = eigh____ h) 19 = nine_____ i) 90 = nine____

Finally, have students write the full number words:

a) 20 = _______ b) 19 = _______ c) 90 = ________ d) 17 = __________

e) 13 = _______ f) 80 = _______ g) 50 = ________ h) 15 = __________

Activity: On the web-site: http://www.funbrain.com/numwords/index.html students can use Method 1

to write the number word in the correct place on the cheque or use Method 2 to read the number word and

write the correct numeral. You may choose between numbers from 0 to 10, 0 to 100, 0 to 1000 or 0 to

10 000, depending on the level of your students.

Extensions:

1. Provide the BLM “Number Word Search.” Encourage students to use the message they find after

finishing the puzzle as a way to check that they did the puzzle correctly.

2. Write the alphabet on the board with enough spacing between the letters to circle some of them.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Write the word “act” on the board and ask a volunteer to circle, in the list, the letters that appear in the

word “act”. ASK: Are the letters in the same order in the word “act” as they are in the alphabet?


Have another student, using a different colour of chalk, circle the letters from the word “sun”.

ASK: Are the letters in the same order in the word “sun” as they are in the alphabet? What order do they

appear in the alphabet? (n-s-u). Have students decide whether or not each of the following words are

written alphabetically: bat, box, cat, mom, snow, most, now, win, lose, knot, knots, stone, ghost.

Challenge students to find the longest alphabetical word that they can.

ASK: Is “dog” alphabetical? Is “doghouse” alphabetical? Fun? Funny? On? One? Pony? Phone? Bone?

Top? Stop? Tops?

Tell your students that you know that since “on” is not alphabetical, you know that the following words

cannot be alphabetical either: pony, money, gone, only. Ask them to explain your thinking.

Then make the connection to number words: Are any of the number words from one to ten written

alphabetically? Eleven to twenty? Did they need to check all the number words from eleven to twenty? Is

there a sequence of letters common to many of the number words? (teen is in many of them and is not

alphabetical, so we don’t even need to check thirteen to nineteen)

Which of the following multiples of ten is written alphabetically?

a) ten b) twenty c) thirty d) forty e) fifty f) sixty

3. Make a chart on the board with headings as follows:

3 letters 4 letters 5 letters 6 letters 7 letters 8 letters 9 letters

Have student volunteers write number words that fit in each column. Students should use number words

from zero to twenty as well as multiples of ten up to ninety (thirty, forty, and so on to ninety). When most

words are on the list, draw the following puzzle on the board:

Tell students that we want to solve this puzzle using number words. Point to the vertical group of 3

squares and ask students if the word FIVE will fit. Why not? How many letters does the word need to

have to fit? Refer your students to the list of 3-letter words they made and ask if there are any they

missed.

THEN SAY: How many letters should the other word have? Repeat the chart for words with 4 letters

(zero, four, five, nine).

Then tell students that one of the letters from the 3-letter word has to be the same as one of the letters

from the 4-letter word. Ask if they can tell which letter from each word needs to overlap the other word.

Have a volunteer circle the second letter from each 3-letter word and have another volunteer circle the

first letter from each 4-letter word. Tell them that the 2nd

letter from the 3-letter word is either n, w, i or e

and that the 1st letter from the 4-letter word is either f, f or n. Tell them that if there are going to be words

that fit in the puzzle, there had better be a letter in both lists. What letter is in both lists? (n) Which 3-letter

word has n as its second letter? (one) Which 4-letter word starts with n? (nine) Write the words into the

puzzle for them. Below are more puzzles (with the answers in brackets) your students can practice with.


(zero, one) (four, one) (one, zero (five, three) (zero, four

or two, four) or nine, three) or five, zero

or nine, zero)

(four, five) (five, nine) (fourteen, one)

(six, sixteen, ten) (seventy thirty, four) (eight, fifty, five, ten)

4. Give students the BLM “Number Words Crossword Puzzle”.

5. Give students the BLM “Crossword Without Clues”.

6. Hand out the BLM “Recognizing Number Words”. The sheet asks students to circle the number words

and to cross out the words that only sound like number words. Have a copy of the BLM on the board or

overhead projector. Read the page out loud and point to the words as you say them. Give lots of hints.

For example, “Eight children ate pie”. What were the people in this sentence doing? Were they sleeping,

playing, eating or working? What were they eating? How many children ate pie?” Repeat the sentence

several times so that all students can see that “eight” is the number word and “ate” only sounds like a

number word. Remind the students that they should circle the number words and cross out the words

that only sound like number words. When a word sounds like a number word other than the one in the

sentence, students will benefit from hearing you read the sentence out loud and then saying some of the

number words from one to ten and then repeating the sentence out loud as often as necessary. When all

students have correctly done this sheet, hand out the BLM “Spelling Number Words” and have students

look at their completed sheet to answer the questions. This sheet will give students a taste of how they

can use the context of words to figure out the correct spelling. It will also show them that some words

that sound the same can be spelled differently.


NS3-4 Writing Numbers

Goal: Students will read and write number words up to nine hundred ninety-nine. Prior Knowledge Required: Reading and writing number words to twenty and

multiples of ten to ninety

Vocabulary: numeral, number word, digit

Write “twenty” on the board and ask a volunteer to write the corresponding numeral. Ask them what number

they think the number word “twenty-three” means. Can they think of an addition sentence from this word?

(20 + 3 = 23) Repeat for twenty-seven and twenty-one. Have students individually write the numbers for the

following words:

twenty-two twenty-five twenty-nine twenty-six twenty-eight twenty-four

Then write: thirty-six. SAY: if thirty means 30 and six means 6, what number do you think thirty-six means?

What addition sentence can you write from that? (30 + 6 = 36) To help them find 30 + 6, provide a number

line or use a metre stick as a number line. Show them where 30 is on the number line so that they just have

to move ahead six places.

Have a volunteer write the number for thirty-five with the addition sentence (35 = 30 + 5), then have students

write the numbers with addition sentences for each number word below:

thirty-three thirty-two thirty-eight thirty-four

Provide them with a number line so that they can see how to add the numbers.

Show them where to find 10, 20 and 30 on the number line and then challenge them to find 40 on the

number line. Have a volunteer write the 2-digit number ” forty-seven” on the board by looking at a number

line and adding the two parts of the number they see. Summarize to the class how the volunteer is finding

the number 40 and then adding 7 to find 47. Repeat: thirty-six, twenty-seven, forty-two, thirty-one, forty-five,

fifty-four.

Write the number sentences on the board:

73 = 70 + 3

seventy-three

15 = 5 + 10

fifteen

32 = 30 + 2

thirty-two

18 = 8 + 10

eighteen

54 = 50 + 4

fifty-four

13 = 3 + 10

thirteen

61 = 60 + 1

sixty-one

16 = 6 + 10

sixteen

If available, use an overhead projector and write the parts in bold in a different colour. Point to each question

and ASK: Where do you see the first digit of the number in the number word – at the beginning or at the

end? Which numbers have the first digit at the beginning? (twenty and higher) Which numbers have the first

digit at the end? (thirteen to nineteen).


When you write twenty-seven, where do you see the first digit in the number word? Where do you see

the last digit? Have them compare this with the number word seventeen. Tell them that number words

for numbers twenty and higher are a bit different from what they’ve seen so far because the first digit

is read first and the last digit is read last. Have students individually write the numbers for the following

number words:

thirty-eight forty-five twenty-six thirty-four fifty-one fifty-four

sixty-seven eighty-nine seventy-four ninety-one eighty-eight forty-two

Then have students write numerals for number words between zero and ninety-nine:

twenty-eight eighteen sixteen four forty

forty-three zero fifty fifty-eight thirteen

twelve nineteen twenty-nine fifty-nine forty-eight

thirty-four thirty-one eleven six fifteen

Have students write number words for numerals between 0 and 99:

a) 41 b) 32 c) 90 d) 9 e) 89 f) 74 g) 99 h) 0 i) 50 j) 25 k) 17 l) 11

Invite students to find any mistakes in the way the following number words are written and to correct them

(some are correct):

forty-zero forty-three twenty-eight thirty nine eight-five seventy-six

Summarize the process for writing numbers between 20 and 99: You can write the 2-digit number by writing

the word for the first digit times ten, a hyphen, and then the word for the second digit, as long as it isn’t zero.

If the second digit is zero, you write only the word for the first digit times ten.

EXAMPLE: 35 = 3 x 10 + 5 and is written as thirty-five, but 30 is written as thirty, not thirty-zero.

ASK: How is writing the number words for 11 to 19 different? (They don’t follow the same pattern.)

Write the number words for 11 through 19 on the board and invite students to look for patterns and

exceptions (eleven and twelve are unique; the other numbers have the ending “teen”).

Once students have mastered writing numbers up to 99, tell them that writing hundreds is even easier.

There’s no special word for three hundreds like there is for three tens:

30 = 10 + 10 + 10 = thirty but

300 = 100 + 100 + 100 = three hundred (not three hundreds)

SAY: You just write what you see: three hundred. There’s no special word to remember.

Have students write the number words for the 3-digit multiples of 100: 200, 300, 400, and so on.

Remind them not to include a final “s” even when there is more than one hundred.

Tell students that they can write out 3-digit numbers like 532 by breaking them down. Say the number out

loud and invite students to help you write what they hear: five hundred thirty-two. Point out that there is no

dash between “five” and “hundred.” Have students practice writing number words for many 3-digit numbers.

EXAMPLES: 134, 761, 898, 903, 740, 500, 601. Emphasize that the word “and” should not appear: 301 is

written as “three hundred one” not as “three hundred and one.”

Write some typical text from signs and banners and have students replace any number words with numerals

and vice versa.


EXAMPLES:

a) Montreal 181 km b) Speed Limit – 110 km/h

c) Max. Height 3 m d) Seventy-Four Queen Street

e) Saskatoon next four exits f) Bulk Sale! Buy ten for the price of five!

g) Highway 61 h) Bus Stop: Route 18

i) Montreal Canadiens – j) Top Racing Broom for Witches and Wizards –

24 Stanley Cup Titles! only $599!

Then have students individually write the correct number words in the following sentences:

a) There are ______ months in a year.

b) There are ______ days in a week.

c) There are (52) ______ weeks in a year.

d) February normally has _______ days.

e) A year normally has ______ days.

f) A leap year has _______ days.

Then have students write number words that make sense:

a) There are _______ girls and _____ boys in grade ______ at my school.

b) My house is about _______ city blocks from my school.

c) I can run ______ km in _______ minutes

d) My teacher is about _______ years old.

e) There are about ______ days in summer vacation.

f) My birthday is in about ______ days from now.


NS3-5 Representation with Base Ten Materials

Goal: Students will practice representing numbers with base ten materials.

Prior Knowledge Required: Place value

Base ten materials

Vocabulary: digit, ones digit, tens digit, hundreds digit, ones block, tens block, hundreds block

Photocopy the BLM “Hundreds Chart and Base Ten Materials” onto a transparency if available. Demonstrate

how to find 3 + 4 by taking 3 ones blocks and then another 4 ones blocks and placing them on the chart in

order, so that the last block is on square 7. ASK: How can I find 13 + 5 by using ones blocks and the

hundreds chart?

ASK: How is the counting already done for them when they put the ones blocks on in order?

Emphasize that they can see the answer by looking under the last ones block.

Tell your students that instead of using ten ones blocks to cover a row, you find it easier just to use one

bigger block. Show them a tens block and ask if anyone remembers what the block is called.

Provide your students with the BLM “Hundreds Charts” as well as 10 tens blocks and 9 ones blocks each.

Have students use 3 tens blocks and 5 ones blocks and cover the squares in order. The hundreds charts

were drawn to be 10 cm by 10 cm so that a ones block will cover a grid square exactly. ASK: How many

squares are covered? How do you know? (They should look under the last ones block to see the number

35.) Repeat for several examples. (41, 23, 59, 74, 99) Then ask your students what number they get if they

use two tens blocks and no ones blocks (20). 5 tens blocks? 7 tens blocks? 10 tens blocks?

Tell your students that we used a tens block instead of ten separate ones blocks. ASK: What can we use

instead of 10 tens blocks? (a hundreds block) Give your students 2 hundreds blocks to add to their 10 tens blocks and 9 ones blocks. ASK: What number do you get if you place a hundreds block on the first hundreds chart and then 3 tens blocks and 7 ones blocks in order on the next hundreds chart? Repeat with:

a) 1 hundreds blocks, 5 tens blocks, 4 ones blocks

b) 1 hundreds block, 6 tens blocks, 2 ones blocks

c) 1 hundreds blocks, 7 tens blocks, 5 ones blocks

d) 1 hundreds blocks, 3 ones blocks

e) 1 hundreds blocks, 2 tens block, 2 ones block

f) 1 hundreds blocks, 1 tens block

g) 1 hundreds block, 3 tens blocks

h) 2 hundreds blocks.

Then show models of base ten blocks without using the hundreds chart and have students tell you what

number is represented. EXAMPLES: 3 hundreds blocks, 4 tens blocks and 2 ones blocks; 5 hundreds blocks

and 8 ones blocks.


Now write only the expanded form and have students tell you what number is represented:

a) 7 hundreds + 5 tens + 3 ones

b) 9 hundreds + 0 tens + 6 ones

c) 8 hundreds + 1 ten + 1 one

d) 4 hundreds + 7 tens + 0 ones

Have your students write out the expanded form from the numerals.

EXAMPLE: 790 = 7 hundreds + 9 tens + 0 ones.

Demonstrate drawing a base ten model for 145 on grid paper:

Shade the blocks and ASK: How many little squares are shaded altogether? (145) Have students draw base

ten models for other 2- and 3-digit numbers: 45, 60, 74, 104, 251, 300, 260.

Activities:

1. Give your students ones, tens, and hundreds blocks. Students might work in teams (with each team

scoring a point for each right answer). Students might also sketch their answers (so you can verify that

they have successfully completed the work):

Hundreds block Tens block Ones block

Instruction:

a) Show 17, 31, 252, etc. with base ten blocks.

b) Show 22 using exactly 13 blocks.

c) Show 31 using 13 blocks.

HINT: for b and c: Start with a standard model and trade for blocks of equal value.

Harder

d) Show 315 using exactly 36 blocks.

Extension: Change the order of the words hundreds, tens and ones and have students fill in the blanks.

EXAMPLE: 793 = ____ tens + ____ hundreds + ___ ones


NS3-6 Representation in Expanded Form

Goal: Students will replace a number with its expanded numeral form and vice versa.

Prior Knowledge Required: Place value (ones, tens, hundreds, thousands)

Vocabulary: digit, numeral

Show the base ten model for 145 again:

ASK: How many little squares are coloured? (145) Point to the hundreds block, then the 4 tens blocks and

finally the ones blocks and ask in turn, how many little squares are coloured from each type of block. Then

write on the board: 145 = 100 + 40 + 5. Ask a volunteer to change this to expanded form using the words:

hundreds, tens and ones (1 hundred + 4 tens + 5 ones).

Have students draw base ten models on grid paper for several numbers and to record the expanded form in

two different ways (using numerals and words or numerals only).

EXAMPLES: 135, 241, 129, 302.

Have students expand several numbers using numerals instead of words.

EXAMPLES: 348 = 300 + 40 + 8, 640 = 600 + 40, 301 = 300 + 1.

Ensure students understand that when we have a 0 digit, we do not include 0 in the expanded sum;

70 is just 70, not 70 + 0.

Have students write the numeral for several sums written in this expanded form.

EXAMPLES: 200 + 70 + 6 = ___, 300 + 50 = ____, 300 + 5 = _____

Bonus:

3000 + 400 + 20 + 7 = _______ 5000 + 40 + 9 = ______

Have students write in the missing numbers:

a) 500 + 30 + ____ = 534 b) 641 = 600 + ____ + 1 c) 812 = ____ + 10 + 2

d) 700 + ___ + 2 = 742 e) 400 + ___ = 420 f) 400 + ____ = 402


Bonus:

54 327 = 50 000 + 4 000 + ____ + 20 + 7

Teach your students to draw rough sketches of base ten models: a large square for a hundreds block, a strip

for a tens block and a small square for a ones block:

Then have students write various numbers in expanded form and then draw a rough sketch of a base

ten model. EXAMPLES: 732, 456, 57, 507, 570.

Tell your students that you read one book with 300 pages, and another book with 70 pages. ASK: How many

pages did I read altogether? Have a volunteer write the corresponding addition statement (300 + 70 = 370)

Repeat with several similar word problems:

a) A store has 100 red bikes, 40 blue bikes and 6 green bikes. How many bikes does the store have

altogether? (100 + 40 + 6 = ______ )

b) On a class field trip, there were 200 children, 10 parent volunteers and 7 teachers. How many people

went on the field trip? (200 + 10 + 7 = ______ )

c) Bonus: In a school in Toronto with 498 children, 400 children were from Canada and 90 children were

from the United States. How many were not from Canada or the United States? (498 = 400 + 90 +

______ )

Activity:

I have ---, who has ----?

Using the BLM “Make Up Your Own Cards,” make enough cards so that everyone in the class can have one

(or for everyone in small groups to have one). Use the expanded sum and base ten materials to make the

cards. For example, if the student has the card,

I have

300 + 40 + 8

----------

Who has

?

they say, “I have 348, who has 213?” and the person with 200 + 10 + 3 then says “I have 213, who has ---?”

depending on which base ten model is on the bottom of their card. Play continues until everyone gets a turn.

Ensure that the bottom of the last card matches with the top of the first card, so that students know when

they get back to the first card.

hundreds block

tens block

ones block


Extensions:

1. Fill in the blanks.

a) 200 + 6 + 90 = ____ b) 30 + 800 + 5 = _____ c) 9 + 300 = _____

d) 854 = 50 + 4 + ____ e) 743 = 3 + ____ + 40 f) 912 = ____ + 900 + 2

2. Tell your students that when we write numbers, we write them as sums of 100s, 10s and 1s.

For example,

723 = 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1. Have your students write

various numbers as a sum of 100s, 10s and 1s. EXAMPLES: 341, 213, 411, 315. Then tell your

students that when we talk about money, we write the money as a sum of coin values. ASK: What are

the values of the coins we have that are less than a dollar? (25¢, 10¢, 5¢, 1¢) Show how 13 can be

written in various ways:

13 = 10 + 1 + 1 + 1 or 5 + 5 + 1 + 1 + 1 or 5 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 or a sum of 13 ones.

Show how each of these corresponds to different ways of making 13 cents. ASK: If we were to make a

standard way of writing money, how would we write 13? Why? Emphasize that the standard way should

use the fewest number of coins just as the standard base ten model uses the fewest number of blocks.

Then have students attempt to find the standard way of making:

a) 21¢ b) 16¢ c) 24¢ d) 29¢ e) 56¢ f) 62

3. Have your students demonstrate expanded form in other ways than base 10.

a) Ask your students to show different ways to make 100 as sums of 25, 10, and 5

(i.e. 100 = 25 + 25 + 10 + 10 + 10 + 10 + 5 + 5 or 100 = 2 twenty fives + 4 tens + 2 fives).

b) Show different ways to make 200 as sums of 100, 50, 10.

c) Show different ways to make 1000 as sums of 500, 250, 100.

4. Which numbers are representations of 352?

a) 300 + 50 + 2 or

b) 1 hundred + 20 tens + 2 ones or

c) 2 hundreds + 15 tens + 2 ones or

d) 34 tens + 12 ones

Make up more problems of this sort.

5. Decompose a number in as many different ways as you can.

EXAMPLE:

312 = 3 hundreds + 1 ten + 2 ones or

2 hundreds + 9 tens + 22 ones, etc…

Students could use base ten blocks if it helps them: they can exchange blocks for smaller denominations

to find different representations.


NS3-7 Representation Numbers - Review

Goal: Students will consolidate the learning in number sense done so far.

Prior Knowledge Required: Expanded form of 2- and 3-digit numbers using words and numerals

(e.g. 2 hundreds + 3 tens + 7 ones)

Expanded form of 2- and 3-digit numbers using numerals alone

(e.g. 200 + 30 + 7)

Reading and writing number words for 2- and 3-digit numbers

Base ten models of 2- and 3-digit numbers, including drawing a rough

sketch of hundreds, tens and ones blocks

Give your students ones and tens blocks. ASK: Which numbers have standard base ten models that

can be arranged as rectangles of width at least 2? That is, if tens blocks are arranged horizontally,

there are at least 2 rows.

EXAMPLES:

60 =

30 + any multiple of 3

up to 39 (33, 36, 39)

22 55

Definitely all numbers with identical digits

(A standard base ten model uses the minimum number of blocks. For example, 35 = 3 tens and 5 ones is

standard, 35 = 2 tens + 15 ones is not.) Note that any number can be modelled as a rectangle of width 1,

hence the restriction!

EXAMPLES:

11 =

35 =

NOTE: This is an open-ended activity: there are many possible answers.

Numbers in the 60s are especially interesting to consider because 2, 3, and 6 divide evenly into the tens digit

(6); if the ones digit is any multiple of these factors, the number can be modelled by a rectangle with width at

least 2:

A good hint, then, is to first make just the tens blocks into a rectangle with at least 2 rows. This is necessary

since the (at most 9) ones blocks cannot rest on top of the tens blocks. The numbers 60, 62, 63, 64, 66, 68,

and 69 can all be modelled this way but 61, 65, and 67 cannot.

Allow students time to make these discoveries on their own.


Extensions:

1. Ask students to explain and show with base ten blocks the meaning of each digit in a number with all digits the same (EXAMPLE: 3333).

2. Have students solve these puzzles using base ten blocks:

a) I am greater than 20 and less than 30. My ones digit is one more than my tens digit.

b) I am a 3-digit number. My digits are all the same. Use 9 blocks to make me.

c) I am a 2-digit number. My tens digit is 5 more than my ones digit. Use 7 blocks to make me.

d) I am a 3-digit number. My tens digit is one more than my hundreds digit and my ones digit is one

more than my tens digit. Use 6 blocks to make me.

3. Have students solve these puzzles by only imagining the base ten blocks. QUESTIONS (a) through (d)

have more than one answer—emphasize this by asking students to share their answers.

a) I have more tens than ones. What number could I be?

b) I have the same number of ones and tens blocks. What number could I be?

c) I have twice as many tens blocks as ones blocks. What 2-digit number could I be?

d) I have six more ones than tens. What number could I be?

e) You have one set of blocks that make the number 13 and one set of blocks that make the

number 22. Can you have the same number of blocks in both sets?

f) You have one set of blocks that make the number 23 and one set of blocks that make the

number 16. Can you have the same number of blocks in both sets?


NS3-8 Comparing Numbers

Goals: Students will use base ten materials to determine which is larger.

Prior Knowledge Required: Naming numbers from base ten materials

Modeling numbers with base ten materials

Vocabulary: hundreds, tens, ones, base ten blocks, greater than, less than

Introduce the phrases “greater than” and “less than”. Emphasize that to say that one number is greater than

another means the first number represents more objects than the second, so 4 is greater than 3 since a

collector of objects 4 objects contains more objects than a collection of 3 objects (4 dollars is more money

than 3 dollars, 4 metres is longer than 3 metres, 4 goals is more than 3 goals, 4 minutes is more time than 3

minutes). It is crucial that students understand that 4 of anything is more than 3 of the same thing and so it

makes sense to compare the numbers 3 and 4 by saying that 4 is “more than” 3. The correct mathematical

expression is 4 is greater than 3, and students should get used to using the expression. Then write on the

board: 4 ____ 5 and have student volunteers say either “greater than” or “less than” as appropriate (in this

case, “less than” is correct). Repeat with several pairs of single-digit numbers to ensure that students are

comfortable with the words “greater than” and “less than.”

If your students are comfortable with base ten blocks and trading a tens block for 10 ones blocks, you may

use base ten materials for this lesson. Otherwise, you might find more convenient to use link-it cubes built

into stacks of 10. This way, instead of trading, students can simply pull apart one stack of 10 if necessary.

Some students may find this more natural than trading.

Make the numbers 25 and 35 using base ten blocks:

Have students name the numbers. ASK: Which number is greater? How can we show which number is

greater using base ten blocks? Explain that 3 tens blocks is more than 2 tens blocks and 5 ones blocks is the

same as 5 ones blocks, so 35 is greater than 25.

Have students use base ten blocks to determine which number is greater:

a) 26 or 28 b) 42 or 32 c) 67 or 57 d) 23 or 83 e) 74 or 78

ASK: Do you need to use base ten blocks to determine which number is greater? (no, we can just look at the

digits) How does looking at the digits tell us which number is greater? (if one of the digits is the same, just

look at the other digit to see which number is greater)

Have students use base ten blocks to determine which number is greater:



ASK: How does looking at the digits tell us which number is greater? (if the tens and ones digit of one

number are both greater than the tens and ones digit of the other number, then that number must be greater)

Make the numbers 43 and 26 using base ten materials.

SAY: Which has more tens blocks? Which has more ones blocks? Hmmm, 43 has more tens blocks,

but 26 has more ones blocks—how can we know which one is bigger?

Show students how trading a tens block for more ones blocks can help them compare the two numbers (or

break apart a stack of 10 link-it cubes):

43 = 4 tens + 3 ones = 3 tens + 13 ones

26 = 2 tens + 6 ones

Now, it is clear that 43 is more than 26—43 has more tens and more ones.

Give students base ten blocks or link-it cubes and ask students make models for each pair of numbers and

then to compare the numbers by trading blocks if necessary (or splitting apart stacks of 10 link-it cubes) so

that the tens and ones in one number are both the same or greater than the tens and ones in the other

number.

EXAMPLES: Which number is greater:


Have students compare several pairs of numbers in their notebook where the tens digit is greater in one

number and the ones digit is greater in the other number. They should draw rough sketches of the base ten

blocks to help them and then trade ten ones blocks for a tens block.

Invite volunteers to show their work on the board. ASK: If two numbers have different tens and ones digits,

which number is greater—the number with greater tens digit or the number with greater ones digit?

Draw pairs of base ten models and have students individually write the numbers modeled and circle the

greater number in each pair. Students should see that the number with more tens is always greater.

EXAMPLE:

34 27

Repeat the exercise, but this time have students write the names for the numbers in words. Make the words

below a regular part of your spelling lessons and have them visible to all students during math lessons: one,

two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, twenty, thirty, forty, fifty, sixty, seventy,

eighty, ninety, one hundred.


When providing pairs of numbers for the students to compare, include examples where both numbers have

the same tens digit and ASK: If both numbers have the same tens digit, how can you tell which number is

greater? (the number with the greater ones digit will be greater) If the numbers have different tens digits, how

can you tell which number is greater? (the number with a greater tens digit will be greater).

Then ask students to make (or draw rough sketches of) base ten models for the numbers 238 and 153.

ASK: Which number has more hundreds blocks? Tens blocks? Ones blocks? Which number do you think is

greater? How can you trade some blocks so that one number has at least as many ones, tens and hundreds

blocks as the other number? Repeat with other pairs of numbers.

a) 345, 169 b) 541, 355 c) 102, 45 d) 600, 497 e) 436, 429 f) 810, 801

ASK: Which number is greater—the number with more hundreds, more tens or more ones? Is a 3-digit

number always greater than a 2-digit number? Is a 2-digit number always greater than a 1-digit number?

How do you know?

Then draw base ten models of several pairs of 2- and 3-digit numbers. Have students write the number word

and the numeral for each number and then to circle the greater number.

Extensions:

1. Create base ten models of a pair of two-digit numbers. Ask students to say how they know which number

is greater. You might make one of the numbers in non-standard form, as shown for the first number

below.

EXAMPLE:

To compare the numbers students could remodel the first number in standard form by regrouping ones

blocks as tens blocks.

2. Ask students to create base ten models of two numbers where one of the numbers…

a) is 30 more than the other

b) is 50 less than the other

c) has hundreds digit equal to 6 and is 310 more than the other

3. Ask students where they tend to see many numbers in increasing order (page numbers, houses,

mailboxes, apartment numbers, line-ups when people need to take a number tag).

First Number Second Number


NS3-9 Comparing and Ordering Numbers

Prior Knowledge Required: One-to-one correspondence

Pairing things up

More and less

Recognizing the number of fingers held up

Write 52 and 42 on the board as follows:

5 2 4 2

Have volunteers write the value of each digit in the appropriate box and then to tell you which number is

greater. Write several similar questions on the board comparing either 2- or 3-digit numbers and have

students determine the greater number in each pair individually in their notebooks.

Repeat the exercises by writing the numbers in expanded form:

52 = 50 + 2

42 = 40 + 2

Since 50 is larger than 40, and 2 is the same as 2, 52 is larger than 42.

Show your students a shortcut for comparing two numbers without having to write the expanded form.

475 = 400 + 70 + 5

465 = 400 + 60 + 5

Since 70 is greater than 60, they know from the expanded form that 400 + 70 + 5 is greater than 400 + 60 + 5.

Without doing the expanded form, we can see this from the digits themselves.

475

465

Since the 7 means 70 and the 6 means 60, we can see the same thing just from the digits in the numbers.

Have students compare two numbers by circling the digit that is different and writing the greater number in

the space below:

a) 475 b) 356 c) 297 d) 493 e) 527

465 358 497 490 507

475


Write 43 and 26 in expanded form and compare them:

43 = 40 + 3 Since 10 is always greater than any 1-digit number, we can split

26 = 20 + 6 43 (the one with more tens) into 30 + 10 + 3:

43 = 30 + 10 + 3

26 = 20 + 6

This makes it clear that 43 is greater than 26, since it wins on all counts. Compare more 2-digit numbers in

this way until it becomes clear that the number with more tens is always greater.

Then move on to 3-digit numbers. Tell students you want to compare 342 and 257. ASK: Which number has

more hundreds? More tens? More ones? Which number do you think is greater, the one with the most

hundreds, the most tens or the most ones? Why? Then write:

342 = 200 + 100 + 42

257 = 200 + 57

Since 100 is greater than any 2-digit number (it has 10 tens and any 2-digit number has at most 9 tens), it

doesn’t matter what the tens and ones are of 257. Ask students to compare more 3-digit numbers with

different hundreds digits in this way (e.g., 731 and 550, 403 and 329). Then look at a pair in which the

hundreds digit is the same (e.g., 542 and 537). This is just like comparing 42 and 37: 542 = 500 + 40 + 2 =

500 + 30 + 12, which is greater than 500 + 30 + 7. ASK: If two 3-digit numbers have different hundreds digit,

how can you tell which one is greater? If they have the same number of hundreds, how can you tell which

one is greater?

Have students compare numbers that do not have the same number of digits (e.g., 350 and 93). Can they

explain why any 3-digit number is always greater than any 2-digit number?

Write the following pairs of numbers on the board:

a) 743 b) 583 c) 392 d) 278

693 591 267 249

Tell your students that 743 has more hundreds and 693 has more tens. ASK: Which number is greater?

Remind your students that the number of hundreds is more important than the number of tens. Circle the

hundreds and tell them that 743 is greater because it has more hundreds. Have volunteers circle the first

digit from the left that is different in each number and then determine the greater number.

Then have students compare numbers given in context. These questions do not have to be given in written

form if some students are uncomfortable reading at this stage. The workbook questions could be read aloud

together before being assigned.

a) Rita’s mother is 43 years old. Anna’s mother is 51 years old. Whose mother is older?

b) Rita has $540. Anna has $259. Who has more money?

c) Montreal is 539 km from Toronto and Ottawa is 399 km from Toronto.

Which city is closer to Toronto?

d) Maurice Richard scored 544 career goals in the NHL. Wayne Gretzky scored 894 career goals. Who

scored more career goals?


Extensions:

1. Use the digits 5, 6, and 7 to create as many 3-digit numbers as you can (only use each digit once when

you create a number). Then write your answers in descending order.

2. List 4 numbers that come between 263 and 527.

3. Name 2 places you might see more than 1000 people.

4. Say whether you think there are more or fewer than 1000…

a) hairs on a dog b) fingers and toes in a class c) students in the school

d) grains of sand on a beach e) left handed students in school

5. Two of the numbers are out of order in each increasing or decreasing sequence. Circle each pair.

a) 28, 36, 47, 42, 95, 101

b) 286, 297, 310, 307, 482

c) 87, 101, 99, 107, 142, 163

6. Introduce students to the > and < notation for “greater than” and “less than”. Have them discover the

notation by using the following trick. Draw a face:

Draw two piles of apples on the board, one with 3 apples and one with 4 apples and have students point

the face towards the pile with more apples because this person is hungry. Repeat with several examples,

always drawing the picture of the apples. Then, instead of the pictures of apples, draw only the number

of apples. ASK: Which number of apples should the hungry person eat? (EXAMPLE: 5 2)

Then erase the face and leave only the mouth. Tell students to just imagine what the mouth will look like

for several pairs of numbers. (EXAMPLE: 7 < 9). Then, for several pairs of numbers, have students write

both the mouth in between the numbers and the words “is less than” or “is more than”. (EXAMPLE: 8 > 7,

8 is more than 7). When students have done this several times, tell them that mathematicians have

invented a symbol to mean “is more than” and another symbol to mean “is less than” and that they just

discovered it. Can they see which symbol (< or >) means more than and which means less than?

Discuss with students why mathematicians may have chosen to invent symbols for these words just like

they did for plus and minus. (Comparing numbers is an important part of mathematics and is done often

enough that they wanted a short form). Ask students if they think that mathematicians could have defined

> to mean “less than” and < to mean “greater than” instead. Emphasize that people often have to make

arbitrary decisions and it is just important to be consistent so that there is no confusion. Brainstorm other

situations where arbitrary decisions like this are necessary (which hand on the clock is longer, which side

of the road do people drive on, which sound does the symbol “b” represent, and so on).

7. For extra practice recognizing differences of 1 and 10, provide the BLM “Hundreds Chart Pieces.” After

students finish, they (or a partner) should check their answers by using an actual hundreds chart.


NS3-10 Differences of 10 and 100

Goals: Students will recognize when numbers are different by 10 or 100.

Prior Knowledge Required: Expanded form

Place Value

3-digit numbers

Vocabulary: difference, expanded form, sum

Have students write “1 more” or “1 less” in the blanks.

a) 8 is _________ than 7 b) 4 is __________ than 5 c) 2 is _________ than 3

d) 10 is _________than 9 e) 11 is _________ than 10 f) 39 is _________ than 40


a) 50 is _________ than 40 b) 30 is __________ than 40 c) 20 is _________ than 10

d) 60 is _________than 70 e) 40 is _________ than 30 f) 70 is _________ than 80


a) 600 is _______than 500 b) 300 is _______ than 400 c) 300 is ______ than 200

d) 600 is _______than 700 e) 500 is _______ than 400 f) 700 is ______ than 600

Have students write each number in expanded form and then tell you how much more or less the first

number is than the second number.

a) 345 = 300 + 40 + 5 b) 259 c) 567 d) 431

335 = 300 + 30 + 5 269 467 432

40 is 10 more than 30, so 345 is 10 more than 335.

ASK: How can you tell how much more or less the number is by looking at which digit is different? Have

students practise this skill with several pairs of numbers which differ in only 1 digit. Students should not use

the expanded form, but should only pay attention to which digit is different.

(EXAMPLES: 756, 746; 430, 431; 542, 442; 542, 552)

Bonus: Have students compare pairs of numbers that differ in only 1 digit, but by more than 1.

(EXAMPLES: 563, 263; 412, 432; 743, 703; 703, 709)

Have students fill in the blanks:

a) ___ is 100 more than 352 b) ____ is 10 more than 352 c) ____ is 1 more than 352.

d) ___ is 100 less than 352 e) ____ is 10 less than 352 f) ____ is 1 less than 352

Provide several problems of this sort with different numbers.

EXAMPLES: ____ is 10 less than 890; ___ is 100 more than 743, ___ is 1 less than 502


Bonus: ____ is 30 less than 572; ____ is 40 more than 641, ___ is 200 more than 738

Have students add or subtract 1, 10 or 100.

a) 354 – 1 b) 653 + 10 c) 987 – 100 d) 432 + 100 e) 581 + 10 f) 436 + 1

Ensure that students can add and subtract 1 comfortably when crossing over a multiple of 10.

EXAMPLES: 59 + 1; 90 – 1; 39 + 1; 20 – 1; 50 – 1; 79 + 1.

Then write on the board: 304 – 10. ASK: How many tens are in 304? (30) If students do not see this

immediately, you may need to ask a series of questions to guide them: How many tens are in 20? 30? 80?

90? 100? 110? 120? 150? 190? 200? 210? How many tens are in 27? 37? 84? 96? 105? 115? 122? 151?

196? 201? 213? Which digit should I cover up to see how many tens are in 304? (cover up the ones digit)

Demonstrate covering up the ones digit. Then tell them that there are 30 tens and ASK: If I take away 1 ten,

how many tens are left? What is 1 less ten than 30 tens? (29 tens) Do I change the ones digit by subtracting

10? (no) What number has 29 tens and 4 ones? (294) Write on the board: 304 – 10 = 294. Similarly,

students can add 496 + 10 by realizing that 49 tens plus one more ten is 50 tens, so 496 + 10 is 506.

Give students practice with crossing over multiples of a hundred when adding and subtracting 10.

(EXAMPLES: 598 + 10; 703 – 10; 392 + 10; 909 – 10; 800 – 10)

Have students state the pattern rules and then extend the patterns:

a) 432, 442, 452, ____, ____, _____ b) 201, 301, 401, ____, ____, ____

c) 947, 847, 747, ____, ____, _____ d) 759, 758, 757, ____, ____, ____

Bonus:

a) 531, 521, 511, ____, ____, _____ b) 703, 723, 743, ____, ____, ____

Activities:

1. Give your students 5 loonies and ask them to show you how much money they would have left if they

took away a dime. Students will see that they have to regroup one of the loonies as ten dimes, and that

there will only be 9 dimes left once one is taken away. (Ask students to translate the result into pennies:

500 pennies becomes 490 pennies.)

2. Ask students to show you in loonies how much money they would have altogether if they had $2.90 and

they were given a dime.

3. Ask students to make a base ten model of 207 and to show you how the model would change if you took

away 10. (Students will see that they have to regroup one of the hundreds as 10 tens and that there

would be 9 tens left)

Extensions:

1. Circle the greater number in each pair: 2. What number is 110 less than…

a) 3 125 or 3 225 b) 4 508 or 4 608 a) 273 b) 860 c) 922 d) 508

2. For more practice adding and subtracting 10, provide the BLM “Hundreds Chart Pieces.”


NS3-11 Comparing Numbers (Advanced)

Goals: Students will order sets of two or more numbers and will find the greatest and smallest number possible when

given particular digits.

Prior Knowledge Required: Comparing pairs of numbers

Vocabulary: greater than, less than

Review reading number words and comparing 2- and 3-digit numbers.

Have students list all the 2-digit numbers they can make using only digits chosen from:

a) 4 b) 4, 5 c) 3, 0 d) 7, 1 e) 8, 5, 2

ANSWERS:

a) 44 b) 44, 45, 54, 55 c) 30, 33 d) 71, 17, 11, 77

e) 88, 55, 22 85, 58, 52, 25, 28, 82

ASK: Which is larger: 54 or 45? 76 or 67? 83 or 38? 91 or 19?

Tell your students that you want to make the largest possible 2-digit number using the digits 3 and 6. Which

number should you make? (63) What if you want the smallest number using the digits 4 and 7? (47)

Have students list the 3-digit numbers that use each digit exactly once:

a) 4, 5, 8 b) 3, 1, 9 c) 5, 3, 2

Which number is greatest? Least? Have students reflect: Could they have found the largest possible number

using the digits 4, 5 and 8 without listing all the possible numbers? How do they know which order to write

the digits in? Should they put the largest digit they have in the ones place, the tens place or the hundreds

place? How do they know? Is any 3-digit number with 4 hundreds more or less than any number with 8

hundreds? Since any 3-digit number with 4 hundreds is less than any 3-digit number with 5 hundreds which

in turn is less than any 3-digit number with 8 hundreds, we should put 8 in the hundreds place.

Have students find the least and greatest 3-digit numbers that use each digit exactly once, this time without

listing all the possible numbers they can make:

a) 3, 4, 1 b) 7, 5, 8 c) 6, 9, 1

Have students first compare pairs of numbers and then order lists of 3 numbers. ASK: Which number is

larger: 74 or 76? 74 or 54? Have students order the 3 numbers starting with the least: 74, 76, 54 (ANSWER:

54, 74, 76). Repeat with other numbers:

a) 64, 91, and 64, 25 b) 543, 523, 543, 743 c) 908, 926 908, 876

Have students order from least to greatest other lists of 3 or 4 numbers by comparing two at a time:

a) 79, 82, 75 b) 872, 864, 587 c) 993, 939, 399 d) 765, 781, 762, 709

Bonus: Provide longer lists of numbers for students to order.


Write on the board the heights of some towers:

Eiffel Tower: 324 m, Tower of Pisa: 56 m, CN Tower: 558 m. Have students list the towers in order from

tallest to shortest.

Bonus: Include other towers in the list as well, such as: Calgary Tower, 191 m; Eureka Tower, 297 m.

Activity: Have students roll a pair of dice 5 times, create a 2-digit number from each roll and try to

achieve the lowest possible sum of their 5 numbers. They win if their total is less than 100.

Variations:

• They win if their sum is at least 200

• Roll 3 dice, create 3-digit numbers and win if their sum is less than 1000

• Roll 3 dice, create 3-digit numbers and win if their sum is at least 2000

NOTE: students can check their sums by using a calculator; they do not need to be able to add 2- or 3-digit

numbers at this time; rather, they should simply be choosing either the least possible or greatest possible

numbers from each roll.

ASK: Which is easier to win: playing with 2 dice or with 3 dice? Can you explain why? (they have more

freedom to make the leading digit as small as possible if there are 3 numbers to choose from than if there

are 2 numbers to choose from, so playing with 3 dice should be easier)


NS3-12 Counting by 2s

Goals: Students will use patterns to skip count by 2s.

Prior Knowledge Required: Counting forwards

Counting backwards

Number lines

Hundreds charts

Repeating patterns

Vocabulary: skip counting, odd, even

Draw a number line on the board. Tell them that to skip count, you have to skip numbers. If you only

count every 2nd

number, you are skip counting by 2. Emphasize the connection between the ordinal number

2nd

and the ordinary number 2. Demonstrate this:

0 1 2 3 4 5 6

Then draw a number line from 0 to 10 and ask if anyone wants to show skip counting by 2 on the

number line.

0 1 2 3 4 5 6 7 8 9 10

Ask them how the arrows show skip counting by 2. Be sure that all students understand that you say the

numbers that the arrows touch and the arrow always points to the number you say next. It tells you to start at

0 and then to say every second number in order.

Then show the first row of a hundreds chart and tell them that we can show skip counting by 2 by colouring

the numbers we say and not colouring (or skipping) the numbers we don’t say. Have a volunteer colour the

right squares to show skip counting by 2 starting from 2:

1 2 3 4 5 6 7 8 9 10

Ask how this is the same as using a number line to skip count and how it is different. Then have 3 rows of a

hundreds chart on the board and ask students how they would show skip counting by 2 on the hundreds

chart. Have a volunteer colour the right numbers in the first row, another volunteer do the second row and

another volunteer do the third row. Have the class read the skip counting out loud as a group.


Ask them if they see anything the same about the numbers in the first row and the numbers from the second

row. If they could remember the numbers they say when skip counting by 2 up to 10, how would this help

them skip count up to 20? Up to 30? Up to 40? When all students understand that the numbers they say

when counting by 2s starting from 2 have ones digit 0, 2, 4, 6 or 8, write a random number on the board

between 0 and 100 and ask students to raise their hand if they think you say that number when skip counting

by 2. Repeat several times. Then, instead of writing the number on the board, say the number out loud.

Have volunteers skip count by 2s from 20 to 30, from 60 to 70, and so on.

Skip count by 2s starting from:

a) 36, ___, ___, ___ b) 48, ___, ___, ___ c) 90, ___, ___, ___ d) 68, ___, ___, ____

Bonus: Skip count by 2s starting from: a) 138 b) 846 c) 896

Tell your students that when you count by 2s starting from 0, the numbers you say have a special name.

ASK: Does anyone know what that special name is? (even numbers) What happens when you count by 2s

starting at an even number—are the numbers you say still even?

ASK: Does anyone know the name for numbers that are not even? (odd) What happens when you count by

2s starting at an odd number—are the numbers you say still odd? What are the ones digits of the even

numbers? Of the odd numbers?

Skip count by 2s starting from:

a) 43, ___, ___, ___ b) 51, ___, ___, ___ a) 67, ___, ___, ___ a) 39, ___, ___, ___

Bonus: Skip count by 2s starting from a) 255 b) 849 c) 897

Ask them why skip counting by 2 might be useful – is there anything they can think of that comes in 2s?

(Feet, hands, shoes, gloves, mittens, etc.) Have students count the number of shoes in the room. Ask them

why skip counting by 2 is a natural way to do so.

Once students learn to count by 2s and to recognize even and odd numbers, they can use these skills as a

foundation for doing mental addition and subtraction: see the Mental Math section of this manual for details.

Activity:

House Numbers

Before doing this activity, have students do the extension below. Have students walk around a residential

neighborhood with an adult and look at house numbers along one side of the street. What are the ones digits

of the numbers they see. Do the numbers seem to be skip counting by any specific number? Are any

numbers missed? Which numbers are missed? Why might numbers be missed? (sometimes, there used to

be more houses in between two houses and then they got torn down). ASK: Why might some houses get

torn down? Why wouldn’t they change all the addresses to make the numbers go up by 2s again? What

problems would this cause?


Extension: Have students find the missing number in each pattern:

35, 37, 39, ___, 43, 45,

62, 64, ___, 68, 70

78, 80, ___, 84, 86

432, 434, ____, 438, 440

Now don’t tell them where the missing term is:

706, 708, 712, 714, 716

56, 58, 62, 64, 66, 68

32, 34, 36, 40, 42, 44

55, 57, 59, 63, 65, 67

317, 321, 323, 325, 327

836, 838, 840, 842, 846

Literacy Connection:

“Two of Everything”, L.T. Hong

(A Chinese folktale where everything gets counted by 2s.)


NS3-13 Counting by 5s and 25s

Goals: Students will use patterns to skip count by 5s and 25s.

Prior Knowledge Required: Number lines Repeating patterns Counting forwards and backwards

Vocabulary: skip counting

Draw a number line on the board:

0 10 20 30 40

ASK: What number is the at? (4) If I skip count by 5 starting at 4, what will be the next number I say?

Have a volunteer draw an at that point. Have another volunteer continue marking all the numbers you say

when skip counting by 5 starting at 4. Underline the ones digits for them and ASK: Do you see a pattern in

the ones digits? Have students describe the pattern rule: 4, 9, then repeat. Have students predict the next

three numbers you say when counting by 5: 4, 9, 14, 19, 24, 29, 34, 39, __, __, __.

Bonus: Is there a pattern in the tens digits? Challenge them to describe it. (start at 0, then increase by 1,

stay the same and repeat). Ask them to be clear about what is repeating (the increase by 1 and the stay the

same).

Show them other sequences formed by counting by 5s and have them find the pattern in the ones digits:

a) 23, 28, 33, 38, 43, 38 b) 79, 84, 89, 94, 99, 104 c) 281, 286, 291, 296, 301, 306

Bonus: Describe the pattern in the number of tens (EXAMPLE: c) start at 28, stay the same, increase by 1,

then repeat).

Have students extend each sequence above. Discuss the patterns in the ones digits in more detail. How are

the patterns the same and how are they different? (Similarities include: the core always has length 2, the first

two terms always differ by 5, differences include: the two starting numbers can be different in each case)

Draw a number line on the board:

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Have a volunteer start at 0 and circle the numbers they would say when counting by 25.

Then ASK: If I know how to count by 25s from 0 to 100, how can I count by 25s up to 200?


0 25 50 75 100

125 150 175 200

Discuss the patterns in a) the number of hundreds and b) the remaining digits (the tens and ones together).

a) 0, 0, 0, 0, 1, 1, 1, 1, 2, ____, ____, ____

b) 0, 25, 50, 75, 0, 25, 50, 75, 0, ____, ____, _____

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

Have students complete the number sequences by skip counting by 25:

50, 75, 100, ____, ____, ____.

125, 150, 175, ____, ____, ____,

275, 300, 325, ____, ____, ____,

800, 825, ____, ____, ____.

650, 675, ____, ____, ____.

Activities:

1. Skip Counting Machines

Tell students that they are going to make a machine to help them skip count by 5 from any number on a

hundreds chart. Give them the BLM “Counting by 5s” and ensure that students understand all

instructions before allowing them to start. Emphasize that students must not cut out the grid; they should

only cut along the lines suggested on the BLM; otherwise they will be left with pieces. When they are

finished counting by 5, provide students with the BLM “Hundreds Charts up to 200”. Ask students to

place their new skip counting machine so that they can start counting at various numbers. Then have

them say the counting sequences out loud with you. EXAMPLE: start at 5, start at 3, start at 9, start at

14, start at 20, start at 102, start at 134, start at 109, start at 145. When students are comfortable with

this, have them make a machine to count by 2s or by 10s.

Students may wish to investigate whether flipping their sheet over either vertically or horizontally will

result in a machine that also works (as long as the mirror line is vertical, the flipped sheet will still work).

Bonus:

Make a machine to count by 25s. Does their tool work for starting at any number? Suppose that they

make their holes arranged as shown by the shaded squares:


This works for counting by 25s starting at 13 (13, 38, 63, 88) by placing the top left shaded square over

the 13. However, it does not work for the sequence starting at 18 (18, 43, 68, 93); not even rotating or

flipping the sheet will work. In order to make it work, the shaded squares should appear in 5 rows, so that

the sequence can start at either the top or the second top hole.

2. Quarters and Nickels

Give students play money (quarters only or nickels only) and have students count the money they have

(in cents, not dollars) by skip counting by 5 or by 25. Then demonstrate counting a combination of

quarters and nickels, first in random order and then in organized order. Have volunteers count the

money first in random order and then in organized order. ASK: Which way is easier? Why do you think

that is? Students may wish to investigate any differences between counting first by 25 and then by 5 and

counting first by 5 and then by 25. Does it matter whether they count the nickels or the quarters first?

(no, as long as they count all the quarters first or all the nickels first)

Tell your students that when they skip count just by 25 and then just by 5 (or vice versa), they can use

the patterns discussed in class, but if they count in random order, there is no pattern they can use, so

each time they have to think separately. That’s why it takes longer to count in random order than in an

organized order.

Extensions:

1. Repeat the exercises above, starting at 10 instead of 0, first having a volunteer circle the correct numbers

and then finding the same patterns:

a) 0, 0, 0, 0, 1, 1, 1, 1, 2, ____, ____, ____.

b) 10, 35, 60, 85, 10, 35, 60, 85, 10, ____, ____, ____.

ASK: What is similar about the patterns for starting at 0 and the patterns for starting at 10? What is the

length of the core in each case?

Have students complete the number sequences by skip counting by 25:

a) 20, 45, 70, 95, ____, ____, ____, ____.

b) 310, 335, ____, _____, ____, ____.

c) 560, 585, _____, _____, _____, ____.

Bonus: 312, 337, ____, ____, ____, ____.


2. Show your students how they can group objects into groups of 2 or 5 to make it easier to count them.

Draw the following picture on the board:

1 2 3 4 5 6 7 8

2 4 6 8

Tell your students that you are counting the dots twice, once by counting normally and once by counting

by 2. Ask your students to compare the two ways of counting. ASK: Did I get the same answer both

ways? How many dots are there? Which way do you find easier? Why? (Some may find counting by 1s

easier because they know the next number to say more easily and others may find counting by 2s easier

because they have to say less numbers – both answers are good answers). Which way is faster? If you

had a lot of things to count, which way would you be done sooner?

3. Have students count the number of letters in the alphabet by counting by 5s. Then provide the BLM

“Foreign Alphabets.” Students will need to count the number of letters in foreign alphabets by counting

by 5s and then counting on by 1s. Before assigning this BLM, it is a good idea to hand out the BLM and

allow students time to familiarize themselves with the different alphabets. Discuss various differences

and similarities between foreign alphabets and our own. Discuss how, in Russian, letters that are

different actually look very similar. Have students write pairs of very similar-looking Russian letters on the

board. Are there any pair of letters like that in English? (e.g. O and Q, C and G, P and R) Students may

wish to discuss symmetry in letters as well, or how b and d are backwards from each other. Are there

any foreign alphabets on the BLM that are like that?

Ask students to look carefully at the Russian alphabet – are there any symbols where they aren’t sure if

the symbol represents a single letter or more than one letter? Correct any such misconceptions before

they start grouping by 5s.

Draw your students’ attention to the Cherokee alphabet. Are there any letters that look like our letters?

Are there any letters that look like our numbers? Tell them that the Cherokee alphabet was made up by

someone who saw English writing, was amazed by the way people could communicate through symbols,

and decided to assign sounds in their language to these symbols. They had no idea that 4 didn’t mean a

sound or even which sound D or R meant, but they just assigned sounds from their own language to

these symbols arbitrarily. As soon as one person invented the writing system, everyone in the

community got excited about it and virtually everyone learned to read very quickly.

If you have a diverse class, ask if anyone’s first language is on the sheet. Ask if anyone’s first language

is missing. Have volunteers write their own alphabet on the board so that other volunteers can group by

5 and count the letters in those alphabets.

Notice that students may group fairly arbitrarily and might have more than 5 single letters left over at the

end if they grouped randomly. That is okay. They might, for example, count the Korean letters as 5, 10,

15, 20, 25, 30, 31, 32, 33, 34, 35 instead of 5, 10, 15, 20, 25, 30, 35. Both are faster than counting by 1s.


4. Ask students to estimate how many dots are on a pre-made page. For example,

Hold a card with dots arranged randomly like this up and ask students to estimate how many they think

there are. Ask how many groups of 5 they think there are. Then circle a group of 5 and ask if anyone

wants to change their guess. Then circle another 5 and again ask if anyone wants to change their guess.

Continue in this way. Ask them if showing a group of 5 made it easier to estimate how many there are.

Then finish putting all groups of 5 and ask them how grouping by 5 made it easier to count them all. One

strategy could be to tally the groups of 5 as you cross them out.

5. Ask the following 2 questions to your class:

a) To make $1.75 in quarters, how many quarters would you need?

b) Which term (1st, 2

nd, 3

rd, etc.) in the sequence 25, 50, 75, …is 175?

Discuss the similarities and the differences between these two questions.


NS3-14 Counting by 2s, 3s and 5s

Goals: Students will count by 3s. Students will decide whether skip counting by 2, 3 or 5 should be done

given endpoints on a number line with a specified number of places.

Prior Knowledge Required: Skip counting by 2 and 5

Number lines

Vocabulary: skip counting

Review skip counting by 2 and by 5 and then introduce skip counting by 3. Draw two rows of skip counting

as follows:

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51,

Have students continue the pattern for 2 more rows by skip counting and then discuss patterns in the rows

and the columns. Challenge them to find the next 2 rows by extending the pattern rather than by skip

counting.

Have students describe the pattern in the ones digits: 0, 3, 6, 9, 2, 5, 8, 1, 4, 7 and then repeat

Have students practice skip counting by 3, starting at various numbers. (EXAMPLES: start at 13, 27, 34, 55,

201, 836) ASK: When you skip count by 2s starting at 13, what can be the ones digits of the numbers you

say? (only 1, 3, 5, 7 and 9). When you skip count by 5s starting at 13, what can be the ones digits of the

numbers you say? (only 3 and 8). When you skip count by 3s starting at 13, what can be the ones digits of

the numbers you say? (any digit)

Write the following number line on the board:

5 7

Ask a volunteer to write what goes in the missing place. Ask how they can check their answer.

Make sure students understand that by finding what comes after their guess, they can make sure it’s 7.

Continually leave more and more spaces in the number line.

Then show them a number line like this:

10 14


Tell your students that you want to skip count from 10 to 14 and only say one number in between. If you

count by 1s, you know that 11 comes right after 10. Does 14 come right after 11?. What should I skip count

by if I want the same number to come right after 10 and right before 14?

Have students choose between what to skip count by in the following order:

1. Skip counting by 1 or skip counting by 2.

2. Skip counting by 2 or by 5.



5. Skip counting by 2, by 3 or by 5.

Always begin by leaving only one space between the numbers and then progress to leaving more spaces.

When choosing between counting by 2, 3 or 5, a good strategy is to start counting by 3 and then either guess

a higher or lower number based on checking their first guess. Does counting by 3 get them too far or not far

enough? Should their next guess be higher or lower? Do they want to go further or less far with their next

guess?

Extension: For the sequence of skip counting by 3 starting at 0, have students describe the pattern in

the number of tens. ANSWER: The number of repetitions is 4, 3, 3 and then repeat. The number being

repeated starts at 0 and increases by 1 each time. (0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, …)


NS3-15 Counting Backward by 2s and 5s

Goals: Students will count backwards by 2s from any number and by 5s from any multiple of 5.

Prior Knowledge Required: Counting forwards

Counting backwards

Skip counting forwards by 2 and 5

Vocabulary: skip counting, counting forwards, counting backwards

Say number sequences and have students tell you what you’re counting forwards by when you say the

numbers – students might hold up 2 or 5 fingers to show their answer. Then repeat by writing number

sequences on the board instead of saying them. ASK: How can you tell what I’m counting by – what do the

ones digits tell you?

Then make it a bit harder. They not only have to tell you what you’re counting by, they have to find the

missing number; the first step is to find what they’re counting by.

Demonstrate: 10 12 14 16 18 20 22 _____ 26 28

Ask them what they are counting by and how they know. Ask what comes next after 22 when they count by

2. Repeat with several examples of counting by 2 or 5, always asking first what they are counting by and

then what comes next. Do not always start with multiples of 2 or 5.

Then tell them that you are going to make it even harder for them by counting backwards instead of

forwards. Write on the board:

20 18 16 14 12 10 8

Ask them if they can tell what number you are counting back by. To help them, tell them to read the numbers

in backwards order – what number are they counting forwards by? Ask them how you could tell what to say

next after saying 8. To help them, ASK: What would you say before 8 when counting forwards by 2? What

operation do you use to get from 6 to 8? (add 2) To get from 8 to 6? (subtract 2) Students should see that

they always need to subtract 2 to find the next number. Repeat with several sequences of counting back by 2

and 5; have students extend the sequences to find the next 3 terms.

Now put a blank in the middle of the sequence: 20 15 _____ 5.

Ask what you’re counting back by and then what goes after the 15 – ask what they would say before 15

when counting forwards by 5. ASK: What is 15 – 5? Then ask them to check their answer by asking

themselves what comes after their answer – What comes right after 10 when counting back by 5? If their

answer of 10 is correct, then 5 should be the next number. Is it? So do they think their answer of 10 is right?

Do several examples of this, counting back by 2 or 5.


Teach students to count back by 10s by skipping every second number they would say when counting back

by 5s. A number line is a good visual for this:

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Activities:

Countdown by 2s

The teacher counts back from 20 to 0 by 2s. Tell the students that they have to stand up before you get to 6.

The last person to stand up before you get to 6 wins. Then they have to focus on which number comes

before 6 when you count backwards by 2s. They won’t want to stand up when you say 10 because someone

else might stand up when you say 8. They also can’t wait until you say 6 because then they won’t win. Vary

this at first only by changing what number you count down from. E.g. Count down from 16 to 6, then from 14

to 6, then from 18 to 6, etc. When everyone in the class understands that 8 comes right before 6 when

counting back, then you can vary the number you count down to as well as from. Eventually, you can ask for

volunteers to count back while the rest of the class plays the game.

Variation: Count back by 5s from 100 or by 10s from 200.

Zero

Have students start at 10 and take turns saying the next number when counting back by 2s and the one to

say 0 wins. Does the person who starts win or lose? Start at higher numbers as students become ready.

Which numbers could I start at if I want to win?

A Strategy Game for Counting Back

Tell your students that they have to say the numbers counting back by 2s from 20, taking turns as in the

game “Zero”, but they have to decide whether to say one or two numbers. The winner is again the person

who says 0. Challenge your students to find the strategy for this game.

EXAMPLE:

Player 1 Player 2 P1 P2 P1 P2 P1

20 18,16 14,12 10 8,6 4 2,0 Player 1 wins

The main strategy is to try to say 12, but not 10, since this will also allow you to say 6 regardless of whether

your partner says both 10 and 8 or just 10. Whatever the case, do not say 4. Make your partner say either 4

or both 4 and 2. Then you can say 2 and 0 or just 0.

Then change the rules so that the person who says 0 loses.

Extension: Give students a sequence counting back by 2, 5, or 10 with a missing number, but don’t tell

them where the missing number is – challenge them to find it. For example:

28 26 24 20 18 16 14 12 10



Goals: Students will extend patterns that count by 10.

Prior Knowledge Required: Adding 10

Subtracting 10

The relation between adding and counting on

The relation between subtracting and counting back

Review adding 10 from section NS3-10: Differences of 10 and 100. Have students practice adding 10 to

extend the patterns in the following order. Patterns that consist of:

• 2-digit numbers with ones digit 0 (EXAMPLE: 40, 50, 60, …)

• 3-digit numbers with ones digit 0 and do not cross multiples of 100

(EXAMPLE: 230, 240, 250, …)

• 3-digit numbers with ones digit 0 and do cross multiples of 100 (EXAMPLE: 270, 280, 290, …)

• 2-digit numbers with non-zero ones digit (EXAMPLE: 26, 36, 46, …)

• 3-digit numbers with non-zero ones digit and do not cross multiples of 100

(EXAMPLE: 217, 227, 237, …)

• 3-digit numbers with non-zero ones digit and do cross multiples of 100

(EXAMPLE: 479, 489, 499, …)

Repeat the sequencing for counting back by 10.

Extension: Complete the sequences

785, 795, _____, 815, _____

675, 685, _____, _____, 715

365, 375, 385, _____, _____


NS3-17 Counting by 2s, 3s, 4s, 5s and 10s

Goals: Students will count by 4s and then will choose between counting by 2s, 3s, 4s, 5s or 10s.

Prior Knowledge Required: Counting by 2s, 3s, 5s and 10s

Write the following pattern on the board: 0 4 8 12 16

Have volunteers finish the second, third and fourth rows. Discuss any patterns they see in the rows and

columns. Have your students individually predict the next two rows in their notebooks. ASK: What is the

pattern in the ones digit? (0, 4, 8, 2, 6, then repeat) What is the pattern in the number of tens? (The number

of times each term is repeated is 3, 2, then repeat. The terms start at 0 and then increase by 1)

Have students practice skip counting by 4, starting at various numbers.

(EXAMPLES: start at 13, 27, 34, 55, 201, 836)

Review choosing between skip counting by 2, 3 or 5 (See NS3-14).

Then write the following number line on the board:

30 26

Tell your students that you want to skip count back from 30 to 26 and only say one number in between. If you

count by 1s, you know that 29 comes right after 30. Does 26 come right after 29?. What should I skip count

by if I want the same number to come right after 30 and right before 26?

Have students choose between what to skip count backwards by in the following order:

1. By 1 or 2

2. By 2 or 5

3. By 2 or 3

4. By 3 or 5

5. By 2, 3 or 5

6. By 3 or 4

7. By 2, 3 or 4

8. By 2, 3, 4 or 5

9. By 5 or 10

10. By 3, 5 or 10


11. By 2, 3, 5 or 10

12. By 2, 3, 4, 5 or 10

Always begin by leaving only one space between the numbers and then progress to leaving more spaces.

When choosing between counting by 2, 3 or 5, a good strategy is to start counting by 3 and then either guess

a higher or lower number based on checking their first guess. Does counting by 3 get them too far or not far

enough? Should their next guess be higher or lower? Do they want to go further or less far with their next

guess?

Provide examples of numbers where missing numbers are filled in incorrectly, so that students need to find

the error.

EXAMPLE:

57 54 52 48

Students will need to first determine what number to skip count back by.

Literacy Connection:

What Comes in 2s, 3s and 4s? S. Aker

(Describes everyday situations where items naturally come in 2s, 3s and 4s.) Have students think of other

things that come in 2s, 3s and 4s. (EXAMPLES: wheels on a bike, eyes on a face, legs on a person, wheels

on a tricycle, sides on a triangle, tennis balls in a can, legs on a dog, legs on a chair, wheels on a car)

Discuss which is easiest to find: things that come in 2s, 3s or 4s and which is hardest to find.




Prior Knowledge Required: Adding 100

Subtracting 100

The relation between adding and counting on

Review adding 100 from section NS3-10: Differences of 10 and 100. Have students practice adding 100 to

extend the patterns in the following order. Patterns that consist of:

• 3-digit numbers with ones and tens digit 0 (EXAMPLE: 400, 500, 600, …)

• 4-digit numbers with ones and tens digit 0 and do not cross multiples of 100

(EXAMPLE: 2300, 2400, 2500, …)

• 4-digit numbers with ones and tens digit 0 and do cross multiples of 100

(EXAMPLE: 2700, 2800, 2900, …)

• 3-digit numbers with non-zero ones and tens digits (EXAMPLE: 267, 367, 467, …)

• 4-digit numbers with non-zero ones and tens digits and do not cross multiples of 100

(EXAMPLE: 3217, 3227, 3237, …)

• 4-digit numbers with non-zero ones digit and do cross multiples of 100

(EXAMPLE: 5479, 5489, 5499, …)

Repeat the sequencing for counting back by 100.

Activities:

The two activities below are from “A Guide to Effective Instruction in Mathematics, Kindergarten to

Grade 3.”

1. This is a game for pairs. Player A thinks of a starting number. Player B may add either 10 or 100 to the

starting number. Player A, in turn, adds either 10 or 100 to the number given by player B. Continue

taking turns in this fashion. The winner is the player who gets to 500 or closest to 500 without going over.

Variation: Use 1000 as the target number. More advanced students may choose between skip counting

by 10 and skip counting by 25. Weaker students may need to begin the counting at 0. Another variation

is for students to skip count backwards starting from 500 or 1000 with the winner being the player who

gets to 0 or closest to 0.

2. Students stand in a circle. The teacher chooses a volunteer to start the game. The first student begins by

saying 25, the next student says 50, then 75, and so on. Whenever a multiple of 100 (100, 200, 300,

etc.) is said, the student must sit down. The counting continues with only students who remain standing.

The game ends when only one student is standing.

Extension: Complete the sequences

723, 823, 923, _____, 1123 5833, 5933, _____, _____, 6233

1780, 1880, 1980, _____, 2180 7981, _____, 8181, 8281, 8381


NS3-19 Regrouping


Prior Knowledge Required: Link-it cubes

Ones blocks, tens blocks

A tens block is exactly like ten ones blocks, only easier to handle

because there is only one of them instead of ten of them

A 2-digit number is written according to the number of tens and ones

Vocabulary: tens blocks, ones blocks, trading

Show students a pile of 12 link-it cubes. Have a volunteer count them. Then group ten of them in a stack so

that you have 2 left over. ASK: How many are in this stack? How do you know? Tell them that you have one

stack of 10 and 2 more ones, so you have 12 link-it cubes altogether. ASK: How many link-it cubes would I

need to build 3 stacks of 10? 7 stacks of 10? Tell them that you have 4 stacks of 10 and 6 more blocks and

draw a model of this on the board:

ASK: How many link-it cubes did I use? (46)

Show students a pile of 27 link-it cubes. Tell them that you find them too hard to handle because there are

too many loose link-it cubes. Ask for strategies to deal with this. Suggest grouping some of them together in

ways that make it easier to count. Demonstrate grouping by 2s, 5s, 10s and 25s and use skip counting to

find out how many you have. ASK: Which way was easiest? When you group into 10s, did you need to skip

count at all to find the answer or is there another way? Remind them that the tens digit of a 2-digit number is

the number of tens and the ones digit tells how many more ones, so 2 stacks of ten with 7 more ones can be

easily read as 27. Draw several stacks of 10 on the board with ones left over and ask students to identify the

number of blocks used.

Bring out the base ten materials and show students that these blocks are just like link-it cubes except that we

don’t have to stack the blocks ourselves; the tens blocks are already together. Instead of grouping together

ten link-it cubes, they can trade 10 ones blocks for a tens block. Demonstrate with piles of 13 link-it cubes

and 13 ones blocks. Group 10 of the 13 link-it cubes and exchange 10 of the 13 ones blocks for a tens block.

Emphasize that you end up with a stack of 10 and 3 leftover blocks in each case, so trading ten ones blocks

for a tens block is just like stacking ten link-it cubes together.


Then show how to model this process on the board.

Tell your students that this process of grouping ten single blocks into one stack of ten is called regrouping

because you are rearranging the blocks into groups.

ASK: How many ones blocks are left after you trade the ten ones for a tens block? How many ones blocks

were there before trading? How does breaking it up into tens and ones make it easier to count them?

Emphasize that 1 ten + 6 ones is the same thing as 16 ones and we use the blocks to show the numbers.

Then do examples where the number of blocks is more than 20, so that 2 tens blocks are required.

Draw base ten models with more than ten ones and have students practice trading ten ones blocks for a tens

block. They should draw models to record their trades in their notebooks. EXAMPLE:

4 tens + 19 ones = 5 tens + 9 ones

Students should also be comfortable translating these pictures into number sentences:

40 + 19 = 40 + 10 + 9 = 50 + 9 = 59

ASK: What number is 6 tens + 25 ones? How can we regroup the 25 ones to solve this question?

25 = 2 tens + 5 ones = 10 + 10 + 5, so 6 tens + 25 ones = 60 + 20 + 5 = 80 + 5 = 85

Tens Ones

6 25

6 + 2 = 8 25 – 20 = 5

Students can practice using such a chart to regroup numbers. Then have them regroup numbers without

using the chart:

3 tens + 42 ones = _________ tens + __________ ones

Draw several arrays with 10 rows of dots in each row:


Bonus:

Have students count by tens to find out how many dots there are and then to say how many tens there are.

(EXAMPLE: 50 ones = 5 tens)

Progress to arrays that have only partial rows of 10, so that students need to count by 10s and then continue

to count by 1s to find out how many dots there are. Have them write how many tens and ones there are.

EXAMPLE:

____ ones = ____ tens + ____ ones

Bonus: Include examples with more than 10 rows.

Then have students fill in the blanks without using arrays:

a) 43 = ___ tens + ___ ones b) 75 = ___ tens + ___ ones

c) 80 = ___ tens + ___ ones d) 19 = ___ tens + ___ ones

e) 27 = ___ tens + ___ ones f) 98 = ___ tens + ___ ones

g) 63 = ___ tens + ___ ones h) 36 = ___ tens + ___ ones

Bonus: a) 126 = ___ tens + ___ ones b) 874 = ___ tens + ___ ones

Activity:

Pick-up Straws

You will need straws and elastics for this activity.

Cut up several straws into thirds. Make sure there are enough pieces so that everyone can have more than

10, preferably an average of about 20. Hide them around the room. Have everyone pick up as many as they

can. Have them count how many straws they have. Then have them pair up with a partner and find how

many straws they got together. Tell them they might have to group their leftover single straws together. Then

have each pair group with another pair. At the end, ask them how did grouping them in tens make it easier to

count the total number they had with a partner?

Extension: Regroup: 5 ten thousands + 7 thousands + 12 hundreds + 15 tens + 6 ones NOTE: You will have to regroup twice.


NS3-20 Regrouping (Advanced)

Goals: Students will regroup base ten materials and coins to make standard models of numbers.

Prior Knowledge Required: Trading ten ones for a tens block

Show your students a hundreds block and ASK: How many tens blocks would I need to glue together to

make this block? How many ones blocks would I need? Tell them that they can trade a hundreds block for 10

tens blocks or for 100 ones blocks. Tell your students that you have 7 tens blocks ASK: How many more

ones blocks would you need to be able to trade for a hundreds block? Tell your students that you could trade

7 tens blocks and 30 ones blocks for a hundreds block.

Introduce the hundreds, tens and ones chart and continue with problems that require:

1. regrouping ones to tens (EXAMPLE: 4 hundreds + 5 tens + 22 ones)

2. regrouping tens to hundreds (EXAMPLE: 5 hundreds + 34 tens + 2 ones)

3. regrouping ones to tens and tens to hundreds (EXAMPLE: 3 hundreds + 14 tens + 25 ones)

4. regrouping ones to tens and tens to hundreds, but you don’t realize you need to regroup the tens to

hundreds until you have regrouped the ones to tens (EXAMPLE: 2 hundreds + 9 tens + 14 ones)

Give students play coins (pennies and dimes) so that each student has more than 10 pennies. Have

students record the number of each coin they have in a “dimes and pennies” chart:

Then have students trade coins so that they have the fewest number of coins but still the same amount of

money. They should give you 10 pennies in exchange for a dime. They should then record the number of

dimes and pennies they have after regrouping.

Repeat this exercise several times. ASK: How is this similar to regrouping tens and ones blocks? Which coin

is like the ones block? Which coin is like the tens block?

Repeat the exercise, but using loonies, dimes and pennies and including a hundreds column in your chart.

Ensure that each student has either more than 10 dimes or more than 10 pennies.

dimes pennies

3 16

dimes pennies

3 16

4 6


Extensions:

1. Ask students to show the regrouping in QUESTIONS 1 to 3 with base ten blocks and with play money.

2. If you taught your students Egyptian writing (see Extension for NS3-1: Place Value – Ones, Tens, and

Hundreds) you could ask them to show regrouping using Egyptian writing.

EXAMPLE:


NS3-21 Adding 2-Digit Numbers

Goals: Students will add 2-digit numbers without regrouping.

Prior Knowledge Required: Base ten materials

Adding 1-digit numbers

Vocabulary: sum

Have volunteers draw base ten models of the numbers 15 and 43 on the board. Students can draw sticks for

tens and dots for ones to make it easier. If students have trouble modelling 15 (or 43), ASK: Which digit is

the ones digit, the 1 or the 5? How many ones do we have? How many tens?

Tell students you want to add these two numbers. Write the following sum on the board:

15

+ 43

ASK: If we add 15 and 43, how much do we have in total? Prompt students to break the problem down into

smaller steps and to refer to the base ten models: How many tens are there altogether? How many ones

altogether? What number has 5 tens and 8 ones? What is 15 + 43?

Now draw a tens block and five ones:

Ask your students to count all the little squares, or ones, including those in the tens block. Tell students that

we use tens blocks because it’s easier to count many objects when we put them in groups of 10. ASK: When

would we use hundreds blocks?

ASK: How does using tens and ones blocks make it easier to add 2-digit numbers? Would using fives blocks

be just as easy? (No.) Why not? (Because we don’t have a fives digit, we have a tens digit. If we make and

count groups of ten, we can immediately write the tens digit in a number. If we make and count groups of

five, we have to do more work to translate the fives into a tens digit and maybe a ones digit. For example,

3 fives = 5 + 5 + 5 = 10 + 5 = 1 ten and 5 ones = 15.)

In their notebooks, have students draw base ten models to add more 1- and 2-digit numbers where

regrouping is not required (EXAMPLES: 32 + 7, 41 + 50, 38 + 21, 54 + 34, 73 + 2).

When students have mastered this, write on the board:

57

+ 21

ASK: How many tens are in 57? How many tens in 21? How many are there altogether? Did we need base

ten blocks to find out how many there are altogether? (no, there are 5 + 2 = 7 tens altogether) There are 5

tens and how many more ones in 57? There are 2 tens and how many more ones in 21? Do we need base

ten models to find out how many extra ones there are in total? (No, there are 8 in total. We can just add the 7

and the 1.) There are 7 tens and 8 more ones altogether—what number is that?


As students answer your questions, write the digits in the correct position beneath the line, to demonstrate

the standard algorithm for addition:

57

+ 21

78

ASK: Why did I write the 8 under the 7 and the 1? Why did I write the 7 under the 5 and the 2? Ensure that

students understand that the ones digit of the answer is written under the ones digits of the addends, and

similarly for the tens digit. Have students add several pairs of 2-digit numbers that requires no regrouping.

(EXAMPLES: 24 + 32, 14 + 73, 25 + 41, 34 + 44, 26 + 13)

Bonus:

Have faster students add three 2-digit numbers (EXAMPLES: 41 + 23 + 15, 30 + 44 + 23, 21 + 21 + 21).

Extension:

Have students use base ten materials to add 2- and 3-digit numbers. Include only questions that

do not involve regrouping.

EXAMPLE: Find the sum:

132

+ 45

STEP 1: Create base ten models for 132 and 45.

132 = 45 =

STEP 2: Count the base ten materials you used to make both models:

132 and 45 = 1 hundred, 7 tens, 7 ones

STEP 3: Now that you know the total number of base ten materials in both numbers, you

have the answer to the sum:

132 + 45 = 177 (since 1 hundred + 7 tens + 7 ones = 177)

STEP 4: Check your base ten answer by solving the question using the standard algorithm

for addition (EXAMPLE: line up the two numbers and add one pair of digits at a time).

132

+ 45

177

Remind students that when they use the standard algorithm for addition, they are simply combining the ones,

tens, and hundreds as they did when they added up the base ten materials above.


NS3-22 Adding with Regrouping (or Carrying)

Goals: Students will add 2-digit numbers with regrouping.

Prior Knowledge Required: Adding 2-digit numbers without regrouping

Vocabulary: regrouping, carrying

Tell students you want to add 27 and 15. Begin by drawing base ten models of 27 and 15 on the board:

27 = 15 =

Then write the addition statement and combine the two models to represent the sum:

27

+ 15 =

ASK: How many ones blocks do we have in the total? How many tens? Replace 10 ones with 1

tens block. ASK: Now how many ones do we have? How many tens? How many do we have altogether?

27

+ 15 =

Use a tens and ones chart to summarize how you regrouped the ones:

After combining the base ten materials

After regrouping 10 ones blocks for 1 tens block

Have students draw the base ten materials and the “tens and ones” charts for:

36

+ 45

28

+ 37

46

+ 36

19

+ 28

Bonus:

32

46

+ 13

29

11

+ 34

Tens Ones

2 7

1 5

3 12

4 2


Show the tens and ones chart for the first question you gave the students (36 + 45).

Show your students the first step of the standard algorithm alongside a tens and ones chart.

ASK: Which step or steps from the addition is being shown on the right? (adding the ones digit and

regrouping the 16 ones as 1 ten and 6 ones). ASK: How does this step show the regrouping? Tell your

students that when we regroup 10 ones for a ten, we put the 1 on top of the tens column. Mathematicians

call this process “carrying the 1.” Ask students for reasons why this name is appropriate for the notation.

ASK: How are the tens and ones shown separately in the chart? How are the tens and ones shown

separately in the algorithm? Emphasize that the ones digit of the sum is always lined up with the ones digit of

the addends and the tens digit of the sum is lined up with the tens digit of the addends.

Have volunteers do the first step of the standard algorithm for several problems.

1

3 5

+ 2 8

3

4 4

+ 1 9

3 7

+ 4 7

4 8

+ 3 9

5 6

+ 2 5

3 8

+ 5 8

Explain to your students that even though we no longer have a tens column and a ones column in a chart

format, we still keep the ones in the same column and the tens in another column. This is why the tens digit

of the total number of ones is put on top of the tens column instead of on top of the ones column. Show them

how it is easier to line up the digits if they are using grid paper. Then have students practice lining up the

digits properly and doing the first step of the algorithm in their notebooks for several more problems.

Add each digit separately

Regroup 10 ones with 1 ten: 70 + 11

= 70 + 10 + 1

= 80 + 1 = 81

Tens Ones

3 6

4 5

7 11

8 1

Tens Ones

2 7

1 9

3 16

4 6

1

27

+ 19

6


Tell them to leave enough space below their work so that they can finish the addition later. When they are

done, show the second step of the algorithm for the initial problem you showed them:

1

2 7

+ 1 9

4 6

Discuss the differences and similarities of these two methods of adding.

Similarities: you are still adding the ones and tens separately; you still get the same answer, you are still

trading the 10 ones from the 16 to get 1 ten and 6 ones. Differences: you don’t have to write the 3 at all; you

put the 1 still in the tens column, but this time it’s on top of the other tens; there is less writing. Emphasize

that you do not need to re-write the ones digit from the number of ones as you do using the chart, and that

you are just doing 2 steps at the same time. When using the standard algorithm, instead of finding 2 + 1 = 3

and then 1 + 3 = 4, you are doing both at the same time by finding: 1 + 2 + 1 = 4.

ASK: What is wrong with the following addition:

2 3

+ 2 5

2 5 5

Have a volunteer do the problem correctly. Discuss the importance of lining up the digits properly when adding.

Tell them that it is not just to make it look better; it actually makes it easier to add the numbers correctly.

ASK: How does grid paper make it easy to line up the digits?

Give students many problems to practice with, using grid paper. Include problems where the numbers add

up to more than 100 (EXAMPLES: 85 + 29, 99 + 15).

Tell your students that you had a student once who always added the tens before the ones. Show them the

student’s work and challenge them to find the answers that the student got wrong:

1 1

+ 5 8

6 9

1

1 7

+ 2 7

3 4

1

2 6

+ 2 6

4 2

4 3

+ 2 5

6 8

Lead a discussion on why it is important to add the ones first – if they add the tens first, they will forget to add

the extra 1 that was traded for 10 ones. Tell them that it is a bit tricky because they have to add from right to

left instead of from left to right. Tell them that even many grade 4 students will sometimes have trouble

remembering to add from right to left because it is so different from what they are used to, so that’s why it’s

important to practice a lot.

Tens Ones

2 7

1 9

3 16

4 6


Give students more problems to practice with.

NOTE: Here is another exercise that will help your students visualize regrouping two digit numbers.

For this exercise you will need real or play money (dimes and pennies), and cards that have been divided

into ten squares, using the layout of tens frames.

Add: 38 + 25

STEP 1: Make a model of the two numbers by placing dimes on the tens cards and pennies on

the ones cards.

38 =

25 =

STEP 2: Move as many pennies from the lower ones-card as you need to fill the upper ones-card.

38

+ 25

STEP 3: Exchange the ten pennies on the upper ones card for a dime and place the dime in the

upper tens card. (This is the equivalent to the carrying step.) Notice that there are only 3

pennies left on the lower ones car:

tens ones

10¢ 1¢ 10¢ 10¢

1¢

1¢

1¢

1¢

1¢

1¢ 1¢

10¢ 10¢ 1¢ 1¢ 1¢ 1¢ 1¢

tens ones

10¢ 10¢ 10¢ 1¢

1¢

1¢

1¢

1¢

1¢

1¢ 1¢

1¢ 1¢

10¢ 10¢ 1¢ 1¢ 1¢

tens ones

1

38 + 25 3

1¢ 1¢ 1¢

10¢ 10¢ 10¢

10¢ 10¢

10¢


STEP 4: Move all the coins to the upper cards and count the total (63).

Addition Rummy

This activity was adapted from: “A Companion Resource for Grade Two Mathematics,” by Saskatchewan

Learning. This activity will consolidate the students’ understanding of 2-digit addition and will prepare them

for 3-digit addition.

Give your students the BLMs “Addition Rummy Preparation” and “Addition Rummy Blank Cards”.

Explain to your students three different ways of adding and how they all correspond to the same problem.

For example:

1

23 2 tens + 3 ones +

+ 39 + 3 tens + 9 ones

62 5 tens + 12 ones

Always make sure to draw the model on the board as well as using the actual base ten materials so that

students can see how the symbol looks on “paper”.

Then have them do the BLM “Addition Rummy Preparation”. They have to create the two matching cards

themselves. They could then make up their own addition questions, solve them using the standard algorithm

and show them in the two different ways on the BLM “Addition Rummy Blank Cards”. Suggest to the

students that they fill the left-hand column with the standard algorithm, the second column with __ tens + ___

ones and the third column with how they would show it with base ten materials. Actually give the students the

base ten materials if it helps them.

Once all students are done the worksheet, they can cut out their cards and they are ready to play Addition

Rummy. They should play with a partner; together they will have made a total of 36 playing cards (12 new

cards each and 12 shared that they can only use one copy of). They should deal out 8 cards to each player

and the remaining cards will be in a pile, face down, between them.

tens ones

1 38 + 25

63

10¢ 10¢ 10¢ 10¢

10¢

10¢ 1¢ 1¢ 1¢


The top card should be placed face up in a separate discard pile. The first player decides to either pick

up the top card from the face up discard pile or to pick up the top card from the face down pile. He or she

then discards a card face up for the next player. It is then the next player’s turn. Once a player has 3

complete sets of 3 matching cards, they win. Note that, on their last turn, the student will not need to discard

any cards.

Estimating Game

You will need dice for this game. Draw on the board:

If you do not have dice, you can have students make their own dice (see below).

Either roll one die 4 times, 2 dice twice, or 4 dice once. Write the digits you rolled on the board for the

students to see. Tell them that you want to place your roll in the top 4 squares so that the sum of the 2-digit

numbers is as close to 100 as possible. Let several volunteers guess and add their 2-digit numbers on the

board. Ask which one is the closest.

You do not need to make a big deal out of who was the closest, but make sure all students agree on which

answer is the closest. Then have them try to make those same numbers as close to 70 as possible and

finally as close to 40 as possible.

Then have a student roll 4 times and write their numbers on the board. Repeat the exercise with those 4

numbers.

Provide students with the BLM “Estimating Game”. Students can work in groups if there is not enough dice

for everyone. They could each have their own sheet, and take turns recording their numbers before anyone

begins the actual page.

As an extension, you could make it harder by having them record the number in a square after each roll and

not allowing them to change their minds based on their next roll.

How to make your own dice:

Students could use the nets from the BLM “Cubes” to make dice. Another more fun (but time-consuming)

way to make dice is to have your students bring in 6-pack egg cartons, but bring in a few extra in case some

students forget. It is a good idea to start collecting them several weeks before actually doing the activity. Let

them know that a 12-pack cut in half will not work for the activity, they actually need a closed 6-pack that you

could shake a coin in and the coin won’t fall out.

To make the dice, have them write different numbers in each egg-hole in the carton. You could have them

write the numbers on paper first and tape or glue them to the carton. Then they put four counters into the

carton and shake. This is their “roll”.


I have ---, who has ----?

Using the BLM “Make Up Your Own Cards,” (see NS3-6: Representation in Expanded Form) make cards,

depending on the number of students in the class. Example,

I have

28 ----------

Who has

17 + 25

The student must read the answer from the bottom (the student says, “I have 28, who has 42?” and the

person with 42 then says “I have 42, who has ---?” depending on what question they have on the bottom of

their card. Play continues until everyone gets a turn.

Dominoes

This is a variation of the “I have --- who has ---?” game. Have one student go up to the board or and tape

their card. Then the person whose top matches the bottom of the other goes to play theirs in domino fashion.

Or have two teams, randomly distributing the cards to two teams and the team that can make the longest

chain wins.


NS3-23 Adding with Money

Goals: Students will add dimes and pennies.

Prior Knowledge Required: Adding 2-digit numbers with regrouping

Using cents notation properly

Vocabulary: fewer, fewest, ¢

Write on the board: 51 = tens + ones. Have volunteers fill in the blanks with all the possible

combinations (EXAMPLE: 51 = 4 tens + 11 ones).

ASK: Which combination uses the most tens? Which combination uses the most ones? Which is the

standard way of writing 51 as tens and ones? (5 tens + 1 one) Why do you think that is? (It uses the fewest

number of tens and ones, i.e. the fewest number of blocks in a base ten model.)

SAY: “I have 2 quarters and 1 penny, so 51¢ in total. I want to trade my 51¢ for dimes and pennies. How

many dimes and pennies could I trade for? Have volunteers provide different answers to the question:

51¢ = dimes + pennies

ASK: Which combination uses the most dimes? Which combination uses the most pennies? Which

combination uses the least number of coins? If you were to invent a standard way of writing amounts of

money as sums of dimes and pennies, how would you do it? Would you use the most dimes or the most

pennies? Why?

Remind your students that you asked how they could write 51 as a sum of tens and ones and also how they

could write 51¢ as a sum of dimes and pennies. ASK: What is the same about these two questions?

(Answers include: the same numbers work in both; a ten is worth te n ones and a dime is worth ten pennies;

the way that uses the fewest blocks also uses the fewest coins.) Have students answer several questions

using both tens and ones, and dimes and pennies. Students should use standard form.

EXAMPLE: 36 = tens + ones 36¢ = dimes + pennies

Then take away the context of tens and ones and have students do examples using only dimes

and pennies.

Draw a dimes and pennies chart and ask students if

they’ve seen another chart that looks similar.

ASK: What do you think the second row is for? What would you do if the headings were “tens” and “ones”?

Tell students that they can regroup 10 pennies as a dime just like they can regroup 10 ones as

a ten. Complete the second row in the chart.

Have students complete more such charts independently. Ensure that every student can regroup 10 pennies

for 1 dime, 20 pennies for 2 dimes, and so on, before moving on. Then have students use charts to add

dimes and pennies the same way they added 2-digit numbers. Finally, have them add dimes and pennies

using regrouping and carrying. Encourage them to write the cents sign (¢) when adding money.

Dimes Pennies

2 12


NS3-24 Adding 3-Digit Numbers

Goals: Students will add 3-digit numbers with and without regrouping.

Prior Knowledge Required: Adding 2-digit numbers

Place value Base ten materials

Vocabulary: algorithm

Have volunteers draw base ten models for 152 and 273 on the board and tell students that you want to add

these numbers.

ASK: How many hundreds, tens, and ones are there altogether? Do we need to regroup? How do you know?

How can we regroup? (Since there are 12 tens, we can trade 10 of them for 1 hundred.) After regrouping,

how many hundreds, tens, and ones are there? What number is that?

Write out and complete the following statements as you work through the EXAMPLE:

152 hundred + tens + ones

+ 273 hundred + tens + ones

hundred + tens + ones

After regrouping: hundred + tens + ones

Have students add more pairs of 3-digit numbers. Provide examples in the following sequence:

• the ones need to be regrouped (EXAMPLES: 238 + 147, 426 + 165)

• the tens need to be regrouped (EXAMPLES: 456 + 381, 277 + 392)

• either the ones or the tens need to be regrouped (EXAMPLES: 349 + 229, 191 + 440).

• both the ones and the tens need to be regrouped (EXAMPLES: 195 + 246, 186 + 593).

• you have to regroup the tens, but you don’t realize it until you regroup the ones (EXAMPLES: 159 +

242, 869 + 237) Use this to emphasize the importance of regrouping the ones first.

Now show students the standard algorithm alongside a hundreds, tens, and ones chart for the first example

you did together (152 + 273):

Hundreds Tens Ones

1 5 2

2 7 3

3 12 5

3 + 1 = 4 12 – 10 = 2 5

1

1 5 2

+ 2 7 3

4 2 5

Point out that after regrouping the tens, you add the 1 hundred that you carried over from the tens at the same time as the hundreds from the two numbers, so you get 1 + 1 + 2 hundreds.


Demonstrate both the chart and the algorithm for adding a 3-digit number to a 2-digit number and emphasize

the importance of lining up the digits – ones with ones, tens with tens and hundreds with hundreds.

Activities:

1. Play addition rummy as in NS3-21, except use 3-digit numbers in place of 2-digit numbers.

2. Play “I have ---, who has ---?” From NS3-6, but this time use pairs of 3-digit numbers that sum to at most

999. These cards can also be used to play the dominoes game introduced in NS3-22.

Extensions:

1. Game for two players:

Each player makes a copy of the grid shown.

Players take turns rolling a die and writing the number rolled on one of their grid boxes.

The winner is the player who creates two 3 digit numbers with the greatest sum.

2. If you have already taught your students Egyptian writing (see Extension for NS3-1: Place Value –

Ones, Tens, and Hundreds) you could ask them to show adding with regrouping using Egyptian writing.

EXAMPLE:

3. Have students add more than 2 numbers at a time:

a) 427 + 382 + 975 + 211

+

+


NS3-25 Subtracting 2- and 3-Digit Numbers

Goals: Students will subtract without regrouping.

Prior Knowledge Required: Subtraction as taking away

Base ten materials Place value Separating the tens and the ones

Vocabulary: standard algorithm

Tell your students that you want to subtract 48 – 32 using base ten materials. Have a volunteer draw a base

ten model of 48 on the board. SAY: I want to take away 32. How many tens blocks should I remove? (3)

Demonstrate crossing them out. ASK: How many ones blocks should I remove? (2) Cross those out, too.

ASK: What do I have left? How many tens? How many ones? What is 48 – 32?

Have student volunteers do other problems with no regrouping on the board (EXAMPLES: 97 – 46, 83 – 21,

75 – 34). Have classmates explain the steps the volunteers are taking. Then have students do similar

problems in their notebooks.

Give students examples of base ten models with the subtraction shown and have them complete the tens

and ones charts:

48 – 32: Tens Ones

4 8 – 3 2

1 6

Have students subtract more 2-digit numbers (no regrouping) using both the chart and base ten models.

When students have mastered this, have them subtract by writing out the tens and ones (as in question 2 on

the worksheet):

46 = 4 tens + 6 ones

– 13 = 1 tens + 3 ones

= 3 tens + 3 ones

= 33

Then have students separate the tens and ones using only numerals (as in QUESTION 3 on the worksheet):

36 = 30 + 6

– 24 = 20 + 4

= 10 + 2

= 12


Now ask students to subtract:

54

– 23

ASK: Which strategy did you use? Is there a quick way to subtract without using base ten materials, or tens

and ones charts, or separating the tens and ones? (Yes—subtract each digit from the one above.) What are

you really doing in each case? (Subtracting the ones from the ones and the tens from the tens, and putting

the resulting digits in the right places.)

Have students draw a base ten model of 624 and show how to subtract 310. Have them subtract using the

standard algorithm (i.e. by lining up the digits) and check to see if they got the same answer both ways.

Repeat with several 3-digit numbers that do not require regrouping.

Extensions:

1. Subtract using the base ten blocks:

a) 7 2 9 b) 8 9 5 c) 5 2 4 d) 3 9 8 e) 5 9 2

– 3 1 6 – 2 5 4 – 4 0 1 – 1 6 3 – 1 7 0

Bonus: Have students subtract from many-digit numbers:

9964387 678439841

– 2541263 – 138210731

Students may subtract from right to left or left to right. If you teach borrowing from right to

left as many teachers do, you may wish to develop the habit of subtracting from right to left

at this stage.

2. Write on the board: 100 – 36. ASK: How is this problem different from problems they have seen so far?

Challenge them to change it to a problem they already know how to do. After letting them work for a few

minutes, suggest that they think of a number close to 100 that does have enough tens and ones to

subtract directly and then adjust their answer. (Students can use any of 96, 97, 98 or 99; for example,

98 – 36 = 62, so: 100 – 36 = 64.)

3. Teach students to subtract numbers like 100 – 30 by counting the number of tens in each number:

10 tens – 3 tens = 7 tens = 70. Give several practice problems of this type and then ask:

What is 100 – 40? What is 40 – 36? (students can count up to find this answer) How does this help to

find 100 – 36? (show a number line to help them see the addition they need to do)

4. Have students subtract by changing to a problem they already know how to do and then adjusting their

answers:

a) 61 – 28 b) 34 – 15 c) 68 – 39

Example solution: 58 – 28 = 30, so: 61 – 28 = 30 + 3 = 33.


NS3-26 Subtracting by Regrouping

Goals: Students will subtract with regrouping using base ten materials and using the standard algorithm.

Prior Knowledge Required: Subtraction without regrouping

Base ten materials Counting by 10s from any number

Vocabulary: standard algorithm, regrouping, “borrowing”

Ask students how they learned to subtract 46 – 21. Have a volunteer demonstrate on the board. Ask the

class if you can use the same method to subtract 42 – 28. What goes wrong? Should you be able to subtract

28 from 42 If you have 42 things, does it make sense to take away 28 of them? (Sure it does). Challenge

students to think of a way to change the problem to one that looks like a problem they did last time. ASK:

What is a number close to 42 that has a larger ones digit as 28? (39) What is 39 – 28? (11) How is this

problem more like the ones from last class? How can we use 39 - 28 to find 42 – 28? Show a number line on

the board to help them.

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Show students how to subtract by adding: 11 + 3

28 39 42

The difference between 42 and 28 is 11 + 3 = 14. ASK: Is there another way to change the problem to make

it more like the problems we did last time? (finding 38 – 28 and then adding 4 would also work) Have

students do more problems of this type by finding a number that is close to the number they are subtracting

from but has the same or larger ones digit as the number they are taking away.

a) + b) + c) +

37 81 38 63 56 72

EXAMPLE: Students could find 81 – 37 by finding 79 – 37 = 42 and then counting up from 79 to 81 to

find 81 – 79 = 2, so 81 – 37 = 44. Students could also find 78 – 37 or 77 – 37 and then count up and add

the differences.

Now change the rules: tell students they are not allowed to use adding anymore; they need to think of a way

to change 46 – 28 into a problem just like last time where 46 has more tens and more ones than 28. ASK:

How can you trade tens and ones so that 46 has more ones than 28? Have pairs work on 46 – 28 using

these new rules. Give them a few minutes to think about the problem, and then have volunteers share their

strategies with the group.


ASK: Which number required trading—the larger number or the smaller number? Why does it help to have

more ones blocks in the number 46? Use base ten materials to illustrate—make a standard model of 46 and

then regroup 1 ten as 10 ones:

46

– 28

SAY: Now there are 16 ones, so we can take away 8 of them. Since we didn’t change the value of 46—we

just traded blocks—we will get the right answer. Removing the shaded blocks (2 tens and 8 ones), we are

left with 1 ten and 8 ones. So: 46 – 28 = 18.

Tell your students that mathematicians have come up with a standard notation to say that you are borrowing

a ten from the tens digit and trading it for ten ones:

4 14

54 = 5 tens + 4 ones 4 tens + 14 ones

Emphasizing that you are taking 1 ten away and replacing it with 10 ones. Have volunteers come to the

board to show this standard notation for various examples alongside the trading of base ten materials. (First,

have students show each number using base ten materials and then have another volunteer show the

trading and the notation for trading at the same time).

When students have mastered this, move on to subtracting using the standard algorithm.

Draw on the board:

7 5 – 4 8

ASK: Can you take away 48 from 75? (yes, 75 is bigger than 48, so if I have 75 objects, I can just remove 48

of them and count how many are left) Can you take away 8 ones from the 5 ones: Do they need to borrow?

How can they change the 5 ones so that they have enough ones? What can they borrow from so that the 75

still has the same value? Have a volunteer show using the standard notation how they would borrow a ten

from the tens digit and replace it with ten ones – emphasize that the 48 stays the same; only the 75 is

changing. When they are done the borrowing should look like:

Tens Ones 4 6

5 4

7 5 6 0 8 9 5 3 3 0


6 15

7 5 – 4 8

Students can check their borrowing by adding 60 + 15 = 75. Then ask if we can take 8 ones from 15 ones?

Can we take 4 tens away from 6 tens? Have a volunteer show this and say the answer to 75 – 48.

Do several examples of this where borrowing is always required and then give them lots of practice where

borrowing is always required.

Then tell them that sometimes borrowing is not required and they have to decide when to use borrowing.

Show them what would happen if you borrowed when you didn’t need to.

6 15

7 5

– 5 2

1 13

They could still do it this way if they remember to trade the ten ones back for a ten to end up with 2 tens and

3 ones, but it is a lot faster not to borrow in the first place:

Give several examples where students only need to decide whether to borrow or not. Have several

subtraction questions on the board and have them raise their hand if they need to borrow and not raise their

hand if they don’t need to borrow as you point to each question. Encourage students to explain how they

know. Emphasize that if there are more ones in the first number, they don’t need to borrow, but if there are

more ones in the second number, they do need to borrow.

Then give them several examples where they need to decide whether or not to borrow and then do the

subtraction. Have student volunteers explain at each step what they are doing.

Activity: Ask students to show the regrouping in QUESTIONS 1 a) to d) with base ten blocks.

7 5 – 5 2 2 3


Extensions:

1. Have students subtract by adding and use the standard algorithm for several problems.

EXAMPLE: 54 – 26 = + + =

26 30 50 54

When is the standard algorithm easiest? (when there is no borrowing necessary) When is subtraction by

adding easiest? (when the adding does not require carrying) Notice that the addition will require carrying

precisely when the subtraction does not require borrowing. For example, 54 – 26 = 4 + 20 + 4 = 20 + 8 =

28 and no carrying is required, whereas 54 – 23 = 4 + 20 + 7 = 20 + 11 = 31 and there is carrying

required.

2. Make a 2-digit number using consecutive digits (EXAMPLE: 23). Reverse the digits of your number to

create a different number, and subtract the smaller number from the larger one (EXAMPLE: 32 – 23).

Repeat this several times. What do you notice? Some students may wish to investigate what happens

when we don’t use consecutive digits (EXAMPLE: 42 – 24 = 18, 63 – 36 = 27, 82 – 28 = 54; the result in

this case is always from the 9 times tables).

Journal:

Have students explain which way of subtracting (subtracting by adding or the standard algorithm) they like

better and why.


NS3-27 Subtracting by Regrouping Hundreds

Goals: Students will subtract 3-digit numbers by borrowing when necessary.

Prior Knowledge Required: subtracting by using base ten materials and regrouping

subtracting 2-digit numbers by borrowing (using the standard algorithm) standard notation for regrouping

Vocabulary: subtraction, borrowing, regrouping, standard notation, standard algorithm.

Have students subtract 3-digit numbers using base ten materials and transfer their results to

standard notation:

For example, to subtract 543 – 365, they would trade a hundreds block for ten tens blocks and a tens block

for ten ones blocks and the borrowing, in standard notation would look like:

13

4 14 4 14 13

5 4 3 5 4 3

3 6 5 3 6 5

Students have to keep trading until all numbers in the top row are larger than all numbers in the bottom row.

They could also trade the tens and ones first instead:

13

3 13 4 3 13

5 4 3 5 4 3

3 6 5 3 6 5

1 7 8

In any case, the answer is 178. Students should be encouraged to check that their borrowing is done

correctly by adding: 4 hundreds + 13 tens + 13 ones = 400 + 130 + 13 = 543. Students should also check

that their subtraction is done correctly by adding 178 + 365 = 543.

Notice that if you teach students to subtract each digit as it becomes available so that they are borrowing

then subtracting then borrowing then subtracting and so on, then you must teach them to borrow from right to

left. If students borrowed the ten tens from the hundreds digit first, their subtraction would start with hundreds

digit 1 and tens digit 8. When they then borrow from the tens, they would find that the 8 is incorrect.


If, however, you teach students to set up all the borrowing first so that the problem looks just like one without

borrowing, then students may borrow in any order. The advantage of this approach is that it reinforces the

idea of turning the problem into one they already know how to do; the disadvantage is that it is less

“standard” in that later teachers might expect your students to borrow from right to left. Even if you teach the

standard method, it will deepen the students’ understanding if you show them that they will get the correct

answer by doing the subtraction after all regrouping is done.

To teach the standard method, provide problems in the following sequence:

• regrouping a ten as ten ones (EXAMPLE: 754 – 216)

• regrouping a hundred as ten tens (EXAMPLE: 754 – 281)

• regrouping a hundred as ten tens and regrouping a ten as ten ones (EXAMPLE: 754 – 467) Show

students how this will obtain the incorrect answer if they borrow from left to right and subtract the

digits as they borrow.

• regrouping a hundred as ten tens is required, but only after regrouping a ten as ten ones (EXAMPLE:

754 – 357) Use this to again emphasize the importance of borrowing from right to left.

Activities:

1. Ask students to make a base ten model for the number 81. Then have students take away any 3 blocks.

Have volunteers draw their model for the number they have left on the board. ASK: What number did the

student take away? What number is left?

Have other volunteers show different answers—how many answers are there?

Give students base ten materials and an individual number with sum of digits at least 5 (81, 63, 52, 23,

41, 87) Ask students to make base ten models for their number and to find as many numbers as they

can by taking away exactly 3 blocks. They should make a poster of their results by drawing models of

base ten materials and showing how they organized their answers. This can be done over 2 days, giving

2-digit numbers the first day and 3-digit numbers the second day.

2. Ask students to show the regrouping in QUESTIONS 1 a) to d) with base ten blocks.

Extensions:

1. Teach your students the following fast method for subtracting from 100, 1 000, 10 000 (it helps them

avoid regrouping).

You can subtract any 2-digit number from 100 by taking the number away from 99 and then adding 1 to

the result. EXAMPLE:

100 = 99 + 1 – 42 = – 42 = 57 + 1 = 58

You can subtract any 3-digit number from 1 000 by taking the number away from 999 and adding 1 to

the result. EXAMPLE:

1000 = 999 – 423 = – 423 = 576 + 1 = 577


2. (See Extension 2 from Worksheet NS3-26: Subtracting by Regrouping) Create a 3-digit number using

consecutive digits (EXAMPLE: 456). Reverse the digits of your number to create a different number, and

subtract the smaller number from the larger one. Repeat this several times. What do you notice? (The

result is always 9 for 2-digit numbers and 198 for 3-digit numbers). Have them find the result for 4-digit

numbers and predict the result for 5-digit numbers.

3. Have students determine which season is the longest if the seasons start as follows (in the northern

hemisphere):

Spring: March 21

Summer: June 21

Fall: September 23

Winter: December 22

Assume non-leap years only.

Students will need to be organized and add several numbers together as well as use some subtraction.

For example, the number of days in spring is March: 31 – 20 (since the first 20 days are not part of

spring) April: 30 May: 31 June: 20.

Total: Students may either add: 11 + 30 + 31 + 20 or notice that they are adding and subtracting 20 and

simply add 31 + 30 + 31 = 92 days. Summer has 10 + 31 + 31 + 22 = 94 days, fall has 8 + 31 + 30 + 21

= 90 days and winter has 10 + 31 + 28 + 20 = 89 days, so in order from longest to shortest: summer,

spring, fall, winter.

Students should be encouraged to compare this ordering with temperature. It is premature to discuss

scientific reasons for this; just noticing the pattern is enough.

Students may wish to examine this question for their own region and the current year.


NS3-28 Mental Math

Goals: Students will decompose numbers into sums in an organized way and will use pairs making 10 to

add single-digit numbers. Students will add multiples of 10 to 1-digit numbers.

Prior Knowledge Required: Decomposing numbers in different ways

Models for addition

Missing addend problems

Addition is commutative

Vocabulary: pair, making 10

Draw 6 rows of 7 circles, with a vertical line between them as shown.

Write 1 + 6 = 7 beside the first row and have different volunteers come up and write number sentences for

each row of circles. Ask your students what number is the same in each addition sentence and what

numbers change. Ask how the numbers change each time. What happens to the first number? (It increases

or grows by 1) What happens to the second number? (It decreases or shrinks by 1).

Ask a volunteer to show 7 + 0 or 0 + 7. Tell them that when finding all the number sentences that add to 7,

counting those two would almost be cheating, because they’re too easy.

Then write 8 = 1 + ____

8 = 2 + ____

8 = 3 + ____

8 = 4 + ____

8 = 5 + ____

8 = 6 + ____

8 = 7 + ____

Ask them for strategies to fill in the rest of the numbers. Possibilities include:

• Draw 8 circles and a vertical line between them at various points and then count the circles to the

right of the vertical line.

• Start at the bottom and write the numbers 1, 2, 3, 4, 5, 6 and 7 in order from the bottom up;

• Start at the top, and write the numbers in backwards order, starting at 7;


Emphasize that from one addition sentence to the next, you are adding one to the first number and

subtracting one from the second number, so you are not changing the total number. That is just like moving

the line one more circle to the right.

Tell them that you don’t need all of the number sentences because some of them have the same numbers,

and you don’t care what order the addends are in. Show how you can erase the bottom three equations and

still have all the pairs that add to 8.

Ask your students how many number sentences they think they’ll need to write all the ways you can sum to

6. (Don’t include 0 + 6 or 6 + 0 because it’s too easy of a sum, so you don’t care about that one).

Now teach students how to find pairs that make 10. Hold up all your fingers on both hands. ASK: How many

fingers do I have up? Then hold up 7 fingers and say, “How many fingers do I have up? How many do I not

have up? What is 7 + 3? How do you know?” Repeat with several examples. Then say, “I want to know what

number makes 10 with 4,” and write on the board: 4 + ____ = 10. ASK: How could I use my 10 fingers? How

many fingers should I hold up? What does the number of fingers I’m not holding up tell me?

Write 3 numbers on the board: 4 5 6. Ask them if 4 makes 10 with either of the next two numbers. Yes it

does, so we can circle the 4 and the 6.

4 5 6

Then try: 2 3 7 and ask if 2 makes 10 with any of the other numbers. What number would have to be there

for 2 to make 10 with it? If some students aren’t sure, remind them that they can hold up 2 fingers and count

the number of fingers that they are not holding up. So we can cross out the 2 because we know it’s not one

of the numbers we have to circle:

2 3 7

Then look at the last 2 numbers – do they make 10? Yes they do, so circle them. Do several examples of

this, either circling the first number with one of the other 2, or crossing out the first number and circling the

other 2.

Bonus: Use longer lists of numbers. (EXAMPLES: 3 4 5 6, 2 3 4 7 9)

Write 9 + 4 on the board. ASK: How can we write 4 as a sum of two numbers? (4 = 1 + 3 or 2 + 2).

Show students how to use this to add 9 + 4: 9 + 4 = 9 + 1 + 3 OR 9 + 4 = 9 + 2 + 2.

ASK: Does one of these make it easier to add 9 + 4? Why? Is it easier to add 10 + 3 or 11 + 2? Why?

Emphasize that we don’t have just one number on either side of the equal sign. Both sides have more than

one number. But that’s okay – it’s still true that if I have 9 “anythings” and add 4 more “anythings”, I get the

same number as if I started with 9, added 1 and then added 3.

Some students might find it helpful to have 3 piles of counters (of 9, 1 and 3) and then put the last 2 piles

together to see that they now have 4 in the second pile and still 9 in the first. Stress that the counters neither

disappear nor appear out of the air, so their number does not change.

Or you could draw three groups on the board, one of 9 circles, one of 1 circle and one of 3 circles. Write the

number sentence on the board as review: 9 + 1 + 3 = 13. Then circle the two last groups to show that you


are grouping them together and write: 9 + 4 = 13. Emphasize that you have not changed the total number of

circles just by grouping the last two piles.

Then ask students how to separate the second number to make 10 with the first number for several

problems, following the sequence below:

• Problems with first addend 9

(EXAMPLES: 9 + 5 = 9 + 1 + 4 = 10 + 4 = 14, 9 + 7, 9 + 3, 9 + 6, 9 + 8, 9 + 4)

• Problems with first addend 8

(EXAMPLES: 8 + 5, 8 + 6, 8 + 4, 8 + 7, 8 + 9, 8 + 8)

For each problem above, students should show what is happening with a model. For example, to show

9 + 5 = 9 + 1 + 4 = 10 + 4 = 14, draw a row of 9 beside a row of 5 separated as shown:

To the left of the line there are 10 dots and to the right, there are 4 dots, so this picture, before the line was

drawn, showed 9 + 5 and now shows 9 + 1 + 4 or 10 + 4. So 9 + 5 = 10 + 4 = 14.

Now provide many problems with first addend ranging from 6 to 9

(EXAMPLES: 6 + 7, 9 + 5, 7 + 7 , 8 + 5, 9 + 8, 7 + 4, 8 + 5, 8 + 7, 9 + 6) Have one volunteer decompose the

second addend using all the number sentences possible and another volunteer find the number that makes

10 with the first addend. Students can then fill in the blanks for problems of the form:

6 + 7 = 6 + ____ + ____

These make 10 Left over

Since 6 + 4 makes 10, students need to think: 7 = 4 + _____.

Then have students use this strategy to add pairs of single-digit numbers. Finally, have them add single-digit

numbers to 2-digit numbers using this strategy:

36 + 7 = 36 + _____ + _____

These make 40 Left over

Give students word problems which require them to use this new skill. (EXAMPLE: 57 students went on a

field trip. 4 teachers went with them. How many people went on the field trip?)

NOTE: The next lessons will begin to teach students how to solve word problems. If some students are

struggling with the word problems on this sheet, go back to these problems after they are more comfortable

with word problems. This will allow students to see their progress if they will later find easy what they

struggle with now.


Activities:

Magic Number Dice

Make a die from a photocopy of one of the nets on the BLM “Cubes”. Copy the numbers from a die onto your

cube. Be sure that opposite sides of the die add to 7 as is true for a regular die, so that 1 is opposite 6, 2 is

opposite 5 and 3 is opposite 4. Tell them that you made a bigger die by copying a smaller one. Tell them

there is a magic number on every die that many people don’t even know about and their goal is to find that

magic number. Tell them that instead of rolling two dice and adding the total, you are just going to roll one

and add the top and the bottom. Roll the die, show the top face and then the bottom face. Ask the students

what the total you rolled is. Ask a volunteer to come write the addition sentence on the board. Then ask

another volunteer to roll the die and write their addition sentence on the board by looking at the top and

bottom numbers. Continue having volunteers do this. Ask if anyone sees a magic number. Ask them if they

think it would make sense, if they didn’t have two dice, to just roll one and add the top and bottom numbers.

Why wouldn’t it make sense?

Tell them that you would like to make a different pair of dice, but this time, you want the magic number to be

10. Ask them what pairs of numbers they could put on the top and bottom to make a total of 10. Ask

volunteers to write addition sentences on the board such as 1 + 9 = 10, 2 + 8 = 10, etc.

Then give each student a copy of the BLM “Cubes” and have them make their own die with a magic number

of 10. Demonstrate cutting out a cube first, being careful not to cut out the tabs. If your students use tape

instead of glue, this is not as important.

Once their cubes are made, they can practice finding the missing number that makes 10 with the top number

and checking the bottom to see if they’re right. They can switch dice with different partners as well to practice

different addends, since some might have chosen different addition sentences that add to 10. Have students

roll their dice onto a plate or a shoebox lid, so that they don’t throw them across the room.

Tens

This is a solitaire game. You will need a deck of cards – remove the 10, J, Q and K. The player shuffles the

cards and turns over the first 10, putting them in 2 rows of 5 cards. The goal of the game is to find cards that

make 10 with the top card of the remaining pile. Each card they turn over, they either place it on top of

another card it makes 10 with or they discard it. After they run through the pile, they can take the discard pile

and use them to make 10 with cards that are at the top of one of the 10 piles that are face up. They can go

through the discard pile as many times as they want. They then count the number of cards in the discard pile

after they cannot place any more in piles. This is the number of points they get. The fewer points they get,

the better.

Game: Modified Go Fish

This game requires students to hold 6 cards and find a pair that makes 10. Details can be found in the

Mental Math section of this guide.

The activities below are designed to give students practice adding single-digit numbers that add to more than

10. They are all adapted from “A Companion Resource for Grade Two Mathematics,” by Saskatchewan

Learning.


Adding on a 9 × 9 grid

Give each student a copy of the BLM “Addition Table (Ordered)” and a small counter. Have the students toss

their counters as many times as they can in 2 minutes and write the answers to the addition: if their counter

lands on the column numbered 4 and the row numbered 9.

They write the answer to 4 + 9 in that square. In this way, students randomly generate questions for

themselves. Repeat this over several periods so that students can see how they are improving, then suggest

that they challenge their parents to a competition—can they add more pairs than their parents can?

Number Sentence Practice

You will need the BLM “Number Sentence Practice,” enough two-colour counters to give each pair of

students 30 of them. Photocopy the BLM enough times so that each pair of students can have one game

card between them. If you don’t have two-colour counters, you can use coins with heads and tails as the two

“colours”.

Group students in pairs. Give each pair of students their own game card and 20 counters. Photocopy a game

card onto a transparency so that you can demonstrate on the overhead projector. If you do not have an

overhead projector, copy a game card onto the board. Tell students that their goal is to get four in a row

before their partner does and that you’ll demonstrate playing against the class. To show them, you will go

first, but you want their help. Tell them that you are allowed to choose how many counters to use and then

you want to shake them to find a number sentence. Demonstrate this with 7 counters and tell students that

you will get a number sentence from the red counters and the yellow counters. Ask them what number

sentence you got. For example, if you roll 5 red and 2 yellow, 5 + 2 = 7 and 2 + 5 = 7 both work. Ask them if

your number sentence is on the card. Ask them if any number sentence on your card uses a total of 7

counters. How can they tell? Ask them if it was a good choice to use only 7 counters? Why not? Then

challenge them to use a better number of counters, one that they can find a lot of number sentences on the

game card for. Are there a lot of number sentences on the card that add to the same number? Which

number is a good number of counters to try? Does someone want to come up and shake that many counters

and see what number sentence they get? If they get a number sentence that is on the card, they get to put

their colour of counter on the board. So if we decide that the teacher is red and the class is yellow, then the

student puts a yellow counter on the game board. Repeat this several times so that students understand,

always demonstrating good strategy on your turn. Make sure they know that once a square is covered, it

cannot be used again. Then let them play against each other in pairs.

Also, since they only have 20 counters, they will need to fill the places that add to 17 and 18 quickly, or they

never will.

Extensions:

1. To practice other missing addend or subtraction problems, students can make dice with other magic

numbers, anywhere from 5 to 16.

2. For extra practice, provide the BLM “Ten-Dot Dominoes”.


NS3-29 Parts and Totals

Goals: Students will use pictures and charts to solve word problems.

Prior Knowledge Required: Subtraction as take away

Subtraction as comparison (how much more than, how much shorter than, and so on)

Addition as how much altogether, or how many altogether, or how long altogether

Vocabulary: difference, total, altogether, how many more than

Write on the board:

Red apples

Green apples

Tell students that each square represents one apple and ASK: How many red apples are there? How many

green apples? Are there more red apples or green apples? How many more? (Another way to prompt

students to find the difference is to ASK: If we pair up red apples with green apples, how many apples are

left over?) How many apples are there altogether?

Label the total and the difference on the diagram, using words and numerals:

Difference: 4 apples

Red apples Total: 8 apples

Green apples

Do more examples using apples or other objects. You could also have students count and compare the

number of males and females in the class. (If yours is a single-gender class, you can divide the class by age

instead of by gender). ASK: How many girls are in the class? How many boys? Are there more boys or girls?

How many more? What subtraction sentence could you write to express the difference? How many children

are there altogether in the class? What addition sentence could you write to express the total?


NS3-30 Parts and Totals (Advanced)

Goals: Students will recognize what they are given and what they need to find in a word problem.

Prior Knowledge Required: Subtraction as take away

Subtraction as comparison (how much more than, how much shorter than, and so on) Addition as how much altogether, or how many altogether, or how long altogether

Vocabulary: equation, fact family

Then draw the following chart and have volunteers fill in the blanks:

Red Apples Green Apples Total Number of

Apples

How many more of

one colour of apple

2 3

7 5

3 6 more green than red

5 3 more red than green

9 2 more green than red

7 1 more red than green

3 11

5 8

Then ask students to fill in the columns of the chart from this information:

a) 3 red apples, 5 green apples

b) 4 more red apples than green apples, 5 green apples

c) 4 more red apples than green apples, 5 red apples

d) 11 apples in total, 8 green apples

e) 12 apples in total, 5 red apples

Have students find the total number of apples and the difference between how many of each colour.

a) 2 red apples and 4 green apples b) 7 red and 3 green c) 3 red and 8 green

Have students find the number of green apples and the total number of apples.

a) 2 red and 6 more green than red b) 7 red and 3 more red than green.

Have students find the number of green apples and how many more of one colour:

a) 2 red and 8 altogether b) 6 red and 10 altogether c) 7 red and 12 altogether


Draw on the board:

Ask your students to suggest number sentences (addition and subtraction) represented by this model.

Ensure that either you or the students state what the numbers in each number sentence represent. Continue

until all possible answers have been identified and explained: 3 + 4 = 7 because 3 white circles and 4 dark

circles make 7 altogether; 4 + 3 = 7 because 4 dark circles and 3 white circles make 7 altogether; 7 – 4 = 3

because removing the 4 dark circles leaves 3 white circles; 7 – 3 = 4 because removing the 3 white circles

leaves 4 dark circles. Tell them that these number sentences all belong to the same fact family because they

come from the same picture and involve the same numbers. Tell them that these number sentences are all

examples of equations because they express equality of different numbers, for example 3 + 4 and 7 or 7 – 4

and 3. Write on the board:

a) 3 + 5 = 8 b) 43 – 20 > 7

c) 9 – 4 < 20 – 8 d) 5 + 12 + 3 = 11 + 9 = 31 – 11

Ask which number sentences are equations (parts a and d only).

Then have students list all the possible equations in the fact family for different pictures. Include pictures in

which more than one attribute varies.

EXAMPLE: (dark and white, big and small)

NOTE: The fact families for 4 + 5 = 9 and 6 + 3 = 9 can both be obtained from this picture, but they are not

themselves in the same fact family.

When students are comfortable finding the fact family of equations that correspond to a picture, give them a

number sentence without the picture and have them write all the other number sentences in the same fact

family. Use progressively larger numbers, sometimes starting with an addition sentence and sometimes with

a subtraction sentence (EXAMPLES: 10 + 5 = 15, 24 – 3 = 21, 46 + 21 = 67, 103 – 11 = 92).


Green

Grapes

Purple

Grapes

Total

Number of

Grapes

Fact Family for

the Total Number

of Grapes

How many more

of one type

of grapes?

7 2 9 7 + 2 = 9 2 + 7 = 9

9 – 7 = 2 9 – 2 = 7 5 more green than purple

5 3

7 3 more purple than green

7 3 more green than purple

5 12

6 10

Look at the completed first row together, and review what the numbers in the fact family represent.

Demonstrate replacing the numbers in each number sentence with words:

7 + 2 = 9 becomes: green grapes + purple grapes = total number of grapes

2 + 7 = 9 becomes: purple grapes + green grapes = total number of grapes

9 – 7 = 2 becomes: total number of grapes – green grapes = purple grapes

9 – 2 = 7 becomes: total number of grapes – purple grapes = green grapes

Have students complete the chart in their notebooks, and ask them to write out at least one of the fact

families using words.

Look at the last column together and write the addition sentence that corresponds to the first entry, using

both numbers and words:

5 + 2 = 7: how many more green than purple + purple grapes = green grapes

Have students write individually in their notebooks the other equations in the fact family, using both numbers

and words:

2 + 5 = 7: purple grapes + how many more green than purple = green grapes

7 – 5 = 2: green grapes – how many more green than purple = purple grapes

7 – 2 = 5: green grapes – purple grapes = how many more green than purple

Make cards labelled:

• # of green grapes

• # of purple grapes

• Total number of grapes

• How many more purple than green

• How many more green than purple

Stick these cards to the board and write “=” in various places.

EXAMPLE: # green grapes # purple grapes = how many more green than purple

Have students identify the missing operation and write in the correct symbol (+ or –).

For variety, you might put the equal sign on the left of the equation instead of on the right.

Be sure to create only valid equations.


For example: How many more purple than green # purple grapes = # green grapes is invalid, since

neither addition nor subtraction can make this equation true.

Teach students to identify which information is given and what they need to find out. They should underline

the information they are given and circle the information they need to find out.

For example:

Cari has 15 marbles. 6 of them are red. How many are not red?

Teach students to write an equation for what they have to find out in terms of what they are given.

EXAMPLE:

not red = marbles red marbles

+ or −

not red = 15 − 6 = 9

They then only need to write the appropriate numbers and add or subtract.

Finally, do a simple word problem together. Prompt students to identify what is given, what they are being

asked to find, and which operation they have to use. Encourage students to draw a picture similar to the bars

drawn in exercise 1 of worksheet NS5-14: Parts and Totals.

EXAMPLE:

Sera has 11 pencils and Thomas has 3 pencils. How many more pencils does Sera have?

I know: # of pencils Sera has (11), # of pencils Thomas has (3)

I need to find out: How many more pencils Sera has than Thomas

How many more Sera has = # of pencils Sera has – # of pencils Thomas has

= 11 – 3

= 8


NS3-31 Sums and Differences

Draw:

10 m

13 m

Tell your students that you asked two people to tell you how much longer the second stick is than the first

stick. One person answered 3 m and the other person answered 23 m. How did they get their answers?

Which person

is right? How do you know?

Emphasize that something measuring only 13 m long cannot be 23 m longer than something else.

Ask if anyone has a younger sister or brother. Ask the student their age, then ask how old their sister or

brother is. Ask them how much older they are than their sister or brother. How did they get their answer? Did

they add the ages or subtract? Why? Emphasize that when they’re asked how much older or how much

longer that they’re being asked for the difference between two numbers so they should subtract. Write:

Smaller Larger How much more?

10 m 13 m 3 m longer

5 years 9 years 4 years older

Give several word problems like this and continue to fill in the chart. Try to wait until most hands are raised

before allowing a volunteer to come to the board. EXAMPLE:

1. Katie has 13 jelly beans. Rani has 7 jelly beans. Who has more jelly beans? How many more? 2. Sara weighs 29 kg. Ron weighs 34 kg. Who weighs more? How much more? 3. Anna’s pet cat weighs 15 kg and her new kitten weighs 7 kg. How much more does the cat weigh than

the kitten?

Continue to use the chart, but this time have students insert a box ( ) when they aren’t given the quantity.

Ask questions such as: Are you given the larger number or the smaller number? How do you know?

EXAMPLES:

1. Sally’s yard is 21 m long. Tony’s yard is 13 m longer than Sally’s. How long is Tony’s yard?

2. Teresa read 14 books over the summer. That’s 3 more than Randi. How many books did

Randi read?

3. Mark ran 5 100 m on Tuesday. On Thursday, he ran 400 m less. How far did he run on Thursday?


Write on the board: larger number – smaller number = how much more 1. – = 2. – = 3. – = Have students write the correct numbers and box under each heading: 1. – 21 = 13 2. 14 – = 3 3. 5100 – = 400

Remind students that there are 4 different equations in the fact family: 7 – 2 = 5 and have a volunteer write

the other three on the board. Then show your students how you can change the first equation to find all the

equations in its fact family.

– 21 = 3 – 3 = 21 3 + 21 = 21 + 3 =

ASK: Do you think that these four problems will have the same missing number? Emphasize that all three

numbers in the equation will be the same for any equation in the same fact family, so the missing number

should be the same. To help them see this, draw on the board: 7 – 2 = 5 7 – 5 = 2 5 + 2 = 7 2 + 5 = 7

ASK: How do you use the numbers 2 and 5 to find the 7? Do you add or subtract? How do you use the

numbers 21 and 3 to find the missing number? Do you add or subtract? How do you know? Emphasize that

they should find the equation where the missing number is by itself; in this case, they add 21 + 3 (or 3 + 21).

Have a volunteer write the fact family for the second equation and have another volunteer find one of the

equations so that the box is by itself. Do they need to add or subtract to find the missing number? Repeat for

the third equation.

Students should be aware that phrases like “in all” or “altogether” in a word problem usually mean that one

should combine quantities by adding.

Similarly, phrases like “how many more”, “how many are left” and “how many less” indicate that one should

find the difference between quantities by subtracting.

However the wording of a question doesn’t always tell you which operation you should use to solve the

question. The two questions below have identical grammatical structure, but in the first question you must

add to find the answer and in the second you must subtract:

1. Paul has 3 more marbles than Ted. Ted has 14 marbles. How many marbles does Paul have?

2. Paul has 3 more marbles than Ted. Paul has 14 marbles. How many marbles does Ted have?

In solving a problem, it helps to make a model, draw a simple diagram, or make a mental model of the

situation. Students should start by getting a sense of which quantity in a question is greater and which is

smaller. This will usually help them decide whether they should add or subtract in a question and whether

their answer makes sense. In the first question, they should recognize that the 14 is the smaller quantity,

so they have – 14 = 3, which they can change to 14 + 3 = . In the second question, they should

recognize that 14 is the larger number, so 14 – = 3, or 14 – 3 = .

When students have mastered this, combine the problems and use a chart similar to the following:


First

number

Second

number

How many

altogether?

How much more

is larger than

smaller?

Have students put the given numbers and the box for the unknown in the right place, and then put an X in

the part that doesn’t apply for each question. Then have them solve the problem by expressing the as the

sum or difference, then adding or subtracting.

Extensions:

1. Here are some 2 step problems your students could try:

a) Alan had 352 stickers. He gave 140 to his brother and 190 to his sister. How many did he keep?

b) Ed is 32 cm shorter than Ryan. Ryan is 15 cm taller than Mark. If Ed is 153 cm high, then how

tall is Mark?

2. Ask your students to say what information is missing from each problem. Then ask them to make up an

amount for the missing information and solve the problem.

a) Howard bought a lamp for $17. He then sold it to a friend. How much money did he lose?

b) Michelle rode her bike 5 km to school. She then rode to the community center. How many km did

she travel?

c) Kelly borrowed 3 craft books and some novels from the library. How many books did she borrow

altogether?

Make up more problems of this sort.

3. Students should fill in the blanks and solve the problems.

a) Carl sold _____ apples. Rebecca sold _____ apples. How many more apples did Carl sell than

Rebecca.

b) Sami weighs _____ kg. His father weighs _____ kg. How much heavier is Sami’s father?

c) Ursla ran _____ km in gym class. Yannick ran _____ km. How much further did Ursla run?

d) Jordan read a book _____ pages long. Digby read a book _____ pages long. How many more pages

were in Jordan’s book?


NS3-32 Larger Numbers

Goals: Students will apply the concepts they have learned so far to numbers with 4 or more digits.

Prior Knowledge Required: Place value for ones, tens, and hundred

Writing number words for numbers with up to 3 digits

Comparing and ordering numbers with up to 3 digits

Writing numbers in expanded form

Addition and subtraction

Review NS3-1 and NS3-2 by asking volunteers to identify the place value of each digit in these numbers:

a) 7 b) 37 c) 237

Write on the board: 4 156. ASK: How is this number different from most of the numbers we have worked with

in this unit? (It has 4 digits.) Underline the 4 and tell students that its place value is “thousands.” Then have a

volunteer identify the place value of each digit in d) 4 237.

Write all the place value words on the board for reference: ones, tens, hundreds, and thousands.

Then write many 4-digit numbers, underline one of the digits in each number, and have students identify and

write the place value of the underlined digits. Be sure to include the digit 0 in some of the numbers.

Have students write the numbers in a) to d) in expanded form using numerals. Then have them write more 4-

digit numbers in expanded form. Again, assign numbers with more digits as a bonus.

Read these numbers aloud with students and write the number words for each one:

4 000, 8 000, 1 000.

Then write on the board: 4 502 = 4 000 + 502. Tell students that to read the number 4 502, they read both

parts separately: four thousand five hundred two.

Have students read these numbers aloud with you: 8 430, 5 001, 3 500, 4 782. Then have them write

corresponding addition sentences for each number (EXAMPLE: 8 430 = 8 000 + 430) and the number words

(EXAMPLE: eight thousand four hundred thirty). Have students write the number words for more 4-digit

numbers. Include the digit 0 in some of the numbers.

Review comparing and ordering numbers with up to 3 digits, then write on the board:

a) 7 834 b) 987 c) 9 050

1 963 4 802 8 950

ASK: What is the largest place value in which these pairs of numbers differ? Why do we care only about the

largest place value that differs? SAY: 9 050 has 1 more in the thousands place, but 8 950 has 9 more in the

hundreds place. Which number is greater? How do you know? Do a few addition and subtraction questions

together, to remind students how to add and subtract using the standard algorithm (EXAMPLES: 354 + 42,

52 + 119, 401 – 259) Then teach students to extend the algorithm to larger numbers by adding and

subtracting 4-digit numbers.


Activity: Give each student a pair of dice, one red and one blue. Have them copy the following diagram

into their notebooks:

<

Each student rolls their dice and places the number from the red die in a box to the left of the less-than

sign and the number from the blue die in a box to the right. If, after 4 rolls, the statement is true (the

number on the left is less than the number on the right) the student wins. Allow students to play for several

minutes, then stop to discuss strategies. Allow students to play again. Variation: Use 5 digit numbers and roll

the die 5 times:

<

Extensions:

1. What is the number halfway between …

a) 42 and 50? b) 42 and 60? c) 42 and 70?

d) 42 and 100? e) 420 and 1000? f) 9 420 and 10 000?

g) 9 870 and 10 000? h) 9 160 and 10 000?

Bonus: What is the number halfway between 99 330 and 100 000?

2. Write out the place value words for numbers with up to 12 digits:

ones tens hundreds

thousands ten thousands hundred thousands

millions ten millions hundred millions

billions ten billions hundred billions

Point out that after the thousands, there is a new word every 3 place values. This is why we put spaces

between every 3 digits in our numbers—so that we can see when a new word will be used. This helps us

to identify and read large numbers quickly. Demonstrate this using the number 3 456 720 603, which is

read as three billion four hundred fifty-six million seven hundred twenty thousand six hundred three.

Then write another large number on the board—42 783 089 320—and ASK: How many billions are in

this number? (42) How many more millions? (783) How many more thousands? (89) Then read the

whole number together.

Have students practise reading more large numbers, then write a large number without any spaces and

ASK: What makes this number hard to read? Emphasize that when the digits are not grouped in 3s, you

can’t see at a glance how many hundreds, thousands, millions, or billions there are. Instead, you have to

count the digits to identify the place value of the left-most digit. ASK: How can you figure out where to put

the spaces in this number? Should you start counting from the left or the right? (From the right, otherwise

you have the same problem—you don’t know what the left-most place value is, so you don’t know where

to put the spaces. You always know the right-most place value is the ones place, so start counting from

the right.)

Write more large numbers without any spaces and have students re-write them with the correct spacing and

then read them (EXAMPLE: 87301984387 becomes 87 301 984 387).


NS3-33 Concepts in Number Sense

Goals: Students will review the number sense concepts learned so far.

Prior Knowledge Required: Ordering numbers

Addition and subtraction

Word problems

The standard algorithm for addition and subtraction

Worksheet NS3-33 is a review of number sense concepts. It can be used as an assessment.

Extensions:

1. Place the numbers 1, 2, 3, 4 in the top four boxes to make the least possible sum and the least possible difference.

+ –

ANSWER: least possible sum: 23 + 14 or 24 + 13; least possible difference; 31 – 24 = 7

2. Find a question where Bob’s method from question 4 on the worksheet NS3-33, would still (albeit

accidentally) give the correct answer (ANSWER: Any pair of numbers whose ones digits add to 11.

EXAMPLES: 47 + 24, 36 + 45, 28 + 53)

3. Ask students to explain…

a) Could 200 people fit in 2 school buses?

b) Could you fit 50 eggs into 4 cartons?

c) Could you take 5 friends to see a movie with $20?

4. A palindrome is a number that looks the same written forwards or backwards: i.e. 212; 37 873, etc.

a) Find as many palindromes as you can using at most 9 tens blocks and 12 ones blocks.

b) Find as many palindromes as you can where the sum of the digits is 10.


NS3-34 Arrays

Goals: Students will model multiplication using arrays.

Prior Knowledge Required: Counting

The multiplication sign

Vocabulary: times, multiplication, product, array, row, column

Draw the following array on the board and circle each row:

ASK: How many rows are there? How many dots are in each row? If you wanted to know how many dots

there are altogether, would you have to count them one by one? How can you take advantage of there

being 5 in each row to find out how many there are altogether?

Then write on the board: 4 5 20 and ask if anyone knows what signs we can put between the numbers

to make a multiplication sentence. Prompt students by suggesting that you have 4 sets with 5 objects in

each set. If no one suggests it, write in the multiplication and equal signs: 4 × 5 = 20.

Draw several arrays on the board and ASK: How many rows? How many dots in each row? How many dots

altogether? What multiplication sentence could you write? Emphasize that the number of rows is written first

and the number in each row is written second. Include arrays that have only 1 row and arrays that have only

1 dot in each row.

In the arrays you’ve drawn, point out that each row contains the same number of dots. Show students that

there are 3 ways to arrange 4 dots so that each row contains the same number of dots:

Then give students 6 counters each and have them create as many different arrays as they can. They

should draw their arrays in their notebooks and write the corresponding multiplication sentences (or

“products”). As students finish with this first problem, give them 2 more counters for a total of 8, then 4 more

for a total of 12, then 3 more for 15, 1 more for 16, and finally 8 more for 24 counters. Some students will

work more quickly than others; not all will get to 24 counters. Allow enough time for every student to make

and record arrays using 6, 8, and 12 counters. Then ask students to share their answers. For example,

some students may have 12 as 4 rows of 3 (4 × 3) or 3 rows of 4 (3 × 4), others as 6 rows of 2 or 2 rows of

6, and some as 12 rows of 1 or 1 row of 12.


Now give students multiplication sentences and have them draw the corresponding arrays (EXAMPLE: 2 ×

4 = 8; 5 × 2 = 10, 3 × 3 = 9). Remind your students that the first number is the number of rows and the

second number is the number in each row.

Finally, give students multiplication word problems that refer directly to rows and the number in each row.

Mix it up as to which is mentioned first:

EXAMPLES:

There are 3 rows of desks. There are 5 chairs in each row.

There are 4 desks in each row. There are 2 rows of chairs.

How many desks are there altogether? How many chairs are there altogether?

(3 × 4 = 12) (2 × 5 = 10)

Have students draw corresponding arrays in their notebooks (they can use actual counters first if it helps

them) to solve each problem. Ensure that students understand what the dots in each problem represent. (In

the examples above, dots represent desks and chairs.)

Demonstrate the commutativity of multiplication (the fact that order doesn’t matter—3 × 4 = 4 × 3). Revisit

the first array and multiplication sentence you looked at in this lesson (4 × 5 = 20):

Then have students count the number of columns and the number of dots in each column and write a

multiplication sentence according to those (5 × 4 = 20):

ASK: Does counting the columns instead of the rows change the number of dots in the array?

Are 4 × 5 and 5 × 4 the same number? What symbol do mathematicians use to show that two numbers or

statements are the same? Write on the board: 4 × 5 5 × 4 and have a volunteer write the correct sign in

between (=). Tell students this is a short way of saying they are both the same number without saying which

number that is (20). ANALOGY WITH LANGUAGE: Instead of saying “The table is brown” and “Her shirt is

brown” we can say “The table is the same colour as her shirt” or “The table matches her shirt.”

To emphasize the commutativity of multiplication (that order doesn’t matter), you could draw several arrays

on chart paper, record the multiplication statements your students observe on the board and then rotate the

large sheet of paper and ask what multiplication statements they see after the rotation. Then record these

new statements underneath the corresponding statement for each array.


Have students write similar multiplication sentences for the arrays they drew in their notebooks earlier. Then

have them fill in the blanks in multiplication sentences that include large numbers (EXAMPLES:

4 × 7 = 7 × , 19 × 27 = × 19, × 5 = 5 × 102, 387 × , = 2 501 × 387)

Remind students about how they can change the order when they add and now when they multiply.

Ask them if they can do that when they subtract.

Activities:

1. Different perspectives. Put 10 counters on the table in 2 rows of 5. Have a pair of students come up

and look at the array from different directions, so that one student sees 2 rows of 5 and the other sees

5 rows of 2. Have pairs of students view different arrays from different directions at their own desks:

I see 2 rows of 5.

I see 5 rows of 2.

2. Tables in Microsoft Word. If your students have access to Microsoft Word, have them create tables.

Draw the tables on the board that you want them to draw in their computer file. They then have to know

which are rows and which are columns when they enter them into the Microsoft Word format. After they

create the tables, they can write addition and multiplication sentences to match. For example, to create

the following table:

They would have to:

• click on Table in the bar on top of the screen

• click on Insert Table

• Put 2 in the “Number of rows” section

• Put 5 in the “Number of columns” section

• Click OK

If they accidentally do 5 rows and 2 columns, their table will not match, so they will have to re-do

their table.

Once they have made this table correctly, they can type on the screen:

2 + 2 + 2 + 2 + 2 = 10 5 + 5 = 10

5 × 2 = 10 2 × 5 = 10


3. Build a multiplying machine. Teach students to make a multiplying machine for up to 6 × 6.

STEP 1: Draw a 6 × 6 array of dots.

STEP 2: Cut out a square corner from a sheet of paper so that the corner is large enough to

cover the array. Then throw away that corner and use the remaining part of the sheet.

STEP 3: Place the remaining part of your sheet so that you see the first number of rows and the second

number of dots in each row.

2 × 5

STEP 4: Count the dots you see. (In the example, we find that 2 × 5 = 10)

Extensions:

1. Students can use counters to model the arrays in QUESTIONS 6 and 7.

2. Use tiles or counters to create arrays for:

a) 3 × 3

b) 2 × 3

c) 1 × 3

d) 0 × 3

3. Create a story problem that you could solve by making 3 rows of 4 counters.

4. Teach students how to draw arrays to represent statements that involve addition and multiplication. For

instance 3 × 5 and 2 × 5 may be represented by:

3 × 5

2 × 5

(3 + 2) × 5 = 5 × 5


Students should see from the above array that: 3 × 5 + 2 × 5 = (3 + 2) × 5

(This is the distributivity law.)

To practice, have students draw arrays for:

a) 2 × 2 + 3 × 2

b) 3 × 4 + 2 × 4

c) 4 × 5 + 2 × 5

5. After doing Activity 3 from this section, teach your students how to make a multiplication table by

counting squares to the left and above each corner:

× 1 2 3

1 6 goes here (there are 2 rows of 3 squares and 6 in total, so 2 × 3 = 6)

2 6

3

Students can use this method to fill in a whole 3 × 3 times table chart. Provide the BLM “Arrays in the

Times Tables” for practice with this skill.

6. Using that order doesn’t matter, complete the multiplication tables:

a) × 1 2 3

1 1 2 3

2 4 6

3 9

b) × 1 2 3 4

1 1

2 2 4

3 3 6 9

4 4 8 12 16

c) × 1 2 3 4 5

1 1 2 5

2 4 8 10

3 3 6 9 15

4 4 12 16

5 20 25

Then have students shade the square that has the same number as the bolded square.

× 1 2 3

1

2

3

× 1 2 3

1

2

3

× 1 2 3 4

1

2

3

4


Bonus: use a 7 × 7 multiplication table instead. If your students are familiar with symmetry, ASK: where

is the symmetry line.

Literature Connection:

One Hundred Hungry Ants, E. J. Pinczes

(Rhymes describes one hundred ants marching toward a picnic. Efficiency dictates that the ants divide into

two lines of fifty, then four lines of twenty-five, and finally ten lines of ten.)

Read the story up until the ant makes the suggestion of dividing up and give pairs of students 100

manipulatives and allow them to discover different combinations/arrays to make the walk quicker. They can

record their findings in their journals and encourage them to write the matching multiplication sentence.)

Spunky Monkeys on Parade, S.J. Murphy

(This book is a fun approach to introducing multiplication. Count by twos, threes, and fours as the monkeys

parade down the street. A list of suggested activities is included in the book.)

Journal:

Have students write about when they can change the order of numbers and when they cannot. Which

numbers can they change the order of? When adding or multiplying, we can change the order of the

addends or the factors and the sum or product will remain the same, as in 3 + 4 = 7 and 4 + 3 = 7 or as in

3 × 4 = 12 and 4 × 3 = 12. Some students might notice that, when subtracting, we can change the order of

the difference and the subtrahend, as in 6 – 1 = 5 and 6 – 5 = 1. This is very different from addition and

multiplication, since the numbers they are changing the order of are not on the same side of the equal sign.

They don’t get the same answer by switching numbers. Although 6 – 1 = 5, it doesn’t even make sense to

write 1 – 6 since if we have only one object, we cannot take away 6 of them.


NS3-35 Adding Sequences of Numbers and NS3-36 Multiplication and Repeated Addition

Goals: Students will understand multiplication as determining how many altogether when objects are

divided into equal sets.

Prior Knowledge Required: The multiplication sign

Addition by counting on

Vocabulary: multiplication, product, sum, group, set, array, row

Write on the board: 2 + 5 + 3 + 4 + 6 + 3 + 3 + 2.

Tell your students that this is a lot of adding to do, so we’re going to break it down and do one step at a time.

ASK: What is 2 + 5? Draw a box over the 5 and write 7 in it:

7

2 + 5 + 3 + 4 + 6 + 3 + 3 + 2

ASK: What is 7 + 3? Count aloud as a class from 8 until you have 3 fingers up. Draw a box over the 7 and

write 10 in it. Continue in this way until the sum is found:

7 10 14 20 23 26

2 + 5 + 3 + 4 + 6 + 3 + 3 + 2 = 28

Then ask volunteers to do smaller problems:

a) 2 + 4 + 3 = b) 3 + 5 + 3 = c) 4 + 2 + 4 = d) 2 + 5 + 1 =

e) 3 + 6 + 3 + 4 = f) 4 + 2 + 2 + 3 = g) 3 + 3 + 3 + 3 =

Give students similar problems to do individually in their notebooks.

Bonus: 3 + 7 + 3 + 4 + 1 + 3 + 5 + 2 + 6

Draw a 2 × 3 array on the board.


ASK: What multiplication sentence does the array show? If most students do not raise their hands, prompt

them by asking: How many rows are there? How many dots are in each row? Which number do we write

first? How many are there altogether? Then write 2 × 3 = 6 next to the array.

Tell your students that instead of arranging the dots in rows, we can arrange them in sets or groups. (Use

the words “group” and “set” interchangeably.) Rearrange the above array as follows:

ASK: How many sets are there? How many dots are in each set? Tell students that we can write a

multiplication sentence for any grouping of objects as long as there is an equal number in each group. Have

students write both addition and multiplication sentences for the following, or similar, pictures:

Then draw the following pictures on the board and ASK: Can you write multiplication sentences

for these pictures? Why not? (There are not an equal number in each group.)

Draw a few more pictures, with both equal and unequal numbers in each group, and have volunteers write

the corresponding addition and, where possible, multiplication sentences.

Then, without using pictures, have students rewrite addition sentences as multiplication sentences and vice

versa. (Given 4 + 4 + 4, they should be able to write 3 × 4) It is important that students understand that the

second number in a multiplication statement is the number that is repeated and the first number is the

number of times the second number is repeated.

First have students write the addition statement given the multiplication statement.

(EXAMPLES: 4 × 3 = 3 + 3 + 3 + 3, 2 × 5, 5 × 2, 1 × 7, 7 × 1, 5 × 4, 4 × 6)

Bonus: 3 × 117, 4 × 204, 2 × 80147.

Then have students write the multiplication statement given the addition statement.

(EXAMPLES: 1 + 1 + 1 + 1 + 1 = 5 × 1, 4 + 4 + 4, 2 + 2 + 2 + 2, 6 + 6, 3, 9 + 9)

Bonus: 217 + 217 + 217, 1047 + 10 47)

Show students how to use circles and dots to model different groups and the objects in them. For example,

to model 3 cars with 4 people in each car, ASK: What are the groups? (the cars) What are the objects in

each group? (people) How many groups are there? How many circles should we draw? How many objects

are in each group? How can we show this using dots? (Draw 3 dots in each circle.) Then have a volunteer

write the multiplication and addition sentences for the picture. Repeat with several examples.


Then show students a pile of 15 counters on the overhead projector (or use circles taped to the board).

Arrange the counters into 5 piles ( 3, 4, 3, 3 and 2). Have a volunteer write the addition statement that the

picture shows: 3 + 4 + 3 + 3 + 2 = 15.

ASK: Can we write this addition statement as a multiplication statement? Can we move one of the counters

so that we can? Have a volunteer show how to do this. Have another volunteer write the new addition

statement and then a multiplication statement. Repeat with several examples and then move on to examples

where they need to move two counters from the same pile: 3 + 5 + 3 + 1 + 3.

When students are comfortable with this, have volunteers change two numbers so that each addition

statement can be turned into multiplication statements.

a) 4 + 3 + 4 + 4 + 4 + 5 b) 3 + 2 + 2 + 2 + 1 c) 6 + 8 + 6 + 6 + 4

Activities:

1. Give each pair of students 20 cards (2 of each card marked 1 through 10) and have them shuffle the

deck and deal out 10 cards to each student. The student with the highest total of cards wins. Students

should add their cards by using either method discussed in Extension 1 below.

2. In groups of 3, have students write an addition sentence and a multiplication sentence for the number of

shoes people in their group are wearing (2 + 2 + 2 = 6 and 3 × 2 = 6). Then have them write sentences

for the number of left and right shoes in the group (3 + 3 = 6 and 2 × 3 = 6). Have them repeat the

exercise for larger and larger groups of students.

3. Instead of left shoes and right shoes, students could group and count circles and triangles on paper:

They can count this as 2 + 2 + 2 = 6 (3 × 2 = 6) or they can count 3 circles + 3 triangles = 6 shapes

(2 × 3 = 6). Then provide students with the 2-page BLM “Multiplication and Order”.

Extensions:

1. Have students find each product by adding:

a) 6 × 1 = ______ Add 6 ones: 1 + 1 + 1 + 1 + 1 + 1

b) 5 × 1 = ______ Add 5 ones: 1 + 1 + 1 + 1 + 1

c) 4 × 1 = ______ Add 4 ones: 1 + 1 + 1 + 1

d) 3 × 1 = ______ Add 3 ones: 1 + 1 + 1

e) 2 × 1 = ______ Add 2 ones: 1 + 1

f) 1 × 1 = ______ Add 1 one: 1

Have students predict: 8 × 1, 11 × 1, 23 × 1, 89 × 1, 9063 × 1.

g) 3 × 3 = ______ Add 3 threes: 3 + 3 + 3

h) 2 × 3 = ______ Add 2 threes: 3 + 3

i) 1 × 3 = ______ Add 1 three: 3

j) 3 × 5 = ______ Add 3 fives: 5 + 5 + 5

k) 2 × 5 = ______ Add 2 fives: 5 + 5

l) 1 × 5 = ______ Add 1 five: 5


Have students predict: 1 × 8, 1 × 17, 1 × 11, 1 × 20, 1 × 304, 1 × 2 987, 1 × 3 446 035

For extra practice, provide the BLM “Multiplying by 1”. Remind students about the commutativity of

multiplication (that order doesn’t matter) and ASK: Does this apply to 1 as well? Is 7 × 1 = 1 × 7? What

are they both equal to?

2. Have students find each product by adding:

a) 5 × 0 = ______ Add 5 zeroes: 0 + 0 + 0 + 0 + 0

b) 4 × 0 = ______ Add 4 zeroes: 0 + 0 + 0 + 0

c) 3 × 0 = ______ Add 3 zeroes: 0 + 0 + 0

d) 2 × 0 = ______ Add 2 zeroes: 0 + 0

e) 1 × 0 = ______ Add 1 zero: 0

f) 0 × 0 = ______ Add no zeroes at all:

Have students predict: 11 × 0, 8 × 0, 14 × 0, 23 × 0, 732 × 0, 79 846 × 0.

g) 5 × 3 = ______ Add 5 threes: 3 + 3 + 3 + 3 + 3

h) 4 × 3 = ______ Add 4 threes: 3 + 3 + 3 + 3

i) 3 × 3 = ______ Add 3 threes: 3 + 3 + 3

j) 2 × 3 = ______ Add 2 threes: 3 + 3

k) 1 × 3 = ______ Add 1 three: 3

l) 0 × 3 = ______ Add no threes at all:

Have students predict: 0 × 5, 0 × 9, 0 × 6, 0 × 32, 0 × 97, 0 × 436, 0 × 50 980

For extra practice, provide the BLM “Multiplying by 0”. ASK: Is 7 × 0 = 0 × 7? What are they both

equal to?

3. Have students find pairs that add to 10 to add longer sequences of numbers.

(EXAMPLES: 3 + 6 + 7 + 2 + 4 + 8 + 1 + 5 + 9)

Students might be given cards with the numbers from 1 to 10 on them, arranged randomly, and asked to

find the sum by recording each step of the process as in the lesson and then rearranging the order so

that pairs adding to 10 are together. Students can then record the new order of numbers and check that

they get the same answer using both methods.

4. Find the number of days in a (non-leap) year by adding the sequence of 12 numbers:

31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 =

J F M A M J J A S O N D

Start by writing the sequence: J F M, and so on, on the board. ASK: Which months have 30 days?

Have a volunteer write 30 above those months. ASK: Which months have 31 days? Have a volunteer

write 31 above those months. Then have another volunteer fill in the missing number. Then write “+”

between the numbers and have students add the sequence of numbers.


5. Have students add strings of numbers that are “almost” arrays using multiplication and adjusting their

answers as needed. EXAMPLE: 3 + 3 + 3 + 2 is not quite 3 + 3 + 3 + 3, but is one less, so you can

solve it by multiplying 4 × 3 (12) and then subtracting 1 (11). Other examples include:

4 + 4 + 4 + 4 + 4 + 5 = 4 × 6 + 1 = 24 + 1 = 25

2 + 2 + 2 + 2 + 3 = 5 × 2 + 1 = 11

6 + 6 + 6 + 6 + 5 = 5 × 6 – 1 = 30 – 1 = 29

6. Students could rewrite the problems in QUESTIONS 4 on the worksheet as complete word problems.

For example, QUESTIONS 4 a) might become: Sally has 3 boxes of pencils. There are 2 pencils in

each box. How many pencils does she have altogether?

7. Teach students to multiply by 10 by using arrays or rows of a hundreds chart. Extend by having them

multiply really large numbers by 10; after they are comfortable with 3 × 10 = 30, 7 × 10 = 70, 2 × 10 = 20,

9 × 10 = 90, have them find 11 × 10, 12 × 10, 23 × 10, 47 × 10, 60 × 10, 64 × 10, 82 × 10, 90 × 10,

93 × 10, 100 × 10.

8. Teach multiplication of 3 numbers using 3-dimensional arrays. Give students blocks and ask them to

build a “box” that is 4 blocks wide, 2 blocks deep, and 3 blocks high.

When they look at their box head-on, what do they see? (3 rows, 4 blocks in each row, a second layer

behind the first) Write the math sentence that corresponds to their box when viewed this way:

3 × 4 × 2.

Have students carefully pick up their box and turn it around, or have them look at their box from a

different perspective: from above or from the side. Now what number sentence do they see? Take

various answers so that students see the different possibilities. ASK: When you multiply 3 numbers,

does it matter which number goes first? Would you get the same answer in every case? (Yes, because

the total number of blocks doesn’t change.)

9. You could also teach multiplication of 3 numbers by groups of groups:

This picture shows 2 groups of 4 groups of 3. ASK: How would we write 4 groups of 3 in math

language? (4 × 3). How could we write 2 groups of 4 groups of 3 in math language? (2 × 4 × 3). If I

want to write this as a product of just 2 numbers, how can I do so? Think about how many groups of 3

hearts I have in total. I have 2 × 4 = 8 groups of 3 hearts in total, so the total number of hearts is 8 × 3.

SO: 2 × 4 × 3 = 8 × 3.


Invite students to look for a different way of grouping the hearts. ASK: How many hearts are in each of

the 2 groups? (There are 4 × 3 = 12 hearts in each of the 2 groups, so 2 × 4 × 3 = 2 × 12.) Do I get the

same answer no matter how I multiply the numbers? (Yes; 8 × 3 = 2 × 12).

Put up this math sentence: 2 × 4 × 3 = 2 × .

ASK: What number is missing? (12) How do you know? (because 4 × 3 = 12). Is 12 also equal to 3 × 4?

Can I write 2 × 4 × 3 = 2 × 3 × 4? Remind students that 2 × 4 × 3 is the same as both 8 × 3 and 2 × 12.

Have students explore different orderings of the factors. Tell them that just like we use the term addend

in sums, we use the term factor in products. Have them try this with 2 × 6 × 5. Ask if there is any way to

order the factors that make it particularly easy to find the product. What is the answer? (2 × 6 × 5 =

2 × 5 × 6 = 10 × 6 = 60). Then challenge them to find more products of 3 numbers: 2 × 9 × 5, 3 × 5 × 2,

5 × 7 × 2, 2 × 4 × 5.


NS3-37 Multiplying by Skip Counting

Goals: Students will multiply by skip counting.

Prior Knowledge Required: Number lines

Multiplication

Skip counting

Vocabulary: product, sum, skip counting

Show a number line on the board with arrows, as follows:

0 1 2 3 4 5 6 7 8 9 10

Ask students if they see an addition sentence in your number line. ASK: What number is being added

repeatedly? How many times is that number added? Do you see a multiplication sentence in this number

line? Which number is written first—the number that’s repeated or the number of times it is repeated? (The

number of times it is repeated.) Then write “5 × 2 = ” and ASK: What does the 5 represent? What does

the 2 represent? What do I put after the equal sign? How do you know?

Have volunteers show several products on number lines.

(EXAMPLES: 6 × 2, 3 × 4, 2 × 5, 5 × 1, 1 × 5)

Then have a volunteer draw arrows on an extended number line to show skip counting by 3s:

Have students find 5 × 3, 2 × 3, 6 × 3, 4 × 3, and other multiples of 3 by counting arrows.

Then draw 2 hands on the board and skip count with students by 3 up to 30 (you can use your own hands

or the pictures):

3

6 12 9

15 18

21 27 24

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


Have students skip count on their hands to find 5 × 3, 2 × 3, 6 × 3, 4 × 3, and other multiples of 3.

ASK: Which is easier to use to multiply by 3, number lines or your hands? Why? (Since they probably don’t

need to count fingers to tell when 6 fingers are held up but they do need to count 6 arrows, students are

likely to find it easier to read answers on their hands.)

For extra practice, provide your students with the BLM “Multiplication Practice”.


5 pizzas.

2 pieces in each pizza.

Tell your students that you want to know how many pieces there are altogether. ASK: What is a multiplication

statement for this? An addition statement? What should we skip count by to find the answer?

(skip count by 2).

Repeat with similar questions, but have your students draw the pictures.

Extension: Use skip counting and that order doesn’t matter to fill in the unshaded parts of the

multiplication table.

× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7

1 5 1

2 10 2

3 15 3

4 4

5 5

6 6

7 7

× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7

1 1

2 2

3 3

4 4

5 5

6 6

7 7


NS3-38 Multiplying by Adding On

Goals: Students will turn products into smaller products and sums.

Prior Knowledge Required: Arrays

Multiplication

Vocabulary: product, sum, skip counting

Draw the following three pictures on the board:

Have students write a multiplication sentence for each picture. Then have them draw their own pictures and

invite partners to write multiplication sentences for each other’s pictures. Demonstrate the various ways of

representing multiplication sentences by having volunteers share their pictures.

Then draw 2 rows of 4 dots on the board:

ASK: What multiplication sentence do you see? (2 × 4 = 8) (Prompt as needed: How many rows are there?

How many in each row? How many altogether?) What happens when I add a row? Which numbers change?

Which number stays the same? (The multiplication sentence becomes 3 × 4 = 12; we added a row, so there

are 3 rows.) How many dots did we add? Invite a volunteer to finish writing the math sentence that shows

how 2 × 4 becomes 3 × 4 when you add 4. (2 × 4 + ____ = 3 × 4).

Look back at the number line you drew above:

0 1 2 3 4 5 6 7 8 9 10

ASK: What multiplication sentence do you see? What addition sentence do you see? (Prompt as needed:

Which number is repeated? How many times is it repeated?) Ensure all students see 5 × 2 = 10 from this

picture. Then draw another arrow:

0 1 2 3 4 5 6 7 8 9 10 11 12

0 1 2 3 4 5 6 7 8 9 10

1 2

3


ASK: Now what multiplication sentence do you see? What are we adding to 5 × 2 to get 6 × 2? Have a

volunteer show how to write this as a math sentence: 5 × 2 + 2 = 6 × 2.

Repeat using the 4 sets of 3 hearts and adding another set of 3 hearts. Have students use arrays to practise

representing products as smaller products and sums. Begin by providing an array with blanks (as in

QUESTION 1 on the worksheet) and have volunteers come up and fill in the blanks, as has been done in the

following EXAMPLE:

3 × 5

4 × 5

+ 5

In their notebooks, have students draw an array (or use counters) to show that:

a) 3 × 6 = 2 × 6 + 6 b) 5 × 3 = 4 × 3 + 3 c) 3 × 8 = 2 × 8 + 8

Have students do the following questions without using arrays:

a) If 10 × 2 = 20, what is 11 × 2? b) If 5 × 4 = 20, what is 6 × 4?

c) If 11 × 5 = 55, what is 12 × 5? d) If 8 × 4 = 32, what is 9 × 4?

e) If 6 × 3 = 18, what is 7 × 3? f) If 8 × 2 = 16, what is 9 × 2?

g) If 2 × 7 = 14, what is 3 × 7?

Finally have students turn products into a smaller product and a sum without using arrays.

Begin by giving students statements with blanks to fill in:

a) 5 × 8 = 4 × 8 + b) 9 × 4 = × 4 + c) 7 × 4 = × +

Extensions:

1. Ask students to circle the correct answer:

2 × 5 + 5 = 2 × 6 or 3 × 5

2 × 5 + 2 = 2 × 6 or 3 × 5

4 × 3 + 3 = 4 × 4 or 5 × 3

4 × 3 + 4 = 4 × 4 or 5 × 3

3 × 9 + 9 = 4 × 9 or 3 × 10

3 × 9 + 3 = 4 × 9 or 3 × 10

4 × 6 + 4 = 5 × 6 or 4 × 7

4 × 6 + 6 = 5 × 6 or 4 × 7

2. Use adding on to fill in the bolded squares of the multiplication tables.

× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7

1 1

2 8 2

3 3 9

4 12 4 20

5 30 5 10

6 12 6

7 7 28


NS3-39 Doubles

Goals: Students will use doubles and doubling to multiply mentally.

Prior Knowledge Required: Counting by 2s

Doubling numbers from 1 to 5

Relationship between skip counting and multiplying

Arrays

Vocabulary: double

ASK: What does the word “double” mean? Allow several students to give definitions and explanations in their

own words. Make sure students understand that to “double” a number means to add the number to itself, or

to multiply the number by 2. Then ASK: How would you find the double of “3”? If students say they “just

know” it’s 6, ask them how they would explain it to someone in grade 2. Some may add 3 + 3, others may

skip count by 3s until they have 2 fingers up. Take several answers.

Then draw 2 rows of 6 dots on the board and pair them up as follows and demonstrate counting the number

in the first row and the total number:

1 2 3 4 5 6

2 4 6 8 10 12

Ask if they can tell from this picture what the double of 4 is. What is the double of 6? Of 3? Of 2? 5? 1?

Then ask if anyone notices a pattern in the second row. What are they counting by to get the numbers in the

second row? Then demonstrate counting by 2s to double a number. Show that that if you count by 2s on

your fingers, then to double 3, for example, they can count by 2s until they have 3 fingers up. Make the

connection to products and ask students to find each product by doubling:

2 × 3, 2 × 4, 2 × 2, 2 × 5, 2 ×1.

Ask students to find the double of 9 by skip counting. ASK: What is another way to find the double of 9? If no

one says it, suggest splitting 9 into 5 + 4 and doubling each number separately. Use the following array to

illustrate this solution:

ASK: What is the double of 5? What is the double of 4? What is the double of 9? Why is it convenient to use

5 to find the double of 9?


Use the above method (splitting a number into 5 + something) to find the doubles of 6, 7, and 8. Be sure all

students know, or can quickly calculate, the doubles of 6–9 before proceeding. Make the connection to

multiplying by 2 and have students find the following products: 2 × 7, 2 × 6, 2 × 9, 2 × 8.

SAY: I want to find the double of 12, so I want to split it up into 5 and something the same way I did before.

Write: 12 = 5 + ____

12 + 12 = 10 + ____ = ____

Have a student volunteer fill in all the blanks. ASK: Is there another way that we could split the 12 to make it

easier to double? Is there a number that is even easier to double than 5? If anyone suggests 10, write out

the split (12 = 10 + 2) and ASK: What is the double of 10? The double of 2? The double of 12? If no one

suggests 10, ASK: What does the 1 in the number 12 represent? How much is it worth? What is 10 doubled?

What is left after I take 10 away from 12?

Then show:

12 + 12 = 20 + 4 = 24.

ASK: How does splitting the number 12 into 10 and 2 make it very easy to double? Did we get the same

answer as when we used 5 to double 12?

Then ask students to help you find the double of 14. Draw 2 rows of 14 dots on the board. Have a student

volunteer explain where you should draw a line to help you see the double of 14 as a sum of two numbers.

Then write:

14 = 10 + ____

14 + 14 = 20 + ____ = _____

Have a volunteer fill in the blanks. Repeat with the number 13, but this time don’t draw the dots. Just write

13 = 10 + ___, so 13 + 13 = 20 + ____ = ____. Finally, double some numbers between 16 and 19. Then

have students multiply several numbers by 2: 2 × 13, 2 × 17, 2 × 14, 2 × 18, 2 × 16. When all students are

able to double numbers less than 20, ask them how they would split 23 to find its double.

ASK: What does the 2 in 23 stand for? Is that number easy to double? Solve the problem together:

23 = 20 + 3 23 + 23 = 40 + 6 = 46

Have students double more 2-digit numbers by splitting the numbers into tens and ones, as in the above

examples. Give students numbers in the following sequence:

– both digits are less than 5 (EXAMPLES: 24, 32, 41)

– the tens digit is less than 5 and the ones digit is more than 5 (EXAMPLES: 39, 47, 28)

– the tens digit is more than 5 and the ones digit is less than 5 (EXAMPLES: 81, 74, 92)

– both digits are more than 5 (EXAMPLES: 97, 58, 65)


Bonus: Students who master this quickly can double 3- and 4-digit numbers.

Then have students multiply several numbers by 2: 2 × 31, 2 × 43, 2 × 22, 2 × 61, 2 × 84, 2 × 49, 2 × 89.

When all students are able to double 2-digit numbers, show them how they can use doubling when they

multiply. Draw 1 row of 6 on the board. ASK: What number is this? How would I show this number doubled?

(Draw another row of 6.) Then draw 3 rows of 8 and ASK: How many dots do I have here?

What multiplication sentence does this show? (3 × 8 = 24) I have 3 rows of 8. If I want to double that, how

many rows of 8 should I make? Why? (I should make 6 rows of 8 because 6 is the double of 3.) Add the

additional rows and SAY: Now I have twice as many dots as I started with. How many dots did I start with?

(24) How many dots do I have now—what is the double of 24? Ask students to give you the multiplication

sentence for the new array: 6 × 8 = 48.

Ask students to look for another way to double 3 rows of 8. ASK: If we kept the same number of rows, what

would we have to double instead? (the number of dots in each row) So how many dots would I have to put in

each row? (16, because 16 is the double of 8) What is 3 × 16? (It is the double of 3 × 8, so it must be 48.)

How we can verify this by splitting the 16 into 10 and 6? Have someone show this on the board:

3 × 16 = 3 × 10 + 3 × 6 = 30 + 18 = 48.

Have students fill in the blanks:

a) 4 × 7 is double of 2 × 7 = 14 , so 4 × 7 = 28

b) 5 × 8 is double of , so 5 × 8 =

c) 6 × 9 is double of , so 6 × 9 =

d) 3 × 8 is double of , so 3 × 8 =

e) 4 × 8 is double of , so 4 × 8 =

Encourage students to find two possible answers for the first blank in part e) (2 × 8 or 4 × 4 are both possible).

Ask volunteers to solve in sequence: 2 × 3; 4 × 3; 8 × 3: 16 × 3; What is double of 16? What is 32 × 3? What

is double of 32? What is 64 × 3? Repeat with the sequences beginning as follows. Take each sequence as

far as your students are willing to go with it.

a) 3 × 5, 6 × 5 b) 2 × 6, 4 × 6 c) 3 × 4, 6 × 4

d) 2 × 7, 4 × 7 e) 2 × 9, 4 × 9 f) 3 × 9, 6 × 9

Activities: Toothpick Tic Tac Toe

1 2 3

4 6 8

9 12 16

1 2 3 4

1 2 4

5 8 10

16 20 25

1 2 4 5

1 2 3 4 5

6 7 8 9 10

12 14 15 16 18

20 21 24 25 28

30 35 36 42 49

1 2 3 4 5 6 7


For each game board, you will need two toothpicks and enough tokens of different colours for each

player (5 each for the smaller game board and 12 each for the larger game board). In each case, the

goal for each player is to place 3 tokens of their colour all in a row horizontally, vertically or diagonally, as

in tic tac toe. There are two toothpicks, which Player 1 can begin by playing on any two of the numbers

below the grid (they may put both toothpicks on the same number if they wish). They can then place their

token on the product of the two numbers. For example, if they placed the toothpicks both on 3, they can

place their token on 9. Player 2 must then move only 1 toothpick. In this case, Player 2 is forced to move

their token to some multiple of 3, but can choose between 3, 6 and 12 (9 is already taken and cannot be

used again). Play continues until one player gets 3 in a row.

Variation A: Change the game boards by arranging the numbers in random order.

Variation B: Use the multiplication table as a game board. This time, if a student places their toothpick

on 2 and 3, they may choose which 6 to cover with their token. This adds an added element of strategy.

With larger game boards, students may aim for 4 in a row.

1 2 3

2 4 6

3 6 9

1 2 3

1 2 3 4

2 4 6 8

3 6 9 12

4 8 12 16

1 2 3 4

1 2 3 4 5

2 4 6 8 10

3 6 9 12 15

4 8 12 16 20

5 10 15 20 25

1 2 3 4 5

Students might also use 6 × 6 and 7 × 7 multiplication tables.

Extensions:

1. Can a double ever be an odd number?

2. Use doubling to fill in the blank squares in the multiplication tables:

× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7

1 1 2

2 2 4 6 8 10 12 14 2 4

3 3 6

4 4 8

5 5 10

6 6 12

7 7 14


× 1 2 3 4 5 6 7

1 3

2 6

3 3 6 9 12 15 18 21

4 12

5 15

6 18

7 21

3. Show students how the doubling strategy in this lesson can be combined with adding on from the

previous lesson (NS3-38).

× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 1 3

2 double 2 6

3 then add on 3 9

4 4 12

5 5 15

6 6 18

7 7 21

double

then add on

4. Review the strategies students have seen so far for finding multiplication facts up to 7 × 7 (skip counting,

adding on, doubling) Challenge students to fill out a blank 7 × 7 times table by combining all the

strategies they have learned so far.

5. Tell students that knowing their multiplication facts (or “times tables”) will help them solve problems

which would otherwise require a good deal of work adding up numbers. For instance, they can now

solve the following problems by multiplying.

a) There are 4 pencils in a box. How many pencils are there in 5 boxes?

b) A stool has 3 legs. How many legs are on 6 stools?

c) A boat can hold 2 people. How many people can 7 boats hold?

6. Introduce sports scores. Tell them that some professional leagues award 2 points for every game you

win. So, for example, if a team has 3 wins and 4 losses, they don’t get any points for the 4 games they

lost, but they get 2 points for each of the 3 games they win. Ask a volunteer to write an addition sentence

for the number of points they will have: 2 + 2 + 2 = 6.

Ask if anyone knows a multiplication sentence that means the same thing (3 × 2) and then encourage

them to think of a different multiplication sentence that will still get the same answer (2 × 3).

Then encourage them to write an addition sentence that means the same thing as the multiplication

sentence (3 + 3).


Encourage students to think of a word for adding 3 to itself. ASK: What are you doing to the 3 – remind

them that there’s a special word for that (doubling it).

Then repeat with other examples until all students understand that the number of points is double the

number of wins. When students understand this, ask volunteers to find the total number of points each

team has:

Blue: 3 wins and 1 loss Blue: 7 wins and 3 losses

Red: 1 win and 3 losses Red: 6 wins and 4 losses

Yellow: 2 wins and 2 losses Yellow: 8 wins and 2 losses

Which team has the most points? Does the team with the most points have the most wins?

Encourage students to double higher numbers: 23 wins and 38 losses means you double

23 = 20 + 3, so you get 40 + 6 = 46.

To double numbers such as 28, students can either write:

28 = 20 + 8 40 + 16 = 56 or 28 = 20 + 5 + 3 40 + 10 + 6 = 56.

When students are comfortable with this, introduce ties. In many sports leagues, a team gets 2 points for

a win and 1 point for a tie (no points for a loss). Then have students use the doubling strategy and then

the standard algorithm for addition to find point totals for teams with the following records:

Team Wins Losses Ties Points

Red 23 38 19 46 + 19 = 65

Blue 41 30 9

Yellow 32 29 19

Green 38 25 17

Orange 28 40 12


Anno’s Magic Seeds, M. Anno

(Two seeds are given and one is planted to grow. Explores the concept of doubling while engaging readers –

links to patterns in skip counting as well.)


NS3-40 Topics in Multiplication

Prior Knowledge Required: Multiplication

Introduce the phrase “times as many.” Tell students that Jenny has 3 stickers and Ryan has 12 stickers.

Draw a diagram to illustrate this situation, and group the larger number of items (12) by the smaller

number (3):

Jenny:

Ryan:

SAY: Since 12 is 4 times 3, we say that Ryan has 4 times as many stickers as Jenny.

Draw similar diagrams, but don’t group the larger number of items. Have volunteers group the larger number

by the smaller number to find out how many times as many items one person has than the other.

Then draw the following partial pictures and have students finish the pictures:

a)

There are three times as many triangles as stars.

b)

There are twice as many circles as stars.

c)

There are 4 times as many squares as stars.

Extension: I am a 2-digit number.

a) Use 6 blocks to make me. Use twice as many tens blocks as ones blocks.

b) Use 12 blocks to make me. Use twice as many ones blocks as tens blocks.

c) Use 12 blocks to make me. Use three times as many ones blocks as tens blocks.

d) Use 10 blocks to make me. Use four times as many tens blocks as ones blocks.


NS3-41 Concepts in Multiplication

Goals: Students will consolidate their learning done on multiplication so far.

Prior Knowledge Required: Multiplying by 1 or 0

Multiplication strategies: repeated addition, skip counting, arrays,

doubling, adding on

Patterns

Vocabulary: sum, product

This worksheet can be used as review or as an assessment.

Activities:

1. Give your students tiles or counters and ask them to find the numbers by making models.

I am less than ten. You can show me with:

2 equal rows of tiles


Solution: The number is 6.

I am between 15 and 25. You can show me with:



Solution: The number is 20

2. Ask your students to solve these questions by making base ten models (using only ones or tens blocks).

I am a multiple of 5. You can make me with 6 base ten blocks (there are two answers: 15, 60).

I am a multiple of 3. You can make me with 3 base ten blocks (there are three answers: 12, 21, 30).

I am a multiple of 4. You can make me with 5 base ten blocks (there is one answer: 32).

Extensions:

1. Write the same number in both boxes to make the multiplication statement true:

a) × = 1

b) × = 25

c) × = 9

d) × = 4

e) × = 16

f) × = 36

2. Look at the numbers from Extension 1 (1, 25, 9, 4, 16, 36). Where do these numbers appear in the times

table in QUESTION 9? Can you describe the position of these numbers?


NS3-42 Pennies, Nickels and Dimes

Goals: Students will be able to count any combination of pennies, nickels and dimes (two coin types)

up to a dollar.

Prior Knowledge Required: Skip counting by 5 and 10

Adding

Vocabulary: penny, nickel, dime, skip counting

Begin by reviewing the names and values of pennies, nickels and dimes. Try asking the class, “What coin

has a maple leaf on it? What is its value?” etc.

Hold up each coin and list its name and value on the board. It’s also a good idea to have pictures of the

coins on the classroom walls at all times while studying money.

Review skip counting by 1, 5 and 10 with your students. Tell them that adding money is really easy when you

just have pennies because each penny is worth 1 cent. Demonstrate counting out a pile of pennies (use the

BLM “Cut Out Coins” if you don’t have actual coins). Say the word “cent” after every penny: 1 cent, 2 cents,

3 cents, etc. Then give each student a copy of the BLM “Cut Out Coins”. Have them cut out paper coins –

they can cut out the squares if it is easier. Have them separate the pennies from the other coins. Then put

different prices on the board and tell them to put that much money to the side of their desk (or an envelope)

as though they were going to buy something for that price.

Repeat the counting activity with nickels and dimes. When you ask your students to put some money aside

as if they were going to buy an item, ask them first to put aside enough coins of the same denomination.

For example, for an item that costs 69¢, the students should put aside 7 dimes. Explain that different coins

can add up to the same amount. Ask students how you could make 5¢ using different coins. (5 pennies, or

1 nickel.) Then ask how you could make 10¢. (10 pennies, 2 nickels, 1 dime and a bonus answer: 1 nickel

and 5 pennies.)

Pick one of the items that the students set money aside for (for example, a pencil for 37 cents).

ASK: can you make exactly 37 cents with dimes? With nickels? This means you need pennies to make the

exact amount. Ask your students to remove a dime from the pile and to replace it with the necessary

amount of pennies. Invite a volunteer to count the coins. Repeat with other items. Write several more prices

on the board and ask your students to put aside the exact amount of money using two types of coins.

45¢ (dimes and nickels) 78¢ (dimes and pennies) 23¢ (nickels and pennies) and so on.

Assessment:

Count the given coins and write the total amount:

a) 10¢, 5¢, 5¢, 5¢ Total amount = b) 5¢, 1¢, 1¢, 1¢, 1¢, 1¢ Total amount =

c) 10¢, 10¢, 10¢, 1¢, 1¢, 1¢ Total amount =


NS3-43 Quarters

Goals: Students will be able to identify 25¢ in any combination of coins.

Prior Knowledge Required: Skip counting by 5, 10 and 25

Adding

Penny, nickel, dime

Vocabulary: penny, nickel, dime, quarter, skip counting

Introduce the quarter the same way you introduced dimes, nickels and pennies last lesson. Ask your

students to find several ways to make 25 cents using nickels, dimes and pennies. Remind them how they

made various amounts of money last lesson and suggest using the same technique.

ASK: Can you make a quarter with dimes only? Why? Can you make a quarter with nickels only? How would

you do that?

Review skip counting by 25s. Repeat the activity involving setting money aside with prices such as 35 cents

(quarters and dimes, quarters and nickels), 27 cents, 30 cents. Use larger amounts of money, like 55 cents,

85 cents, and even more than a dollar.

Activity: Ask students to estimate the total value of a particular denomination (EXAMPLE: quarters,

pennies, etc.) that would be needed to cover their hand or book. Students could use play money to test their

predictions. This activity is a good connection with the measurement section.

Extension: When skip counting by 25, leave out some numbers without telling them where you are

leaving out the numbers (example: 0 25 50 100 125 150) and see if students can figure

out where to insert the missing number and what the missing number is.


NS3-44 Counting by 2 or More Coin Values

Goals: Students will be able to count any combination of coins up to a dollar.


Adding

Penny, nickel, dime, quarter

Counting on by 1,5 and 10


Review the two previous lessons. Complete the first page of the worksheet together. Review using the

finger counting technique to keep track of your counting. It might help to point to a large number line when

skip counting with numbers over 100. It’s a good idea to keep this large number line on the wall while

studying money. Let your students continue with the worksheets independently. Stop them after

QUESTION 4. Give each student a handful of play money coins. Ask the students to sort them by

denomination—putting all the pennies together, all the nickels together, etc. Once they are sorted,

demonstrate how to find the value of all the coins by skip counting in different units starting by tens. Do

some examples on the board (EXAMPLE: if you had 3 dimes, 2 nickels and 3 pennies you would count 10,

20, 30, 35, 40, 41, 42, 43). Have the students practice counting up the play money they have. They can

also trade some coins with other classmates to make different amounts each time.

NOTE: Allow your students to practice the skill in QUESTIONS 5 AND 6 of the worksheet with play money.

Do not move on until they can add up any combination of coins up to a dollar. Another way to help

struggling students is to cross out the counted money on the worksheet.

Assessment: Count the given coins and write the total amount:

a) 25, 1, 10, 5, 5, 25 Total amount = b) 10, 1, 1, 25, 25, 5 Total amount =

Activities:

1. Place ten to fifteen play money coins on a table. Ask students to estimate the amount of money, and

then to count the value of the coins. (Students could play this game in pairs, taking turns placing the

money and counting the money.)

2. Adding Money Game – Adapted from “A Companion Resource for Grade One Mathematics” by

Saskatchewan Learning.

Each pair should have the BLM “Adding or Trading Game” as a game board and each player should

have a different token to use as their playing piece. They will also need a die to know how many pieces

to move forward. When they roll, they move forward the correct number of squares and receive the coin

shown on the board. When both players are at the end of the board (not necessarily by the exact amount

shown on the die), they count up their money – the player with the most amount of money wins.


NS3-45 Counting by Different Denominations

Goals: Students will identify how many cons of any particular denomination would need to be added to an

amount to make a certain total.


Adding

Penny, nickel, dime, quarter

Counting on by 1,5 and 10


Begin by reviewing how to skip count by 10s and 5s from numbers not divisible by 10 or 5. (i.e. count by 5s

from 3. Count by 10s from 18).

Have students practice writing out skip counting sequences in their notebooks. Assign start points and what

to count by. Some sample problems:

a) Count up by 10s from 23

b) Count up by 10s from 27

c) Count up by 5s from 7

d) Count up by 5s from 41

Have students notice the number patterns. When counting by 10s, all the numbers will end in the same

digit. When counting by 5s, all the numbers will end in one of two digits. If you used the “skip counting

machines” in section NS3-13, review the activity with the students.

Give your students some play coins (cents only), and let them play the following game—one player takes 3

or 4 coins of two denominations, counts the money he has and says the total to the partner. Then he hides

one coin and gives the rest to the partner. The partner has to guess which coin is hidden. Make the task

harder allowing more denominations.

Next, modify the above game. Give the students additional play coins. One player will now hide several

coins (at most three coins that must be of the same denomination). He then says the total value of all the

coins, as well as the denomination of the hidden coins. The other player has to guess the number of hidden

coins (of each denomination).

Assessment:

Draw the additional money you need to make the total. You may use only 2 coins (or less) for each

question.

a) 25¢, 10¢, 5¢, 1¢ Total amount = 61¢

b) 25¢, 25¢, 10¢, 1¢, 1¢ Total amount = 92¢


Activities:

1. Bring in flyers for some local businesses that sell inexpensive items (i.e. the grocery store, the drug

store, a convenience store).

Give each student a handful of play money change including quarters, dimes, nickels and pennies.

Have students count up their change. How much money do they have? What item or items could they

afford to buy?

Ask the students consult the flyers and to select one item that they would like to ‘buy’. This item must

cost less than $1, but it must also cost more money than they currently have. Ask them to figure out

how many additional coins of any one denomination they would need to have enough money to buy

their item (EXAMPLE: ask how many nickels would you need in addition to what you have now to afford

this item? How many dimes would you need?) Students are allowed to go over the amount and get

change back.

2. Extend the activity above. Ask students if it would be possible to pay for their item using only four coins.

Have them determine which four coins they would need. Then ask if it would be possible to buy the item

with three or two coins. For some students, it will not be possible to make their total

with only four coins or fewer. Ask them to determine how many coins they would need to make

their target amount.

Extensions:

1. Quan says he can make 76¢ using only nickels, dimes and quarters. Is he correct?

Explain.

2. Lynne says she can make 80¢ using only quarters and dimes. Is she correct?

Explain.

3. Kirsten says she can make $4.18 using only quarters and dimes. Is she correct?

Explain.

4. Rennish says he can make $5 using only nickels. Is he correct?

Explain.

5. What coins am I counting if I say …

a) … 10, 20, 30, 35, 40, 45, 50?

b) … 25, 50, 75, 85, 95, 100?

c) … 100, 200, 300, 325, 350, 351, 352, 353?

d) … 200, 400, 500, 600, 700, 710, 715, 720?


NS3-46 Least Number of Coins

Goals: Students will make specific amounts of money using the least number of coins.

Prior Knowledge Required: Creating amounts of money using a variety of coins.


Begin with a demonstration. Bring in $2 worth of each type of coin (EXAMPLE: 4 rolls of pennies, 2 rolls of

nickels, etc.). Let students pick up each $1 amount to compare how heavy they are. Explain that it is

important to figure out how to make amounts from the least number of coins, because it is too heavy to

carry extra coins around.

Tell the class that the easiest way to do this is to start with the largest possible denomination and then

move to smaller ones.

Ask students how they could make exactly 10¢ with the smallest number of coins. They will probably give

the right answer immediately. Next, ask them to make up a more difficult amount, such as 35¢.

A quarter could be used, so put one aside (you can use the large pictures of the denominations to

demonstrate). We have 25¢, so what else is needed? Check a dime. Now we have 25¢ + 10¢ = 35¢. That’s

just right. Then look for another combination. Ask the students how to make 35¢ using dimes, nickels and

pennies, but no quarters. Compare the arrangements and the number of coins used in each.

Ask students to compare how many dimes they need compared to how many nickels they need to make the

same amount of money, example: 12¢, 14¢, 13¢, 32¢, 51¢. Give students either dimes and pennies or

nickels and pennies and have them work in pairs to see who used the smaller number of coins, the person

with dimes and pennies or the person with nickels and pennies.

Demonstrate how they can keep track of how many of each coin they’ve used by using a chart similar to the

one on the worksheet:

Dimes Nickels Pennies

8¢ � ��

11¢ � �

14¢ � ��

20¢ ��

19¢ � � ��

Let your students practice making amounts with the least number of coins.

Try totals such as 15¢, 17¢, 30¢, 34¢, 50¢, 75¢, 80¢, 95¢.


Ask if they noticed what coin is best to start with (the largest possible without going over the total).

When giving larger sums, you might use more than one volunteer—ask the first to lay out only the

necessary quarters, the next one—only the dimes, etc.

Show a couple of examples with incorrect numbers of coins, like 30¢ with 3 dimes, 45¢ with 4 dimes and a

nickel, 35¢ with a quarter and two nickels. Ask if you laid the least number of coins in the right way. Let your

students correct you.

Assessment:

Make the following amounts with the least number of coins:

a) 9¢ b) 45¢ c) 53¢ d) 17¢ e) 3¢ f) 80¢

Activities:

1. Use play money to trade the following amounts for the least number of coins.

a) 7 quarters b) 9 quarters c) 7 nickels d) 12 dimes e) 7 dimes and 1 nickel

2. Money Memory

Use the first page of the BLM “Money Memory” to play concentration by matching equivalent sums of

money. When students excel at this game and are ready for a greater challenge, you can add the

second page of the BLM.

Extension: Have students look at their answers for QUESTIONS 9 and 10, and ask them if there are any

answers where they use more than one nickel. Ask them to explain why there should never be more than

one nickel in these kinds of problems.


Smart by Shel Silverstein, contained in “Where the Sidewalk Ends”.

Read this poem to your class. With each stanza, stop and ask the students how much money the boy in the

story has. Give students play money to use as manipulatives so they can each have the same amount of

coins as described in the poem. This is an excellent demonstration that having more coins does not

necessarily mean having more money.


NS3-47 Dimes and Pennies

Goals: The students will represent amounts up to $1 in dimes and pennies.

Prior Knowledge Required: Dimes and pennies

Count by two denominations

Skip counting by 10

Vocabulary: dimes, pennies, tens, ones, tens digit, ones digit

Give your students some dimes and pennies and ask them to make various amounts of money using only

these two types of coins. Ask your students to write their answers in a T-table:

Dimes Pennies Amount in ¢

Ask your students if they notice a pattern. Is the number of dimes the same as the tens digit or the ones

digit? Which digit is the same as the number of pennies? Write several money amounts in the third column of

the table and ask your students to fill in the other two columns without using play money.

You may also use the activity that involved hiding coins from the lesson NS3-45: Counting by Different

Denominations.

Extensions:

1. I have 35¢ in dimes and pennies. How many coins do I have?

2. I have 3 coins (dimes and/or pennies), How much money can I have? Find all the possible answers.


NS3-48 Making Change Using Mental Math

Goals: Students will make change for amounts less than $1 using mental math.

Prior Knowledge Required: Counting by 1s and 10s

Subtraction

Vocabulary: skip counting, penny, nickel, dime, quarter, loonie, cent, dollar, change

Pose a sample problem. You would like to buy a pen which costs 18¢, but you only have a quarter to pay

with. How much change should you get back?

Rather than starting with a complicated subtraction question that includes decimals and carrying, show

students how to ‘count up’ to make change. Demonstrate that you can first count up by 1s to the amount.

In this case you would start at 18 and count up: 19, 20, 21, 22, 23, 24, 25. You would need 7¢ as change.

Review the finger counting technique, keeping track of the number counted on their fingers.

Model more examples, getting increasingly more volunteer help from the class as you go along.

Have students practice this skill with play money, if helpful. They should complete several problems of this

kind before moving on. Some sample problems:

a) Price of an orange = 39¢ Amount paid = 50¢

b) Price of a hair band = 69¢ Amount paid = 75¢

c) Price of a sticker = 26¢ Amount Paid = 30¢

Pose another problem: You would like to buy a flower that costs $80¢, but you only have a loonie to pay

with. How much change should you get back? How many cents are in $1? Before the students start to

count by 1s to 100, tell them that you will show them a faster way to do this.

Explain that instead of counting up by 1s, you can count by 10s to 100. Starting at 80 you would count up:

90, 100 to $1.

Give students several problems to practice. Some sample problems:

a) Price of a newspaper = 50¢ Amount Paid = $1.00

b) Price of a milk carton = 80¢ Amount Paid = $1.00

c) Price of a paper clip = 10¢ Amount Paid = $1.00

Pose another problem: You would like to buy a postcard that costs 55 cents, but you only have a loonie to

pay with. How much change should you get back?

Explain that you can count up by 1s to the nearest 10, then you can count by 10s to $1 (or the amount of

money paid). In this case you would start at 55 and count up to 60: 56, 57, 58, 59, 60.


Then count up by 10s. Starting at 60 you would count up: 70, 80, 90, 100. At the end, you will have counted

five ones and four 10s, so the change is 45¢.

Demonstrate the above problem using the finger counting technique. Explain to the students that this can

be clumsy, because it is easy to forget how many 1s or how many 10s you counted by.

How could you use this method to solve the problem without the risk of forgetting?

Model another example. You would like to buy a candy bar costs 67¢, but you only have a loonie

to pay with.

STEP 1: Count up by 1s to the nearest multiple of 10 (Count 68, 69, 70 on fingers).

STEP 2: Write in the number that you have moved forward and also the number that you are now ‘at’ (Write

“counted 3, got to 70¢”).

STEP 3: Count up by 10s to $1.00. Write in the number that you have counted up.

(Count 80, 90, 100, write 30¢)

STEP 4: Add the differences to find out how much change is owing (3¢ + 30¢ = 33¢).

Let a couple of volunteers model this method with sample problems. Give the students another several

problems to complete in their notebooks, such as:

a) Price of a bowl of soup = 83¢ Amount Paid = $1.00

b) Price of a stamp = 52¢ Amount Paid = $1.00

c) Price of a pop drink = 87¢ Amount Paid = $1.00

d) Price of an eraser = 45¢ Amount Paid = $1.00

e) Price of a candy = 9¢ Amount Paid = $1.00

Assessment:

a) Price of a pencil = 80¢ Amount Paid = $1.00

b) Price of a chocolate bar = 62¢ Amount Paid = $1.00

c) Price of a candy = 17¢ Amount Paid = $1.00

Bonus:

a) Price of an action figure = $9.99 Amount Paid = $10.00

b) Price of a book = $4.50 Amount Paid = $5.00

Activity:

Play Shop Keeper

Set up the classroom like a store, with items set out and their prices clearly marked. The prices should be

under $1.00. Tell the students that they will all take turns being the cashiers and the shoppers. Explain that

making change (and checking you’ve got the right change!) is one of the most common uses of math that

they will encounter in life!


Allow the students to explore the store and select items to ‘buy’. Give the shoppers play money to ‘spend’

and give the cashiers play money to make change with. Ask the shoppers to calculate the change in their

heads at the same time as the cashiers when paying for the item. This way, students can double check and

help each other out. Reaffirm that everyone needs to work together, and should encourage the success of

all of their peers.

Allow the students a good amount of time in the store. Plan at least 30 minutes for this exploration.

Extensions:

1. Once students have calculated the change for any question, have them figure out the least number of

coins that could be used to make that change.

2. This would be a great lead up to an actual sale event. If your community or school has an initiative that

your class wanted to support, you could plan a bake sale, garage sale, etc.

that would raise funds for the cause. Students would find the pretend task even more

engaging if it was a ‘dress rehearsal’ for a real event.

3. How much change should you get back if…

…you have 4 dimes and a quarter and you want to buy something for 58¢?

…you have 3 quarters and 4 pennies and you want to buy something for 66¢?

…you have 2 quarters and 3 pennies and you want to buy something for 37¢?

4. Word Problems:

I have 2 quarters, 3 dimes and a nickel. How much more money do I need if I want to buy something

worth 93¢?

I have 1 quarter, a dime and 3 nickels. How much more money do I need if I want to buy something

worth a dollar? Worth 60¢? Worth 84¢?


NS3-49 Lists

Goals: Students will use lists to find numbers which have two properties.

Prior Knowledge Required: Comparing Numbers

Multiplication and skip counting

Even and odd numbers

Vocabulary: even, odd, multiple, greater than, less than

Have a volunteer write all the numbers from 0 to 9 in order. ASK: Which numbers are greater than 7?

Where can you find them? (to the right of 7) Which numbers are less than 5? Where can you find them?

(to the left of 5)

Have students individually write the numbers that are:

a) less than 4

b) greater than 6

c) greater than 3

d) less than 7

ASK: How can you find the numbers that are less than 7 and greater than 3? (find the numbers that are in

both the “less than 7” list and the “greater than 3” list)

Have students find numbers that are:

a) less than 6 and greater than 2

b) less than 9 and greater than 7

c) less than 8 and greater than 5

d) less than 7 and greater than 1

Remind students that 0 is a multiple of any number. For example, 0 = 2 × 0, so 0 is a multiple of 2.

Have students individually write the numbers that are:

a) multiples of 2

b) multiples of 3

c) multiples of 4

d) multiples of 5

e) multiples of 2 and multiples of 3

f) multiples of 3 and less than 7

g) multiples of 4 and greater than 7

Ask if anyone remembers what the words “odd” and “even” mean and have a student explain.

Then have students individually write the numbers that are:

a) odd

b) even


c) odd and less than 6

d) even and less than 5

e) odd and more than 4

f) odd and a multiple of 3

g) even and a multiple of 3

Bonus:

a) List the odd numbers that are multiples of 3 and are less than 5

(odd numbers are: 1, 3, 5, 7, 9; multiples of 3 are: 0, 3, 6, 9; numbers less than 5 are: 0, 1, 2, 3, 4.

The only number that belongs to all 3 lists is 3.)

b) List the even numbers that are multiples of 3 and are greater than 5 (even numbers are: 0, 2, 4, 6, 8;

multiples of 3 are: 0, 3, 6, 9; numbers greater than 5 are: 6, 7, 8, 9. The only number that belongs to all

three lists is 6.)

Extensions:

Introduce combinations of properties where no number has both or all properties.

a) Which numbers are odd multiples of 2? (none)

b) Which numbers are greater than 6 and less than 4? (none)

c) Which numbers are odd multiples of 5 that are less than 4? (none)

Bonus: Find another combination of properties which gives a list of no numbers.


NS3-50 Organized Lists

Goals: Students will make various amounts of money using specific coins by creating an organized list.

Prior Knowledge Required: Canadian coins

Vocabulary: penny, nickel, dime, quarter, loonie, cent, dollar, list

Present the following problem: You need to program a machine that will sell candy bars for 45¢. Your

machine accepts only dimes and nickels. It does not give change. But you have to teach your machine to

recognize when it’s been given the correct change. The simplest way to do this is to give your machine a list

of which coin combinations to accept and which to reject. So you need to list all combinations of dimes and

nickels that add up to 45¢.

Explain that there are two quantities—number of dimes and number of nickels. The best way to find the

possible combinations is to list one of them (usually the larger—dimes) in increasing order. You start with

no dimes, then with one dime, etc. Where do you stop? How many dimes will be too much? Then next to

each number of dimes in the list, you write down the number of nickels needed to make 45¢. Make a list

and use volunteers to fill it in. You can also use a table, as shown below.

Dimes Nickels

0

1

2

3

4

Let your students practice making lists of dimes and nickels (for totals such as 35¢, 75¢, 95¢), nickels and

pennies (for totals such as 15¢, 34¢, 21¢), quarters and nickels (for totals such as 60¢, 85¢, 75¢).

Bonus:

Use dimes and nickels to make 135¢, 175¢.

Assessment:

List all the combinations of dimes and nickels to make 55¢.

Now let your students solve a more complicated riddle:

Dragons come in two varieties: the One-Headed Fearsome Forest Dragons and the Three-Headed Horrible

Hill Dragons. A mighty and courageous knight is fighting these dragons, and he has slain 5 of them. There

are 9 heads in the pile after the battle. Which dragons did he slay?


Remind your students that it is convenient to solve such problems with a list and to start with the largest

number. How many 3-headed dragons could there be? Not more than 3—otherwise there are too many

heads. How many 3-headed dragons could there be? There are 5 dragons in total. Make a list and ask the

volunteers to fill in the numbers:

Ask another volunteer to pick out the right number from the table—there were two 3-headed dragons and

three 1-headed dragons slain.

Give your students additional practice, with more head numbers: 6 heads, 2 dragons; 10 heads, 4 dragons.

What is the least and the most number of heads for 10 dragons?

Extensions:

1. List all the combinations of quarters and dimes to make 60¢, 75¢.

2. Monsters come in two varieties: Three-Headed Danger-of-Dale Monsters and Ever-Quarrelling-Nine-

Headed-Terror-of-Tundra Monsters. A mighty and courageous knight fought a mob of these monsters

and cut off all of their heads. There were 24 heads in the pile after the battle (and none of them

belonged to the knight). How many of each type of monster did the knight slay? Find all possible

solutions (ANSWERS: two 9-headed and two 3-headed, one 9-headed and five 3-headed, no 9-headed

and eight 3-headed).

3. The monsters from the previous question all have 7 tails each. When the mighty knight next got into a

fight, he produced a pile with 35 tails and 27 heads. Which monsters were destroyed? (ANSWER: two

9-headed and three 3-headed monsters)

3-Headed Horrible

Hill Dragons

1-Headed Fearsome

Forest Dragons

Total Number

of Heads

0 5 5

1

2

3

Workbook 3 - Number Sense, Part 1 1BLACKLINE MASTERS

Adding or Trading Game _________________________________________________2

Addition Rummy Blank Cards _____________________________________________3

Addition Rummy Preparation _____________________________________________4

Addition Table (Ordered) _________________________________________________5

Arrays in the Times Tables ________________________________________________6

Counting by 5s _________________________________________________________7

Crossword Without Clues _________________________________________________8

Cubes ________________________________________________________________9

Cut Out Coins _________________________________________________________10

Estimating Game ______________________________________________________11

Foreign Alphabets _____________________________________________________12

Hundreds Chart and Base Ten Materials ____________________________________13

Hundreds Chart Pieces __________________________________________________14

Hundreds Charts_______________________________________________________16

Hundreds Charts up to 200 ______________________________________________17

Make Up Your Own Cards ________________________________________________18

Money Memory _______________________________________________________19

Multiplication and Order ________________________________________________21

Multiplication Practice __________________________________________________23

Multiplying by 0 _______________________________________________________24

Multiplying by 1 _______________________________________________________25

Number Sentence Practice ______________________________________________26

Number Word Search ___________________________________________________27

Number Words Crossword Puzzle _________________________________________28

Place Value Cards ______________________________________________________29

Recognizing Number Words _____________________________________________30

Spelling Number Words _________________________________________________31

Ten-Dot Dominoes _____________________________________________________32

NS3 Part 1: BLM List



Adding or Trading Game

END 1¢ 5¢ 1¢

1¢ 5¢ 1¢ 5¢10¢ 5¢

1¢ 5¢ 1¢25¢ 10¢ 1¢

START 5¢ 10¢ 1¢ 10¢25¢ 1¢ 1¢

1¢

1¢

10¢5¢

1¢

1¢

10¢ 25¢

1¢

1¢

10¢1¢10¢25¢

Addition Rummy Blank Cards


Addition Rummy Preparation

1

37

+ 17

54

1

23

+ 39

62

2 tens + 3 ones

+ 3 tens + 9 ones

5 tens + 12 ones

+

4 tens + 7 ones

+ 2 tens + 6 ones

6 tens + 13 ones

+



+ 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Addition Table (Ordered)


Arrays in the Times Tables

Count the squares in the rectangle.

Fill in the blanks.

2 × 3 = ____ 3 × 1 = _____ 4 × 2 = _____

Now draw the rectangle yourself and write the product in the

bottom right square of the rectangle.

3 × 2 = 6 1 × 3 = _____ 3 × 4 = _____

Now complete the whole chart.

BONUS:

× 1 2 3

1

2

3

× 1 2 3

1

2

3

× 1 2 3 4

1

2

3

4

× 1 2 3

1

2

3

× 1 2 3 4 5 6 7

1

2

3

× 1 2 3

1

2

3 6

× 1 2 3

1

2

3

× 1 2 3 4

1

2

3

4



Counting by 5s

How to Make a Skip-Counting Machine:

1. Fold along line A.

2. Cut along lines B, C, D, and E.

3. Cut along lines F and G.

4. Unfold.

5. Place this sheet over a hundreds chart. What numbers do you see?

B F C D

G E

A


Crossword Without Clues

eighty

fifteen

forty

nine

one

seventeen

seventy

six

ten

three

twenty

two

zero

1. Group the words according to the number of their letters.

2. Which word is by itself in a group? Where does it fit?

3. Solve the puzzle. HINT: Cross out the words as you use them.

3 letters 4 letters 5 letters 6 letters

one

six

ten

two

7 letters 8 letters 9 letters



Cubes




Cut Out Coins

Estimating Game

I rolled , , , and .

+

100

+

40

+

70

I rolled , , , and .

+

100

+

40

+

70


Foreign Alphabets

Count by 5s and then by 1s to # nd the number of letters or symbols in

each foreign alphabet.

Hawaiian:

A, E, I, O, U, H, K, L, M, N, P, W, ‘

5, 10, 11, 12, 13

Russian:

Korean:

Spanish:

a, b, c, ch, d, e, f, g, h, i, j, k, l, ll,

m, n, ñ, o, p, q, r, s, t, u, v, w, x, y, z

Greek:

Cherokee:



Hu

nd

red

s C

har

t an

d B

ase

Ten

Mat

eria

ls

12

34

56

78

91

0

11

12

13

14

15

16

17

18

19

20

2 r

ow

s o

f a

hu

nd

red

ch

art

:

Ten

s b

lock

:

On

es

blo

cks:


Hundreds Chart Pieces

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 204 + 10 =

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 5038 + 10 =

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 10081 + 10 =

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

7 + 10 =

18 + 10 =

23 – 10 =

30 + 10 =

47 – 10 =

Shade the next 10 numbers after the bolded square. Add 10.

Move down a row to add 10.

Move up a row to subtract 10.



Hundreds Chart Pieces (continued)

The boxes below are pieces from a hundreds chart.

Add 1 or 10 to find the missing numbers.

3 24 1939 48 60

Subtract 1 or 10 to find the missing numbers.

19

36

34

70

47 70

Find the missing numbers.

32

32

32

32

29

24

1826

57 74


Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



Hundreds Charts Up to 200

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200


Make Up Your Own Cards

I have I have

Who has Who has

I have I have

Who has Who has




Money Memory



Money Memory (continued)

Add before multiplying.

3 + 3 + 3 + 3 =

4 × 3 =

4 + 4 + 4 =

3 × 4 =

2 + 2 + 2 + 2 + 2 =

5 × 2 =

5 + 5 =

2 × 5 =

3 + 3 + 3 + 3 + 3 =

5 × 3 =

5 + 5 + 5 =

3 × 5 =

4 + 4 + 4 + 4 + 4 + + +

× = ×

Make a prediction. Write the same number in each box.

Check your answer by adding.

Fill in the blanks.

+ + + + = 5 × =

+ = 2 × =

Multiplication and Order


Fill in the blanks.

Which number sentences make sense?

5 + 6 = 6 + 5 5 – 6 = 6 – 5 5 × 6 = 6 × 5

4 × = 6 ×

3 × = 5 ×

2 × = 7 × =

Multiplication and Order (continued)



Multiply.

5 10 15 20 25 30

35 40 45 50

Count by 3s.

3 × 5 = 8 × 5 = 6 × 5 = 7 × 5 = 10 × 5 =

Multiply.

4 × 3 = 1 × 3 = 2 × 3 = 9 × 3 = 5 × 3 =

Now use your own # ngers.

7 × 2 = 4 × 4 = 9 × 2 = 6 × 4 = 8 × 2 =

3 6

30

Multiplication Practice


Multiplying by 0

0 + 0 + 0 =

3 × 0 =

0 + 0 + 0 + 0 + 0 =

5 × 0 =

12 × 0 =7 × 0 = 23 × 0 =

213 × 0 = 0 × 174 = 11 × = 0

15 × = 0 × 38 = 0 × 94 = 0

When you add 3 zero times, you are not adding it at all!

So, 0 × 3 = 0.

Fill in the blanks:

0 × 1 =

0 × 4 =

0 × 7 =

0 × 8 =

0 × 16 =

0 × 21 =

0

0 + 0 + 0 + 0 + 0 + 0 =

6 × 0 =

0 + 0 + 0 + 0 =

4 × 0 =



Multiplying by 1

1 + 1 + 1 + 1 + 1 + 1 = so 6 × 1 =

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = so 9 × 1 =

1 + 1 + 1 + 1 = so 4 × 1 =

Make a prediction.

7 × 1 = 26 × 1 = 583 × 1 =

When you add 3 once, you just get 3. So 1 × 3 = 3.

Fill in the blanks.

1 × 2 = 1 × 7 = 1 × 98 =

1 × = 4 1 × = 8 1 × = 67

× 5 = 5 × 6 = 6 × 93 = 93

Isobel says: 1 × 71 = 71 × 1. Is she right?

Explain:

Fill in the blanks.


8 + 4 9 + 5 7 + 6 5 + 8

4 + 7 8 + 8 6 + 9 7 + 7

9 + 7 6 + 6 9 + 8 9 + 9

8 + 7 8 + 6 9 + 3 4 + 9

9 + 8 8 + 7 6 + 5 6 + 6

8 + 6 8 + 5 6 + 7 9 + 7

8 + 4 3 + 9 7 + 4 5 + 9

6 + 6 9 + 4 5 + 7 7 + 7

5 + 9 7 + 4 8 + 9 9 + 9

9 + 6 7 + 8 7 + 7 8 + 4

9 + 2 8 + 8 6 + 5 8 + 3

8 + 6 9 + 3 5 + 7 6 + 6

Number Sentence Practice



Number Word Search

Find:

one ten eleven two twenty

twelve three thirty four forty

fifty zero seventeen eight

Use the leftover letters to finish the message.

The four seasons are fall, ,

.

This puzzle was made using the Internet tool at http://www.superkids.com/aweb/tools/words/search

w t i t w e n t y

n w t e o e r t s

f o u r v n y w p

s e v e n t e e n

z t l e r r i l f

e e f i f t y v o

r n h g n g a e r

o t t h r e e n t

d s u t m m e r y


3

4

1 2

6

5

7

8 9

10

11

Across

2. Four less than ten

4. Rhymes with fine

7. Ten + Seven

8. Fifty + Thirty

10. Twenty + Twenty

11. Nothing

Down

1. Eleven – Ten

2. Two more than sixty-eight

3. Twenty – Five

5. Two tens

6. Seven + Three

9. Seven – Four

Number Words Crossword Puzzle



Place Value Cards

Ones

Hundreds

Tens


Recognizing Number Words

1. Eight children ate pie.

2. Ravi ate eight cookies.

3. She won two games.

4. He only won one game.

5. Four friends played soccer for fun.

6. She had to fix six bikes.

Circle the number words.

Cross out the words that only sound like number words.



Spelling Number Words

Circle the spelling of the number words.

HINT: Look at the words you circled.

one won wun

to too two

for four fore

six sicks siks

ate eight ait

1

2

4

6

8


Ten-Dot Dominoes

Draw the missing dots on the blank side.

Finish the number sentence.

All of these dominoes have a total of 10 dots.

8 + = 10 9 + = 105 + = 10

10 = + 2 10 = + 710 = 6 +

3

+

10

1

+

10

+ 4

10



Measurement Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

ME3-1 Estimating Lengths in Centimetres Goals: Students will estimate measurements in centimetres.

Prior Knowledge Required: The ability to measure

Units of measurements

Linear measurement

The ability to estimate

Skip counting

Vocabulary: measurement, centimetre, estimate, index finger

Point out the centimetre marks on a ruler, and explain to your students that a centimetre is a unit of

measurement. Write the word “centimetre” (circle the letters c and m) and the abbreviation “cm,” and explain

that these are the two ways to write centimetre.

Give your students a number of small items (tacks, buttons, blocks, etc.) that range in length from 1 mm to

10 cm—several of them should measure 1 cm long. Ask them to estimate if the item measures 1 cm exactly,

or longer or shorter than 1 cm. Remind students that “estimate” means close to or about, and explain that

they won’t be able to determine exact lengths without measuring. Then have each student present an item

with an explanation for his or her estimate.

Now give your students a centimetre measure that they can take anywhere and always have. Can they

guess what it is? Show them that their index finger is approximately 1 cm wide.

Have students compare the small items measuring 1 cm with the width of their index fingers. How close are

they to matching in length? Then have students measure the width of their index fingers with a ruler to

determine how close it is to a centimetre.

Collect all of the items given out at the beginning of the lesson and put the items measuring 1 cm in a box

that will be stored somewhere prominent.

As your students learn the many units of measurement, they will add respectively measured items to the box.

Then demonstrate to your students how they can use their index fingers, measuring 1 cm, to measure

something (a book, an eraser, etc.).

Have students select two objects from their desks or backpacks, measure the objects with their index fingers

and then complete the following sentence for both objects.

___________________ is approximately_________ cm long.

Show your students, by having a volunteer measure with his or her index finger, that the width of a penny is

about 2 cm. Then ask if they can measure their two objects with a penny instead of their index fingers.

Explain that it can be done by skip-counting by two. Distribute play-money pennies, have students measure

(in pennies) the length of their hands, notebooks, pencils, etc., and then convert their measurements into cm.


Activity: Create a box to collect benchmark items that correspond with each unit of measurement

introduced. Everyone can then refer to and compare these items as the lessons progress.

Extensions:

1. Students can use benchmarks, such as the width of their thumb or index finger (approximately 1 cm) or

the width of their hand with fingers spread (approximately 10 cm) to estimate lengths.

Ask students to estimate the length of various objects in the classroom using their hands or thumbs as

benchmarks before they measure the actual lengths with a ruler.

2. Students could use non-standard units, such as the width of a penny (which is about 2 cm) to estimate or

measure lengths. Give students play money pennies and have their measure the length of their hand, a

notebook, a pencil, etc…in pennies. Students should convert their measurements into cm. (This is a

good exercise in skip counting by 2s or in ‘doubling’.)

3. Estimate the height of a classmate in cm. Then measure their height using your hand. (Your hand with

fingers spread should be about 10 cm wide.) Finally, use a metre stick to check your result. How close

were you?

HINT: Measure them against a wall to get an accurate result.

Estimate ____ cm Hand Measurement ____ cm Actual Measurement ____ cm


0 cm 1 2 4 3 5 6 8 7 10

9

1 2 3 4 5

1 2 3

0 cm 1 2 4 3 5 6 8 7 10

9

ME3-2 Measuring in Centimetres Goals: Students will take measurements in centimetres on a ruler.

Prior Knowledge Required: The ability to count using a number line



Skip counting

Centimetres

Vocabulary: measurement, centimetre, estimate, index finger, number line

Ask your students what they always have with them that can be used as a rough measure of a centimetre.

Then ask them how they could measure a fairly large item (a poster, the blackboard, etc.). They might

suggest using their hand with fingers spread slightly (which is about 10 cm wide) or some other referent.

What if the measurement had to be exact, not approximate? They should know that a ruler will make exact

measurements.

Draw a number line and explain that counting on a number line is just like using a ruler. Demonstrate this by

asking a volunteer to count (by “hopping”) to five using a number line. Then ask another volunteer to

demonstrate how to measure 5 cm with the ruler.

Ask your students how they can find a measurement without starting at the zero mark of a ruler.

Demonstrate this by starting at the 2 cm mark and hopping to the 5 cm mark, measuring 3 cm.

Have your students practice this by assigning them the worksheet.

Assessment:

Record the distance between two points.

a) from 3 to 8 b) from 2 to 11 c) from 5 to 15 d) from 12 to 14


0 cm 1 2 4 3 5 6 8 7 10

9 0 cm 1 2 4 3 5 6 8 7 9

ME3-3 Rulers and ME3-4 Measuring in Centimetres with Rulers Goals: Students will take measurements in centimetres using a ruler.

Prior Knowledge Required: The ability to count using a number line.


Linear measurement


Skip counting

Centimetres

Vocabulary: measurement, centimetre, estimate, index finger, number line, ruler

Present a variety of objects and ask your students to estimate their length. Remind your students that they

can make a better estimate if they measure the objects with their built-in benchmarks: for instance, they

might use their index fingers or their hand with their fingers spread slightly. Ask your students to record

their estimates.

Hold up an object about 10 cm long and explain that you need to know if it fits into a box that is 10 cm long.

Ask your students to explain why estimating may give them an incorrect answer. What other method could they

use to determine whether the object will fit into the box? Ask your students to explain why they need to know

how to measure things. In which situations in life they might need to take exact measurement?

Draw several objects on the board together with rulers so that some pictures show a correct way to measure

the lengths of the objects and some do not (as shown below):

Ask your students if measurements taken in the way shown would be correct. Review the method of

measuring from the previous lesson.

Ask your students to measure some objects whose lengths they have estimated.


Activities:

1. Have students use rulers to measure items in the classroom. Have them find an item that:

a) is 6 cm long b) is 10 cm long c) is between 12 and 15 cm long

d) is 1 cm long e) is greater than 15 cm long f) is about 35 centimetres long

2. Draw a triangle on grid paper and measure its sides to the nearest cm. Calculate the total distance

around the shape. This distance is called the perimeter of the shape. (see sections ME3-11 to ME3-14).

As a project, students could find examples of optical illusions (in books or on the Internet) where the objects

appear to have different lengths, but actually to have the same length when measured. Students might try to

find an explanation of how the brain is tricked by the illusion.


ME3-5 Drawing to Centimetre Measurements Goals: Students will measure and draw items to a specific length in centimetres.


Using a ruler

Centimetres

Vocabulary: measurement, centimetre, ruler, grid paper

Give your students several objects of standard length (paper clips, unused pencils, etc.) and ask them to

measure the lengths.

Demonstrate how to draw a line 2 cm in length with a ruler. Have your students practice these steps—

separately, if necessary—in their notebooks.

STEP 1: Find the zero mark on a ruler/number line. Draw a vertical line to mark the zero.

STEP 2: Count forward from zero by two hops.

STEP 3: Draw a vertical line to mark the two.

STEP 4: Draw a line connecting the two vertical marks.

Have your students draw lines of several lengths with a ruler. If the class does not have a standard set of

rulers, have students trade rulers with their classmates so that each measurement is made using a different

ruler. This reinforces the idea that rulers have equal measurement markings, even when they look different.

Ask your students to draw another line 2 cm in length, but to start it at 3 on the number line. Following the

steps taught earlier in this lesson, but beginning at 3 and ending at 5 on the number line, have a volunteer

demonstrate how the line is drawn. Have students reproduce the problem in their workbooks and solve

several others using this method (EXAMPLE: a line 8 cm in length beginning at 3 on the number line,

a line 11 cm in length beginning at 2 on the number line, a line 14 cm in length beginning at 1 on the

number line, etc.).

Distribute centimetre grid paper to your students and have them estimate the width of the squares. Then

have them measure the width with their rulers. Demonstrate how to draw items of specific length by using the

centimetre grid as a guide. On your grid, draw a pencil (or some other item) that is 5 cm long. Explain that

you selected a starting point and then hopped forward by 5. Draw vertical marks at the start and end points,

then draw the item to fill the length between the marks.

Assessment:

Draw a line 4 cm long and a pencil 8 cm long.


Activities:

1. Draw each object to the given measure.

a) A shoe 6 cm long b) A tree 5 cm high c) A glass 3 cm deep

2. Draw a collection of long items.

a) Draw a collection of alligators, each one being 1 cm longer than the previous.

b) A pencil shrinks when it is sharpened. Draw a collection of pencils, each one being 1 cm shorter than

the previous.

c) Draw a sequence of toboggans where each one is 2 cm longer than the last.

d) A carrot shrinks when it is eaten. Draw a collection of carrots, each one being 2 cm shorter

than the previous.

3. Write a story about one of the growing or shrinking items in Activity 2 (or invent your own!). Tell the story

of how the item grows or shrinks. How does the carrot get eaten? Who wants to ride the toboggans?

Write the story to go with the pictures that you have drawn.


ME3-6 Estimating in Centimetres Goals: Students will measure and draw items to a specific length in centimetres


Using a ruler

Centimetres

Vocabulary: measurement, centimetre, ruler, grid paper

Draw a line on the board which measures 20 cm. Have students volunteer a non-standard unit to measure it

with. The units (link cubes, unit cubes, paperclips, string, shoelace, link its, etc.) can be displayed where the

students can easily see them.

Show the unit to the students and have them compare its size to the line’s size. Ask how many of the unit

they think it would take to measure the line. Discuss, focusing on differentiating between “informed” guesses

and “wild” guesses. Write students responses on board. Test the predictions.

Choose another unit of measurement (preferably one that is distinctly larger or smaller than the first) and

have students estimate how many non-standard units it would take to measure the line. Test the predictions

again and record the new measurement. Assuming that the predictions are more accurate the second time

around, discuss why this is so. If it is not, then ask students how they think they might achieve closer

estimates.

Ask students to look at the two measurements.

NOTE: You may want to record estimates and real measurements on a chart to get students use to

organizing their data. Discuss why it takes more of one unit and less of the other to measure the same line.

Students should see the connection between size of unit and number of units required.

Suggest the students to measure one of the non-standard units of measurement they used with a ruler and

to obtain the estimate for the length of the line in cm.

Ask for three student volunteers. Assign each a task, two students to find objects which are longer than the

line, and the other, an object which is shorter than the line. Then have them order themselves beside the line

on the board, from shortest to longest object, including the line itself. Have each student explain how they

knew where to stand.


ME3-7 Estimating in Metres Goals: Students will estimate lengths in metres.

Prior Knowledge Required: Using a ruler

Centimetres

Vocabulary: measurement, centimetre, metre, metre stick, ruler

Identify the length and marking of a metre on a ruler or tape measure, and explain to your students that a

metre is another unit of measurement. Write the word “metre” and the abbreviation “m”, and explain that

these are the two ways to write metre. Explain that there are 100 centimetres in one metre. Explain to your

students that they have a natural benchmark in their body that is close to 1 m. Their arm span is close to

1 m, and so is a large step.

As a class, select an assortment of items to measure with a metre stick. First estimate if each item is closer

in length to 0 m, 0.5 m or 1 m, then measure each item to verify its length. Hold the item up to the zero mark

of a metre stick and note how far the end of the item is from the zero mark, the 50 cm mark and the 1 m

mark. The shortest distance between either of these marks and the end of the item is the correct estimate for

the item’s length.

Help your students to develop an estimate for the length of the blackboard. First, ask your students to guess

the length of the blackboard in metres. You might wish to record their estimates in a table. Place a metre

stick along the board. Ask your students to compare the length of the board to the stick. Draw a line that

extends from one end of the board to the other and mark of a length of 1 metre at one end of the line. Ask

your students if they wish to adjust estimates for the length of the board. Record any new estimates in the

table. Continue to mark off lengths of 1 metre on your line and allow your students to adjust their estimates.

When the last unmarked section of the line is clearly less than 1 metre, ask your students to guess whether

the section is more or less than half a metre (50 cm) or one quarter of a metre (25 cm) long.

Activities:

1. Ask students to measure and compare the lengths of various body parts using a string and a ruler.

EXAMPLE:

a) Is your height greater than your arm span?

b) Is the distance around your waist greater than your height?

c) Is your leg longer than your arm?

2. Encourage your students to predict the answers before they perform the measurements. Estimate the

width of the school corridor: First, try to compare the length to some familiar object. For example, a

minivan is about 5 m long. Will a minivan fit across the school corridor? Then, estimate and measure the

width with giant steps or let several students stand across the corridor with arms outstretched. Estimate

the remaining length in centimetres. Measure the width of the corridor with a metre stick or measuring

tape to check your estimate.


3. Measure your shoe. Use this natural benchmark to measure long lengths: the classroom, the width of the

hallway, or the length of the whole school!

4. Your students should know how to make and record measurements in various forms; for instance they

might write a measurement as 1 metre and 25 centimetres, or 1 m and 25 cm, or 125 cm. When

students measure various objects in the classroom, ask them to record their measurements in these

three ways shown above.

Extension: Some natural benchmarks you can use to make estimates are the index finger (for 1 cm), the

hand with fingers slightly spread (for 10 cm), the arm span and a giant step (for 1 m). Will these benchmarks

represent approximately the same lengths in 5 years? Compare the length of your “benchmarks” with those

of some adult members of your family.


ME3-8 Estimating in Metres (Advanced) Goals: Students will estimate lengths in metres.


Metres

Addition, multiplication

Skip counting

Vocabulary: measurement, centimetre, metre, metre stick, ruler

Ask your students to think of objects that can act as benchmarks for estimating lengths, heights, and

distances larger than the ones they used in the previous lesson. Would you use paper clips to measure a

height of a tree? You may measure the actual length or height of some of these objects if they are

available. Here are some possible benchmarks:

• A (very tall) adult and a door are about 2 m tall.

• A level, or storey, in a building (viewed from outside the building) is about 2 doors tall,

so about 4 m tall.

• A school bus is about 10 m long.

• A typical car is about 3 m long.

Invite students to use these benchmarks to estimate greater lengths and heights, such as:

• A basketball field is about 9 cars long. How long is a basketball field in metres?

• Two minivans are as long as 1 school bus. How long is each minivan?

• A playground is about 10 cars long. How long is the playground in metres? How many minivans can

be parked along the playground?

• Daniel lives in an apartment building with 8 storeys. About how many metres tall is the building?

About how many school buses standing end-to-end, one on top of the other, are as tall as Daniel’s

building?

• Ileana wants to estimate the height of a tree that grows near the school building. The tree is

3 storeys tall. How tall is the tree?

• A storey is about 4 m tall. How tall is 5-storey-high building? How many storeys are in a building

that is about 100 m high?

Students can use either multiplication or skip counting to solve these problems. Ask your students to

explain their solutions and encourage them to try different ways of thinking. For example, the last problem

could be solved in different ways, apart from skip counting by 4s to 100:

1. Skip count by 4s: 4, 8, 12, 16, 20. The building is 20 metres tall. I can skip count by 20 to 100: 20,

40, 60, 80, 100. I said 5 numbers. Each 20 metres is 5 storeys, so I can skip count by 5s 5 times to

get the height in storeys: 5, 10, 15, 20, 25. The building is 25 storeys high.


2. Five storeys are 20 metres high. Make a T-table:

Height in

storeys

Height in

metres

5 20

10 40

15 60

20 80

25 100

3. Each storey is 4 metres high. 5 storeys are 4 x 5 = 20 metres high. What should I multiply 20 by to

get 100? I should multiply by 5. The 100 m-tall building is 5 times higher than 20 m-tall building (of 5

storeys). This means I have to multiply 5 by 5 to get the height of 100 m-tall building in storeys.

4. If your students are familiar with division, they could use the following method:

Each storey is 4 m high. 5-storeys are 20 m tall. What do you do to get 20 from 4 and 5?

(Multiply them.) What do you do to get 5 from 20 and 4? (Divide 4 into 20) The building is 100 m

high. What should I multiply 4 by to get 100? I divide 4 into 100, and get 25. This is the height of

the building in storeys.

Extension: Use a number line to solve the next question: A swimming pool is 5 m deep. If you place a

school bus upright into this swimming pool, and place a car upright near this swimming pool, which will be

taller? Hint: Put the zero mark at the bottom of the swimming pool.


ME3-9 Kilometres Goals: Students will estimate lengths in metres.


Centimetres

Vocabulary: measurement, kilometre, metre, metre stick, ruler

Explain to your students that a kilometre is another unit of measurement. Write the word “kilometre” (circle

the letters k and m) and the abbreviation “km”, and explain that these are the two ways to write kilometre.

Explain that there are one thousand metres in one kilometre.

A kilometre can then be represented in the measurement box (see ME3-1) with a spool of fishing line. [The

packaging always labels the length of fishing line on each spool in metres; one spool of line should suffice.]

A kilometre is about the same distance that we can walk in fifteen minutes, the length of about ten football

fields, or the length of about ten small city blocks. Ask your students to name a place that they think is about

1 km from the school.

As a class, skip-count by 100s to 1000, by 10s to 100, by 25s to 100. Ask your students: A soccer field is

about 100 m long. How many soccer fields make a kilometre? A school bus is 10 m long. How many school

buses can be parked along the soccer field? Along 1 km? A tennis court is about 25 m long. How many

tennis courts make 100 metres? How can you use this information to find out how many tennis courts will fit

in 1 km?

Tell your students a story: A university student planning a cheap trip to Saskatchewan can only afford

enough gas to travel a total of 500 km. View the map and determine the most interesting places to visit.

Display an overhead of the BLM “Map of Saskatchewan” and ask students to identify the distances between

sets of two adjacent points (EXAMPLE: Weyburn to Regina, and Regina to Saskatoon). Then ask them to

calculate the distance between three points (EXAMPLE: Saskatoon to Moose Jaw to Regina).

Assessment:

1 Jenny drives from Monkton to Yarmouth. How long is her trip?

2. A minivan is 5 m long. How many minivans parked in a line will it take to equal…

a) 10 m? b) 20 m? c) 100 m? d) 1 km?


Activities:

1. Kilometre for a Cause

If your school is planning any charitable events, such as a food or toy drive, set a goal to raise a

kilometre of goods. Have students measure a variety of items that might be collected in the drive

(canned goods, bags of pasta, etc.), and calculate the average length. If the items are lined up end to

end, how many will need to be collected to create a line a kilometre long? Set that as a goal. When

the drive is done, try to snake a line of all the items in a large open space (maybe the gym). How long is

the line?

2. Have students use a web application like Google Maps or MSN Maps to see how far their house is from

the school.

Extensions:

1. An adult Chinese Alligator is about 2 m long. How many crocodiles lined end to end will equal kilometre?

What about a…

• Saltwater Crocodile, 5 m long

• Newborn Saltwater Crocodile, 20 cm long (How many equal 1 m? 10 m? 100 m?)

2. Create a BLM map of a place that the class has actually visited. Bring in photographs of attractions to

enrich the experience for your students. Alternately, if the class is planning a trip, bring in a map of the

destination and ask students to help plan a good route.

3. If you lined up the following objects would they be: (i) close to 1 km, (ii) less than 1 km, or (iii) more

than 1 km? Explain your answer on a separate piece of paper.

a) 1000 paper clips b) 1000 bikes c) 1000 JUMP books d) 1000 baseball bats

HINT: First decide if the individual object is close to a metre, less than a metre or more than a metre.


ME3-10 Ordering and Assigning Appropriate Units

Goals: Students will estimate lengths in metres.


Centimetres

Metres

Kilometres

Vocabulary: measurement, centimetre, metre, kilometre, appropriate

Individually present the items from the measurement box (see ME3-1) and have your students express the

measurement that each item represents. Review the full name and abbreviation for each unit.

Ask your students to tell you how far it is from Halifax to Calgary in centimetres. Then ask them to calculate

the width of a chocolate bar in kilometres.

Explain that it is important to choose the appropriate unit of measurement for the length/distance being

measured. Ask your students to tell you which unit of measurement will best express the distance from

Halifax to Calgary. Which unit of measurement will best express the width of a chocolate bar?

Practice this by displaying a variety of objects (a book, a stapler, a coin, etc.) and asking your students to tell

you which unit of measurement will best express the length of the object. Is the length of a stapler expressed

best by a centimetre or a metre?

Have your students select five items in the classroom and guess which unit of measurement will best

express each item’s height, width or length.

Have a volunteer measure one of the practice items (the blackboard, for example), but do not specify a unit

of measurement. After the volunteer relates the measurement to the class, identify the unit of measurement

used and ask your students why the volunteer chose that unit of measurement. Students will likely respond

that metres were used because the board is about the right size. It would be many centimetres long, and it’s

much smaller than a kilometre.

Have your students measure their five objects. Which units of measurement do they automatically use?

Do these units of measurement lend themselves easily to the task? Would alternate units of measurement

offer simpler measurements?

Ask your students to list in their notebooks five things that could be measured and are not in the room

(a bicycle, a video game, a rocket ship, anything). Ask students to arrange the five things in order from

smallest to largest. Then ask them to indicate which unit of measurement will give the simplest measurement

for each item.


Play a game with the class. State a length and have your students identify the best unit of measurement

for it. Allow your students to identify the best unit of measurement by displaying the appropriate benchmark

item from the measurement box (see ME3-1).

Round 1: Centimetres and Metres

Which unit of measurement will make these statements correct?

1. Your index finger is about 1 wide.

2. Blue whales are an average of 25 long.

3. A pencil is about 20 long.

4. A car is about 4 long.

5. A paper clip is about 3 long.

Round 2: Metres and Kilometres

Which unit of measurement will make these statements correct?

1. An average-sized adult is about 1 and .5 tall.

2. The distance between Edmonton and Calgary is about 300 .

3. An average alligator is about 4 and .5 long.

4. A marathon course is about 42 long.

5. A male African elephant may grow to be as tall as 4 .

Round 3: Bonus Round

Which unit of measurement will best express the length (or height) of each item?

1. Length of an ice cube

2. Length of an ice rink

3. Length of Antarctica

4. Length of a maple leaf

5. Height of a polar bear

Ask your students how many centimetres are in a metre. Write “100 centimetres in a metre, 100 cm = 1 m.”

Ask them to identify the bigger number, 35 or 2. Then ask them to identify the bigger distance, 35 cm or 2 m.

Have them explain their reasoning. Remind them that there are 100 cm in every metre. Then have your

students measure out both lengths with the centimetre and metre benchmarks from the measurement box

(see ME3-1).

Explain that the easiest way to compare measurements that are expressed in different units is to convert all

measurements to the small unit. For instance, to compare 3 m and 250 cm, convert 3 m to 300 cm, and it

becomes clear that 3 m is greater than 250 cm.

Remind students to multiply by 100 when converting metres into centimetres. Solve a couple example

problems (1 × 100, 2 × 100, etc.) together as a class. It may help them to skip-count by 100s when

converting from metres to centimetres.


Let your students practice with questions like:

Circle the greater amount:

1 m or 80 cm 6 m or 79 cm 450 cm or 5 m 230 cm or 2 m

Ask students to order these lengths from shortest to longest and ask them to mark these lengths on a

number line:

0 cm 100 cm 200 cm 300 cm 400 cm 500 cm 600 cm

Assessment:

Thickness of a book: 3 Length of a chocolate bar: 20 _____

Height of the school: 8 Height of a tree: 12 ____

Height of a mountain: 4 or 4000 Distance from a window to a door: 4 ____

Distance from your school to your home: 700 Distance from your nose to your toes: 113 ____

Distance from the Earth to the Moon: 385 000 Distance between your ears: 20 _____

Convert the distances to cm and then order them from greatest to smallest.

a) 75 cm b) 85 m c) 230 cm d) 7 m e) 4 cm

Activity: Divide your class into three groups and assign one unit of measurement (cm, m or km) to each

group. Give each group a sheet of chart paper, and ask them to write the full name of their unit of

measurement and the abbreviation at the top of the page. Have them list as many things as they can that

could be measured with that unit of measurement. Set a target quantity (maybe twenty) and ask students to

try and list more than that quantity. Have each group share their ideas.

Extensions:

1. Give students extra practice exercises like the ones in QUESTIONS 9 to 11 on the worksheet. For more

advanced work, students could compare measurements in mm and cm, or in mm and m. For example,

which is larger:

a) a piece of string 150 mm long or one that is 17 cm long?

b) a piece of string 2 m long or one that is 1387 mm long?

2. Could you…? Challenge:

Here are some questions you could ask your students to help them practice estimating.

(Younger students could use a calculator to multiply with or you could teach them the standard method

for multiplying larger numbers.) You might also have students round all numbers to the leading digit and

then count by 10s, 100s, or 1000s to estimate.

QUESTION 1 (Warm Up): Could you…fit all the students in your school onto three school buses?

QUESTION 1 Solution:

ASK: How many students fit in one school bus?

ASK: How many students in the school?

ESTIMATE: Round the numbers.


ANSWER: – Unlikely (unless it is a very small school) that you could fit all the students in your school

onto three school buses.

QUESTION 2 (Warm Up): Could you…walk up 100 steps from your classroom to the main office?


Have the students try this. Estimate how many steps it takes to get to a certain location, then use this

information to decide if it is possible to walk 100 steps from your classroom to the office.

QUESTION 3 (Harder): Could you…reach as high as the CN Tower with all the rulers in your school?


RESEARCH: What is the height of the CN Tower? The CN Tower is about 533 metres tall. This is equal

to 53 300 cm.

What is the length of the ruler most commonly used in the school? Usually 30 cm.

How many rulers are in your classroom? Equal to the number of students. Multiply by the number of

classrooms and round.

Multiply the estimate for the number of rulers by the height of one ruler.

EXAMPLE: For a school with 600 rulers, the height of all the rulers is 18 000 cm. Far shorter than

the CN Tower.

Here are some other questions your students could try.

QUESTION 4. Could you…stack 100 pennies to be as high as the school?

QUESTION 5. Could you…read 50 books in a month?

QUESTION 6. Could you…fit 100 boxes of macaroni and cheese into a suitcase?

QUESTION 7. Could you…place 100 oranges in a line as long as your classroom?

QUESTION 8. Could you…walk 10 kilometres in a day?

QUESTION 9. Could you…fill a fish tank with 100 bottles of juice?

QUESTION 10. Could you…find a rhinoceros as heavy as everyone in your class put together?


ME3-11 Measuring Perimeter Goals: Students will estimate and measure perimeters using non-standard units.

Prior Knowledge Required: What measurement means

Practice and comfort with other forms of linear measurement and non-

standard unit use

How to count

Add strings of 1-digit numbers

Vocabulary: perimeter, around, area

Hold up a piece of artwork. Ask students what parts of the artwork they would need to measure in order to

frame the picture. Then, ask how many link cubes or pattern block triangles it might take those

measurements.

Draw a rectangle on the board (22 cm × 30 cm) and demonstrate how to line up the link cubes along the

border of the rectangle. Note the importance of only counting the units whose edges touch the edge of the

object being measured. Review the importance of ensuring that units are lined up in a straight line and that

they touch sides but do not overlap.

Next, ask students what they would do if they only had one link cube to measure the distance around the

rectangle. Show students how to use only one unit and make marks to show where the unit starts and

finishes so that they can keep track of what they have measured.

As you line up the link cube, create a number sentence which encompasses the length and width of each

side, e.g., ____ + ____ + ____ + ____ = ____

Ask students if they know what the mathematical term is for measuring distance around an object. If they do

not say perimeter, introduce the term and explain that when measuring perimeter, they are measuring the

“outside edge of any area.” As you explain, use the rectangle from before and with a marker/chalk, go over

the outside edge of the shape to reinforce the concept. You may also want to write this on a sentence strip

and post it somewhere in the classroom for easy student reference.

Challenge students to come up to the board to create two shapes, one which would have a smaller

perimeter, and one with a larger perimeter than the rectangle used to demonstrate the concept of perimeter.

Other students can predict what the perimeter of each is and then test and measure the shapes. Encourage

students to write corresponding number sentences to find each shape’s perimeter.

Be prepared to address concerns about half units and get students to think of solutions. Possible solutions

could be to write that the perimeter is about X units, while some students may realize that if they have two

halves, that makes a whole and they would add it to the total of units.


Activities:

1. Have students work in partners to measure the distance around various objects in the classroom using

link and/or unit cubes and check each others measurements for discrepancies.

2. Give students link cubes and ask them to create various shapes with a perimeter of 12 cubes.

3. Have students create fences for fields by using 4 pieces from a tangram set (two small triangles, medium

triangle, and square). Challenge them to make different shaped fields with the same four pieces. Have

them measure the perimeter with a link cube. Does the P remain the same? Why or why not?

Literature/Cross Curricular Connection:

How big is a foot? R. Myller

(The King wants to order a bed for his Queen but beds have not yet been invented. Begin the story and stop

at the point of figuring out how big the bed should be. Have students brainstorm how to solve this dilemma.

They should be focusing on figuring what perimeter the bed should be. Encourage them to use actual size to

solve the problem, and then give them grid paper to record a solution with a partner. They then will write a

letter to the King’s apprentices to explain their work and thinking. Have a group discussion to compare pairs’

solutions and then finish reading the story to them to find out how the characters solved the problem.)

Extensions:

1. What unit of measurement would students use to measure the distance around the classroom? What

would be most effective and efficient? Find out the perimeter of the classroom. (Giant steps?)

2. Have students draw a square that has a perimeter of 12 cubes. Have them figure out the length of each

side. Next, tell them to draw a rectangle that has a perimeter of 12 cubes and tell what the length of each

of the sides will be. Finally, have them draw a triangle with a perimeter of 12 cubes and have them figure

out the length of each of the sides.

3. Paul bought 13 bushes to place around the perimeter of his yard (shown in the diagram below).

For each edge of the diagram, he planted one bush. He measured the perimeter before going to the

nursery but he thinks he made a mistake because he doesn’t have enough bushes. Can you help him?


ME3-12 Perimeter Goals: Students will estimate and measure perimeters using the standard and non-standard units.

Prior Knowledge Required: What measurement means.

Practice and comfort with linear measurement and

non-standard unit use.

How to count.

Vocabulary: perimeter, around, area, cm

Write the word “perimeter” and explain to your students that perimeter is the measurement around the

outside of a shape. Illustrate the perimeters of some classroom items; run your hand along the perimeter

of a desk, the blackboard or a chalkboard eraser. Write the phrase “the measurement around the outside

of a shape.”

Draw this figure:

1 cm

Explain that each edge of the squares represents 1 cm, and that perimeter is calculated by totalling the

outside edges. Demonstrate a method for calculating the perimeter by marking or crossing out each edge

as it is counted. Demonstrate this several times.

Examine the figure again. Squares have four sides, so why is the perimeter only 10 cm and not 16 cm

(4 × 4)? Remind your students that perimeter is the measurement around the outside of a shape.

The squares on the ends each have three outside edges. The squares in the middle have two outside

edges. The inside edges—sides that touch—are not totalled in the perimeter. Refer your students to the

definition again.

Ask them why they might want to know the perimeter of an object. (To border a picture with ribbon? To wrap

a present? To fence in a garden?)

Demonstrate another method for calculating perimeter by counting the entire length of one side, instead

of counting one edge at a time, then adding the lengths. Remind your students that addition is a quick way

of counting.


Activity: Distribute one piece of string, about 30 cm long, and a geoboard to each student. Have them tie

the ends of the string together to form a loop, and then create a variety of shapes on the geoboard with the

string. Explain that the shapes will all have the same perimeter because the length of the string, which forms

the outside edges, is fixed. How many different shapes can all have the same perimeter?

Extension: Distribute Pentamino pieces (a set of twelve shapes each made of five squares – see the

BLM) to your students and have them calculate the perimeter of each shape. Create a table and order the

perimeters from smallest to greatest. Have students also calculate the amount of square edges inside each

shape. Can they notice a pattern emerging in the table?

Shape Perimeter Number of Inside Edges


ME3-13 Exploring Perimeter Goals: Students will measure perimeters of given and self-created shapes.

Prior Knowledge Required: Perimeter


non-standard unit use

How to count


Vocabulary: perimeter, around, grid paper

Review the perimeter, its definition and how it is calculated by totalling the outside edges of a figure.

Demonstrate the method for calculating perimeter by counting the entire length of each side and creating an

addition statement. Write the length of each side on the picture. Draw several figures on a grid and ask your

students to find the perimeter of the shapes. Include some shapes with sides one square long, like the shape

in the assessment exercise, as students sometimes overlook these sides in calculating perimeter. Ask your

students to draw several shapes of their own design on grid paper and exchange the shapes with a partner.

For the last exercise, suggest that students draw a letter, or simple word (like CAT), or their own names.

Assessment:

Write the length of each edge beside each edge and count the perimeter of this shape.

Do not miss any sides—there are ten!

2 cm

Extensions:

1. Explain that different shapes can have the same perimeter. Have your students draw as many shapes as

they can with a given perimeter (say ten units). (From the Western Curriculum)

2. Can a rectangle be drawn with sides that measure a whole number of units and have a perimeter…

a) of seven units?

b) with an odd number of units? [Both are impossible.]

Students can use a geoboard rather than grid paper, if preferred.

3. Create three rectangles with perimeters of 12 cm. (Remember: A square is a rectangle.)


ME3-14 Investigations Goals: Students will estimate and measure perimeters of given and self-created shapes in standard units.

They will also predict the perimeter of figures in growing patterns.

Prior Knowledge Required: Perimeter


non-standard unit use

Patterning


Vocabulary: perimeter, around, grid paper

Draw this figure:

and have your students demonstrate the calculation of the perimeter by totalling the outside edges.

Then draw the same rectangle without the inside edges:

and ask them how they could calculate the perimeter again. Explain that it can be measured, and then have

volunteers measure each side with a metre stick.

Draw several shapes on the board, as shown below:

Ask your students to estimate the sides of the shapes using natural benchmarks (remind them that the

widths of their hands are about 10 cm). Then ask your students to measure the sides and to find the

perimeter of each shape.

Review appropriate units. Have students determine the best units of measurement for calculating the

perimeters of the schoolyard, a chalkboard eraser, a sugar cube, etc. Point out to your students that if you

had used 1 cm to represent a kilometre in the drawings above, their measurements would have to be

expressed in kilometres.


Draw a growing pattern on the board:

As a class, find the perimeter of each shape in the sequence. Examine the number of squares and then ask

your students how the perimeters change with the addition of each square. [The perimeter increases by two.]

Why does the perimeter only change by two, even though each added shape has four sides? Reiterate that

perimeter is the measurement of outside edges only. Every time a new square is added to the sequence it

covers one of the edges that had previously been on the outside.

Invite volunteers to extend the sequence and have your students predict the perimeter of the three

subsequent shapes. Draw each shape to check the predictions.

Draw the following shape:

Ask a volunteer to find the perimeter. Ask another volunteer to add a square to the figure (so that at least

one edge of the square is coincident with an edge of the figure) and calculate the new perimeter. Ask

students to repeat this exercise by adding squares in different positions. Summarize the results in a table. (It

is also good to mark congruent shapes. Do they have the same perimeter?) Why does the perimeter change

the way it does?

How many edges that had previously been on the outside of the figure are now inside? Are there any

positions where you can add a square so that the perimeter does not change?

Present a word problem:

Sally wants to arrange eight square (1 m × 1 m) posters into a rectangle. How many different rectangles can

she create? She plans to border the posters with a trim. For which arrangement would the border be least

expensive? Explain how you know.

Ask your students to create different rectangles from 8 squares. They can use blocks, grid paper or

geoboards. What should Sally do to find how much material she needs for the border? (Find perimeter) Ask

your students to find the perimeter of each rectangle. Tell your students that material for the boarder costs 5

cents per metre.

How much does the border cost for each arrangement?

Assessment:

1. Add a square to this shape so the perimeter’s measurement…

a) …increases by two. b) …remains the same.


2. Measure the sides of the shape and find its perimeter.

TEACHER: Draw the lengths of all sides in whole centimetres.

Extensions:

1. A hexagon has equal sides 5 cm long. What is its perimeter?

2. On grid paper, draw 3 different figures with perimeter 8. (The figures don’t have to be rectangles.)

3. The sides of a regular pentagon are all 7 m long. What is the pentagon’s perimeter?

4. Will 500 toothpicks, placed end to end be enough to cover perimeter of your school? First, estimate the

length and the width of the school with giant steps. Draw the shape of the school building and find the

perimeter. Estimate and measure the length of a toothpick. What is the length of 100 toothpicks? Of 500

toothpicks? As a challenge, approximate the perimeter of the school in toothpicks.


ME3-15 Measuring Mass Goals: Students will estimate and measure perimeters of given and self-created shapes in standard units.

They will also predict the perimeter of figures in growing patterns.

Prior Knowledge Required: What is mass

Gram

kilogram

Ordering from greatest to least, from least to greatest

Vocabulary: gram, kilogram, mass

NOTE: Mass is a measure of how much substance, or matter, is in a thing. Mass is measured in grams and

kilograms. A more commonly used word for mass is weight: elevators list the maximum weight they can

carry, package list the weight of their contents, and scales measure your weight. The word weight however,

has another very different meaning. To a scientist, weight is a measure of the force of gravity on an object.

An object’s mass is the same everywhere—on Earth, on the Moon, in space—but it’s weight changes

according to the force of gravity. When we use the term weight in this and subsequent lessons, we use it as

synonym for mass.

Remind students that mass (which we often call weight) is measured in grams (g) and kilograms (kg). Give

several examples of things that weigh about 1 gram or about 1 kilogram:

1 g: a paper clip, a dime, a chocolate chip

1 kg: 1L bottle of water, a bag of 200 nickels, a squirrel.

List several objects on the board and ask students to say which unit of measurement is most appropriate for each one—grams or kilograms:

• A whale • A cup of tea

• A table • A workbook

• A napkin • A minivan

Ask students to match these masses to the objects above:

2000 kg 50 000 kg 10 g 150 g 400 g 10kg

Have students order these objects from heaviest to lightest.

Ask students think of three other objects that they would weigh in grams and three objects that would

demand kilograms.


Activities:

1. Let students feel the weight of objects that are close to 1 g (EXAMPLE: paperclip) or 1 kg (EXAMPLE: a

1 L bottle of water). They can use these referents to estimate the weight of the objects in the classroom,

such as books, erasers, binders, games and calculators. (You could ask students to order the objects

from lightest to heaviest.) Students should use scales to weight the objects and check their estimates.

2. Weigh an empty container, then weigh the container again with some water in it. Subtract two masses to

find the mass of the water. Repeat this with a different container but the same amount of water. Point out

to students that the mass of a substance doesn’t change even if its shape does.

Extension: Jane wants to estimate the weight of one grain of rice. She weighs 100 grains of rice and

divides the total by 100. Try to weigh 1 grain of rice. Explain why Jane uses this method above. Use Jane’s

method to estimate the weight of a bean or a lentil.


ME3-16 Measuring Capacity Goals: Students will determine the capacity of various containers.

Prior Knowledge Required: Litres

Millilitres

Half, quarter

Vocabulary: litre, millilitre, capacity

Explain that the capacity of a container is how much it can hold. Write the term on the board. Explain that

capacity is measured in litres (L) and millilitres (mL). Explain that there are 1 000 millilitres in 1 litre.

Write on the board: 1 litre = 1 000 millilitres 1 L = 1 000 mL

Put out several containers (EXAMPLES: milk and juice boxes, medicine bottles, measuring cups, cans of

paint, cans of pop) with capacities clearly marked on them. Invite students to help you separate the

containers into two groups: those that can hold 1 or more litres and those that can hold less than 1 litre. Then

ask students to help you order the containers by capacity, from least to greatest. The containers that hold 1L,

500 mL or 250 mL can also act as “capacity benchmarks” that you can keep in a class measurement box.

Write the following on the board and ask students whether they would measure the capacity of each

container in millilitres or litres:

• a glass of juice

• a bowl of soup

• a pail of water

• a pot of soup

• an aquarium

• a backyard pool

Ask students to think of three more quantities that are measured in litres and three that are measured in

millilitres.

Review the concepts of halves and quarters. ASK: How many halves make one whole? How many quarters

make one whole? Show your students an empty 1 L milk carton but don’t tell them that its capacity is 1 L.

Ask a volunteer to find the capacity written on the carton. Next show your students an empty 500 mL milk

carton with the capacity marking covered up. Tell students that you want to find the capacity of this carton

using the larger carton. Fill the smaller carton with water and empty it into the 1 L carton. Invite a volunteer to

look into the larger carton and indicate on the outside (using a finger or marker) how much water is inside.

ASK: About how much water is now in the carton? Refill the small carton and empty if into the larger carton

again. Now the carton is full. ASK: How many smaller cartons did we need to fill the 1 L carton? What is the

capacity of the smaller carton in terms of parts of a litre? (one half) Repeat with a 250 mL carton in place of

the 500 mL carton.


Show students a small (200 mL) glass. ASK: How can we determine the capacity of the glass in millilitres

using the 1 L carton? You might ask a volunteer to check how many times they can fill the glass with water

from the carton. Then ASK: What is the capacity of the milk carton in millilitres? How many times did we fill

the glass? What should we do to the capacity of the large carton to get the capacity of the glass? What is the

capacity of the glass? (NOTE: You will need a large bowl or pot in which to empty out the glass each time it

is filled.)

After you have determined the capacity of the cartons and the glass, ask your students to order the

containers according to capacity, from smallest to largest.

Activities:

1. Measure the capacity of several glasses or containers in your classroom. Estimate the capacity of the

containers before you measure their capacity. Students should select and justify appropriate units to

measure the capacity of a container. NOTE: Students will need a measuring cup and several containers

for this activity.

2. Ask students to fill a measuring cup marked in millilitres and estimate how many cups they would need to

fill a 1 L container. Then, knowing the number of millilitres their cup holds, they should estimate how

many millilitres are in a litre. Students could then test their prediction by filling up the litre container and

keeping track of how many millilitres they need.

3. Give students several glasses of containers on which the capacities have been covered or removed. Ask

them to estimate the capacity of each container and then check their answers. Remind students to

include the appropriate units (mL or L) with their estimates.

4. Weigh a measuring cup. Pour 10 mL of water in the cup. Calculate the mass of the water by subtracting

the weight of the cup from the weight of the water with the cup. How much do 10 mL of water weigh?

(How much does 1 mL of water weigh?)


ME3-17 Measuring Temperature Goals: Students will learn to read a thermometer and will identify given temperatures as hot or cold.

Prior Knowledge Required: Thermometer

Proficiency in using number lines

Degree Celsius

Vocabulary: thermometer, degree Celsius

Hold up a large thermometer and ask students to identify what it is. ASK: What do we measure with a

thermometer? (temperature) Where and when do people use thermometers? (to measure the temperature of

a home, a greenhouse, the outdoors, the fridge or freezer, the body, and so on)

Explain how a thermometer works. The liquid inside the bulb goes up and down depending on the

temperature. It goes up as the temperature gets warmer, and down as it gets colder. You can demonstrate

this using cups of water at different temperatures. Fill one cup with hot water, one with water at room

temperature, and another with cold water and ice. Invite a volunteer to touch the cups and to order them from

coldest to hottest. Then insert the thermometer into each cup and let students observe the movement of the

liquid in the thermometer.

Explain to your students that temperature is measured in units called degrees. In Canada, we use degrees

Celsius, which can also be written like this: °C. Ask your students to observe the numbers alongside the liquid

in thermometer. What do they notice? Explain that the zero on the thermometer indicates the freezing mark.

This is the temperature at which water freezes. When the liquid in the thermometer drops below the zero, we

say the temperature is “below freezing.” You might point out that water becomes ice below 0°C, but other

liquids—like the liquid inside the thermometer—stay unfrozen.

Draw several thermometers on the board and shade them in to show different temperatures. (Only the zero

mark needs to be indicated.) Ask students to tell whether the thermometers are showing temperatures above

or below freezing.

Draw a thermometer on the board and divide it into five sections as shown. Ask your

students to think of words that describe different air temperatures (words they may

suggest include: freezing, cool, warm). Record the words and have students order

them from coldest to hottest. ASK: When it’s so cold outside that water turns to ice,

how would you describe the temperature? (possible answers: very cold, freezing

cold) Ask a volunteer to identify the part of the thermometer where the temperatures

are below freezing. The temperatures in this section can be labelled “freezing cold.”

Ask student volunteers to label the temperatures in the other sections on the

thermometer as cold, cool, warm, and hot.

As the liquid in the thermometer rises above zero, the temperature get warmer, so we count up. The larger

the number above zero, the warmer it is. As the liquid in the thermometer drops below zero, the temperature

0°C


gets colder, so we put a minus sign in front of the number and count up. The larger the number below zero,

with the minus sign, the colder it is.

Ask students to help you add a scale to the thermometer on the board. Ask them to count up by tens from

zero, and mark the degrees on the thermometer so that each mark sits on the boundary between two

sections. Write the range of temperatures beside each descriptive word: hot 30°C and up; warm 20°C to

30°C, cool 10°C to 20°C, cold 0°C to 10°C, and freezing cold 0° and below. Remind students that these

ranges and descriptions refer to the temperature of air.

Point to different temperatures on the thermometer (between the ten degree increments marked) and ASK: If

the liquid in the thermometer was here, how would you describe the temperature? Then ask students to think

about the temperature that is appropriate or necessary for various outdoor activities. For each activity, draw

two thermometers showing very different temperatures and ask students which one shows the temperature

that is more appropriate for the activity. EXAMPLES: water skiing: –10°C or 30°; berry picking: 0°C or 17°C;

ice hockey: 15°C or –3°C.

Draw part of a vertical number line on the board and mark every fifth increment: 0°C, 5°C, 10°C, 15°C.

Explain to your students that this is the enlarged scale of the thermometer. On a real thermometer there is no

room to write the other numbers, but you will teach them how to read the temperature even when these

numbers are missing.

Invite a volunteer to count up by fives from 0 to 10 and by ones from 10 to 15. Then point at the mark

showing 12°C and explain that, to read this temperature, people first count by fives and then continue by

ones until they reach the mark. Invite students to count up with you until you reach 12°C. Repeat with several

examples until all of your students are able to count first by fives then by ones to any mark you indicate. If

necessary, separate the tasks – ask students who have trouble counting by both fives and then ones to

count by fives only or to count by ones only from the nearest marked increment. Ask your students to tell

whether the weather outside is hot, cold, warm, or cool for each of the temperatures you show.

Explain that water feels differently than air at different temperatures. For instance, what feels colder – the air

on your face when it is 0°C outside or a piece of ice that you hold in your bare hand? Students can do the

activity below to compare the temperatures of water and air.

ASK: Is –5˚C hot or cold? Is water at this temperature a liquid or is it ice? Explain that if you have a piece of

ice at –5˚C, you have to heat it for some time to make it melt. How much do you have to heat it? (enough to

raise its temperature by 5˚C, to 0˚C)

Write several temperatures on the board. Ask students to describe each temperature and to tie it to familiar

events or activities, as well as states of water, such as:

–10˚C – freezing cold; skating, skiing

0˚C – cold, water freezes

5˚C – cold, water is nearly frozen, ice has melted

20˚C – air is warm, water still cold; bike riding

36˚C – air is very hot, near to normal human body temperature; swimming, sitting in the shade

50˚C – very hot, desert (You can bake eggs in the sand!)


Add to this list the range of temperatures for each season where you live, to give students the information

they need to complete the worksheet.

Have students solve several problems in which they have to add or subtract to find the temperature.

SAMPLE PROBLEMS:

• The temperature today is 15˚C. Yesterday it was 5˚C higher. What was the temperature yesterday?

• The average temperature in winter is –10 ˚C. The average temperature in summer is 25˚C. What is the

difference between the average temperatures in winter and in summer? (Let students use a

thermometer to solve this problem.)

Assessment:

a) What was the temperature on Monday?

b) What was the temperature on Wednesday?

c) How much warmer was in on Monday than on Wednesday?

d) A jar of water was left outside. On which day did the water

turn to ice?

Activity: Students will need thermometers. Give students some hot tap water. (NOTE: The water

temperature should not be greater than 49°C for safety reasons.) Let your students measure its temperature.

Add some cold tap water, let students feel the water to estimate its temperature, and then have students

measure the temperature. Repeat so that students can feel and measure the temperature of water at

different temperatures. Student can describe the various temperatures of water as they did the temperatures

of air during the lesson, and then compare them. For example, how does at air at 36˚C compare to water at

or near that temperature?

Extension: Explain that the scale of the thermometer is based on the properties of water, so that

anyone can build his or her own thermometer. Explain that 0˚C is the temperature at which water freezes

and 100˚C is the temperature at which water boils. So to make a new thermometer, you need to freeze some

water to find the 0˚C mark and then boil some water to establish the 100˚C mark. Then you divide the scale

into 100 parts and your thermometer is ready. You can illustrate the process while drawing a thermometer on

the board.

– 25

0°C

10

20

30

5

15

25

– 15

– 5

– 20

– 10

– 25

0°C

10

20

30

5

15

25

– 15

– 5

– 20

– 10

Monday Wednesday

Workbook 3 - Measurement, Part 1 1BLACKLINE MASTERS

Map of Saskatchewan ___________________________________________________2

Pentamino Pieces _______________________________________________________3

ME3 Part 1: BLM List



Map of Saskatchewan


Pentamino Pieces

Probability & Data Management Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

PDM3-1 Introduction to Classifying Data

Goals: Students will group data into categories. Students will identify attributes shared by all members

of a group.

Prior Knowledge Required: The difference between “and” and “or”

Vocabulary: data, classify, attribute, and, or

Have eight volunteers stand up. Ask the students to suggest ways in which to classify them (for example,

long or short hair, boy or girl, nine or ten years old, wearing jeans or not wearing jeans, wearing yellow or

not wearing yellow). Then have one student classify the eight volunteers into two groups without telling the

class how he or she chose to classify them. The student tells each of the eight volunteers which side of the

room to stand on. Each remaining student then guesses which group he or she belongs to. (If the student

guesses incorrectly, the student who classified the volunteers moves that student into the right group but

does not reveal the classification.) Stop when five consecutive students have guessed correctly. The last

student to guess correctly appoints each remaining student in the class to either of the two groups and is

told if they’re right or not.

Repeat this exercise several times, with different students doing the classifying. Note that the student

doing the classifying never reveals the classification. To make guessing the classification harder, students

may decide to combine attributes, such as grouping “boys not wearing yellow” and “boys wearing yellow

and girls.”

Write the following words on the board:

J.K. Rowling lion Alberta Ottawa Anna Klebanov

dog cat Canada mouse Rita Camacho

Ask students to put the words into the following categories:

People Places Animals

Have students add more words to each category. Then tell students that the category “People” could be

divided further. For example: adults and children, girls and boys, first language English or first language

other. Have students suggest other ways to categorize people and have volunteers put their own names in

the appropriate categories. Then do the same for Places and Animals.

Have students think of categories for the following data:

Weather (e.g., sunny, cloudy, rainy)

Time of day (e.g., morning, afternoon, evening, night)

Foods

Fruits

Vegetables


Have students identify attributes shared by all members of the following groups:

a) grey, green, grow, group (EXAMPLE: starts with “gr,” one-syllable word)

b) pie, pizza, peas, pancakes (EXAMPLE: food, starts with p)

c) 39, 279, 9, 69, 889, 909 (EXAMPLE: odd number, ones digit 9, less than 1000)

d) 5412 9807 7631 9000 9081 (EXAMPLE: number, 4 digits, greater than 5000)

e) hat, cat, mat, fat, sat, rat (EXAMPLE: three-letter word, ends in “at”) Have students write 2 attributes for the following groups:

a) 42, 52, 32, 62, 72 b) lion, leopard, lynx

Activity: Show students cards from the game SetTM

and ask students to say what categories can be used

to sort the cards (e.g. shape, colour, number, shading) and then to sort the shapes using those categories.

Have groups of students play the card game SetTM

. Include only the solid shapes (no stripes or blanks) until

students are very comfortable finding sets.

Extensions:

1. Which 2 attributes could have been used to sort the items into the groups shown?

Group 1:

Group 2:

� at least 1 right angle � 4 sides or more � 2-D shapes � quadrilaterals

� no right angles � 3-D objects � 3 sides � hexagons

Which categories do all shapes in both groups belong to? Which categories do no shapes in either group

belong to? What categories do some shapes in group 1 and some shapes in group 2 belong to? What

category do all shapes in group 1 and no shapes in group 2 belong to? What category do no shapes in

group 1 but all shapes in group 2 belong to? How were these groups categorized?

2. Give pairs of students the BLM “Shapes” (it shows 8 shapes that are either big or small, triangles or

squares) or the Set™ card game. The BLM has 2 of each shape, so that students can work individually.

Ask students to separate the small triangles from the rest of the shapes, then challenge them to

describe the remaining group of shapes (big or square). Encourage them to use the words “or” or

“and” in their answers, but do not encourage use of the word “not” at this point since, in this case,

“not small” simply means “big.” If students need help, offer them choices: big or triangle, big and

square, big or square.

Challenge students to describe the remaining group of shapes when they separate out the:

a) small squares e) shapes that are big or triangular

b) big squares f) shapes that are square or big

c) big triangles g) shapes that are both square and big

d) shapes that are small or triangular h) shapes that are both triangular and small


PDM3-2 Venn Diagrams

Goals: Students will sort data using Venn diagrams.

Prior Knowledge Required: Identifying attributes

Classifying according to attributes The words “and,” “or,” and “not”

Vocabulary: Venn diagram, or, and, not

Label a cardboard box with the words “toy box,” then ask individual students if various items (some items

toys and some items not toys) belong inside or outside the box. Then tell students that you want to classify

items as toys without having to put them in a box. Draw a circle, write “toys” in the circle, and tell your

students that you want all of the toy names written inside the circle and all of the other items’ names written

outside the circle. Ask your students to tell you what to write inside the circle. Make sure to crowd the words.

ASK: Is there a way we can save space and not write the whole word in the circle? If students suggest you

write just the first letter, ask them what will happen if two toys start with the same letter. Explain to them that

you will instead write one letter for each word. (EXAMPLE: A. blocks B. bowl C. toy car D. pen)

ASK: Which letters go inside the circle, and which letters go outside the circle?

Draw several shapes and label them with letters.

A. B. C. D. E. F.

Draw:

shapes

ASK: Do all the letters belong inside the box? Why? Which letters belong inside the circle? Which letters

belong outside the circle but still inside the box? Why is the circle inside the box? Are all of the triangles

shapes? Students should understand that everything inside the box is a shape, but in order to be inside the

circle the shape has to be a triangle. Change the word inside the circle and repeat the sorting exercise.

triangles

toys


[Suggested words to use include: dark, light, quadrilaterals, polygons, circles, dark triangles, dark circles.]

Ask your students why the circle is empty when it’s labelled “dark circles.” Can they think of another property

that would empty the circle of all the shapes? Remind them that the word inside the circle should reflect the

fact that the entire box consists of shapes, so “rockets” probably isn’t a good word to use, even though the

circle would still be empty.

Lay two hula hoops on a table or on the floor. Clearly label one hula hoop “pens” and another “blue,” then

ask your students to assign several coloured pens (black and red) and pencils (blue, red, and yellow) to the

proper position—inside either of the hoops or outside both of them. Do not overlap the hoops at this point.

Then present your students with a blue pen and explain that it belongs in both hoops. ASK: How can we

move the hoops so that the blue pen is circled by both hoops at the same time? Ask a student volunteer to

show the others how it’s done. If the student moves the hoops closer together without overlapping them, and

positions the blue pen such that part of it is in the “blue” hoop and part of it is in the “pens” hoop, be sure to

ask them if it makes sense for part of the pen to be outside of the “pens” hoop. Shouldn’t the entire pen be

circled by the “pens” hoop?

Draw:

A. B. C. D. E. F.

And draw:

Dark Triangles

Explain to your students that this is called a Venn diagram. Ask them to explain why the two circles are

overlapping. Have a volunteer shade the overlapping area of the circles and ask the class which letters go in

that area. Have a second volunteer shade the area outside the circles and ask the class which letters go in

that area. Ask them which shape belongs in the “dark” circle and which shape belongs in the “triangle” circle.

Change the words representing both circles and repeat the exercise. [Suggested words to use include: light

and quadrilateral, light and dark, polygon and light, circles and light.]

This is a good opportunity to tie in concepts from other subjects. For example, students can categorize words

by the beginning or ending letters, by rhyme patterns, or by the number of syllables. (EXAMPLE: “rhymes

with tin” and “2 syllables”: A. begin; B. chin; C. mat; D. silly). Start with four words, then add four more words

that belong to the same categories. Encourage students to suggest words and their place in the diagram.

[Cities, provinces, and food groups are also good categories.]

Activity: Create a big circle on the floor with masking tape and label it “8 years old”. Have students stand

in the circle if they are 8 years old and have them stand outside the circle if not.


Repeat with several other properties:

Wearing Girl Blue Takes the

yellow eyes bus to

school

Have your students stand inside the circle if the property applies to them. Have them suggest properties and

try to move appropriately (either to the circle or away from it) before you finish writing the words. Students

should learn to strategically pick properties that take a long time to write. Repeat the activity by overlapping

two circles.


PDM3-3 Introduction to Tallying Data

Goals: Students will switch between numbers and their corresponding tallies.

Prior Knowledge Required: Ability to count

Number recognition

Counting by 5

Vocabulary: tally marks, tallies, diagonal, vertical

Explain to students that experts in mathematics and other subjects often use tally marks to keep track of

what they are counting. Tell students that you want to count the number of boys in the room. Draw a vertical

stroke each time you say a boy’s name:

IIIIIIIIII

Then repeat using diagonal lines for each fifth boy:

IIII IIII

ASK: Which set of marks is easier to read? Explain that in a tally, each vertical stroke represents one, but

every fifth stroke is drawn diagonally across the first four. This makes it easy to count by fives.

Then write the numbers 1–5 on the board, making sure they are well spaced out. Under the number 1, draw

the corresponding tally: one vertical stroke. Ask students what they think the tally for the number 2 will look

like. Have a volunteer write that on the board. Repeat with the numbers 3 and 4. For number 5, remind

students that tally marks are grouped in fives and we draw the fifth line diagonally across the other four—the

fifth line “bundles” the other four together.

Next, write the numbers 6–10 and show students the tally for the number 6. Ask volunteers to write the

tallies for the numbers 7–9. For the number 10, ask students to predict what the tally will look like and then

show them.

Repeat the process with numbers 11–15 and then 16–20.

Now show students the tally for the number 4 and ask them to identify it. Then do 5, 8, 12, 15, and others

until all students can quickly and easily read the tallies.

Next, draw 5 apples on the board and remind students that tallies are useful for tracking data. Cross out an

apple, and draw a tally mark. Ask a volunteer to continue the process. Repeat with an array of 12 circles.

Bonus Questions: Who can count quickly? Show 15 as a tally. Then show 20, 25, 35… and 100!


Activity:

Paper Clip Search. Place enough paper clips around the room to average about 10 per student. Have

students find and collect as many paper clips they can. Show students how to use 1 paper clip to bundle

another 4:

Ask students how many paper clips they have in total. They should use the bundles of 5 to count by 5s. Then

have students pair up and find out how many the pair has in total; they may need to bundle once more.

Extension: Show students a penny and ask them to show what the penny is worth with tally marks.

Continue showing tallies for a nickel, a dime, a quarter, a loonie, and a toonie.


PDM3-4 Reading Data from a Tally Chart

Goals: Students will compare, order, add, subtract or multiply data to find new information

Prior Knowledge Required: Tallies

Addition, subtraction, and multiplication by small numbers

Vocabulary: add, subtract, multiply, twice as many, three times as many

Take a survey of your students and ask what type of fruit is their favourite among: apples, bananas,

oranges, grapes, and peaches. Record the tallies yourself and have students complete the chart by writing

the appropriate number for each tally.

Fruit Apples Bananas Oranges Grapes Peaches

Tally

Number

Ensure that students can read the data directly from the tally chart. ASK: How many students like grapes

best? How many like apples best? Oranges? Bananas? Peaches?

Tell students that they can compare and order the data to find new information. ASK: Did more people

like oranges or bananas better? Which fruit was the most popular? Which fruit was the least popular?

Working independently, have students list the fruits in order from most popular to least popular. Ask them

to explain their thinking. (PROMPTS: How did you know which fruit to put first? How did you decide which

one came next?) Tell students that they can add the data values together. ASK: How many students

answered the survey? Was it everyone in the class? How many students are in the class? How many

students liked either apples or grapes best? Bananas or oranges? Peaches or oranges or apples?

Bonus: Combine comparing and ordering with addition. For example, ASK: Did more people like bananas

best than liked peaches and grapes best?

Tell students that sometimes they need to subtract data to find new information. ASK: How many more

students liked apples best than peaches best? (Or “peaches best than apples best” if appropriate). Make

more such comparisons. Then ASK: Of all the people who answered the survey, how many did not choose

oranges? What number do we need to subtract from what to figure this out? How many students did not

choose bananas? Can we answer these questions without using subtraction? (Yes, we can add the other

four data values.)

Bonus: How many chose students neither bananas nor oranges? Did you use addition or subtraction or both?

Tell students that sometimes they need to multiply data to find new information. ASK: Is any fruit twice as

popular as another fruit? Is any fruit three times as popular (or almost three times as popular) as another

fruit?


PDM3-5 Introduction to Pictographs

Goals: Students will read and interpret pictographs that have a scale of 1.


One-to-one correspondence

Understanding of symbols

Comparing and ordering numbers

Addition, subtraction, multiplication

Vocabulary: pictograph, key, symbol, more, less, least

ASK: What is a symbol? Allow students to discuss this in pairs and then debrief as a class. If students need

hints, remind them of maps—what pictures/symbols they have seen there? What are some of our country’s

symbols? (beaver, maple leaf, loon, etc.)

Tell students that they will learn how to show data in a pictograph today. Explain that pictographs use

symbols which represent the data in order to show how many of each set of data there are. Explain that the

symbol should match who is being asked the question or what is being asked about.

Display this data on the board:

What time do students go to bed during the week?

Before 8:30 x

8:30 x x x x

9:00 x x x x x x x

9:30 x x x x

After 9:30 x x

Ask students what symbols they could use to show the above data and choose one. (Suggestions include: a

stick person, a pillow.) Invite a volunteer to replace the Xs with the new symbol.

Next, introduce the word “key” to students. Draw the symbol chosen for the above example and an equal

sign next to it. Ask students what each symbol in the chart represents (one student). Write the number 1 next

to the equal sign. Explain that this is the key and it tells us what each symbol represents. In this case, each

symbol represents or is equal to one student. Tell students that all pictographs usually include a key.

Encourage students to talk about what the data tells us. Encourage them to make comparisons and to ask

questions. ASK: How many students go to bed at 8:30? How many go to bed after 9:30? What is the most

popular bedtime? How many students were surveyed? (Show students how to count the symbols to find out.)

How many more students go to bed at 9:00 than after 9:30? How many more students go to bed at 9:00 or

earlier than at 9:30 or later? How many students go to bed at 8:30 or later? To answer this last questions, did


they add all the data values or did they subtract from the total number of students surveyed that they found

earlier?

Discuss good symbols to show various data. Have students first choose the best possible symbol from a list

and then move to having them choose their own.

a) Number of books Tanya read in each season:

b) Number of sunny days in each season: c) Number of rainy days in each season. d) Number of people whose favourite fruit is apples, bananas, oranges, grapes or peaches.

ERRATA NOTE: The chart in the workbook should say “Number of Rainy Days,” not “Number of Sunny Days.”

Activities:

1. Have students create concrete graphs by collecting small items (e.g., leaves, shapes, beads, buttons)

and sorting them. A partner can interpret the data and write a few sentences about the items collected.

(EXAMPLE: Most of the buttons are big. There are more red beads than blue beads.)

2. Place students into small groups and give them a package of Smarties, M&Ms, or jelly beans. Ask

students to sort the candies (you can choose categories or they can), count how many are in each

group, record the data, and display it. Encourage students to analyze the data—what does it tell them?

What are there more of and less of? Why do they think that happened?


PDM3-6 Pictograph Scale

Goals: Students will read pictographs that have a scale larger than 1.

Prior Knowledge Required: Pictographs that have scale 1

Vocabulary: symbol, key, scale, pictograph

Tell your students that you have a garden but you are very secretive about how many flowers are in it. You

want to make a pictograph of how many of each kind of flower you have. ASK: What is a good symbol to use

for the key? Tell your students that instead of using one symbol to mean one flower, you will use one symbol

to mean many flowers—that way, your students won’t know exactly how many flowers you have without

knowing your key.

Draw on the board:

Daffodil F F F F

Buttercup F F F F F

Daisy F F F

Tell your students that each symbolic flower could mean any number of actual flowers, but it’s always the

same number for each symbol. If the first symbol means 2 flowers, then all the symbols mean 2 flowers.

ASK: If each symbol means only 1 flower, how many flowers do I have in my garden? What numbers did you

add together to find the answer? What if each symbol means 2 flowers—then how many flowers would I

have? What strategy did you use to find the answer? (Allow several students to explain how they found the

answer, to illustrate the diversity of strategies. ) What if each symbol means 3 flowers? 4 flowers? 5 flowers?

ASK: If each symbol means 3 flowers how many daisies do I have?

If each symbol means 100 flowers, how many buttercups do I have?

If each symbol means 5 flowers, how many daffodils do I have?

If each symbol means 4 flowers, how many daisies do I have?

Bonus: “Accidentally” tell your students that you have 12 daffodils. Can they figure out the key?

Ask students whether or not it would make a difference if the data in the pictograph was presented vertically

instead of horizontally. Re-create the graph so that the symbols are stacked vertically, starting from the

bottom and moving upwards. Then erase each flower and draw an “x” in its place.

Ask students if this reminds them of any type of graph they saw last year. It should remind them of line plots.

Remind your students that they learned pictographs and line plots last year, but they always used the symbol

to mean only one object. Now they are using a symbol to mean more than one object.


Tell your students that sometimes people will pick a symbol just because it’s easier to draw, and circles are

often a good choice. Draw the following pictograph and tell students that each circle means 4 flowers:

Number of Flowers

Daffodil

Buttercup Key: 1 means 4 flowers

Daisy

ASK: How is the circle easier to draw than the flower? Use the following questions to address the half circle:

If means 4 people, how many people does mean? What if means 6 people—then how many people

does mean? Repeat with the circle meaning a variety of even numbers. Then return to the pictograph

above and ask how many flowers of each type there are.

Draw the following pictograph, and ask how many people like each colour:

Number of People with Each Favourite Colour

Red

Blue

Yellow

Orange Key: 1 means 10 people

Green

Purple

Other

Write the number of people who like each colour as students tell you the answers. Then have students draw

a pictograph of the same data using a different key: 1 means 5 people. Discuss the similarities and

differences between the two pictographs. How would you tell from each graph which colour was most often

picked as favourite?

Explain that the data sometimes makes it easy to choose a scale. Ask students to choose a scale of two, five, or ten for the following data:

a) 12, 4, 10 b) 5, 25, 35 c) 40, 20, 70 d) 35, 50, 20 e) 16, 8, 14

Choose a scale of two, three, or five for the following data:

a) 3, 9, 18 b) 20, 10, 25 c) 6, 12, 15 d) 8, 18, 6 e) 40, 25, 30

Bonus:

f) 18, 15, 9, 21, 27, 30 g) 40, 105, 35, 70, 60, 95, 35 h) 40, 20, 36, 18, 24, 16, 32


PDM3-7 Displaying Data on a Pictograph

Goals: Students will select a symbol for a pictograph that is easy to draw based on the choice of scale.

Prior Knowledge Required: The symbol on a pictograph can mean more than one object.

Vocabulary: scale, key, pictograph, symbol

Take a survey of your students’ favourite colours among the choices: blue, red, or yellow. Tally the results on

the board and then write:

Favourite Primary Colour

Blue

Red

Yellow

1 means 2 students

Have a volunteer complete the pictograph, then demonstrate how difficult it would be if you chose the same

symbol to mean 5 students—it is hard to draw one fifth of the happy face!

Brainstorm symbols that are easy to draw when you need to draw half a symbol and symbols that are easy

to draw when you need to show a fifth of a symbol. (EXAMPLES: flower with 5 petals, star with 5 points)

As in the last lesson, ask your students which scale they would use (5 or 10) for each data set below:

a) 80, 90, 95 b) 20, 10, 25 c) 120, 45, 90 e) 40, 25, 30

To guide your students, ASK: For the scale you used, do you need to draw a half symbol? How many whole

symbols would you need to draw? Emphasize that students don’t want to have to draw too many symbols.

It’s okay to use half symbols! Students should choose the symbol carefully so that they can draw half of the

symbol easily. Have students practice drawing a pictograph using the data from part c) above, a scale of 10,

and the symbol of their choice.

Discuss with your students what is wrong with the following pictograph:

Tell your students that one happy face represents one

student who picked that sport as their favourite.

ASK: Which sport is the most popular? Which sport has

the longest row of faces? Why is it easier to read the

graph when all the faces are the same size?

Tell your students that it is easier to make all the happy

faces the same size if they draw on grid paper. That

way, they can draw each happy face in one grid square.

Students should use 2 cm grid paper to make drawing

the objects easier.

Favourite Sport of Students in Class A

Soccer

Hockey

Basketball


PDM3-8 Introduction to Bar Graphs

Goals: Students will read and interpret bar graphs.

Prior Knowledge Required: Pictographs

Scale of a pictograph

Vocabulary: bar, graph, bar graph, common

Review pictographs. Tell your students that there are other ways, besides pictographs, to show data. Write

the words ‘bar graph’ and ‘pictograph’ on the board side by side. Underline the word ‘graph’ in each. A

pictograph uses pictures to display data. Ask students how they think a bar graph will display data.

Create a pictograph and a bar graph for the same data ahead of time, or use the examples below. Identify

the parts of the bar graph: the two axes (vertical and horizontal), the scale, the labels (including the title), and

the data (shown in the bars).

What time do students go to bed during the week?

Before 8:30 x

8:30 x x x x

9:00 x x x x x x x

9:30 x x x x

After 9:30 x x

1 x = 2 students

Student Bed Times During the Week

0

2

4

5

8

10

12

14

16

Before

8:30

8:30 9:00 9:30 After 9:30

Times

Nu

mb

er

of

Stu

de

nts


Remind students that a pictograph can be drawn vertically as well and have a volunteer draw a vertical

pictograph that shows the same data as the horizontal pictograph.

Allow your students to compare the bar graph and the vertical pictograph, then ask them if they think the

graphs show the same data. Explain to them that the scale of the bar graph expresses how much each

mark on the grid represents. Ask them what scale was used for the bar graph. Was the same scale used in

their pictograph? Instead of using a symbol, how does the bar graph represent two students? How would

the bar graph represent one student? How does the pictograph represent one student? Emphasize that a

bar graph is like a pictograph that uses one grid square as a symbol, and that grid square can mean any

number of objects, just like the symbol on a pictograph. The nice thing about the grid square as a symbol, is

that it is easy to draw half a symbol, or a third of a symbol, or a quarter of a symbol—just use a ruler to

measure half the height or a third of the height or a quarter of the height.

Half a symbol A third of a symbol A quarter of a symbol

Just as there is usually space between the pictures on a pictograph, there is always a space between bars

on a bar graph.

Draw students’ attention to the height of the bar and the number where the bar stops. Explain that this shows

how many students answered in each of the categories. Ask students how many students responded in each

category. Change the data and repeat to ensure that students are able to read the vertical axis correctly.

Then move on to information that follows more indirectly from the bar graph, such as: How many students go

to bed at 9:00 or earlier? How many students do not go to bed before 8:30? How many more students go to

bed at 9:00 than at 9:30? Have students indicate the concepts and/or operations they used to answer the

questions—addition, subtraction, comparing, and/or ordering. Challenge students to think of a question that

would require multiplication.

Next, introduce ‘common’ by writing it on the board and ask students to define it. Then, ask them what the

most and least common answer was to the question “What time do you go to bed?” Prompt discussion by

asking why 9:00 is the most common bedtime? In other cases, students may use the word ‘popular’ to

describe data such as favourite foods or sports. Discuss this as a group and explain why it would not be

accurate to say that the most common bedtime is the same as the most popular bedtime.

Activity: Ask students to discuss, and then write everything they can about, the data displayed in the

following bar graph. There is no title on the graph. This is done on purpose, so that students may interpret

the data in different ways. Discuss the importance of a title on a graph and how it clarifies what the data

represents even further than the labels on the bars.


0

1

2

3

4

5

6

7

8

dogs cats fish none


Tikki Tikki Tembo by A. Mosel

(This is a Chinese folktale with a moral which lends itself well to student participation through chanting.

A boy’s long name causes a dilemma.)

Survey the class about how many letters there are in each of the students’ names. Graph the data.


PDM3-9 Bar Graphs and PDM3-10 Bar Graphs (Advanced)

Goals: Students will read, interpret, and create bar graphs.

Prior Knowledge Required: Bar graphs and pictographs

Scale

Vocabulary: data, bar graph

Using a scale of 10, draw a bar graph for the following data:

Skis sold by a sports store during each season

Fall: 120 Winter: 60 Spring: 45 Summer: 15

ASK: In which season did the store sell 3 times as many skis as in another season? In which season did

the store sell twice as many skis as in another season? In which season were there 15 more skis sold than

in Spring?

Then draw a partially completed bar graph, with only the “tennis” bar shown:

Favourite sports of people in a Grade 3 class

Tennis: 4 Hockey: 5 Basketball: 12 Soccer: 10

Tell your students that hockey was chosen one more time than tennis was and have a volunteer draw

the bar. Then tell them that basketball was chosen three times as often as tennis and have a volunteer add

that bar. Finally, tell them that soccer was chosen twice as often as hockey and have a volunteer add the last

bar. Ask students what other conclusions they can draw from the graph.

Activities:

1. You will need a large open space for this activity. Create a ‘human’ birthday graph. Place cards with the

names of the months of the year along a horizontal line. Students should line up in rows behind their

birth month. If a camera is available, take a photo of the graph. After determining which months have the

most and least birthdays, ask students why it is easy to tell this. (The ‘bars’ are the rows of students—

some rows are long and some are short.) Students can then transfer the data into a graph on paper.

2. Give students a paragraph of text and ask them to create a graph that shows the number of words on

each line.

3. Some students might enjoy the Internet game found at:

http://pbskids.org/cyberchase/games/bargraphs/bargraphs.html

Students need to be quick with their hands to catch bugs. A bar graph charts the number of each colour

caught. The game gives practice at changing scale.


4. Encourage students to look at bar graphs in books, in magazines, on the Internet, or on television

(such as on The Weather Network). Have them record the number of markings on the scales and the

number of bars. About how many markings do most bar graphs use? About how many bars do most bar

graphs use?


PDM3-11 Collecting Data

Goals: Students will ask good survey questions.

Prior Knowledge Required: Ability to distinguish between a statement and a question

Ability to distinguish between “and” and “or”

Vocabulary: survey, other, none

Explain to the class that the quality of a survey question determines the quality of the data collected. For

students to be successful at conducting surveys and collecting data, they must learn how to ask a good

question.

Conduct a survey with your students by asking them what their favourite flavour of ice cream is—do not limit

their choices at this point.

Tally the answers, then ask students how many bars will be needed to display the results on a bar graph.

How can the question be changed to reduce the number of bars needed to display the results? (Remind

them of what they learned from Activity 4 in PDM3-9,10). How can the choices be limited? Should choices

be limited to the most popular flavours? Why is it important to offer an “other” choice?

Explain to your students that the most popular choices to a survey question are predicted before a survey is

conducted. Why is it important to predict the most popular choices? Could the three most popular flavours

of ice cream have been predicted?

Have your students predict the most popular choices for the following survey questions:

• What is your favourite colour?

• What is your favourite vegetable?

• What is your favourite fruit?

• What is your natural hair colour?

• What is your favourite animal?

Students may disagree on the choices. Explain to them that a good way to predict the most popular choices

for a survey question is to ask the survey question to a few people before asking everyone.

Emphasize that the question has to be worded so that each person can give only one answer. Which of the

following questions are worded so as to receive only one answer?

a) What is your favourite ice cream flavour?

b) What flavours of ice cream do you like?

c) Who will you vote for in the election?

d) Which of the candidates do you like in the election?

e) What is your favourite colour?

f) Which colours do you like?


Have your students think about whether or not an “other” category is needed for the following questions:

What is your favourite food group?

Vegetables and Fruits Meat and Alternatives

Milk and Alternatives Grain Products What is your favourite food?

Pizza Burgers Tacos Salad

Then ask students how they know when an “other” category is needed. Discuss which of the following questions would require an “other” category and why:

• What is your favourite day of the week? (List all seven days.)

• What is your favourite day of the week? (List only Friday, Saturday, and Sunday.)

• What is your favourite animal? (List horse, cow, dog, pig, cat.)

• How many siblings do you have? (List 0, 1, 2, 3, 4 or more)

• Who will you vote for in the election? (List all candidates.)

Then bring up the point that an “other” category may not be an option. For example, if the teacher wants to

bring 2 movies to show on the last day before Christmas holidays and she has only 5 movies at home, she

would give only those 5 movies as choices and would bring the 2 most popular ones to class.

Ask students what they think most people in the class would prefer to do for a party: go to a movie or go on

a skating trip. Ask students what kind of question you should ask to gather this data. Record all of their

suggestions on the board. Once this is done, review all the questions and determine all the possible answers

for each one.

Questions and answers might include the following:

1. Do you want to have a party? Yes/no

2. If you had a party, would you like to go to a movie? Yes/no

3. If you had a party, would you like to go on a skating trip? Yes/no

4. If you had a party, would you like to go to a movie or go on a skating trip? Movie/skating trip/neither/both

5. If you had a party, choose one of these things to do: go to a movie, or go skating? Movie/skating trip

Now, discuss with students which is the best questions to ask. Questions 1, 2, and 3 are limiting and will not

capture all data. Question 4 makes it difficult to make a decision (i.e., determine what is the most popular

choice) but Questions 5 makes it clear which activity is preferred. (NOTE: Make sure that you emphasize the

positive in each suggestion, showing how it’s a good start and demonstrating where to go from there to get to

the targeted question.)

Activity: Discuss the difference between ‘and’ and ‘or’ in a question and what they mean. For example,

the questions “Do you have a cat and a dog?” and “Do you have a cat or a dog?” will produce different

answers. For the first question, respondents can answer no (I don’t have a cat AND a dog) or yes (I have

both a cat AND a dog). For the second, respondents can answer yes and no but the meaning changes:

yes means I have a cat OR a dog; no means I don’t have either.

If possible, bring in pictures (perhaps cut from magazines) of people with cats and/or dogs: some with a cat

and a dog, some with only a cat, some with only a dog, and some with neither. Hold up the pictures one at a


time and ask students how each person would answer these two questions: Do you have a cat or a dog? Do

you have a cat and a dog? Tally the results. For example, depending on the pictures you brought, your tallies

might look like:

YES NO

I have a cat or a dog

I have a cat and a dog Then have students summarize the answers in a separate bar graph for each question.


PDM3-12 Practice with Surveys and PDM3-13 Blank Tally Chart and Bar Graph

Goals: Students will create a survey question, tally the data and present their data in a bar graph

Prior Knowledge Required: How to show data (graphs)

How to collect data

How to ask a question

What choices to give survey takers

How to analyze data

Vocabulary: survey, tallies, pictograph, bar graph

Tell students you want to find out what they will be doing on their summer holidays. Then discuss the

question and choices: What will you be doing over the summer holidays? Camp, family trip, summer school,

staying at home, other. (Modify the choices according to your students’ interests and activities.)

Ask each student to identify their summer activity by raising their hand when you call out the choice. Record

the data in a tally chart with the choices listed.

Count the tallies for each category to determine how much space you will need for the bar graph. Draw a grid

with a fixed number of markings, say 8. ASK: What scale should we use?

Remind students that there is a space between the bars in a bar graph and have students independently

create a bar graph for the data collected. Remind them to include a title and labels on their graphs.

Analyze the data together. Ask students what the data is telling them. Record those statements.

Activity: Ask students to name some of their favourite authors. (Have a selection of books which are

popular during read alouds and independent reading time on hand for students to refer to.) Then, select

the top 3 authors as well as “other” for categories. Ask students to identify their favourite author (explain that

they can only raise their hand once when ‘voting’ and record data using tally marks) and collect the data

from the class.

When you’re finished, ask how students can tell if everyone voted or not. Do they think anyone voted twice?

Does the number of votes equal the number of students in the class? What would happen if someone didn’t

vote? If someone voted twice?

ASK: If we want no more than 5 markings on our graph (have a graph with 5 markings on the vertical axis on

the board), what scale should we use? Would it make sense to have only 1 marking on the graph? How

would that make the graph hard to read? Decide on the number of markings and the scale for the graph.

Then complete the graph.


Tell your students that they will be designing their own survey and then surveying their classmates.

Everyone can ask a different question, so suggest several topics if they have trouble getting started.

(EXAMPLES: How do you get to school? What is your favourite colour? How many people are there in your

family? What time do you wake up on weekdays? How long do you take to get ready for school? Does your

jacket have a hood? What pizza toppings do you like? What is your favourite meal? What is your favourite

cereal? What is your favourite season? Who is your favourite person? What type of home do you live in?

What is your favourite summer or winter activity?)

Extensions:

1. After completing their survey, students can transfer their graph data to KidPix to create a computer-

generated graph.

2. Ask student how they could find out the favourite colour of every teacher in the school. How would they

collect the data? How would they organize the data? Students can work on this in pairs or small groups

after they have a plan of action. After collecting and representing the data, students can report their

findings orally.


So you want to be president? by J. St. George

(A Caldecott winner. Anyone can be president, no matter what they look like or where they are from.)

After reading the book, have students work in small groups to collect data from their classmates. They can

ask the following question: What do you want to be when you grow up? Students should be encouraged to

organize the data in more than one way and be prepared to present their final work.


PDM3-14 Collecting and Interpreting Data

Goals: Students will understand when surveys are appropriate to find out information and when other

methods are required.

Prior Knowledge Required: Asking a good survey question

Vocabulary: survey, measuring, researching, observing

Conduct a survey of your students to determine how many of them were born before noon and how many of

them were born after noon. It is likely that very few, if any, students will raise their hands for either option.

Explain that deciding when to conduct a survey, or when to use another method to collect data, depends on

whether or not the people being surveyed can answer the question. Let’s look at the example above:

Question: When were you born?

Choices: Before or at noon / After noon

The question is very clear and there is no “other” category because no other choice is possible. But if

people don’t know the answer, the survey is pointless.

Have your students think about which topics from the list would be good survey topics:

• What are people’s favourite colours? (good)

• Are people left-handed or right-handed? (good)

• Can people count to 100 in one minute or less? (not so good)

• How many sit-ups can people do in one minute? (not so good)

• What are people’s resting heart rates? (not so good)

• What are people’s favourite sports? (good)

• How fast can people run 100 m? (not so good)

• How do people get to school? (good)

• How many siblings do people have? (good)

Is a survey needed for the following topics or can the data be obtained by observation?

• Do students in the class wear eyeglasses?

• Do students in the class wear contact lenses?

• What are adults' hair colours?

• What are adults' natural hair colours?

Sometimes a measurement or calculation is needed to obtain data. Have your students decide if

observation, a survey or a measurement is needed to obtain the data for the following topics:

• What are people’s heart rates?

• What are people’s favourite sports?

• How do people get to school?

• How many sit-ups can people do in one minute?

• How long are people’s arm spans?


• What colour of shirt are people wearing?

• How tall are people?

Sometimes students cannot find the information themselves but need to rely on measurements or

observations that other people have taken. Have students brainstorm examples of such situations (e.g.,

sports data from last year’s championships, world records, how fast different types of birds can fly, the time

of year that different animals hibernate, and so on).

Workbook 3 - Probability & Data Management, Part 1 1BLACKLINE MASTERS

Shapes ________________________________________________________________2

PDM3 Part 1: BLM List



Shapes

Geometry Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

G3-1 Sides and Vertices Goals: Students will identify polygons, sides and vertices, and distinguish polygons according to the

number of sides.


Distinguish straight line

Vocabulary: polygon, sides, vertex, vertices, triangle, quadrilateral, pentagon, hexagon

Draw and label a polygon with the words “polygon,” “sides” and “vertex/vertices”. Remind your students of

what a side and vertex are and explain that a side has to be straight. Show students how to count sides—

marking the sides as you count—then have them count the sides and the vertices of several polygons. Ask

them if they can see a pattern between the number of vertices and the number of sides. Be sure that all

students are marking sides properly and circling the vertices, so they don’t miss any sides or vertices.

Construct a large triangle, quadrilateral, pentagon and hexagon using construction paper or bristol board.

Label each figure with its name and stick them to the chalkboard. Explain that “gon” means angle or corner

(vertex), “lateral” means sides. You might want to leave these figures on a wall throughout the geometry unit.

Explain that “poly” means many, and then ask your students what the word polygon means (many angles or

vertices). Explain that a polygon is a shape that only has straight sides. Draw a shape with a curved side and

ask if it is a polygon. Label it as “not polygon”.

Draw several shapes on the board and ask students to count the sides and sort the shapes according to the

number of sides. Also ask them to draw a triangle, a pentagon, a figure with six sides, a figure with four

angles, and a figure that is not a polygon but has vertices.

Bonus:

Draw a figure that has:

a) Two curved sides and three straight sides

b) Two straight sides and three curved sides

Assessment:

1. Draw a polygon with seven sides.

2. Draw a quadrilateral. How many vertices does it have?

3. Draw a figure that is not a polygon and explain why it is not a polygon.

Possible answers: it has a curved side, a circle is not a polygon, a rectangle with rounded edges does

not have proper vertices.


Activity:

Give each student a set of pattern blocks or several tangram pieces with the following instructions:

a) Group your pieces according to how many sides they have.

How many of each type do you have?

b) Can you make a shape with four sides using two triangles? Three triangles?

c) Can you make a large triangle using four triangles?

d) Can you make a triangle from two small tangram triangles and a square?

e) Can you make a pentagon with pattern blocks?

f) Can you make a seven-, eight- or nine-sided figure with pattern blocks or tangram pieces?

Extension:

1. How many sides does each group of shapes have?

a) 2 pentagons b) 3 pentagons c) 4 pentagons

Students should see the connection with multiplication: 4 pentagons have 4 × 5 = 20 sides.

2. How many sides would 2 pentagons and 3 hexagons have?

3. Count the sides of a paper polygon. Count the vertices. Cut off one of the vertices. Count the sides and vertices

again. Cut off another vertex. Repeat the count. Do you notice a pattern? (The number of sides will increase by

one and the number of vertices will increase by one.)

4. “Word Search Puzzle (Shapes)” in the BLM section.

S

S


smaller larger

G3-2 Introduction to Angles Goals: Students will identify right angles in drawings and objects.

Vocabulary: right angle

Ask your students if they know what a right angle is—a right angle is the corner of a square, but there is no

need to define it in terms of degrees at this stage. Ask them where they can see right angles in real life

(EXAMPLE: of a sheet of paper corners, doors, windows, etc.). Draw a right angle and then show them how

to mark right angles with a small square. Explain to your students that not all angles are right angles; some

are sharper than a right angle, some are less sharp. Tell them to think of corners; the sharper the corner is,

the smaller the angle is. NOTE: You may want to perform Activity 1 here.

Draw two angles.

Ask your students which angle is smaller. Which corner is sharper? The diagram on the left is larger, but the

corner is sharper, and mathematicians say that this angle is smaller. The distance between the ends of the

arms is the same, but this does not matter. What matters is the “sharpness”. The sharper the angle is, the

narrower the space between the angle’s arms. Explain that the size of an angle is the amount of rotation

between the angle’s arms. The smallest angle is closed—with both arms together. Draw the following picture

to illustrate what you mean by smaller and larger angles.

With a piece of chalk you can exhibit how much an angle’s arm rotates. Draw a line on the chalkboard then

rest the chalk along the line’s length. Fix the chalk to one of the line’s endpoints and rotate the free end

around the endpoint to any desired position.

You might also illustrate what the size of an angle means by opening a book to different angles.

Draw some angles and ask your students to order them from smallest to largest. EXAMPLE:

a)

A B C


b)

A B C

Students that have trouble comparing the angles could use a book and open its pages so that the cover

halves coincide with arms of the angles.

Discuss whether flipping the third angle changed its size.

Bonus: Provide longer lists of angles.

Draw a polygon and explain that the polygon’s angles are inside the figure, not outside the figure. Draw

several polygons and ask volunteers to mark the smallest angle in each figure.

Note that the smallest angle in the rightmost figure is A and not B—B is actually the largest angle, since the

angles are inside the polygon. (You might hold up a pattern block or a cut-out of a polygon so students can

see clearly which angles are inside the figure.)

Do some of the activities before proceeding to the worksheets.

Draw several angles and ask volunteers to identify and mark the right angles. For a short assessment, you

can also draw several shapes and ask your students to point out how many right angles there are. Do not

mark the right angles in the diagram.

Activities:

1. Make a key from an old postcard. Have students run their fingers over the corners to identify the

sharpest corner. The sharper the corner is, the smaller the angle is.

2. Ask students to use any object (a piece of paper, an index card) with a square corner to identify various

angles in the classroom that are “more than," “less than” or “equal to” a right angle.

• corner of a desk • angle made by an open door and the wall

• window corners • corners of base ten materials

A

B


3. Use a geoboard with elastics to make …

a) a right angle

b) an angle less than a

right angle

c) an angle greater than

a right angle

4. Use a geoboard with elastics to make a figure with…

a) no right angles

b) 1 right angle

c) 2 right angles

5. Have your students compare the size of angles on pattern blocks by superimposing various pattern

blocks and arranging the angles in order according to size.

Students may notice that there are two angles on the trapezoid that are greater than the angles in the

square, and that there are two angles that are less than the angles in the square.


G3-3 Equilateral Shapes Goals: Students will classify polygons according to the number and the lengths of sides and the number of

right angles. Students will also identify equilateral shapes.

Prior Knowledge Required: Sides, vertices

Right angles


Write the word “equilateral” on the board. Ask the students what little words they see in it. What might the

parts mean? Where did they meet these parts? (“Equi” like in “equal” and “lateral” like in “quadrilateral”). Give

your students the set of shapes below. Students can either cut out the shapes and fold them to check if the

sides are equal or measure the sides with a ruler. Ask your students to extend the following chart to classify

the shapes.

Number of

Right Angles

Number of

Sides Equilateral

4 4 Yes

Include the word “equilateral” on your next spelling test.

Extensions:

1. Pick a property (i.e. same number of sides, vertices, right angles, etc.) and find three shapes in

QUESTION 3 that all have that property in common.

2. Find a group of four shapes where three share a common property and one doesn’t belong.


G3-4 Quadrilaterals and Other Polygons Goals: Students will distinguish the quadrilaterals from other polygons.

Prior Knowledge Required: Count sides of polygons

What is a polygon

Measure straight lines with a ruler


Draw a quadrilateral on the board. Ask how many sides it has. Write the word “quadrilateral” on the board

and explain what it means. Explain that “quad” means “four”, “lateral” means “sides” in Latin. Ask if students

have ever encountered any other word having either of these parts in it. (EXAMPLES: quadrangle,

quadruple). You might also mention that “tri” means “three” and ask if they know what the French words are

for 3 and 4.

Emphasize the similarities: “tri” and “trois”, “quad” and “quatre”. Draw several polygons on the board and

ask whether they are quadrilaterals. Write the number of sides for each and mark the answer on the board.

Make two columns (quadrilaterals and non-quadrilaterals) on the board.

Use the polygons from the shape game (see the BLM section) with tape on the back side. Invite volunteers

to come and affix the shapes at the right column on the board. You may ask them to explain their choice.

You may also ask the students to sort the pattern block pieces into quadrilaterals and non-quadrilaterals.

You may also ask your students to sort shapes according to different properties—those that have right

angles and those that do not, number of sides or angles or right angles, equilaterality, etc. Include the word

“quadrilateral” on your next spelling test.

Activities:

1. Give each of your students a set of 4 toothpicks and some model clay to hold them together at vertices.

Ask to create a shape that is not a quadrilateral. (This might be either a 3-dimensional shape or a self-

intersecting one). Ask also to make several different quadrilaterals.

2. Provide each of your students a set of 10 toothpicks. Ask them to check how many different triangles

they can make with these toothpicks. Each figure should use all the picks. How many different

quadrilaterals you can make using 10 toothpicks? The answer for the second problem is infinity—a slight

change in the angles will make a different shape.


Extension: Explain that a kite is a quadrilateral with two pairs of equal adjacent sides. Draw a kite and

ask a volunteer to mark the equal pairs of sides. Point out that a kite has no indentation – illustrate the

meaning with a picture.

Draw several polygons on the board, some kites, some not, some resembling kites, and ask your students to

measure the sides of the shapes and to determine which of these shapes are kites.

Hold up a cut-out paper kite and ask your students how they could check whether this shape is a kite

without measuring the sides. Would folding help? Invite a volunteer to fold the shape and to check whether

the sides are equal. Ask: Are all the angles in a kite different? How do you know? (Folding the kite along the

diagonal that is also a line of symmetry will show that the opposite angles between the non-equal sides are

also equal.)

What about the other pair of opposite angles? Ask a volunteer to check whether these two angles are equal

by folding the kite along the other diagonal. (The Atlantic curriculum)

Indentation, so this is not a kite.

These angles are equal

Fold


G3-5 Tangrams Goals: Students will become familiar with polygons and develop the use of the vocabulary.

Prior Knowledge Required: Quadrilateral, square, rectangle, pentagon, hexagon


Give your students copies of the Tangram sheet in the BLM section of this manual. Have them cut out the

pieces and follow the instructions on the worksheet.

Activity:

Communication Game. Two players are separated by a barrier that prevents each player from seeing the

table directly in front of the other player. Player 1 makes a simple shape using a limited number of tangram

pieces. (For instance ). Player 1 then tells player 2 how to build the shape. For instance: “I used the

small triangle and the square. I put the triangle on the left side of the square.”

NOTE: There are several ways to carry out this instruction: or or or so Player 1

would have to give more precise instructions such as “Put one of the short sides of the triangle against the

side of the square so the right angle of the triangle is at the bottom”.

As students play this game (and as they see how difficult it is to describe their shapes) teach them the

terms they will need (“right”, “left”, “short side”, “long side”, “horizontally”, “vertically”, etc.) and show them

some situations in which more precise language would be needed. For instance, in the example above you

might say that the short side of the triangle is adjacent to the square and the long side goes from top left

corner right and down.

NOTE: Each player is allowed to ask questions but students should only use only geometric terms in the

game, rather than describing what a shape looks like (EXAMPLE: NOT “It looks like a house.”) Here are

some simple shapes students could start with.

S = small triangle M = medium triangle L = large triangle

S S

S S

L M

S S

M M

L

L

S S

S S

S S S

S


Extensions:

1. How many rectangles can you make using the tangram pieces? (REMEMBER: squares are

rectangles.)

EXAMPLES:

2. How many ways can you find to construct a square as in QUESTION 5 on the worksheet? Which way

uses the smallest number of shapes? The greatest number of shapes? (The Ontario curriculum)

3. Try to make various polygons using all the tangram pieces. Sample shapes may be found at

www.tangrams.ca. SEE ALSO:

http://tangrams.ca/puzzles/asso-02.htm and

http://tangrams.ca/puzzles/asso-02s.htm

M S S S S

S

M

S S

M L


G3-6 Congruency Goals: Students will become familiar with polygons and develop the use of the vocabulary.


Vocabulary: congruent shapes

Explain that two shapes are congruent if they are the same size and shape. A pair of congruent two-

dimensional figures will coincide exactly when one is placed one on top of the other. Have students actually

do this with tangrams or pattern blocks. As it isn’t always possible to check for congruency by superimposing

figures, mathematicians have found other tests and criteria for congruency.

A pair of two-dimensional figures may be congruent even if they are oriented differently in space (see, for

instance, the figures below):

As a first test of congruency, your students should try to imagine whether a given pair of figures would

coincide exactly if one were placed on top of one another. Have them copy the shapes onto grid paper.

Trace over one of them using tracing paper and try to superpose it. Are the shapes congruent? Have your

students rotate their tracing paper and draw other congruent shapes. Let your students also flip the paper!

You might also mention the origin of the word: “congruere”—“agree” in Latin.

Assessment: Circle the pair of shapes that are congruent:

a) b)


G3-7 Congruency (Advanced) Goals: Students will identify congruent shapes regardless of their position and colour.



Review the previous lesson. Draw two congruent shapes of different colours and ASK: Are the shapes

congruent? Are they of the same size? Are they of the same shape? Remind your students that congruent

shapes are of the same size and shape, and they can have different colors. Give your students several

shapes of different colours and with different patterns (stripes, dots, etc.) and ask them to find congruent

pairs. Increase the number of congruent pairs in the same group of shapes gradually.

Make several shapes of blocks, such as the shapes below and ask your students to explain why these

shapes are not congruent. Repeat with pairs of polygons.

Ask your students to build shapes that are congruent to the shapes above.

Activities:

1. Give each student a set of pattern block shapes and ask them to group the congruent pieces. Make sure

students understand that they can always check congruency by superimposing two pieces to see if they

are the same size and shape.

2. Give students a set of square tiles and ask them to build all the non-congruent shapes they can find

using exactly 4 blocks. They might notice that this is like Tetris game. Guide them in being organized.

They should start with two blocks, and then proceed to three blocks. For each shape of 3 blocks, they

should add a block in all possible positions and to check whether the new shape is congruent to one of

the previous ones.

SOLUTION:


G3-8 Recognizing and Drawing Congruent Shapes

Goals: Students will identify and draw congruent shapes.

Prior Knowledge Required: Count to10

Identify congruent shapes


Review the previous lesson. Draw a group of shapes as shown below and ask your students to find a pair of

congruent shapes among the shapes below. Ask your students also to explain why the other shapes are not

congruent to the two congruent shapes.

Activity: Review the names of the following polygons: triangle, rectangle, square, rhombus, pentagon,

hexagon. Each pair of students will need a spinner as shown. Player 1 spins the spinner and draws a

polygon according to the result of the spinner. Player 2 has to draw two polygons of the same type, so that

one is congruent to the polygon drawn by Player 1, and the other polygon is not.

Advanced: Review the concepts of attributes before starting the activity. Player 1 spins the spinner and

draws a polygon according to the result on the spinner. Player 1 decides on an attribute (such as striped

pattern) and then names the attribute. Player 2 draws a shape congruent to the shape drawn by her partner,

so that it differs in the given attribute (in this case, a different pattern). Then Player 2 draws a shape that is

not congruent to the given shape so that it shares the given attribute (it will still be striped). For example, the

spinner reads “Triangle”: Player 1 says: Pattern Different pattern Same pattern

Rectangle

SquareRhombus

Triangle Pentagon

Hexagon


G3-9 Exploring Congruency with Geoboards Goals: Students will create congruent and non-congruent shapes on a geoboard.

Prior Knowledge Required: Count to10

Identify congruent shapes


Draw a shape on a grid on the board and ask your students to make a copy of it on their geoboards. (This

exercise could also be done on dot paper). Then ask your students to create a new shape, congruent to the

first one, but differently oriented. Repeat with several other shapes. Ask your students also to make shapes

that are not congruent to the given shape, and to explain why these shapes are not congruent.

Activities:

1. Make 2 shapes on a geoboard. Use the pins to help you say why they are not congruent.

EXAMPLE:

2. Repeat the activity from the previous lesson with geoboards.

The two shapes are not congruent: one has a larger base

(you need 4 pins to make the base).


G3-10 Exploring Congruency with Grids Goals: Students will create congruent and non-congruent shapes on a grid.

Prior Knowledge Required: Congruent shapes


Ask your students to check how many different squares they can draw on a 4 × 4 grid. Ask your students:

How can you check if the squares are congruent? What do you have to check to make sure your shape is a

square? Suggest that your students measure the sides with rulers and angles with benchmarks to make sure

that the shapes they created are squares, and then to check congruency.

Include the word “congruent” into your next spelling test.

Bonus: Repeat with the 5 × 5 grid. Can you find eight non-congruent squares? (HINT: some of the squares

may be oriented so that their sides are diagonal to the grid).

Extension: The picture shows one way to cut a 3 × 4 grid into 2 congruent shapes. Show how many

ways you can cut a 3 by 4 grid into 2 congruent shapes.


G3-11 Symmetry Goals: Students will find lines of symmetry using paper folding.


Vocabulary: line of symmetry

Explain that a line of symmetry is a line that divides a figure into parts that have the same size and shape

(i.e. into congruent parts), and that are mirror images of each other in the line of symmetry. You can check

whether a line drawn through a figure is a line of symmetry by folding the figure along the line and verifying

that the two halves of the figure coincide.

Hold up a large paper parallelogram. Mark a line on it as shown below.

Ask your students: Is this line a line of symmetry? Are the halves of the figure on both sides of the line

congruent? Fold the shape along the line and ask: Do the halves of the figure coincide? Students should see

that the two halves do not coincide so they are not mirror images of each other in the line. Hence the line is

not a line of symmetry. Invite volunteers to check if other lines on the shape are lines of symmetry. (They can

try a line that connects the other pair of the diagonals or pairs of sides.) They will find that the shape has no

lines of symmetry.

Let your students cut out various polygons and check how many lines of symmetry the shapes have. You

might suggest that your students predict the number of lines of symmetry first and then check their prediction.

Extension: These shapes are called regular shapes. All their sides and angles are equal. Fill in the

T-table:

Figure

Number of Sides

Number of Symmetry

Lines


G3-12 Lines of Symmetry Goals: Students will find lines of symmetry in pictures and draw mirror images.


Horizontal

Vertical

Vocabulary: lines of symmetry horizontal, vertical

Review the definition of a line of symmetry. Give an example using the human body. You may wish to draw

a symbolic human figure on the board and mark the line of symmetry. Review the meaning of the words

“horizontal” and “vertical”. To help the students remember the word “horizontal”, remind them of how they

might draw the horizon line in art class.

Draw several pictures and ask students to find the horizontal and the vertical lines of symmetry.

Ask them to circle the pictures that have a vertical line of symmetry and to draw a square around the

pictures that have a horizontal line of symmetry. Does every picture have a line of symmetry? Are there

pictures with more than one line of symmetry?

EXAMPLES:

Challenge your students to find all the possible lines of symmetry for the following shapes:

Assessment:

Draw all the possible lines of symmetry:

a) b) c)


Activities:

1. Find a picture in a magazine that has a line of symmetry and mark the line with a pencil. Is it a horizontal

or a vertical line? Try to find a picture with a slanted line of symmetry.

2. Cut out half an animal or human face from a magazine and glue it on a piece of paper, draw the missing

half to make a complete face.

Extension:

Cross-curriculum Connection: Check the flags of Canadian provinces for lines of symmetry.

Possible Source: http://www.flags.com/index.php?cPath=8759_3429.


G3-13 Completing Symmetric Shapes Goals: Students will draw mirror images and find lines of symmetry in figures.


Symmetry lines

Vocabulary: symmetry line, mirror line, mirror image

Draw several simple images such as the ones below and ask your students to draw the missing halves so

that the resulting pictures have a line of symmetry. Ask students to mark the line of symmetry. Is it a

vertical or a horizontal line?

Explain to your students that the halves of pictures that they have drawn are called mirror images of the

original pictures. Can your students explain why the pictures are called so? Let your students put a mirror or

a MIRA along the symmetry line and compare the images in the mirror with their pictures. Explain that the

symmetry line is also often called mirror line.

Activities:

1. Using each pattern block shape at least once, create a figure that has a line of symmetry. Choose one

line of symmetry and explain why it is a line of symmetry. Draw your shape in your notebook.

2. Using exactly 4 pattern blocks, build as many shapes as you can that have at least one line of symmetry.

Record your shapes in your notebook.

Extension: Take a photo of a family member’s face (such as an old passport photo) and put a mirror

along the line of symmetry. Look at the face that is half the photo and half the mirror image. Does it look the

same as the photo?


G3-14 Completing Symmetric Shapes Goals: Students will compare shapes according to a given pattern.


Lines of Symmetry

Sides and vertices

Polygons

Right angles

Equilateral

Vocabulary: line of symmetry, equilateral shapes, polygon, square, rectangle, triangle, pentagon,

hexagon, right angle, vertices, vertex, sides

Draw a regular hexagon on the blackboard. ASK: How many sides and how many vertices does it have?

What is it called? Does it have any right angles? Is it an equilateral shape? How many lines of symmetry

does it have? Have volunteers mark the lines of symmetry. Then draw a hexagon with two right angles and

make the comparison chart shown below. Ask volunteers to help you fill in the chart.

Property Same? Different?

Number of vertices 6 6 �

Number of edges

Number of right angles

Any lines of symmetry?

Number of lines of symmetry

Is the figure equilateral?

Ask students to summarize the information from the table in a short paragraph that describes any

similarities and differences of the shapes.

Use the worksheet for more practice.

Assessment: Write a comparison of the two shapes. Be sure to mention the following properties:

The number of vertices

The number of sides

The number of right angles

Number of lines of symmetry

Whether the figure is equilateral


G3-15 Sorting Shapes by Property Goals: Students will compare shapes according to a given pattern.


Symmetry lines

Sides and vertices

Polygons

Right angles

Equilateral

Vocabulary: symmetry line, equilateral shapes, polygon, square, rectangle, triangle, pentagon, hexagon,

right angle, vertices, vertex, sides

Give each student (or team of students) a deck of shape cards and a deck of property cards from “2-D

Shape Sorting Game” of the BLM section. Let them play the first game in Activity 1 below. The game is an

important preparation for Venn diagrams.

Draw a Venn diagram on the board. Show an example—you may do the first exercise of the worksheets

using volunteers. In process remind your students that any letters that cannot be placed in either circle

should be written outside the circles (but inside the box). FOR EXAMPLE, the answer to QUESTION 1 a)

of the worksheet should look like this:

Also remind students that figures that share both properties, in this case D and A, should be placed in the

overlap. Ask your students where they would put a figure that looks like the one shown below:

Let your students play the game in the second activity. Then draw the following set of figures on the board.

Ask your students to make a list of figures satisfying each of the properties in a given question below they

draw a Venn diagram to sort the figures.

D A

C

A B F G H

E I

Has at least 2 right angles

Quadrilateral


Here are some properties students could use to sort the figures in a Venn diagram.

a) 1. At least two right angles

2. Equilateral

b) 1. Equilateral

2. Has exactly one line of symmetry

(In this case the centre part contains only one irregular pentagon—figure J)

c) 1. Quadrilateral

2. Has at most one right angle

d) 1. Pentagon

2. Equilateral

Assessment: Use the same list of figures to create a Venn diagram with properties:

1. Has at least one line of symmetry

2. Has at least two right angles

Activities:

Students will need a deck of shape cards and a deck of property cards from the “2-D Shape Sorting Game”

of the BLM section.

1. 2-D Shape Sorting Game

Each student flips over a property card and then sorts their shape cards onto two piles according to

whether a shape on a card has the property or not. Students get a point for each card that is on the

correct pile. (If you prefer, you could choose a property for the whole class and have everyone sort

their shapes using that property.) Once students have mastered this sorting game they can play the

next game.

2. 2-D Venn Diagram Game

Give each student a copy of the Venn diagram sheet in the BLM section (or have students create their

own Venn diagram on a sheet of construction paper or bristol board) in addition to the shape cards and

property cards. Ask students to choose two property cards and place one beside each circle of the

Venn diagram. Students should then sort their shape cards using the Venn diagrams. Give 1 point for

each shape that is placed in the correct region of the Venn diagram.

Extension:

A Game for Two: Player 1 draws a shape without showing it to the partner, then describes it in terms

of number of sides, vertices, right angles, lines of symmetry, etc. Player 2 has to draw the shape from

description.

A

B

C

D

E

F

G

H J

K

I


G3-16 Finding Polygons Goals: Students will identify polygons in drawings.


Symmetry lines

Sides and vertices

Polygons

Right angles

Equilateral

Vocabulary: symmetry line, equilateral shapes, polygon, square, rectangle, triangle, pentagon, hexagon,

right angle, vertices, vertex, sides

As a preparation for QUESTION 2 of the worksheet, draw two shapes on the board:

Invite volunteers to make a comparison chart and to write a comparison paragraph for these two shapes.

Properties you might mention:

• The number of vertices

• The number of sides

• The number of right angles

• Number of lines of symmetry

• Whether there are pairs of equal sides

• Whether the equal sides adjacent or opposite

• Whether the figure is equilateral

QUESTION 5 on the worksheet can be done with pattern blocks.

Activity: On a picture from a magazine or a newspaper, ask the students to mark as many polygons as

possible.


G3-17 Problems and Puzzles

This worksheet is the final review and may be used for practice.

Activities:

1. Give students a set of circular tiles of two colours. Ask them to make as many 3-tile triangles as they can

inside a 6-tile triangle.

Solution:

Repeat this exercise with a larger triangle:

2. Take any 2 congruent pattern blocks. Predict the shapes you can make by putting the blocks edge to

edge. (Can you make a quadrilateral? a pentagon? a hexagon? or a shape with more sides?) Trace

around the pattern blocks to show how they combine to make your figure. Repeat this exercise with 3

different pairs of pattern blocks.

3. Using pattern block triangles, try to make the following shapes:

a) A quadrilateral. b) A hexagon. c) A bigger triangle. d) A pentagon.

Extension: In QUESTION 6 students can make a square as shown:

Workbook 3 - Geometry, Part 1 1BLACKLINE MASTERS

2-D Shape Sorting Game _________________________________________________2

Tangram ______________________________________________________________6

Venn Diagram __________________________________________________________7

Word Search Puzzle (Shapes) ______________________________________________8

G3 Part 1: BLM List



2-D Shape Sorting Game


2-D Shape Sorting Game (continued)




Three or

more sides

Four or

more vertices

Three or

more vertices

More than

one line

of symmetry

No lines

of symmetryHexagon


No right

angles


Two or

more sides

One right

angle only

At least

fi ve sides

Equilateral

Quadrilateral



Tangram


Venn Diagram



Word search puzzle—can also be assigned as homework.

WORDS TO SEARCH:

triangle

vertices

quadrilateral

sides

pentagon

angle

vertex

polygon

l a o l g i o l d l r l

a e n e e a e g l g r o

r n i l i s e v c e t e

e l g n a i r t s n n n

t n e o v d v t p e e g

a c l g l e i a y a v p

l g a a r s o i i r e r

i e e t a p e e i i v x

r o e n i p n r t t r n

d x s e c i t r e v n x

a i o p o l y g o n t a

u a q q a e s n q s a e

q l n n x s r g q e s t

Word Search Puzzle (Shapes)


PA3-20 Patterns Involving Time Goal: Students will create T-tables for growing and shrinking patterns and use them to identify the rules for

the patterns.

Prior Knowledge Required: Addition, Subtraction, Skip counting, Number patterns

Days of the week, Months of the year, T-tables

Vocabulary: T-table, chart, term, growing pattern, shrinking pattern

Explain that today you are going to use T-tables to solve problems involving money and time. Students

are sure to encounter similar problems in their day-to-day lives; it is important they know how to approach

such problems systematically. Make sure all of your students know the days of the week and the months of

the year.

Give an EXAMPLE: Jenny started babysitting in the middle of June and earned $10 in June. She continued

babysitting throughout the summer and earned $15 in July and August. She saved all the money. How much

money did she save through the whole summer? Make a T-table and label the columns Month and Savings

($). Fill in the starting amount and add circles at the side, as shown:

Month Savings ($)

June 10

July

August

ASK: How much money did Jenny add to her savings in July? Write “+ 15” in the upper circle. How much

money did Jenny have saved by the end of July? Ask volunteers to continue the pattern and to finish

the problem.

Let your students practice solving similar problems, like the two below. Help your students to create a T-table

for each problem.

1. Lucy’s family spent 9 days hiking the Bruce Trail. They started by hiking 15 km on Saturday. They hiked

20 km every day after that until the next Sunday. Make a chart to show their progress. How many

kilometers did they hike in total over the course of their trip?

2. Natalie started saving for her mom’s birthday present in March. She saved $8 in March and plans to

save $5 every month after that. Her mom’s birthday is July 1. Will she have enough money by the end of

June to buy a present that costs $22? Make a T-table to show her savings.


Invite volunteers to help you solve the following problem:

Ronald received $50 as a birthday present on March 31. Every month he spends $4. How much money

will remain by the end of July?

Allow your students to practise solving more problems that involve decreasing patterns.

Assessment:

1. There are 15 fish in an aquarium on Monday. On Tuesday, Thomas the cat catches and eats two fish.

He eats two more fish every day after that. If Thomas’ owners don’t stop him, how many fish will they find

in the tank on Saturday night? Use a T-table to solve the problem.

2. The zoo is breeding a group of critically endangered Golden Lion tamarins. Every year 6 new monkeys

are born. At the end of 2006, the zoo had 13 tamarins. Make a T-table to show how many tamarins will

be at the zoo in the years 2007–2011.

Activity: Divide your students into pairs, give them play money and some beads or counters, and let them

play a selling and buying game: one player is a seller (starts with all the beads or counters and no money),

the other is a buyer (starts with all the money). The buyer can only buy and the seller can only sell (i.e., the

buyer can’t “sell” anything back to the seller). Have buyers and sellers record the money they have after

each transaction in a T-chart.

After a few transactions, ask the students to look at the patterns in their T-charts. Point out that each buyer’s

amount is a decreasing sequence, or shrinking pattern. ASK: What other patterns do you know? Is the

seller’s money shrinking, growing, or repeating?

Extension: Samia collects 4 stamps every month. She starts collecting at the beginning of November.

How many stamps does she have at the end of February? When will her collection reach 20 stamps? When

will she have 28 stamps?


PA3-21 Calendars Goal: Students will identify patterns in calendars.

Prior Knowledge Required: Addition. Subtraction, Skip counting, Number patterns

Days of the week, Months of the year, T-tables

Vocabulary: T-table, chart, growing pattern, shrinking pattern, vertically, horizontally, diagonally

Remind your students which months have 30 days, 31 days, and 28 days. Mention the leap year.

Give your students blank BLM “Calendars” to work with and draw a large calendar on the board or use the

overhead projector. Solve the following problems as a class, but give students time to solve each problem

independently before calling on volunteers to share their answers on the board.

Fill in the calendar for August, so that August 1st is Tuesday.

a) Steven gets a pet snake on his birthday, August 5th. The snake should be fed every five days. Mark the

days when Steven has to feed his pet with a little snake. How are the snake-filled squares situated—

vertically, horizontally, or diagonally? Review the meaning of these terms with your students.

b) Steven plays the drums every Wednesday. Mark the days he drums on the calendar with a D or small

picture of a drum. Ask your students to describe the pattern of the drum days. Are there any days when

Steven has to feed his pet snake and play the drums? (yes, August 30th)

c) Steven bought his mother an orchid on August 3rd

. The orchid has to be watered every 6 days. Shade

the days when the orchid has to be watered. What pattern do the shaded squares make? (a diagonal

pattern) On which date does Steven play the drums and water the orchid? (August 9th) Are there any

dates when the snake needs feeding and the flower needs watering? (August 15th)

Activity: Ask students to create a calendar for the current month. They should label the days of the week

and mark the dates of personal events, such as lessons, chores, and family activities. Encourage students to

think of both special events, like birthdays or parties, and recurring events, such as feeding a pet, cleaning

their room, going to the library, taking out the garbage, or visiting a relative.

Extensions:

1. Fill in a blank calendar for any month, beginning on any day. Draw a square around any 4 numbers. Add

the pair of numbers on one diagonal. Then add the pair of numbers on the other diagonal. What do you

notice about the two sums? Will this always happen? Can you explain why?


2. PROJECT: Calendars of the World. Explain to students that different calendars are used in other parts of

the world to mark time. Students can learn more about any one of these calendars. Questions to

consider: How many months are in the year in your calendar? How long is the year? How long are the

months? What defines the months and the year (movement of the Sun, the Moon, the Nile)? Is there a

leap year? What is a leap year (additional day or additional month)? How often does a leap year occur?

What patterns can be found in the calendar? (The Chinese calendar is particularly interesting in this

respect.)

POSSIBLE SOURCE: http://webexhibits.org/calendars/calendar.html


PA3-22 Number Lines Goal: Students will use number lines to solve pattern problems.


Days of the week, Months of the year

Vocabulary: number line

Write a sequence on the board: 36, 31, 26, 21. ASK: What is this kind of sequence called? (a decreasing

sequence) What is the difference between successive terms in the sequence? (5) Ask a volunteer to

continue the sequence and to explain how he or she figured out what the next term should be. Students

might say that they counted backwards or used their number facts to subtract. Ask students if they can think

of another way to solve the problem—if, for example, the numbers are so large that counting backwards or

subtracting is difficult. If no one suggests it, tell students they can use a number line and you will now teach

them how.

Present the following EXAMPLE: A snail crawls 4 cm in an hour. It is 24 cm away from the end of the branch.

How far from the end of the branch will it be after four hours of crawling?

Draw a number line and add arrows showing the snail’s progress every hour:

You could ask:

When will the snake reach the end of the branch? (after 6 hours)

A cherry hangs 11 cm from the end of the branch. Invite a volunteer to mark the place where the cherry

is on the number line. When would the snail reach the cherry? (in the 4th hour)

Let your students draw number lines in their notebooks. Here are some problems they could practice with:

A messenger pigeon flies 3 km in an hour. How long will it take this pigeon to carry a letter to a person

who lives 21 km away from the sender? The pigeon sets out at 1:00 p.m. When will it reach the

addressee?

Jonathan has $20. He spends $6 a week for snacks. How much money will he have after three weeks?

When will his money be completely spent?

3rd

hour 2nd

hour 1st hour 4

th hour

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24


Activity: Draw a number line on the floor. It should have at least 12 intervals, but preferably a few more.

Ask a volunteer to demonstrate a hopping pattern: Four hops forward, one backward. Let the volunteer start

from 0 and repeat the core several times. Ask students to draw a number line on paper and record the

movement of their classmate with arrows. ASK: Where will the volunteer be after 6 hops? 7 hops? 9 hops?

12 hops?

As a challenge, repeat the exercise but have the volunteer hop from the other end of the line.

Now ask a volunteer to stand at the 7th mark, take 3 hops backward, then 6 hops forward. Where does the

volunteer end up? Ask your students to predict where the volunteer will end up if he or she hops forward 5

and backward 2. Make up more questions, beginning at different points on the number line. (EXAMPLE:

Start at 4, take 6 hops forward then 1 hop backward.)

Extension: A painter’s ladder has 12 steps. The painter spills red paint on every second step and

blue paint on every third step. Which steps have red and blue paint on them? Which steps will have no paint

on them?


PA3-23 Mixed Patterns Goal: Students will extend patterns of various kinds

Prior Knowledge Required: Patterning, Core of the pattern

Vocabulary: growing pattern, shrinking pattern, repeating pattern

Review the concept of the core of repeating patterns with the students. Draw several repeating patterns on

the board and invite volunteers to circle the core of each pattern and to extend the patterns. Here are some

sample patterns:

A H H T A H H T R U N R U N R U N R U 1 4 6 7 2 2 1 4 6 7 2 2 1 4 6 7 2 2

Draw a pattern with a core that begins and ends with the same symbol and repeat the task. ASK: Was it

harder to find the core and continue the pattern? Why? Give students more such patterns to practice with.

Here are some samples:

A H H A A H H A R U R R U R R U R R U 2 1 4 6 2 2 2 1 4 6 2 2 2 1 4 6 2 2

Then allow your students to practise with a mixed assortment of repeating patterns.

Draw the following pattern on the board:


ASK: Is this a repeating pattern? Does this pattern have a core? (no) Can you continue this pattern? How

many diamonds come next? Explain that this pattern is called a growing pattern and ask your students to

explain why they think this pattern is called that. (One part of the pattern grows, or increases.) Let your

students extend more growing patterns, such as:

S I S I I I S I I I I I S I I I I I I I K a a K K a a a K K K a a a a K K K K a a a a a

Ask your students what they think a shrinking pattern might look like. If they do not guess correctly, prompt:

In growing patterns, one of the parts was growing. What should some part of a shrinking pattern do? If we

start with M U U U U U U M, what might the next group of letters in the pattern be? (The number of Us

should shrink; the next term in the pattern could be M U U U U U M [1 fewer U] or M U U U U M [2 fewer U’s]

or even M U U U M [3 fewer U’s]. You could have students find the third term for each sequence, too.) Invite

students to draw examples of more shrinking patterns on the board. They can create shrinking patterns using

letters, numbers, or shapes. EXAMPLES:

5 5 5 5 5 4 4 4 4 3 3 3 2 2 1 9 9 9 9 8 8 8 7 7 6 A a a a a a a a a B b b b b b b C c c c c D

Invite a volunteer to write the letters of the alphabet on the board and ask your students to continue the

following patterns:

A Z B Y C X D Aa Bbb Cccc D F H J L

If some students have trouble seeing the rule for a pattern, suggest that they circle the letters in the alphabet

as they appear in the pattern. Invite students who see the rule to describe it to their classmates.

Finally, show your students a mixed assortment of patterns—repeating, growing, shrinking, letter—and ask

them to continue the patterns.

Activity: A game for pairs. Each pair of students will need a spinner as shown. Player 1 spins the spinner

so that his partner does not see the result. Player 1 draws a pattern according to the result on the spinner.

Player 2 has to write the next three terms of the pattern.

Repeating pattern

Growing pattern

Shrinking pattern

Letter pattern


PA3-24 Describing and Creating Patterns Goal: Students will identify and describe increasing, decreasing, and repeating patterns. They will also

create increasing, decreasing, and repeating patterns.

Prior Knowledge Required: Patterning, Core of the pattern, Increasing sequences

Decreasing sequences, Repeating patterns, Addition

Subtraction, Difference between pattern terms

Vocabulary: increasing sequence, decreasing sequence, repeating pattern, difference

Write a sequence on the board: 3, 5, 7, 9, 11, 13, 15. Ask your students what kind of sequence this is. (The

numbers grow, or increase, so this is an “increasing sequence.”) Write another sequence: 95, 92, 89, 86, 83,

80, 77, and ask students what this sequence is called. Write “decreasing sequence” on the board.

ASK: Is every sequence an increasing or decreasing sequence, or can a sequence do both—can it increase

and decrease? Give an example: Jennifer saved $80. She receives a $20 allowance every two weeks. She

spends money occasionally:

Date Balance ($)

Sep 1 80

+20 Allowance

Sep 5 100

–11 Book

Sep 7 89

–10 Movie + Popcorn

Sep 12 79

+20 Allowance

Sep 19 99

Jennifer’s balance increases between the first two terms, decreases for the next two terms, and then

increases again.

Write several similar sequences on the board and ask volunteers to put a “+” sign in any circle where the

sequence increases, and a “–” sign where it decreases. Let your students practice this skill, and then ask

them to mark the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence

sometimes increases and sometimes decreases.

Sample Sequences

a) 7 , 8 , 7 , 10 b) 2 , 4 , 7 , 9

c) 10 , 7 , 4 , 2 d) 2 , 5 , 1 , 17

e) 15 , 23 , 29 , 28 f) 22 , 52 , 59 , 62


Explain to your students that now their task will be more difficult. This time they will have to determine if the

sequence is increasing or decreasing and find the magnitude of the difference between successive terms.

Invite volunteers to find differences by counting forwards and backwards using their fingers or number lines.

Sample Sequences

a) 7 , 9 , 11 , 13 b) 2 , 5 , 9 , 12

c) 10 , 7 , 4 , 1 d) 32 , 36 , 40 , 44

e) 15 , 23 , 31 , 39 f) 72 , 52 , 32 , 12

Ask your students to find the differences between the terms in the previous sample sequences. Then list

several possible descriptions of sequences, such as:

A: Increases by different amounts

B: Decreases by different amounts

C: Increases by a constant amount

D: Decreases by a constant amount

Ask your students to match the descriptions with the sequences on the board. When the terms of the

sequence increase or decrease by the same amount, you might ask students to write a more precise

description. For example, the description of sequence (d) above could be “Increases by 4 each time” or,

more precisely, “Start at 32 and add 4 each time.” Encourage your students to give more precise

descriptions.

Next let your students do the activity below.

Assessment:

1. Put a “+” sign in any circle where the sequence increases, and a “–” sign where it decreases. Then mark

the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence sometimes

increases and sometimes decreases. Write the next term in each sequence.

a) 12 , 17 , 14 , 19 ___ b) 1 , 5 , 8 , 3 ___ c) 18 , 13 , 17 , 23 ___

d) 28 , 26 , 19 , 12 ___ e) 17 , 8 , 29 , 25 ___ f) 53 , 44 , 36 , 38 ___

2. Match the descriptions to the patterns. Each description may fit more than one pattern.

17, 16, 14, 12, 11 3, 5, 7, 3, 5, 7, 3 A: Increases by the same amount

10, 14, 18, 22, 26 4, 8, 11, 15, 18, B: Decreases by the same amount

54, 49, 44, 39, 34 4, 5, 3, 2, 6, 2, 4 C: Increases by different amounts

4, 8, 12, 8, 6, 4, 6 11, 19, 27, 35, 43 D: Decreases by different amounts

74, 70, 66, 62, 58 12, 15, 18, 23, 29 E: Repeating pattern

58, 55, 52, 49, 46 67, 71, 75, 79, 83 F: Increases and decreases

For patterns that increase or decrease by the same amount, write an exact rule. Don’t forget to include the

starting point!


Bonus:

1. Put a “+” sign in any circle where the sequence increases, and a “–” sign where it decreases. Then mark

the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence sometimes

increases and sometimes decreases.

a) 257 , 258 , 257 , 260 b) 442 , 444 , 447 , 449 c) 310 , 307 , 304 , 298

d) 982 , 952 , 912 , 972 e) 815 , 823 , 829 , 827 f) 632 , 652 , 649 , 572

2. Match the descriptions to the patterns.

97, 96, 94, 92, 91 A: Increases by the same amount

3, 5, 8, 9, 3, 5, 8, 9 B: Decreases by the same amount

210, 214, 218, 222 C: Increases by different amounts

444, 448, 451, 456 D: Decreases by different amounts

654, 647, 640, 633 E: Repeating pattern

741, 751, 731, 721 F: Increases and decreases by different amounts

For patterns that increase or decrease by the same amount, write an exact rule.

Activity: A game for pairs. Your students will need the spinner shown. Player 1 spins the spinner so that

Player 2 does not see the result. Player 1 has to write a sequence of the type shown by the spinner. Player 2

has to write the rule for the sequence or describe the pattern. For example, if the spinner shows, “Decreases

by the same amount,” Player 1 can write “31, 29, 27, 25.” Player 2 has to write “Start at 31 and subtract 2

each time.”

Extension: Write the differences for the patterns. Identify the rule for the sequence of differences.

Extend first the sequence of differences, then the sequence itself.

a) 7 , 10 , 14 , 19 b) 12 , 15 , 20 , 27

c) 57 , 54 , 50 , 45 d) 32 , 30 , 26 , 20

e) 15 , 18 , 22 , 25 , 29 f) 77 , 72 , 69 , 64 , 61


PA3-25 2-Dimensional Patterns Goal: Students will identify and describe patterns in 2-dimensional grids.

Prior Knowledge Required: Patterning, Core of the pattern, Increasing sequences

Decreasing sequences, Repeating patterns, Addition, Subtraction

Difference between pattern terms

Understanding that reading is done from left to right

and that text wraps onto the next line

Vocabulary: increasing sequence, decreasing sequence, repeating pattern, difference

Draw a 2-dimensional grid on the board. Remind your students that rows are vertical and columns are

horizontal. Point out the diagonals as well. Draw this diagram as a reference:

Explain that in 2-dimensional patterns, you generally count columns from left to right, but there are two

common ways of counting rows. In coordinate systems, which students will learn later, you count from the

bottom to the top. But in this lesson, students will count rows from the top to the bottom. Ask a volunteer

number the columns and rows on your grid.

Ask your students to draw a 4 × 4 grid in their notebooks and to fill it in as shown:

Ask them to shade the first column, the first row, and the third row, and to circle both diagonals. Let them

describe the patterns that they see in the shaded rows and columns and in the diagonals. Then invite

students to do the activities below for more practice.

Write this sentence on the board exactly as shown, i.e., over two lines:

“The big monkey ate the bananas

that were almost too ripe.”

13 17 21 25

10 14 18 22

7 11 15 19

4 8 12 16

D C I R O W A L G U O M N N A L

V

E

R

T

I

C

A

L

H O R I Z O N T A L


Ask students where the sentence starts and where the sentence ends. Ask them to explain how they know.

Address any misconceptions students may have about sentences ending at the end of a line. Explain

that the “reading pattern” rule is that we read from left to right and that a sentence can continue on to the

next line.

Next, ask students to identify the word that comes right before the following words in the sentence: monkey,

ate, ripe, that. Then, ask students to identify the words that come right after these words: too, bananas, big,

almost, ate.

Draw a five-frame on the board and place only the numbers 1 and 5 in the row:

1 5

ASK: Which number comes right after the 1? Which number comes right before the 5? Can you predict

which number comes right after the 2 but right before the 4?

Add a row to the five-frame turning it into a ten-frame:

1 2 3 4 5

Challenge students to write the number that comes next on the next line, reminding them (if necessary) of

the reading pattern and of the number that comes after 5.

Next, draw the following charts on the board and challenge students to use the reading pattern to fill in the

missing numbers.

1 1 1

3 5 3

7 6

Emphasize that students should always start at the top row from the left, then go on to the next row, again

starting at the left. Then give students blank charts and ask them to fill in the charts starting at 1:

Students could also fill the above charts by skip counting by 2s, 3s, and 5s.


Activities:

1. Give students a hundreds chart with the following instructions: Put your finger on the first number given

in each question below. Then move your finger to show the result of the addition. Then describe how

your finger moved.

EXAMPLES:

a) Start at 2. Add 20 (Answer: My finger moved 2 rows down.)

b) Start at 5. Add 21 (My finger moved 2 rows down and 1 column across.)

c) Start at 17. Add 42 (My finger moved 4 rows down and 2 column across.)

Make up more questions of this sort. With practice, students should be able to predict how their finger

will move before they carry out the addition, so they can add simply by moving their finger.

2. Students will need the “Hundreds Charts” BLM. Write the following patterns on the board and ask

students to identify in which rows and columns in the hundreds chart the patterns occur:

After students have had some practice finding patterns in the hundreds chart, ask them to fill in the

missing numbers in patterns. (Challenge them to do this without looking at the hundreds chart.)

EXAMPLES:

Make up more such problems for students who finish their work early.

Extensions:

1. Add the digits of some 2-digit numbers on the hundreds chart. Can you find all the numbers that have

digits that add to ten? Describe any pattern you see.

2. Sudoku is an increasingly popular mathematical game that is now a regular feature in many newspapers.

In the BLM section of this guide, you will find Sudoku suitable for children (“Sudoku – Warm Up” and

“Mini Sudoku”) with step-by-step instructions. Once students master this easier form of Sudoku, they can

try the BLM “Sudoku – The Real Thing.”

3

13

23

33

38

48

58

68

43

54

63

74

12

23

34

45

12

22

32

57

77

87

7

18

38

45

56

67


PA3-26 Patterns in Two Times Tables Goal: Students will identify even and odd numbers.

Prior Knowledge Required: Skip counting by 2s, Rows, Columns, Diagonals

Vocabulary: row, column, diagonal, multiple, even, odd

Explain to your students that the numbers you say when skip counting by 2s, starting from 2, are called

multiples of 2. ASK: What other words similar to “multiple” do you know? (multiplication, multiply) What do

these other words mean? Ask your students to find the sequence of multiples of 2 on a multiplication chart.

What do you do to the numbers 1, 2, 3, 4, 5, … to get 2, 4, 6, 8, 10, …? (You multiply by 2.) What is a

“multiple”? (The multiple of a number is what you get when you multiply that number by any other number.)

Let your students complete the worksheet independently, or guide the class through the worksheet in stages.

You might wish to explain where the name “even” comes from: Give your students 20 small objects, like

markers or beads or base ten unit blocks, to act as counters. Ask them to take 12 blocks and to divide the

blocks into two equal groups. Does 12 divide into two groups evenly? Ask students to repeat the exercise

with 13, 14, 15, and any other number of blocks. When the number of blocks is even, they divide into two

groups evenly. If they do not divide evenly, the number is called “odd.” Write the on the board: “Even

numbers: 2, 4, 6, 8, 10, 12, 14… All numbers that end with .” Ask volunteers to finish the sentence.

After that write: “Odd numbers: 1, 3, 5, 7, 9, 11, 13… All numbers that end with ” and ask another

volunteer to finish this sentence.

Assessment: Circle the even numbers. Cross out the odd numbers.

23, 34, 45, 56, 789, 236, 98, 107, 3 211, 468 021.

Activity: Call up a group of volunteers and give each one a card with a whole number on it.

1. Ask the rest of the class to give the volunteers orders, such as “Even numbers, hop” or “Odd numbers,

raise your right hand.”

2. ADVANCED: Ask a student to skip count by 3s, starting with three. These are the multiples of three. Ask

your “numbers”: Are you a multiple of three? Repeat, skip counting by 5s or other numbers.

3. ADVANCED: Give the volunteers more complex orders, such as “All numbers that are multiples of five

or even numbers, clap” or “All numbers that are odd and multiples of five, stomp.”

Extension: The “Colouring Exercise” in the BLM section of this guide. (ANSWER: The Quebec flag.)


PA3-27 Patterns in the Five Times Tables Goal: Students will identify multiples of five.

Prior Knowledge Required: Skip counting by 5s, Rows, Columns, Diagonals

Vocabulary: row, column, diagonal, multiple

Let your students complete the worksheet independently, or guide the class through the worksheet in stages.

Then let students play the game below (SEE: Activity).

After students have played the game, write several numbers with blanks for the missing digits on the board

and ask your students to fill in each blank so that the resulting number is divisible by 5. Ask students to list all

possible solutions for each number. EXAMPLES:

2___ 32___ 8___ 56___ 5___5 ___35 8___0

Assessment:

1. Circle the multiples of 5.

75 89 5 134 890 40 234 78 4 99 100 205 301 45675

2. Fill in the blanks so that each number become a multiple of 5. In two cases it is impossible to do so. Put

an “X” through these numbers.

3___ 3___5 3___9 ___0 70___ 5___0 ___3

Bonus: Fill in the blanks so that the numbers become multiples of 5. (List all possible solutions.) In two

cases it is impossible to do so. Put an “X” beside these numbers.

3 44___ 34 ___25 13 6___9 6 783 45___ 786 7___0

4 567 70___ 234 5___6 780 13 460 0___3 15 4___0 000

Activity: A game for pairs. One player will be the “pro-5” player, and the other will be the “anti-5” player.

The anti-5 player starts by adding either 3 or 5 to the number 4. The pro-5 player is then allowed to add

either 3 or 5 to the result. Each player then takes one more turn. If either player produces a multiple of 5, the

pro-5 player gets a point. Otherwise, the anti-5 player gets a point. The players exchange roles and start the

game again. Students should quickly see that there is always a winning strategy for the pro-5 player.


Extension: “Who am I?”

1. I am a number between 31 and 39. I am a multiple of five.

2. I am an even multiple of five between 43 and 56.

HINT: Write down all even numbers between 43 and 56.

3. I am an odd multiple of five between 76 and 89.

4. I am the largest 2-digit multiple of five.

5. I am the smallest 3-digit multiple of five.

6. I am the smallest 2-digit odd multiple of 5.


PA3-28 Patterns in the Eight and Nine Times Tables

Goal: Students will identify multiples of eight and nine.

Prior Knowledge Required: Skip counting is a quick way to add

Ones, Tens, Place value, Patterning

Addition, Rows, Columns, Diagonals

Vocabulary: column, diagonal, ones, tens, multiple

The worksheet for this lesson is similar to the worksheets for the previous two lessons—it guides students to

discover the properties of certain multiples. Because the patterns in the eight times table are more

complicated than those seen previously, you might want to guide students through the worksheet in stages

or teach the following lesson and have students complete the worksheet as a review or for reinforcement.

Ask students to skip count orally by 4s to 40. Then, write the corresponding number sentences on the board:

4

4 + 4 =

4 + 4 + 4 =

4 + 4 + 4 + 4 = and so on.

Have a volunteer find the sums. Now, ask students if they see a connection between skip counting by 4s and

repeated addition. Have students extend the pattern beyond the addition sentences written on the board.

Remind your students that the numbers that you say when skip counting by 4s, starting from 4, are called

“multiples of 4.” Review the meaning of the term “multiple” and its connection to multiplication.

Now ask your students to skip count by 8s. Suggest that they write a similar group of addition sentences for

the first five multiples of 8 and write the sums in a column:

8 08

8 + 8 = 16

8 + 8 + 8 = 24

8 + 8 + 8 + 8 = 32

8 + 8 + 8 + 8 + 8 = 40

Ask your students to circle the ones digits in the multiples. ASK: What pattern can you see in the ones digits

of the multiples of 8? What is the rule for this pattern? (Subtract 2.) What is the rule for the pattern in the tens

digits? (Add 1.)

Now write the addition sentences for the next five multiples of 8. List the sums in a column, as before. Look

at the patterns in the ones digits and the tens digit. Have the patterns changed? Have the rules changed?


(The ones digit went back to 8, and you again subtract 2 each time. The tens digit started at 4—the same

digit as the fifth multiple of 8—and you again add 1 each time.)

ASK: Can you use these patterns to write the next five multiples of 8? Invite a volunteer to write the ones

digits, then the tens digits, for the first two terms (88 and 96). ASK: What should the ones digit in the next

term be? (4) What should the tens digit in the next term be, according to the pattern? (9 + 1 = 10) Is 10 a

digit? (No; it’s a 2-digit number.) What should we do? Invite your students to find the next multiple of 8 by

adding 8 to the previous term (96), and point out that the pattern they spotted in the tens digits holds—there

are indeed 10 tens in the next term! (96 + 8 = 104 = 10 tens and 4 ones) ASK: How many tens are in 80? In

88? In 100? In 101? Let a volunteer find the next two multiples of 8. Then ask your students to write the next

five multiples of eight.

Ask your students to find the first five multiples of 9 by addition. Write them in a column (starting from 09).

ASK: Can you see a pattern in the ones digits? What’s the rule? Ask students to continue the pattern. Repeat

with the tens digits.Ask your students to add the digits in any of the multiples of 9. What do they notice?

Activity:

“Eight-Boom” Game Players stand in a circle and count up from one, each saying one number in turn.

When a player has to say a multiple of 8, he says “Boom!” instead: 1, 2, 3, 4, 5, 6, 7, “Boom!”, 9, … If a

player makes a mistake, he leaves the circle.

ADVANCED VERSION: When a number has “8” as one of its digits, the player says “Bang!” instead. If the

number is a multiple of 8 and has 8 as a digit, the player says both. EXAMPLE: “6, 7, “Boom, Bang!”, 9,” or

“15, “Boom!”, 17, “Bang!”, 19.”

Both games are a fun way to learn the eight times table, and can be used for multiples of any other number

you would like to reinforce.

Extensions:

1. Let your students use base ten materials to build multiples of nine in the standard way. Pick a multiple of

nine, say 27. How many blocks are in the standard model? (9: 2 tens and 7 ones) What do you do when

you add 9 to 27? You add 9 ones blocks, so now you have 16 ones blocks. Trade 10 ones blocks for 1

tens block. How many blocks do you have now? (9: 3 tens and 6 ones) When does this pattern break?

(When you add 9 to 90. You do not have 10 ones to trade.)

2. Continue the pattern:

9 × 1 = 09

9 × 2 = 18

9 × 3 = 27

9 × 4 = 36

What do you have to do to the second factor in the multiplication sentence to get the tens digit? Why?

(Nine is one less than ten. To find, say, 9 × 3, you add three tens and remove three ones – one for each

nine (30 - 3). You have to regroup one of the tens to do this. This means the number of tens in a multiple

of nine is one less than the second factor in the multiplication statement.)


3. Show your students a method to remember the multiples of 9 using fingers. Put your hands on the table

with fingers spread. You want to find, say, 9 × 3. Count three fingers from the left and fold the third

finger. The fingers to the left of the folded finger are the tens, and the fingers to the right of the folded

finger are the ones. The answer is 27. Ask your students to use Extensions 1 and 2 to explain why the

trick works.

4. “Who am I?”

a) I am a number between 31 and 39. I am a multiple of eight.

b) I am the largest 2-digit multiple of eight.

c) I am the smallest 3-digit multiple of eight.

d) I am a two digit number larger than 45. I am a multiple of eight and a multiple of five as well.

Twenty seven


PA3-29 Patterns in Times Tables (Advanced) Goal: Students will use Venn diagrams to reinforce their knowledge of multiples of two, five, eight,

and nine.

Prior Knowledge Required: Multiples of two, five, eight and nine, Venn diagrams

Vocabulary: multiple

Students can use patterns in the times tables of various 1-digit numbers to help them learn their

multiplication facts. See “How to Learn Your Times Tables in a Week” in the Mental Math section of this

manual for some effective strategies for using patterns to learn times tables.

Review Venn diagrams (SEE: PDM3-2, empty Venn Diagrams are provided in the BLM section). Remind

your students that any numbers that cannot be placed in either circle should be written outside the circles

(but inside the box). Then draw a Venn diagram with the properties:

1. Multiples of five 2. Multiples of two

Ask your students to sort the numbers between 20 and 30 in the diagram.

Draw another Venn diagram, this time with multiples of five in one circle and multiples of eight in the other.

Ask your students to sort the numbers between 30 and 41, then between 76 and 87, into this diagram. Ask

volunteers to sort some 3-digit numbers in the diagram (for example, 125 and 140).

Draw another Venn diagram:

1. Multiplies of two 2. Multiplies of eight

Ask volunteers to sort into it the numbers: 5, 8, 14, 15, 27, 28, 56, 40, 25, 30, 99. ASK: Why is there an

empty part in the diagram? (All multiples of eight are also multiples of two.) Ask your students to add two

numbers to each part of the diagram that is not empty.

Assessment:

Make a Venn diagram with properties:

1. Multiplies of nine 2. Multiplies of five

Bonus: Add three 3-digit numbers to each part of the last diagram.

Activity: Give students a set of base ten blocks (ones and tens only). Ask them to build base ten models

of the following numbers.

a) You need 3 blocks to build me. I am a multiple of 5. (Solution: 30)

b) You need 6 blocks to build me. I am a multiple of 5. (There are 2 solutions: 15 and 60.)

c) You need 3 blocks to build me. I am a multiple of 4. (Solution: 12)

Challenging

d) You need 6 blocks to build me. I am a multiple of 3. (There are 6 solutions: 6, 15, 24, 33, 42, 51.)


PA3-30 Patterns with Increasing Gaps Goal: Students will extend patterns with increasing gaps

Prior Knowledge Required: Increasing sequences, T-tables

Vocabulary: increasing sequence, difference, gap

Draw the following pattern on the board or build it with blocks:

Figure 1 Figure 2 Figure 3 Figure 4

Ask a volunteer to build the next term of the sequence. Ask your students to complete a T-table for the

number of blocks in each term. Then ASK: How many blocks are added each time? What are you adding to

each figure to make the next figure in the sequence? (A new column is being added on the right side—you

could shade or highlight the new column in each figure, to help students see it.) How many blocks are in the

column added to figure 1? How many in the column added to figure 2? This is the difference in the total

number of blocks at each stage. Ask a volunteer to write the difference in the circles beside the table.

Ask your students if they can see a pattern in the differences. Ask a volunteer to determine the next term in

the pattern of differences, i.e., how many blocks are added to figure 4 to make figure 5. Ask another

volunteer to fill in the next row of the table. Ask a third volunteer to build figure 5 to check the result.

For practice, ask students to find the differences between the terms of these sequences, extend the

sequence of the differences, and then extend the sequence itself.

a) 5 , 8 , 12 , 17 , _____ , _____ c) 11 , 14 , 20 , 29 , , _____

b) 3 , 5 , 9 , 15 , 23 , , _____ d) 6 , 8 , 13 , 21 , 32 , , _____

Figure Number

of blocks

1

2

3

4


Show the following geometrical pattern and ask how many triangles will be in the next design:

Draw a T-table for the pattern. How many triangles do you add each time? (Point out that a new row is being

added each time, and that it is always two triangles longer than the previous row.) The number of triangles in

the new row is the difference, or gap, between the total number of triangles in two successive figures. This

means that the pattern in the gaps follow the rule “Start at 3 and add 2 each time.” Where in the T-table do

you write the gaps? Ask volunteers to extend first the sequence of gaps in the circles beside the T-table,

then the sequence itself.

Assessment:

a) 15 , 18 , 22 , 29 , ______ , ______

b) 13 , 16 , 22 , 31 , 43 , ______ , ______

Bonus:

a) 7 , 17 , 37 , 67 , ______ , ______

b) 88 , 189 , 391 , 694 , 1098 , ______ , ______

Activity: Let your students build growing patterns using pattern blocks. For each sequence of patterns,

find out how many blocks you need for the next term using a T-table.

Extension: Janet is training for a marathon. On Monday she ran 5 km. Every day after that, she ran 1 km

more than on the previous day. How many kilometres did she run in total from Monday to Sunday?

Draw the T-table. The first 3 entries should appear as follows:

Day Km from the beginning

of the week (total)

1. Monday 5

2. Tuesday 11

3. Wednesday 18

6 Each day,

Janet runs

1 km more. 7


PA3-31 Patterns with Larger Numbers Goal: Students will practise extending sequences of larger numbers using T-tables.

Prior Knowledge Required: Addition, Subtraction, Increasing and decreasing sequences, T-tables,

Canadian money


Tell your students that they have done so well with patterns that today you are going to give them patterns

with HUGE numbers. Present several problems and invite volunteers to solve them using T-tables.

EXAMPLES:

A normal heartbeat rate is 72 times in a minute. How many times will your heart beat in five minutes?

There are 60 minutes in an hour. George sleeps for six hours. How many minutes does he sleep?

A sprinting ostrich’s stride is 700 cm long. A publicity-loving ostrich spots a photographer 3 000 cm away and

runs towards him. How far from the camera will the ostrich be after three strides?

An extinct elephant bird weighed about 499 kg. Make a T-table to show how much five birds would weigh.

Do you see a pattern in the numbers? (HINT: Look at the ones, tens, and hundreds separately.) Can you

write the weights of six, seven, and eight birds without actually adding?

Extension: A regular year is 365 days long, a leap year (e.g., 2000, 2004, …) is 366 days long. Tom was

born on January 8, 2002. How many days old was he on January 8, 2003? On January 8, 2005?


PA3-32 Extending and Predicting Positions Goal: Students will extend patterns using T-tables.


Ordinal numbers


Ask the students if they remember what special word mathematicians use for the figures or numbers in

patterns or sequences. Write the word “term” on the board. Review with the students how they to find the

core of a pattern. Give students several patterns and ask them to identify the cores.

Draw a pattern on the board:

Ask volunteers to circle the core of the pattern, identify its length, and continue the pattern. ASK: Which

terms are circles? What will the 20th term be—a circle or a diamond? Write down the sequence of the term

numbers for the circles (1, 3, 5, 7, …) and diamonds (2, 4, 6, 8, …), and ask which sequence the number 20

belongs to.

For more advanced work students could try predicting which elements of a pattern will occur in particular

positions. EXAMPLE: If the pattern below were continued, what colour would the 23rd

block be?

The worksheet for this lesson illustrates a method for solving this sort of problem using a hundreds chart.

Students could also use number lines to solve the problem, as shown below.

STEP 1: Find the length of the core of the pattern

The core is 5 blocks long.

R R Y Y R R Y Y Y Y

R R Y Y R R Y Y Y Y


STEP 2: Mark off every fifth position on a number line and write the colour of the last block in the core above

the marked position.

The core ends on the 20th block and starts again on the 21

st block. Write the letters of the core in order on

the number line starting at 21. The 23rd

block is yellow.

Eventually you should encourage students to solve problems like the one above by skip counting. Students

might reason as follows:

“I know the core ends at every fifth block so I will skip count by 5s until I get close to 23. The core ends at 20,

which is close to 23, so I’ll stop there and write out the core above the numbers 21, 22, and 23.

Students can use the activity (below) to practise predicting terms in patterns and sequences.

Assessment:

What is the 20th term of the pattern: A N N A A N N A A N N A A? What is the 30

th term?

Bonus:

Find the core of the following pattern, then find terms 20, 30, 40, 50, …, 100 (i.e., all multiples of 10 to 100).

R W W R R W W R W W R R W W R W W…

Activity: Each pair of players will need a die and a set of coloured beads or blocks. Player 1 rolls the die

so that Player 2 does not see the result. Player 1 builds a pattern of blocks or beads with a core of the length

given by the die. Then Player 2 throws the die and multiplies the result by 10. Player 2 predicts the bead for

the term he or she got. For instance, if Player 2 rolled three on the die, he or she has to predict the 30th term

of the pattern.

I know the core ends here

19 20 21 22 23

×

R R Y

A new core starts again here

The 23rd

block is yellow

Each × shows where the core ends. The core starts again here.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

× × × ×

Y Y Y Y R R Y Y Y

24 25


PA3-33 Equations Goal: Students will write and solve (by guessing and checking) algebraic equations of the type

a + x = b and x – a = b.

Prior Knowledge Required: Addition, Subtraction

Vocabulary: equation

Tell your students that today they will solve algebraic equations. Let them know that equations are like the

scales people use to weigh objects (such as apples). But in equations, there’s a problem: A black box

prevents you from seeing part of whatever object you are weighing. Solving an equation means figuring out

what is inside the black box. Write the word “equation” on the board. Ask your students if they know any

similar words (EXAMPLES: equal, equality, equivalence). So the word “equation” means “making the same,”

or balancing the scales.

Make a line down the middle of a table (e.g., with tape). Put 7 apples on one side of the line and put 3 apples

and a bag or box containing 4 more apples on the other side. Tell your students that there is the same

number of apples on both sides of the line. (You could represent the same problem with a scale and a

collection of objects of equal weight.) Ask your students to guess how many apples are in the box.

Tell your students that it is easy to represent the problem they just solved by guessing with a picture:

+ =

Invite a student to draw the missing apples in the box.

Ask your students how they could make sure that their guess about the number of apples is right without

looking into the box. Your students might suggest adding the number of apples inside the box and the

number of apples outside the box to check if it is the same as the number of apples on the other side of the

equal sign (or line on the table).

After students have had practice with this sort of problem, explain that it is inconvenient to draw the apples

all the time, so people use numbers to represent the visible quantities. This is called an equation. Let

students write the equation for the picture above:

+ 3 = 7


+ 3 = 7

Ask students to make models for the following equations, using circles for the known numbers of apples

(e.g., 7 and 11 in the first equation) and a square for the unknown number.

7 + = 11 6 + = 13 4 + = 10 + 9 = 12

Then ask your students to solve the equations they have written by guessing and checking. They should

refer to their models as required. As an alternative, your students might use a chart, as shown below for the

equation:

Guess Right side Left side Is my guess good? Why not?

2 2 + 3 = 5 7 no Too low

6 6 + 3 = 9 7 no Too high

Show students how they can solve such equations by removing the same number of items from each side of

the equation. Whatever is left on the right side is the amount on the left (in the box). Use one of the

equations students have already solved to model this process, to illustrate that it produces the same result.

Present this word problem: Sindi has a box of apples. She took two apples from the box and four were left.

How many apples were in the box before she removed the apples?

Draw the box. SAY: There are some apples inside, but we do not know how many. Draw two apples and

cross them to show that they are taken away. Four apples were left in the box, so draw them too. How many

were there from the beginning? (Six)

= =

Explain that when we draw an equation, we draw it differently. We draw the apples that we took out of the

box outside the box, with the “ – ” sign, to show that they were taken away:

– =

So to solve the equation we have to put all the apples into the box—the ones that we took out and the ones

that were left inside.

Remind your students that they also learned to write equations using numbers instead of pictures. Can they

guess what the equation for this problem will look like?

– 2 = 4


Draw several models like the ones on the worksheet, and ask your students to write the corresponding

equations. Ask volunteers to present the answers on the board. Students could use a chart, like the one

used above, to solve the equations by guessing and checking.

Then ask your students to draw models for these equations and to solve them by drawing the original

number of apples in the box:

– 6 = 9 – 7 = 12 – 5 = 3 – 3 = 10

Assessment:

Draw models for the equations. Solve the equations by guessing and checking.

3 + = 9 12 – = 7 – 4 = 11 + 7 = 14

Extensions:

The next two extensions satisfy the demands of the Western Curriculum.

1. What is the same and what is different in the following equations?

3 + = 9 3 + = 9 3 + = 9

Discuss with the students the similarities and the differences. Does the solution depend on the

symbol used in the equation?

2. a) Alina says that 3 and 4 solve the equation: 3 + + = 10. Jane says that 2 and 5 solve

this equation, too. Are both answers correct? Which other solutions can you find for this

equation?

b) Janet says that she can find different solutions for the equation 3 + = 9, too. Is she

correct? Explain.


PA3-34 Adding and Subtracting Machines Goal: Students will solve (by counting forward and backward) algebraic equations of the types:

a + x = b, x + a = b, a – x = b, x – a = b.

Prior Knowledge Required: Addition, Subtraction, Counting forward and backward


Prepare an adding machine. First, make a strip with the numbers from 1 to 20:

For a machine that adds 3 to a number, make a rectangle of length 3 and make two slots with distance 2

between them:

Write “+ 3” in the space between the slots. Pull the strip through the slots so that two numbers are covered:

Fold the strip, so that you can show only the numbers adjacent to the slots. You will need several adding

machines like this one, for different numbers (e.g., + 4, + 6). A subtracting machine can be made the same

way; the only difference is the numbers on the strip are written in decreasing order (20, 19, … 1).

Show your students the adding machine, and explain that it adds the number 3 to any number between 1

and 17. Show them how the machine works with two or three examples, then ask them to predict the sum for

several more examples. Check their answers with the machine. Show them another adding machine, and

ask them to predict some answers.

Show your students the back of the machine, so that they see the strip of paper going through the slots.

ASK: How many numbers are covered by the rectangle? Suggest that the students count up using their

fingers to check how many numbers are covered. (The machine adds 3, and 4 + 3 = 7; I see 4 and 7, but 5

and 6 are covered, so two numbers are covered.)

SAY: I have a machine that adds 3 to a number. The answer is 8. (Show this example on the “+ 3” machine,

covering the 5 with your hand.) What number was fed into the machine? Suggest to students that they count

backward to find the number that was fed into the machine. (The answer is 8, the machine adds 3, two

1 2 3 …

+ 3 3 6 … …


numbers are covered, and the third is shown. I count backward: 7, 6 are covered, 5 is on the other side of

the machine.)

Show your students some adding machines without telling them what number the machine adds, and ask

them to find that out from the numbers that are put through the machine.

Repeat the exercises above with subtracting machines.

To conclude, show your students several machines with the numbers that go “in” and “out,” but do not show

the number that is added or subtracted, or the operation that is performed (i.e., cover the front of the

machine). Ask your students to find out what was added or subtracted.

Activity: A game for pairs. Give your students a deck of cards with signs and 1-digit numbers, such as

+ 3 or - 5 on each. Player 1 takes a card at random without showing it to Player 2. Player 2 gives Player 1 a

number that is more than 12. Player 1 performs the operation shown on the card with the number given by

Player 2 and tells Player 2 the result. Player 2 has to guess what card Player 1 has. For example: Player 1

has the card “- 5,” Player 2 says “13,” Player 1 then says “8.”

Extensions:

1. Robin has an adding machine. Robin enters the number 12 into the machine, takes the answer, and

enters it back into his machine. The answer is 16. What number does Robin’s machine add to the

numbers he enters?

2. Colin has an adding machine that adds 3 to a number. Colin enters a number into the machine, takes the

answer, and enters it into the machine again. He takes the second answer and enters it into the machine

a third time! The machine gives him the number 15. What was the first number that Colin fed into his

machine?


PA3-35 Equations (Advanced) Goal: Students will solve (by guessing and checking) algebraic equations of the types:

a + x = b, x + a = b, a – x = b, x – a = b.

Prior Knowledge Required: Addition, Subtraction, Counting forward and backward


Review the previous lesson. Write the equation + 3 = 7 on the board and ASK: Which adding machine

does this equation remind you of? Which number is added? The result is 7, so how did you find out what

number was fed into the machine? Encourage your students to solve several equations of this type by

counting backward.

Continue with equations of the type 5 + = 8. Explain that this is also an adding machine, but you do not

know what number is added. Ask your students to use finger counting to find what number was added.

Repeat with equations of the types 12 – = 8 and – 3 = 6. Give your students a mix of equations to

solve. They can either use finger counting or guess and check. As a bonus, add several equations of the

type 3 × = 12.

Remind your students about the models they were using for equations in lesson PA3-34. Ask a volunteer to

draw a model for an addition equation. Ask your students to tell a story that fits the model. Invite your

students to draw several different models and tell different stories for the same equation. Repeat with

subtraction equations.

Tell your students that they can also create problems for multiplication equations. Explain that “2 ×” means

that some quantity is taken two times.

For example:

2 × =

Present the equation 2 × = 10 and ask your students to draw the appropriate number of circles

in the box:

2 × =


Explain that this model may represent several problems, such as:

a) Jane has some apples. George has twice as many apples as Jane. He has 10 apples. How many

apples does Jane have?

b) Tim and Tom ate the same number of cookies. They ate 10 cookies together. How many cookies did

each of them eat?

c) How many sets of two are in 10?

Present more equations and ask your students to draw models for these equations. Ask your students to

create short stories that fit their models.

Extension: Provide students with the BLM “Hanji Puzzles.” These 4 worksheets introduce the students

to Hanji Puzzles and then asks them to use what they have learned about finding the missing number in

equations.


PA3-36 Problems and Puzzles PA3-36 is a review worksheet, which can be used for practice.

Extensions:

1. An empty box or a letter can represent a number. Ask students if they can find the solutions to these

equations.

a) + 2 = 5 b) + 5 = 9 c) + 3 = 4

d) a + 4 = 10 e) a + 17 = 25 f) b + 12 = 19

g) – 3 = 2 h) – 5 = 4 i) + 7 = 3

j) a – 2 = 8 k) x – 6 = 12 l) b – 9 = 15

Advanced

m) 25 – 4 = 15 + n) 32 + 6 = – 9 o) 41 + 7 = 50 –

The equation + 2 = 6 could represent the problem: “Peter has six marbles. He has 2 more marbles

than Fran. How many marbles does Fran have?” (The box represents Fran’s amount.)

Ask students to make up a problem and then write an equation that represents the problem. Students

should recognize that it doesn’t matter what symbol they use to represent the unknown.

2. Fill in the missing numbers (from 1 to 25) so that the numbers increase by one:

HINT: The sequence can travel up, down, left, right, or in a combination of directions.

a) b) c)

1 10

4

7

13

17

4 1

7 9

15

7 9

1

5 3

17 13


Calendars _____________________________________________________________2

Colouring Exercise ______________________________________________________3

Hanji Puzzles ___________________________________________________________4

Hundreds Charts________________________________________________________8

Mini Sudoku ___________________________________________________________9

Sudoku—The Real Thing ________________________________________________11

Sudoku—Warm Up ____________________________________________________12

Venn Diagram _________________________________________________________14

PA3 Part 2: BLM List



Calendars

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

NAME OF MONTH:


NAME OF MONTH:


NAME OF MONTH:


Colouring Exercise

Colour the even numbers blue, and leave the odd numbers white.

45

93 91 3 42

5

1038

9872

1670

12

66102

4

40

1845

68

36

24 113

1

34

10620

32

14

3

27

15

17

39

53

29

31

49

120

8

2 14

2626

18

6

42

4

10 917

3

15

7

38

345

19

74

16

76

18

8

44

86

45

58

48

79

37

51

75

278

3373

17

22

46

80 56

86

20

48

52

35

6

54

88

4524

84

1351

6917

4

57

531

528243

55

45

19

57

64

26

21

47

41

23

5925

71

54

68

60

96

94

30

92 2810

90

94

33

73 71

357

9

1

What do you get? _________________________________________________________________________



Hanji PuzzlesCount the shaded squares in each row.

Write an addition sentence to fi nd out how many shaded squares altogether.

Write 2 addition sentences for the total number of shaded squares.

1

+ 0

+ 2

3

1

+ 0

+ 0

3

1

+ 0

+ 0

3

Now count by column.

3 + 1 = + + 1 =

2

1

+ 2

1 + 3 + 1 =

2

1

+ 2

1 + 3 + 1 =

+ + 1 =


Hanji Puzzles (continued)

Circle the full rows and columns.

Shade the full rows and columns. Is the right number shaded in each row and column?

5 04 03 02 0

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1 3 4 23 12 1

2

1

1

2

1

1

1

2

1

2

2

1




Finish solving these puzzles. Start by crossing out the squares you can’t shade.

1 2 3 3 3

4

5

3

4 2 3

3

2

3

1

5 3 2

1

3

1

3

2

BONUS:

4 2 4 2 3 2 2 4 2 3 2 4

12

6

12

4

Find the rows that have enough shaded. Cross out the white squares in those rows.

Then do the columns.

1 4 4 2

2

2

3

4

2 1 3

1

3

2

2 3 0 3 3

4

4

3



This Hanji puzzle is not possible. Can you see why? HINT: Try solving it.

3 4 2 1

4

2

4

0

Solve the Hanji puzzles.

STEP 1: Shade the full rows and columns.

STEP 2: Cross out the squares you can’t shade.

STEP 3: Finish the puzzle. Check your answer.

4 2 5

3

2

1

3

2

1 5 1 4

4

2

2

2

1

2 3 0 3 3

4

4

3

BONUS:

4 3 2

2

3

1

3

3 3 4

4

2

4

1

4

3 + + 2 = 3 + 1 + 3 + 2

3 2

3

1

3

2

Find the missing number. Then solve the puzzle.



Hundreds Charts

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 60

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100


Mini Sudoku

In these mini Sudoku problems, the numbers 1, 2, 3 and 4 are used.

Each number must appear in each row, column and 2 × 2 box.

When solving Sukoku problems:

1. Start with a row or square that has more than one number.

2. Look along rows, columns and in the 2 × 2 boxes to solve.

3. Only put in numbers when you are sure the number belongs there (use a pencil with

an eraser in case you make a mistake).

EXAMPLE:

Here’s how you can fi nd the numbers in the second column:

The 2 and 4 are given so we have to decide where to place the 1 and the 3.

There is already a 3 in the third row of the puzzle so we must place a 3 in the fi rst row of the

second column and a 1 in the third row.

Continue in this way by placing the numbers 1, 2, 3 and 4 throughout the Sudoku. Before you

try the problems below, try the Sudoku warm-up on the following worksheet.

1. a) b) c)

2. a) b) c)

1

2

3

4

1 3

2

1 3

4

1 4

4 3 2

4 1 3

3 4

2 1

1 4 3

4 3 2

3 4

1 4

4 1

2 3

3 2

2 1

3

1 4

1

2

4 3

4

1

3

3 2



Mini Sudoku (continued)

Try these Sudoku Challenges with numbers from 1 to 6. The same rules and strategies apply!

Bonus

3. 4.

5. 6.

1 4 3 5

6 4 5 1

2 3 6

4 1 6 2 3

5 1 2

2 5 3 4 6

5 4 2 1

1 5 6

3 6 4 5

6 4

2 3 1

2 3

4 3

6 4 5 2

6

5 1

1 5 2 3

2

2 3

4 1 2

6 5 1

4 3

6

3 5 1


Sudoku—The Real Thing

Try these Sudoku puzzles in the original format 9 × 9.

You must fi ll in the numbers from 1 through 9 in each row, column and box. Good luck!

Bonus

Super Bonus

For more Sudoku puzzles, check the puzzle section of your local newspaper!

4 2 1 8 3 9

7 1 4 6

6 7 3 2

3 6 9 5

6 2 8 4 5 1 3

8 7 6

9 4 8

1 8 3 6 5 7

5 7 1 4 2

5 3 6 7 8

2 8 6

4 7 3 9

7 1 3 4

1 4 5 2

4 9 1

2 4 6 8 7

8 3 9 5 1

9 1 4



Sudoku—Warm Up

1. Each row, column or box should contain the numbers 1, 2, 3, and 4. Find the missing

number in each set.

a) c) d) e) f)

b)

2. Circle the pairs of sets that are missing the same number.

a) b) c)

3. Find the number that should be in each shaded square below. REMEMBER: In sudoku puzzles

a number can only appear once in each row, column, or box.

a) b) c)

d) e) f)

4. Fill in the shaded number. Remember that each row, column, and box must have the numbers

1, 2, 3, and 4.

a) b) c)

4

1 31

3

2

1 3 4 1

4 2

2

4

34 1 2

3 4

2

3

2

1

2 3 4

3

4 2

4

3

1

1 3 2

2

3

2 4 3

4

4

3

1

1

4 2 1

1 2

3

1

3 4

3

4

4 2

1 2

2 1 3

3

2

4

2 4

1

4

3

4 2


Sudoku—Warm Up (continued)

d) e) f)

Bonus

Can you fi nd the numbers for other empty squares (besides the shaded ones)?

6. Try to solve the following puzzles using the skills you’ve learned.

a) b) c)

d) e) f)

7. Find the missing numbers in these puzzles.

a) b) c)

d) e) f)

Now go back and solve the mini Sudoku puzzles!

1 3

2

1

2

4

1

3

4

3 2 4

4 2 1

2 4

4 1 2

4 2 1

1 2

1 4 2

3 4

3 1

2 4

1 3

4 2

1 3 4

4 3 1

1

2 4

3 2 4

2

1 2

4

4

3 1

3 2 4

1 2

2 3

1 2 4

4 2

1 4

2

4 2

3 1

2 1 4

3 1

3

4



Venn Diagram


NS3-51 Ordinal Numbers Goal: Students will use ordinal numbers correctly.

Prior Knowledge Required: Number lines Counting

Vocabulary: ordinal numbers, position, ordinals from first to tenth.

Teach students that ordinal numbers are used to tell the position of objects.

Write the ordinal words from first to tenth on the board in order. Leave a lot of space between words. Ask

volunteers if they know what the words mean.

Hand out cards numbered 1 to 10 to ten volunteers. Ask the volunteers to line up in the order of their

number, underneath the appropriate word. The student with “1” should stand under the word “first” and so

on. Tell the class to pretend that these students are in line for tickets and ask for examples of what the

tickets might be for (a sporting event, a play, a movie theatre, and so on). Ask various volunteers to

demonstrate that they know where they are in line. EXAMPLES: Will the second person please clap their

hands? Will the 6th person please take one step forward and then one step back? Will the 8th person please

turn around in a complete circle?

Ask the class to tell you who is 3rd in line. Who is 4th? Who is 9th? Who is first? Who is last? How many

people are before the 4th person in line? How many are after the 8th person? Use your students’ names to

ask a question like “How many places before Mark is Sara?” Demonstrate counting back from Mark’s place

in line to Sara’s. SAY: “Tony is 1 place before Sara, Lisa is 2 places before Sara, Bilal is 3 places before

Sara” and so until you reach Mark.

ASK: How many are before the 8th person and after the 4th person? How many people are between the 2nd

and 4th people? Between the 2nd and 5th people? The 2nd and 7th? The 5th and 9th? Tell students that Sally

finds the number of people between the 2nd and 4th people by finding 4 – 2, so she says that there are 2

people between the 2nd and 4th people—is she right? How much is she off by? Then repeat for the other

examples: the 2nd and 5th people, the 2nd and 7th, the 5th and 9th. How much is Sally off by each time? How

can you change Sally’s answer to make it right? Then ask students to see if this strategy works for various

other pairs of small ordinal numbers. (EXAMPLES: 3rd and 8th, 2nd and 9th, 3rd and 6th, 6th and 9th. 7th and 9th,

8th and 9th.) Then challenge students to extend this pattern to larger ordinal numbers. For example, if there

are 30 people in line, have students find the number of people in between the 14th and the 28th by subtracting

and then subtracting 1, or between the 3rd and the 25th, and so on.

Review the words “vowel” and “consonant.” Then ASK: What is the 4th letter in the alphabet? What is the 4th

vowel in the alphabet? What is the 3rd letter in the word “Montreal”? What is the 3rd vowel in the word

“Montreal”? Which two letters are the same in the word “apple”? (Have students phrase their answer in terms

of ordinal numbers, i.e. 2nd and 3rd). Have students find two letters (and describe them by their ordinal

position) that are the same in each word:

a) moon b) sleep c) penny d) counting


Bonus: Make up a word that has 2 letters the same and describe the position of those letters. As an extra

challenge, make up a word with 3 letters the same and describe the position of those letters.

Have students count by 5s, starting at 5. ASK: What is the 3rd number you say? What is the 7th number you

say? Teach students to keep track by using their fingers.

ASK: What is the 4th letter you say if:

a) You say the alphabet starting from H?

b) You say the alphabet starting from V?

c) You say the alphabet backwards starting from T?

d) You say the vowels starting from A?

What is the 7th number you say if:

a) You count by 2s starting at 8?

b) You count by 2s starting at 17?

c) You count by 5s starting at 35?

d) You count by 5s starting at 49?

e) You count by 100 starting at 433?

Then draw a number line from 0 to 30, labelled with only the multiples of 10. Show students how to use the

number line to find the 7th number they say when counting by 3, starting from 3. Students can mark 3 with an

X, mark every third number with an X and then count to find the 7th X.

Activities:

1. Working in Pairs

For a random way to pair up students, give half the class cards with numbers (say 1 to 10 and 21 to 25

if there are 30 students in the class) and give the other half cards with ordinal number endings put st,

nd, rd, and th in quotation marks “st”, “nd”, “rd”, and “th” (st (2 cards), nd (2 cards), rd (2 cards) and th

(9 cards)). Have students find a partner that matches, for example, 2 matches with “nd” because the

corresponding ordinal is 2nd or second, 4 through 10 all match with “th”. Students will see quickly that

“th” is the easiest to match with.

2. Decoding Messages

Teach students how to use skip counting by 5 to find the position of letters in the alphabet quickly:

A B C D E = 5th

F G H I J = 10th

K L M N O = 15th

P Q R S T = 20th

U V W X Y = 25th

Z


ASK: What is the 22nd letter? The 14th? The 11th? The 19th? The 26th? And then, to make sure they’re paying

attention, ask: The 27th?

Have students decode messages using ordinal numbers. For example, the code for “recess” is:

18th 5th 3rd 5th 19th 19th

Some examples of messages you could encode for your students to decode include:

• To make glue, mix water and flour.

• No math homework today.

• Ordinal numbers are used to tell position.

• Ordinal number words are used for fractions too.

Students could make up a message for a partner to decode.

3. Money

The following is a sample problem from an Ontario Ministry of Education Guide for Teachers. Students

could solve the problem using play money:

Remove the 3rd coin.

Move the last coin into the second place.

Remove the 4th coin.

Move the 5th coin into 1st place.

Remove the second coin.

How much money do you have left?

Extension: Ask students to write the third letter of the word “fast,” the second letter of the word “puppy,”

the last letter of the word “mop,” the most common letter in the word “green” and then the fourth letter in the

word “fourth.” Ask students what word they spelled? Encourage them to make up their own similar puzzles,

either by using their classmates’ names as words or making up their own names.

25¢ 5¢ 5¢ 5¢ 5¢ 10¢ 10¢


NS3-52 Round to the Nearest Tens and NS3-53 Round to the Nearest Hundreds Goal: Students will round to the closest ten or hundred, except when the number is exactly half-way

between two multiples of ten or a hundred.

Prior Knowledge Required: Number lines

Concept of closer

Vocabulary: multiple

Show a number line from 0 to 10 on the board:

0 1 2 3 4 5 6 7 8 9 10

Circle the 2 and ask if the 2 is closer to the 0 or to the 10. When students answer 0, draw an arrow from the

2 to the 0 to show the distance. Repeat with several examples and then ask: Which numbers are closer to

0? Which numbers are closer to 10? Which number is a special case? Why is it a special case?

Then draw a number line from 10 to 30, with 10, 20 and 30 a different colour than the other numbers.

Circle various numbers (not 15 or 25) and ask volunteers to draw an arrow showing which number they

would round to if they had to round to the nearest ten.

Repeat with a number line from 50 to 70, again writing the multiples of 10 in a different colour. Then repeat

with number lines from 230 to 250 or 370 to 390, etc.

Ask students for a general rule to tell which ten a number is closest to. What digit should they look at? How

can they tell from the ones digit which multiple of ten a number is closest to?

Then give several examples where the number line is not given to them, but always giving them the two

choices. EXAMPLE: Is 24 closer to 20 or 30? Is 276 closer to 270 or 280?

Tell students that the multiples of 10 are the numbers they say when they start at 0 and skip counting by 10,

namely 0, 10, 20, 30, and so on. ASK: Is 70 a multiple of ten? Is 130 a multiple of 10? What about 37? How

can you tell whether or not a number is a multiple of 10?

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


ASK: Which two multiples of 10 is 37 between? (30 and 40) How can you tell? How many tens are in 37? (3)

What is one more ten? (4) So 37 is between 3 tens and 4 tens. How many tens are in 97? (9) What number

has exactly 9 tens? (90) What is one more ten than 9 tens? (10 tens) What number has 10 tens? (100) What

two multiples of 10 is 97 between? (90 and 100) How many tens are in 794? (79) Show students how they

can cover up the ones digit to find the number of tens. What is one more ten than that? (79 + 1 = 80 tens).

Which two multiples of ten is 794 between? (790 and 800)

Ensure that all students can tell you which number is another way of saying:

a) 54 tens b) 3 tens c) 10 tens d) 99 tens e) 100 tens f) 1430 tens

Have students find which two multiples of 10 the following numbers are between:

a) 53 b) 153 c) 671 d) 809 e) 998 Bonus: f) 789 432 g) 12 349 087

Then have students round each number to the nearest ten. Explain that to round a number to the nearest ten

means to find the multiple of ten that the number is closest to.

STEP 1: Decide which two multiples of ten the number is between.

STEP 2: Look at the ones digit to decide which multiple of ten the number is closest to.

a) 327 b) 411 c) 32 d) 48 e) 196 Bonus: 53 098 006

Tell students that when you round a 3-digit number to the nearest ten, you usually get a 3-digit number.

Challenge your students to find an exception. (The exceptions are 995, 996, 997, 998 and 999—995 is not

closer to either 990 or 1000; this case will be discussed in the next lesson)

Repeat the lesson with a number line from 0 to 100, that shows only the multiples of 10.

0 10 20 30 40 50 60 70 80 90 100

At first, only ask students whether numbers that are multiples of 10 (30, 70, 60 and so on) are closer to 0

or 100. (EXAMPLE: Is 40 closer to 0 or 100? Draw an arrow to show this.) ASK: Which multiples of 10

are closer to 0 and which multiples of 10 are closer to 100? Which number is a special case? Why is it a

special case.

Then include numbers that are not multiples of 10. First ask your students where they would place the

number 33 on the number line. Have a volunteer show this. Then ask the rest of the class if 33 is closer to 0

or to 100. Repeat with several numbers. Then repeat with a number line from 100 to 200 and another

number line from 700 to 800.

Review the word “multiple” and ASK: If a multiple of 10 means “the numbers you say when skip counting by

10s starting from 0,” what do you think a multiple of 100 is? Say various numbers and have students tell you

whether each number is a multiple of 100.

(EXAMPLES: 320; 1 500; 78 000; 341; 12 341; 12 300; 890)

ASK: How can you tell whether or not a number is a multiple of 100?

Remind students that to find the number of tens, we can cover up the ones digit and read the number we

see. ASK: How can we find the number of hundreds in a number? What digits should we cover up? (Cover

up the ones and tens digits.)


Have students find the number of hundreds in various numbers.

(EXAMPLES: 349; 890; 1 954; 39 876 421)

ASK: Once we’ve found the number of hundreds in a number, how can we find the two multiples of a

hundred the number is between?

Have students decide which two multiples of 100 the above examples are between.

Ask your students for a general rule to tell which multiple of a hundred a number is closest to.

What digit should they look at? How can they tell from the tens digit which multiple of a hundred a number is

closest to? When is there a special case? Emphasize that the number is closer to the higher multiple of

100 if its tens digit is 6, 7, 8 or 9 and it’s closer to the lower multiple of 100 if its tens digit is 1, 2, 3 or 4. If

the tens digit is 5, then any ones digit except 0 will make it closer to the higher multiple. Only when the tens

digit is 5 and the ones digit is 0 do we have a special case where the number is not closer to either.

Activity: Attach 11 cards to a rope so that there are 10 cm of rope between each pair of cards. Write the

numbers from 30 to 40 on the cards so that you have a rope number line. Make the numbers 30 and 40 more

vivid than the rest. Ensure that the numbers are stuck to the rope so that they cannot move. Take a ring that

can slide freely on the rope and pull the rope through it.

Ask two volunteers to hold the number line taught. Ask a volunteer to find the middle number between 30

and 40. How do you know that this number is in the middle? What do you have to check? (the distance to the

ends of the rope—make a volunteer do that). Let a volunteer stand behind the line holding the middle.

Explain to your students that the three students with a number line make a rounding machine. The machine

will automatically round the number to the nearest ten. Explain that the machine finds the closest ten. Put the

ring on 32. Ask the volunteer who is holding the middle of the line to pull it up, so that the ring slides to 30.

Try more numbers. Ask your students to explain why the machine works.


NS3-54 Rounding Goal: Students will round whole numbers to the nearest ten or hundred.

Prior Knowledge Required: Knowing which two multiples of ten or a hundred a number is between

Finding which multiple of ten or a hundred a given number is closest to

Vocabulary: rounding, multiple

Review rounding to the nearest ten when the ones digit is not 5.

Then tell your students that when the ones digit is 5, it is not closer to either the smaller or the larger ten,

but we always round up. Give them many examples to practice with: 25, 45, 95, 35, 15, 5, 85, 75, 55, 65. If

some students find this hard to remember, you could give the following analogy: “I am trying to cross the

street, but there is a big truck coming, so when I am part way across I have to decide whether to keep going

or to turn back. If I am less than half way across, it makes sense to turn back because I am less likely to get

hit. If I am more than half way across, it makes sense to keep going because I am again less likely to get

hit. But if I am exactly half way across, what should I do? Each choice gives me the same chance of getting

hit.” Have them discuss what they would do and why. Remind them that they are, after all, trying to cross

the street. So actually, it makes sense to keep going rather than to turn back. That will get them where they

want to be.

Another trick to help students remember the rounding rule is to look at all the numbers with tens digit 3 (i.e.

30-39) and have them write down all the numbers that we should round to 30 because they’re closer to 30

than to 40. Which numbers should we round to 40 because they’re closer to 40 than to 30? How many are

in each list? Where should we put 35 so that it’s fair?

Then move on to 3-digit numbers, still rounding to the nearest tens: 174, 895, 341, 936, etc.

Bonus: Include 4- and 5-digit numbers.

Then move on to rounding to the nearest hundreds. ASK: Which multiple of 100 is this number closest to?

What do we round to? Start with examples that are multiples of 10: 230, 640, 790, 60, 450 (it is not closest

to either, but we round up to 500). Then move on to examples that are not multiples of 10. (EXAMPLES:

236, 459, 871, 548)

SAY: When rounding to the nearest 100, what digit do we look at? (The tens digit). When do we round

down? When do we round up? Look at these numbers: 240, 241, 242, 243, 244, 245, 246, 247, 248, 249.

What do these numbers all have in common? (3 digits, hundreds digit 2, tens digit 4). Are they closer to 200

or 300? How can you tell without even looking at the ones digit?

Then tell your students to look at these numbers: 250, 251, 252, 253, 254, 255, 256, 257, 258, 259. ASK:

Which hundred are these numbers closest to? Are they all closest to 300 or is there one that’s different?

Why is that one a special case? If you saw that the tens digit was 5, but you didn’t know the ones digit, and

you had to guess if the number was closer to 200 or 300, what would your guess be?


Would the number ever be closer to 200? Tell your students that when you round a number to the nearest

hundred, mathematicians decided to make it easier and say that if the tens digit is a 5, you always round

up. You usually do anyway, and it doesn’t make any more sense to round 250 to 200 than to 300, so you

might as well round it up to 300 like you do all the other numbers that have tens digit 5.

Then ASK: When rounding a number to the nearest hundreds, what digit do we need to look at? (The tens

digit.) Then write on the board:

Round to the nearest hundred: 234 547 651 850 493

Have a volunteer underline the hundreds digit because that is what they are rounding to.

Have another volunteer write the two multiples of 100 the number is between, so the board

now looks like:

Round to the nearest hundred: 234 547 651 850 973

200 500 600 800 900

300 600 700 900 1000

Then ask another volunteer to point an arrow to the digit they need to look at to decide whether

to round up or round down. Ask where is that digit compared to the underlined digit? (It is the

next one). ASK: How do you know when to round down and when to round up? Have another volunteer

decide in each case whether to round up or down and circle the right answer.

Tell students that most 3-digit numbers, when rounded to the nearest hundred, will have 3 digits. Which

numbers will be exceptions? (any number from 950 to 999)

Extension: Ask students to round a 3-digit number to all possible places.

EXAMPLE: 1382

1000

thousands

1400

Hundreds

1380

tens


NS3-55 Estimating Sums and Differences Goal: Students will estimate sums and differences by rounding each addend to the nearest ten or

hundred.

Prior Knowledge Required: Rounding to the nearest ten or hundred

Adding and subtracting

Vocabulary: the “approximately equal to” sign ( ≈ ), estimating

Show students how to estimate 52 + 34 by rounding each number to the nearest ten: 50 + 30 = 80.

SAY: Since 52 is close to 50 and 34 is close to 30, 52 + 34 will be close to, or approximately, 50 + 30.

Mathematicians have invented a sign to mean “approximately equal to.” It’s a squiggly “equal to” sign: ≈.

So we can write 52 + 34 ≈ 80. It would not be right to put 52 + 34 = 80 because they are not actually equal;

they are just close to, or approximately, equal.

Tell students that when they round up or down before adding, they aren’t finding the exact answer, they are

just estimating. They are finding an answer that is close to the exact answer. ASK: When do you think it

might be useful to estimate answers?

Have students estimate the sums of 2-digit numbers by rounding each to the nearest ten. Remind them to

use the ≈ sign.

EXAMPLES:

41 + 38 52 + 11 73 + 19 84 + 13 92 + 37 83 + 24

Then ASK: How would you estimate 93 – 21? Write the estimated difference on the board with students:

93 – 21 ≈ 90 – 20

= 70

Ensure that students can add and subtract multiples of 10 (EXAMPLES: 30 + 20, 70 – 40, 130 – 50). Have

students estimate the differences of 2-digit numbers by again rounding each to the nearest ten.

EXAMPLES:

53 – 21 72 – 29 68 – 53 48 – 17 63 – 12 74 – 37

Then have students practise estimating the sums and differences of:

• 3-digit numbers by rounding to the nearest ten (EXAMPLES: 421 + 159, 904 – 219).

• 3- and 4-digit numbers by rounding to the nearest hundred (EXAMPLES: 498 + 123, 4 501 – 1 511).

Ensure that students can add and subtract multiples of 100 (EXAMPLES: 300 – 100, 600 + 300,

800 – 200)


Bonus: Students who finish quickly may add and subtract larger numbers, rounding to tens, hundreds, or

even thousands.

Teach students how they can use rounding to check if sums and differences are reasonable.

EXAMPLE:

Daniel added 273 and 385, and got the answer 958. Does this answer seem reasonable?

Students should see that even rounding both numbers up gives a sum less than 900, so the answer can’t

be correct. Make up several examples where students can see by estimating that the answer cannot be

correct.


NS3-56 Estimating Goal: Students will solve word problems by estimating rather than calculating.

Prior Knowledge Required: Estimating sums and differences by rounding

Reading word problems and knowing when to add or subtract

Vocabulary: estimating

Tell students that you want to estimate how many apples were sold altogether if 58 red apples were sold

and 21 green apples are sold. Tell students that it would be easier to add these numbers if they were

multiples of 10 and ASK: Are these numbers close to multiples of 10? What is the closest multiple of 10 to

58? (60) To 21? (20) What is 60 + 20? (80) Do you think that 80 is a good estimate for 58 + 21? Will

58 + 21 be close to 60 + 20? Why? What is the actual answer to 58 + 21? (79) Was 80 a good estimate?

Have students estimate the total number of apples sold in these situations:

a) 27 red apples were sold and 42 green apples were sold;

b) 46 red apples were sold and 78 green apples were sold;

c) Jenn sold 52 apples and Rita sold 31 apples;

d) Jenn sold 42 apples and Rita sold 29 apples.

Write out some of the answers on the board, using the “approximately equal to” sign

(EXAMPLE: 27 + 32 ≈ 30 + 30 = 60).

Write the following question on the board:

About how many more green apples than red apples were sold in part a)?

ASK: What word in that question tells you I only want an estimate? Does the question ask for the sum of the

numbers of green and red apples or the difference between them? How do you know? What operation

should I use to find the difference—addition or subtraction? Tell students that you would find it easier to

subtract if the numbers were multiples of 10. ASK: What multiples of 10 are closest to the number of red

and green apples? If there are about 30 red apples and about 40 green apples, about how many more

green apples were sold than red apples?

Have students estimate how many more green apples were sold than red apples in b), and then how many

more apples Jenn sold than Rita in c) and d).

Tell students (and write on the board): Greg collected 37 stamps, Ron collected 72 stamps, and Sara

collected 49 stamps. ASK: How many more stamps did Ron collect than Sara? How many more did Sara

collect than Greg? How many more stamps does Ron have than Greg?

Give students similar problems with 3-digit numbers, asking them to round to the nearest ten to estimate the

answer. Then give problems with 3- and 4-digit numbers and ask students to round to the nearest hundred.


Ask students to estimate this sum: 27 + 31. Then ask them to find the actual answer. What is the difference

between the estimate and the actual answer? Which number is larger, the estimate or the actual answer?

How much larger? What operation do they use to find how much more one number is than another? (They

should subtract the smaller number from the larger number.) Repeat with sums and differences of:

2-digit numbers, rounding to the nearest ten

(EXAMPLES: 39 + 41, 76 – 48).

3-digit numbers, rounding to nearest ten

(EXAMPLES: 987 – 321, 802 + 372).

3- and 4-digit numbers, rounding to nearest hundred

(EXAMPLES: 3 401 + 9 888, 459 – 121).

Draw on the board:

ASK: How many balls are in each box? If I want to know how many are in 4 boxes, what is an easier number

to multiply 4 by that is close to 12? (10) What makes that number easy to multiply by? About how many balls

are in 4 boxes? (40).

Repeat with the following pictures, having students estimate how many balls are in 7 boxes:

a) b) c)

NOTE: Encourage your students to use estimating to judge the reasonableness of their answers. Give them

the following questions and ask them to tell you what they would estimate the answer will be before they

perform the operation.

a) 382 + 217 b) 427 + 604 c) 923 – 422 d) 875 – 215


Activities:

1. Count the number of slices in an orange and estimate the number of slices in 6 oranges.

2. Take a handful of counters and drop them onto a sheet of paper folded into 4 equal regions. Count the

number of counters in one of the four regions and estimate the number of counters you dropped.

3. Fill up a clear plastic container with beans or small blocks. Ask students to estimate the number of beans

or blocks in the container. Ask them how they arrived at their estimate. Did they count out 10 blocks to

use 10 as the known quantity? Or 100 blocks? 10 beans or 100 beans?

Extensions:

1. Estimate the number of pages in your JUMP Math Part 2 workbook. Note that the page number on the

last page of the workbook shows the number of pages in both Part 1 and Part 2, so students will need to

round both this number and the number of pages in their Part 1 workbook and then subtract. To guide

students, ASK: What page does Part 2 end at? What page does it start at? So what page does Part 1

end at? How many pages are in both Part 1 and Part 2 together? How many pages are in just Part 1?

How can we find the number of pages in just Part 2? What operation should we use? To make the

numbers easier to subtract, what numbers close to these numbers are easier to work with?

2. Estimate the number of pages in all the JUMP Math Part 2 workbooks in the class. Hint: Round the

number of pages to the nearest hundred and the number of workbooks to the nearest ten.

The following three extensions are adapted from the Atlantic Curriculum Guide (A2):

3. Which estimate is closer to find 46 + 25? 50 + 20 OR 50 + 30 How do you know?

4. Is the following estimate for 82 – 47 too high or too low: 80 – 50? How do you know?

5. If you have a loonie, do you have enough money to buy:

• A pencil for 12¢

• An eraser for 25¢

• A notebook for 29¢

• A pen for 19¢

Explain your strategy.


NS3-57 Mental Math and Estimation Goal: Students will use doubles to add mentally.

Prior Knowledge Required: Counting by 2s

Doubling 2-digit numbers

Relationship between skip counting and multiplying

Arrays

Vocabulary: Double, double plus one, double minus one

Review doubling (see NS3-38) with your students.

Prepare 20 paper circles to use as counters and to tape to the board. Draw 2 circles and put 5 counters in

each of the two circles. ASK: What double have I shown? (Double 5)

Draw two new circles and put 3 counters in one circle and 5 in another. ASK: What addition statement does

this show? (3 + 5 = 8) Challenge students to think of a way to move one of the counters so that there is the

same number in each circle. What double have you shown? (4 + 4 = 8)

Repeat with larger numbers. EXAMPLES: 8 + 6, 7 + 9, 5 + 7, 11 + 9. Bonus: Move two counters to change

7 + 11 into a double.

Teach students to change addition statements into doubles without using counters to help them. Emphasize

that when you move a counter from one pile to the other, you are adding 1 counter to one pile and

subtracting 1 counter from the other pile, so 8 + 6 becomes 8 – 1 + 6 + 1 = 7 + 7.

Have students change 7 + 5 to a double, then 6 + 8, then 10 + 12, then 33 + 31, then 62 + 60.


Tell your students that the four counters are in front of a mirror. Ask a volunteer to draw what they would see

in a mirror on the other side of the dotted line. Have another volunteer write an addition sentence based on

the number of circles on one side of the mirror, and the total number of circles they see altogether. Do they

see a double anywhere?


Do more examples of this and then ask what happens if we put a circle over here outside the range of

the mirror:

Have a volunteer show where to draw the circles we would see in the mirror. Ask how many circles there

appear to be altogether. Suggest two different number sentences: 4 + 4 + 1 = 9 and 5 + 4 = 9 and ask your

students to tell you what you are thinking by each addition sentence. Ask what number is being doubled and

what they think you mean by a double plus one. Repeat with several other examples of doubles plus one.

Then write the sum: 3 + 4 and ask if that can be written as a double plus one. What number would be

doubled? Write on the board:

3 + 4 = + + 1 and tell students that you want to put the same number in each box. Ask them

how writing the sum this way can help them add 3 + 4. Show students how they can make a doubles chart if

they don’t have the doubles memorized:

0 1 2 3 4 5 6 7 8 9 10

0 2 4 20

Have students guide you in finishing this double’s chart. Then demonstrate how to use the chart to find

doubles:

Have students use the method above to find the following sums: 8 + 7; 4 + 5; 9 + 10; 9 + 8.

Bonus: 23 + 24; 35 + 36; 57 + 56;

NOTE: Students might also try finding the sums above by doubling the larger number and subtracting one:

7 + 6 = 7 + 7 – 1 = 14 – 1 = 13.

Teach students to subtract numbers from 100 using the following mental math strategy. Ensure that

students can …

1) Subtract single-digit numbers easily from 10 (EXAMPLE: 10 – 4 = 6) See the Modified Go Fish game in

the Mental Math section of this guide.

2) Subtract 2-digit multiples of 10 from 100 (EXAMPLE: 100 – 40 = 60)

3) Subtract single-digit numbers from multiples of 10 (EXAMPLE: 80 – 4 = 76)

4) Subtract two-digit numbers from 100 (EXAMPLE: 100 – 74 = 100 – 70 – 4 = 30 – 4 = 26)

Bonus: Subtract 2-digit numbers from multiples of 100 (EXAMPLE: 800 – 74 = 726)

Teach students to find sums by adding the digits separately. Ensure that students can …

1) Add 2 single-digit numbers (EXAMPLE: 7 + 8 = 15)

2) Add 3 single-digit numbers (EXAMPLE: 6 + 7 + 9 = 22)


3) Separate the digits (EXAMPLES: 48 = 40 + 8; 132 = 100 + 30 + 2)

4) Add a 2-digit number and a 1-digit number by separating the tens and ones

(EXAMPLE: 48 + 6 = 40 + 8 + 6 = 40 + 14 = 54)

5) Add three 2-digit numbers by separating the tens and ones

(EXAMPLE: 48 + 16 + 23 = 40 + 10 + 20 + 8 + 6 + 3 = 70 + 17 = 87)

6) Add 3 numbers that include a 3-digit number

(EXAMPLE: 532 + 54 + 7 = 500 + 30 + 50 + 2 + 4 + 7 = 500 + 80 + 13 = 593)

Activities:

1. If you have miras available, students can show different examples of doubles plus one, such as 3 + 3 + 1

using counters.

2. Compete (teacher against the class) to see who can come up with the most strategies to find 78 – 29.

Some strategies include:

• 78 – 28 = 50, so 78 – 29 = 49.

• 78 – 29 = 79 – 30 (because if I have two piles of counters, one with 78 counters and one with 29

counters and I add a counter to each pile, the difference stays the same) = 49.

• 78 – 29 = 80 – 31 (adding two counters to each pile instead) = 49.

• 29, 30, 70, 78 has differences 1, 40 and 8, so the total difference is 1 + 40 + 8 = 49.

• Separating the tens and ones and then regrouping:

70 + 8 60 + 18

– 20 + 9 – 20 + 9

40 + 9 = 49.

3. Repeat Activity 2, but with adding 27 + 49. Sample strategies include:

• 27 + 50 = 77, so 27 + 49 = 76

• 20 + 49 = 69, so 27 + 49 = 69 + 7 = 76

• 27 + 49 = 20 + 40 + 7 + 9 = 60 + 16 = 76

• 25 + 50 = 75, so 27 + 49 = 75 + 2 – 1 = 76

Extensions:

1. Teach students to subtract 2-digit numbers from 100 by adding: 100 – 73 = 100 – 80 + 80 – 73. = 20 + 7.

This can be represented as:

7 20 27

73 80 100

2. Find an easy way to add: 299 + 198 + 399.

3. Tell students that the numbers 40 and 60 are partners because they add to 100. Have students find a

number which is its own partner. (50). Then have students make a chart of various multiples of ten and

find their partners. What is an easy way to find the partner of a multiple of ten? Then move on to


examples that are not multiples of ten. Have students make a chart with headings: “Number” and

“Partner that adds to 100.”

Give students various numbers to put in the first column (24, 33, 19, 47, 85, 76, 93, 4, 11). Then have

students use various numbers of their choice to add more pairs to their chart. Challenge students to look

for a pattern so that they can find an easy way to find a number’s partner without using the method

taught in class. In particular, guide students’ attention to the ones digits and the tens digits. Look at the

ones digits of the pairs of numbers. What do they notice about them? (The ones digits add to ten.) Look

at the tens digits of the pairs of numbers. What do they notice about them? (The tens digits add to 9).

Why does this make sense? If the sum is 9 tens and 10 ones, what number is that? (90 + 10 = 100).

Have students use this method to find the partner of various numbers. Start by filling in the tens digit for

them and having the students only find the ones digit, then fill in the ones digit and have them only find

the tens digit. The mix up which digit you give them and which digit they need to find. Finally, have them

find both digits.

The following four extensions were taken from Atlantic Curriculum B6:

4. Why might someone find it easier to subtract 123 – 99 than 123 – 87?

5. Which sum is closest to 500? Explain how you know.

329 + 189 329 + 217 329 + 207

6. Which difference is closest to 50? Explain how you know.

125 – 30 168 – 115 103 – 82

7. You subtracted a number in the 3 hundreds from a number in the 5 hundreds. The answer was about

100. What might the numbers have been?

8. Teach students to estimate by clustering. For example: 23 + 24 + 34 is estimated, by rounding, as

20 + 20 + 30 = 70. But if students notice that 3 + 4 + 4 is about 10, a better estimate is 70 + 10 = 80.

Similarly, 232 + 244 + 322 is about 200 + 200 + 300 + 100.


NS3-58 Sharing – Knowing the Number of Sets ERRATA NOTE: This 2-sheet spread in the workbook is accidentally titled NS3-58: Multiplication and

Division (Review) and NS3-59: Knowing the Number of Sets

Goal: Given the total number of objects divided into a given number of sets, students will identify the

number of objects in each set.

Prior Knowledge Required: Dividing equally

Word problems

Vocabulary: set, divide, equally

Divide 12 volunteers into 4 teams, numbered 1-4, by assigning each volunteer a number in the following

order: 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 1. Separate the teams by their numbers, and then ask your students what

they thought of the way you divided the teams. Was it fair? How can you reassign each of the volunteers a

number and ensure that an equal amount of volunteers are on each team? An organized way of doing this

is to assign the numbers in order: 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. Demonstrate this method and then

separate the teams by their numbers again. Does each team have the same amount of volunteers? Are the

teams fair now?

Present your students with 4 see-through containers and ask them to pretend that the containers represent

the 4 teams. Label the containers 1, 2, 3 and 4. To evenly divide 12 players (represented by a counter of

some sort) into 4 teams, one counter is placed into one container until all 12 counters are placed into the 4

containers. This is like assigning each player a number. Should we randomly distribute the 12 counters and

hope that each container is assigned an equal amount? Students should see that it makes more sense to

place the students’ counters (or name tags) into the container one at a time.

Now, suppose you want to share 12 cookies between yourself and 3 friends. How many people are sharing

the cookies? [4.] How many containers are needed? [4.] How many counters are needed? [12.] What do the

counters represent? [Cookies.] What do the containers represent? [People.] Instruct your students to draw

circles for the containers and dots inside the circles for the counters. How many circles will you need to

draw? [4.] How many dots will you need to draw inside the circles? [12.]

Draw 4 circles.

Counting the dots out loud as you place them in the circles, have your students yell “Stop” when you reach

12. Ask them how many dots are in each circle? If 4 people share 12 cookies, how many cookies does each

person get? If 12 people are divided among 4 teams, how many people are on each team? Now what do

the circles represent? Now what do the dots represent?


If 12 people ride in 4 cars, how many people are in each car? What do the circles and dots represent? Have

students suggest additional representations for circles and dots.

Assign your students practice exercises, drawing circles and the correct numbers of dots in each circle.

If it helps, allow students to first use counters and count them ahead of time so they know automatically

when to stop.

a) 12 cookies, 3 people b) 15 cookies, 5 people c) 10 cookies, 2 people

When students have mastered this, write the following word problem.

5 friends shared 20 strawberries. How many strawberries does each friend get?

Have a volunteer read the word problem out loud and then ask how many dots are needed. What is to be

divided into groups? [Strawberries.] How many circles are needed? What will the circles represent?

[Friends.] What will the amount of dots in each circle illustrate? [The number of strawberries that each friend

will receive.] Have another volunteer solve the problem for the rest of the class. Then assign your students

several word problems. Read all the word problems out loud, and remind students that they can use a

dictionary if they don’t understand a word.

EXAMPLES:

a) 3 friends picked 15 cherries. How many cherries did each friend pick?

b) Joanne shared 15 marbles among 5 people (4 friends and herself). How many marbles

did each person receive?

c) There are 18 plums on 6 trees. How many plums are on each tree?

d) There are 16 apples on 2 trees. How many apples are on each tree?

e) 20 children sit in 4 rows. How many children sit in each row?

f) Lauren’s weekly allowance is $21. What is Lauren’s daily allowance?

g) An egg carton has 12 eggs divided into 2 rows. How many eggs are in each row?

Bonus:

Have students use base ten materials for the following questions.

a) 3 friends picked 69 cherries. How many cherries did each friend pick?

b) Joanne shared 84 marbles among 4 people (3 friends and herself). How many marbles

did each person receive?

c) There are 63 plums on 3 trees. How many plums are on each tree?

d) There are 68 apples on 2 trees. How many apples are on each tree?

Activity: Students might act out their solutions to questions 5 and 6.


NS3-59 Sharing – Knowing How Many in Each Set

Goal: Given a number of objects to be divided equally and the number in each set, students will determine

the number of sets.

Prior Knowledge Required: Dividing objects into equal sets

Word problems

Vocabulary: sets, groups, divide, row, column

Distribute 12 counters to each student and have them divide the counters evenly into 3 piles, then 2 piles,

then 6 piles, and then 4 piles. Then have them divide the 12 counters first into piles of 2, then into piles of 3,

then 6, then 4. Tell your students that, last class, the question always told them how many piles (or sets) to

make. This class, the question will tell them how many to put in each pile (or in each set) and they will need

to determine the number of sets.

Tell your students that Saud has 30 apples. Count out 30 counters and set them aside. Saud wants to

share his apples so that each friend gets 5 apples. He wants to know how many people can get apples.

What can be used to represent Saud’s friends? [Containers.] Do you know how many containers we need?

[No, because we are not told how many friends Saud has.] How many counters are to be placed in each

container? [5.]

Put 5 counters in one container. Are more containers needed for the counters? [Yes.] How many counters

are to be placed in a second container? (5) How do they know? (Each container gets 5 counters because

each friend gets 5 apples.) Will another container be needed? (Yes, because there are at least 5 apples, or

counters, left) Continue until all the counters are evenly distributed. The 30 counters have been distributed

and each friend received 5 apples. How many friends does Saud have? How do they know? [6, because 6

containers were needed to evenly distribute the counters.]

Now draw dots and circles like in the last lesson. What will the circles represent? [Friends.] How many

friends does Saud have? How many dots are to be placed in each circle? [5.] Place 5 dots in each circle

and keep track of the amount used.

5 10 15 20 25 30

6 circles had to be drawn to evenly distribute 30 dots, meaning 6 friends can share the 30 apples. What is

the difference between this problem and the problems in the previous lesson? [The previous lesson

identified the number of sets, but not the number of objects in each set.]


Saud has 15 apples and wants to give each friend 3 apples. How many friends can he give apples to? How

can we model this problem? What will the dots represent? What will the circles represent? Does the

problem tell us how many circles to draw? (no) Or how many dots to draw in each circle? (yes, 3) Do we

know how many dots to draw altogether? (yes, 15) Draw 15 dots on the board and demonstrate distributing

3 dots in each circle:

There are 5 sets, so he can give apples to 5 people.

Have your students distribute 3 dots into each set, and then ask them to count the number of sets.

a)

b)

c)

Repeat with 2 dots into each set:

a)

b)

Have students draw the dots to determine the correct amount of sets.

a) 15 dots, 5 dots in each set b) 12 dots, 4 dots in each set c) 16 dots, 2 dots in each set

Bonus:

24 dots and

a) 2 dots in each set b) 3 dots in each set c) 4 dots in each set

d) 6 dots in each set e) 8 dots in each set f) 12 dots in each set

ASK: As the number of dots in each set gets bigger, what happens to the number of sets?

Activity: Students might act out their solutions to questions 5 and 6.


NS3-60 Sets Goal: Students will identify the objects to be divided into sets, the number of sets and the number of

objects in each set.

Prior Knowledge Required: Sharing equally

Vocabulary: groups or sets, divided or shared

Draw 12 children divided equally into 4 canoes, as illustrated in the worksheet. Ask your students to identify

the objects to be shared or divided into sets, the number of sets and the number of objects in each set.

Repeat with 15 apples divided into 3 baskets and 12 plates placed on 3 tables. Have students complete the

following examples in their notebook.

a) Draw 9 people divided into 3 teams (Team A, Team B, Team C)

b) Draw 15 flowers divided into 5 flowerpots.

c) Draw 10 fish divided into 2 fishbowls.

Note that, in division problems, the word that tells you what is being divided or shared will almost always

come right before the word “each” (“in each”, “on each”, “to each”, “for each”, “at each”). The word coming

after “each” is usually the set.

For instance, in the sentence: “There are 4 kids in each boat,” the word ‘”kids” comes right before the phrase:

“in each boat”. Boats are the sets and kids are being divided into sets.

Students should also think of the set as a kind of container that holds the things that are being divided

or shared.

On the board, write several phrases or sentences with the word “each” in them and ask your students to say

what is being divided or shared, and what are the sets.

a) 5 boxes, 4 pencils in each box (pencils are being divided, boxes are sets)

b) 3 classrooms, 20 students in each classroom (students are being divided, classrooms are sets)

c) 4 teams, 5 people on each team (people are being divided, teams are the sets)

d) 5 trees, 30 apples on each tree (apples are being divided, trees are the sets)

e) 3 friends, 6 stickers for each friend (stickers are being divided, friends are the sets)

f) There are 3 sides on each triangle.

g) There are 6 houses on each block.

h) There are 30 kids on each school bus.

i) There are 3 school buses for each school.

j) There are 6 schools in each neighbourhood.


To ensure contextual understanding, you might ask your students to draw some of the above situations.

Then have students use circles to represent sets and dots to represent objects to be divided into sets.

a) 5 sets, 2 dots in each set

b) 7 groups, 2 dots in each group

c) 3 sets, 4 dots in each set

d) 4 groups, 6 dots in each group

e) 5 children, 2 toys for each child

f) 6 friends, 3 pencils for each friend

g) 3 fishbowls, 4 fish in each bowl, 12 fish altogether

h) 20 oranges, 5 boxes, 4 oranges in each box

i) 4 boxes, 12 pens, 3 pens in each box

j) 10 dollars for each hour of work, 4 hours of work, 40 dollars

Bonus:

k) 12 objects altogether, 4 sets

l) 8 objects altogether, 2 objects in each set

m) 5 fish in each fishbowl, 3 fishbowls

n) 6 legs on each spider, 4 spiders

o) 3 sides on each triangle, 6 triangles

p) 3 sides on each triangle, 6 sides

q) 6 boxes, 2 oranges in each box

r) 6 oranges, 2 oranges in each box

s) 6 fish in each bowl, 2 fishbowls

t) 6 fish, 2 fishbowls

u) 8 boxes, 4 pencils in each box

v) 8 pencils, 4 pencils in each box

Activities:

1. The teacher starts by saying a sentence that describes some objects that are divided into groups.

EXAMPLE: There are 4 kids in each boat. The first student then says a sentence where the sets become

the objects being divided. For example, the first student might say: “There are 7 boats on each river.” Or

“There are 4 boats on each dock.” Or “There are 5 boats in each boathouse.” Students continue in this

way. For example, “there are 7 boathouses in each river” and then “there are 5 rivers in each province.”

2. A student makes up a division-related phrase, for instance, “There are 4 kids in each boat.” They then try

to make up a sentence where things that were previously divided are now the sets and the sets are the

things divided: ie, “Each kid has 4 boats.” (In some cases this will be impossible to do.)


NS3-61 Two Ways of Sharing Goal: Students will recognize which information is provided and which is absent among: how many sets,

how many in each set and how many altogether.

Prior Knowledge Required: Solving problems where either the number of sets or amount of

objects in each set is given

Samuel has 15 cookies. There are two ways that he can share or divide his cookies equally.

1. He can decide how many sets (or groups) of cookies he wants.

If he wants to share his cookies with two friends, he will have to divide

the cookies into 3 sets. He will draw 3 circles and place one cookie

into each circle until all 15 cookies are placed in the circles.

2. He can decide that he wants each person to receive 5 cookies.

He counts out sets of 5 cookies until he has counted all 15 cookies.

Show students how to divide 3 rows of 8 dots into 4 circles.

And so on. Or, instead of crossing out the dots, students might count the total number of dots (or multiply to

find the total number of dots) and then count as they place the dots in the circles.

Assign your students several similar problems.

a) 2 rows of 6 squares into 3 circles

b) 3 rows of 8 hearts into 6 circles

c) 1 row of 18 dots into 2 circles

d) 1 row of 12 vertical lines into 6 circles

NOTE: Some students may find it easier to draw dots instead of triangles, squares or hearts.

Then ask your students if they remember how to group the dots so that there are 4 dots in

each set, and to explain how this is different from the previous problem.

a)

b)

Have students draw 12 dots and group them so that there are…

a) 4 dots in each set

b) 2 dots in each set


c) 6 dots in each set

d) 3 dots in each set

Sometimes the problem provides the number of sets, and sometimes the problem provides the number of

objects in each set. Have your students tell you if they are given the number of sets or the number of

objects in each set for the following problems.

a) There are 15 children. There are 5 children in each canoe.

b) There are 15 children in 5 canoes.

c) Aza has 40 stickers. She gives 8 stickers to each of her friends.

Have your students draw and complete the following table for PROBLEMS a)–i). Insert a box for

information that is not provided in the problem.

What has been shared

or divided into sets?

How many sets? How many in each set?

EXAMPLE 18 6

a) 24 4

EXAMPLE: There are 6 strings on each guitar. There are 18 strings.

a) There are 24 strings on 4 guitars.

b) There are 3 hands on each clock. There are 15 hands altogether.

c) There are 18 holes in 6 sheets of paper.

Make sure your students understand that words such as “book shelves” or “tables” or “containers” or

“vehicles” might refer to the sets or the objects being divided into sets.

Ask students to decide in each statement below whether the words “book shelves” represent sets or objects

being divided into sets.

a) There are 5 books on each bookshelf.

b) There are 6 bookshelves in each room.

c) Each bookshelf has several books on it.

d) Each library has several bookshelves.

NOTE: In question 8 all of the examples tell you the number of containers (or sets). For variety assign your

students several questions which give the number of items in each set. For example:

1. Paul has 15 stamps. He put 5 on each page.

2. 12 kids sit down to dinner. There are 3 kids at each table.


NS3-62 Division and NS3-63 Dividing by Skip Counting

Goal: Students will learn the division symbol through repeated addition and skip counting.

Prior Knowledge Required: Skip counting using fingers or a number line

Sharing or dividing into sets or groups

Vocabulary: division (and its symbol ÷), divided by, dividend, divisor

Ensure that students can tell you, for various pictures: a) how many objects altogether, b) how many sets

and c) how many objects in each set.

Then write two division statements for each picture: 8 ÷ 4 = 2 and 8 ÷ 2 = 4 for the first picture and

15 ÷ 5 = 3 and 15 ÷ 3 = 5 for the second picture.

Explain that 15 objects divided into sets of 5 equals 3 sets. This is written as 15 ÷ 5 = 3 or as 15 ÷ 3 = 5 and

read as “fifteen divided by 5 equals 3” or “fifteen divided by 3 equals 5.”

Distribute 12 counters to each of your students and then ask them to divide the counters into sets of 3. How

many sets do they have? What two division statements can they write?

The following website provides good worksheets for those having trouble with this basic definition of

division:

http://math.about.com/library/divisiongroups.pdf.

Using various symbols, have your students find 12 ÷ 2, 12 ÷ 3, 12 ÷ 4 and 12 ÷ 6. For example:

So 12 ÷ 2 = 6.

Have your students illustrate each of the following division statements with two pictures.

6 ÷ 3 = 2

12 ÷ 3 = 4 9 ÷ 3 = 3 8 ÷ 2 = 4 8 ÷ 4 = 2


Then have your students write two division statements for each of the following illustrations.

A B C

D E F G H

I J K

Challenge students to find another way to group the objects in each illustration that will show the same

division statement. For example, instead of 3 groups of 4 in diagram A, students can draw 4 groups of 3;

instead of 4 groups of 1 in diagram I, students can draw 1 group of 4. You might also draw several

pictures and have students pair up the pictures that show the same division statements. (In the pictures

above, G and I both show 4 ÷ 1 = 4 and 4 ÷ 4 = 1; B and C both show 12 ÷ 2 = 6 and 12 ÷ 6 = 2).

WRITE: 15 ÷ 3 = 5 (15 divided into sets of size 3 equals 5 sets).

3 + 3 + 3 + 3 + 3 = 15

Explain that every division statement implies an addition statement. Ask your students to write the addition

statements implied by each of the following division statements. Allow them to illustrate the statement first,

if it helps.

15 ÷ 5 = 3 12 ÷ 2 = 6 12 ÷ 6 = 2 10 ÷ 5 = 2

10 ÷ 2 = 5 6 ÷ 3 = 2 6 ÷ 2 = 3

Add this number

WRITE: 15 ÷ 3 = 5 This many times.

Ask your students to write the following division statements as addition statements, without

illustrating the statement this time.

12 ÷ 4 = 3 12 ÷ 3 = 4 18 ÷ 6 = 3 18 ÷ 3 = 6

18 ÷ 2 = 9 18 ÷ 9 = 2 25 ÷ 5 = 5

Bonus:

132 ÷ 43 = 3 1700 ÷ 425 = 4 90 ÷ 30 = 3 1325 ÷ 265 = 5

Then have your students illustrate and write a division statement for each of the following

addition statements.

4 + 4 + 4 = 12 2 + 2 + 2 + 2 + 2 = 10 6 + 6 + 6 + 6 = 24

3 + 3 + 3 = 9 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21 5 + 5 + 5 + 5 + 5 + 5 = 30


Then have your students write division statements for each of the following addition statements, without

illustrating the statement.

17 + 17 + 17 + 17 + 17 = 85

21 + 21 + 21 = 63

101 + 101 + 101 + 101 = 404

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 36

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 21

Draw:

What division statement does this illustrate? What addition statement does this illustrate? Is the addition

statement similar to skip counting? Which number could be used to skip count the statement?

Explain that the division statement 18 ÷ 3 = ? can be solved by skip counting on a number line.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

How many skips of 3 does it take to reach 18? (6.) So what does 18 ÷ 3 equal? How does the number line

illustrate this? (Count the number of arrows.) NOTE: Some students might find it helpful if you start with 18

counters and count out 3 at a time, drawing an arrow for each set of 3 counters that you set aside. You

know to stop when all 18 counters are used up, or when you reach 18 on the number line.

Explain that the division statement expresses a solution to 18 ÷ 3 by skip counting by 3 to 18 and then

counting the arrows.

Ask volunteers to find, using the number line: 12 ÷ 2; 12 ÷ 3; 12 ÷ 4; 12 ÷ 6;

0 1 2 3 4 5 6 7 8 9 10 11 12

ASK: If I want to find 10 ÷ 2, how could I use this number line?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

What do I skip count by? How do I know when to stop? Have a volunteer demonstrate the solution.

Ask your students to solve the following division statements with a number line to 18.

8 ÷ 2, 12 ÷ 3, 15 ÷ 3, 15÷ 5, 14 ÷ 2, 16 ÷ 4, 16 ÷ 2, 18 ÷ 3, 18 ÷ 2

Then have your students solve the following division problems with number lines to 20 (see the BLM

“Number Lines to Twenty”). Have them use the top and bottom of each number line so that each number

line can be used to solve two problems. For example, the solutions for 6 ÷ 2 and 8 ÷ 4 might look like this:


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

6 ÷ 2 8 ÷ 4 6 ÷ 3 14 ÷ 2 9 ÷ 3 15 ÷ 3

10 ÷ 5 20 ÷ 4 20 ÷ 5 18 ÷ 2 20 ÷ 2 16 ÷ 4

Provide various solutions of division problems and have your students express the corresponding division

and addition statements. For example, students should give the statements “20 ÷ 4 = 5”

and “4 + 4 + 4 + 4 + 4 = 20” for the following number line.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Draw number lines for 16 ÷ 8, 10 ÷ 2, 12 ÷ 4, and 12 ÷ 3.

Explain that skip counting can be performed on the fingers, as well as on a number line. If your students

need practice skip counting without a number line, see the MENTAL MATH section of this teacher’s guide.

They might also enjoy the following interactive website:

http://members.learningplanet.com/act/count/free.asp

Draw 12 dots on the board. Ask your students if they can figure out how to divide a set of 12 dots into 3

equal sets by skip counting by 3s. As students count up by 3s ask them to imagine putting one dot into each

set (thus placing 3 dots altogether) every time they say a multiple of 3. The skip counting helps them keep

track of the number of dots placed altogether and the number of fingers raised helps them keep track of the

number of dots placed in each set.

“3” I’ve placed one dot in each

set (3 altogether).

“6” I’ve placed 2 dots in each set

(6 altogether).


(9 altogether).


(12 altogether).


To solve 12 ÷ 3, skip count by 3 to 12. The number of fingers it requires to count to 12 is the answer. [4.]

Perform several examples of this together as a class, ensuring that students know when to stop counting

(and in turn, what the answer is) for any given division problem. Have your students complete the following

problems by skip counting with their fingers.

a) 12 ÷ 3 b) 6 ÷ 2 c) 10 ÷ 2 d) 10 ÷ 5 e) 12 ÷ 4 f) 9 ÷ 3

g) 16 ÷ 4 h) 50 ÷ 10 i) 25 ÷ 5 j) 15 ÷ 3 k) 15 ÷ 5 l) 30 ÷ 10

Bonus:

a) 200 ÷ 50 b) 125 ÷ 25

Two hands will be needed to keep track of the count for the following questions.

a) 12 ÷ 2 b) 30 ÷ 5 c) 18 ÷ 2 d) 28 ÷ 4 e) 30 ÷ 3 f) 40 ÷ 5

Bonus:

a) 450 ÷ 50 b) 175 ÷ 25

As an extra challenge, provide problems that require counting beyond the fingers on two hands.

Bonus:

a) 22 ÷ 2 b) 48 ÷ 4 c) 65 ÷ 5 d) 120 ÷ 10 e) 26 ÷ 2

Then have your students express the division statement for each of the following word problems and

determine the answers by skip counting. Ask questions like: What is to be divided into sets? How many sets

are there and what are they?

a) 5 friends share 30 tickets to a sports game. How many tickets does each friend receive?

b) 20 friends sit in 2 rows at the movie theatre. How many friends sit in each row?

c) $50 is divided among 10 friends. How much money does each friend receive?

Have your students illustrate each of the following division statements and skip count to

determine the answers.

3 ÷ 3 5 ÷ 5 8 ÷ 8 11 ÷ 11

Without illustrating or skip counting, have your students predict the answers for the following division

statements.

23 ÷ 23 180 ÷ 180 244 ÷ 244 1 896 ÷ 1 896

Then have your students illustrate each of the following division statements and skip count to determine the

answers.

1 ÷ 1 2 ÷ 1 5 ÷ 1 12 ÷ 1

Then without illustrating, have your students predict the answers for the following

division statements.

18 ÷ 1 27 ÷ 1 803 ÷ 1 6 692 ÷ 1


Extension: Teach students that division is similar to repeated subtraction. Start with 20 counters and

take away 4 each time until there is none left. Symbolize this on the board by writing:

20 – 4 – 4 – 4 – 4 – 4 = 0

ASK: How many times did you subtract 4? What is 20 ÷ 4?

Compare this method to dividing on a number line or by skip counting backwards by 4s:

20 16 12 8 4 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


NS3-64 Division and Multiplication Goal: Students will understand the relationship between division and multiplication

Prior Knowledge Required: Relationship between multiplication and skip counting

Relationship between division and skip counting

Vocabulary: divided by

Write “10 divided into sets of 2 results in 5 sets.” Have one volunteer read it out loud, another illustrate it,

and another write its addition statement. Does the addition statement remind your students of

multiplication? What is the multiplication statement?

10 ÷ 2 = 5

2 + 2 + 2 + 2 + 2 = 10

5 × 2 = 10

Another way to express “10 divided into sets of 2 results in 5 sets” is to write “5 sets of 2 equals 10.”

Have volunteers illustrate the following division statements, write the division statements, and then rewrite

them as multiplication statements.

a) 12 divided into sets of 4 results in 3 sets. (12 ÷ 4 = 3 and 3 × 4 = 12)

b) 10 divided into sets of 5 results in 2 sets.

c) 9 divided into sets of 3 results in 3 sets.

Assign the remaining questions to all students.

d) 15 divided into 5 sets results in sets of 3.

e) 18 divided into 9 sets results in sets of 2.

f) 6 people divided into teams of 3 results in 2 teams.

g) 8 fish divided so that each fishbowl has 4 fish results in 2 fishbowls.

h) 12 people divided into 4 teams results in 3 people on each team.

i) 6 fish divided into 3 fishbowls results in 2 fish in each fishbowl.

Then ask students if there is another multiplication statement that can be obtained from the same picture as

3 × 4 = 12? How can the dots be grouped to express that 3 sets of 4 is equivalent to 4 sets of 3?

3 sets of 4 is equivalent to 4 sets of 3.

ANSWER: The second array of dots should be circled in columns of 3.

ASK: If 12 ÷ 4 = 3 and 3 × 4 = 12 are obtained from the same picture, what division statement comes from

the same picture as 4 × 3 = 12? (12 ÷ 3 = 4)


Draw similar illustrations and have your students write two multiplication statements and

two division statements for each. Have a volunteer demonstrate the exercise for the class

with the first illustration.

Demonstrate how multiplication can be used to help with division. For example, the division statement

20 ÷ 4 = _____ can be written as the multiplication statement 4 × _____ = 20. To solve the problem, skip

count by 4 to 20 and count the number of fingers it requires. Demonstrate this solution on a number line,

as well.

20 divided into skips of 4 results in 5 skips. 20 ÷ 4 = 5

5 skips of 4 results in 20. 5 × 4 = 20, SO: 4 × 5 = 20

Assign students the following problems.

a) 9 × 3 = 27, SO: 27 ÷ 9 = _____

b) 2 × 6 = 12, SO: 12 ÷ 2 = _____

c) 8 × _____ = 40, SO: 40 ÷ 8 = _____

d) 10 × _____ = 30, SO: 30 ÷ 10 = _____

e) 5 × _____ = 30, SO: 30 ÷ 5 = _____

f) 4 × _____ = 28, SO: 28 ÷ 4 = _____

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


NS3-65 Knowing When to Multiply or Divide Goal: Students will understand when to use multiplication or division to solve a word problem.

Prior Knowledge Required: Relationship between multiplication and division

Vocabulary: each, per, altogether, in total

Have your students fill in the blanks.

sets 2 sets

objects per set 5 objects per set

objects altogether 10 objects altogether

When you are confident that your students are completely familiar with the terms “set,” “group,” “for every,”

and “in each,” and you’re certain that they understand the difference between the phrases “objects in each

set” and “objects” (or “objects altogether,” or “objects in total”), have them write descriptions of the

diagrams.

2 groups

4 objects in each group

8 objects

Explain that a set or group expresses three pieces of information: the number of sets, the number of objects

in each set, and the number of objects altogether. For the problems, have your students explain which piece

of information isn’t expressed and what the values are for the information that is expressed.

a) There are 8 pencils in each box. There are 5 boxes. How many pencils are there altogether? (5 groups,

8 objects in each group, how many altogether?)

b) Each dog has 4 legs. There are 3 dogs. How many legs are there altogether?

c) Each cat has 2 eyes. There are 10 eyes. How many cats are there?

d) Each boat can fit 4 people. There are 20 people. How many

boats are needed?

e) 30 people fit into 10 cars. How many people fit into each car?

f) Each apple costs 20¢. How many apples can be bought for 80¢?

g) There are 8 triangles divided into 2 sets. How many triangles are

there in each set?

h) 4 polygons have a total of 12 sides. How many sides are on

each polygon?

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Introduce the following three problem types:

Type 1: You know the number of sets and the number of objects in each set.

Example: You have 4 sets of objects and 2 objects in each set. How many objects do you have in total?

STEP 1: Draw 4 boxes to represent the 4 sets:

STEP 2: Fill each box with 2 objects:

STEP 3: Count the number of objects: 8 objects (or “8 objects altogether” or

“8 objects in total”)

Your student can then write a multiplication statement to represent the solution: 4 � 2 = 8

ASK: If you know the number of objects in each set and the number of sets, how can you

find the total number of objects? What operation should you use—multiplication or division?

Write on the board:

Number of sets × Number of objects in each set = Total number of objects

Other examples you could use:

a) 3 sets b) 3 groups

5 objects in each set 7 objects in each group

How many objects? How many objects in total?

Type 2: You know how many objects there are altogether and how many objects there are in each set.

Example: You have 6 objects altogether and 3 objects in each set. How many sets do you have?

STEP 1: Draw the total number of objects:

STEP 2: Draw a box around three objects

at a time until you’ve put all the

objects in boxes:

STEP 3: Count the number of boxes: 2 boxes (or "sets")

Your student should then write a division statement to represent the solution: 6 ÷ 3 = 2

Again use the same multiplication statement as before.

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If there are 4 objects in each set and 12 objects in total, how many sets are there?

_____ × 4 = 12, SO: 12 ÷ 4 = 3. There are 3 sets.Other examples you could use:

a) 12 objects altogether b) 16 objects

4 objects in each group 2 objects per set

How many groups? How many sets?

Type 3: You know how many objects there are and how many sets there are.

Example: You have 10 objects and 5 sets. How many objects are there in each set?

STEP 1: Draw the total number of sets:

STEP 2: First, put one object in each set.

STEP 3: Check to see if you have placed all

the objects. If not, put one more object

in each set and continue until you have

placed all the objects:

STEP 4: Count the number of objects in each set: 2 objects in each set

Your student should then write a division statement to represent the solution: 10 ÷ 5 = 2


If there are 6 sets and 12 objects in total, how many objects are in each set?

6 × _____ = 12, SO: 12 ÷ 6 = 2. There are 2 objects in each set.

Other examples you could use:

a) 15 objects altogether b) 3 sets

5 sets 12 objects altogether

How many objects in each set? How many objects in each set?

Have students write multiplication statements for the following problems with the blank in the correct place.

a) 2 objects in each set. b) 2 objects in each set. c) 2 sets.

6 objects in total. 6 sets. 6 objects in total.

How many sets? How many objects in total? How many objects in

each set?

[ _____ × 2 = 6 ] [ 6 × 2 = _____ ] [ 2 × _____ = 6 ]

Which of these problems are division problems? [Multiplication is used to find the total number

of objects, and division is used if the total number of objects is known.]

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Assign several types of problems.

a) 5 sets, 4 objects in each set. How many objects altogether?

b) 8 objects in total, 2 sets. How many objects in each set?

c) 3 sets, 6 objects in total. How many objects in each set?

d) 3 sets, 6 objects in each set. How many objects in total?

e) 3 objects in each set, 9 objects altogether. How many sets?

When your students are comfortable with this lesson’s goal, introduce alternative contexts for objects and

sets. Start with point-form problems (EXAMPLE: 5 tennis courts, 3 tennis balls on each court. How many

tennis balls altogether?), and then move to complete sentence problems (EXAMPLE: If there are 5 tennis

courts, and 3 balls on each court, how many tennis balls are on the tennis courts altogether?).

Have your students solve these problems:

a) 20 apples; 4 baskets.

How many apples in each

basket?

b) 3 birds; 5 cages.

How many birds?

c) 18 bottles of water, 3 cases.

How many bottles in each

case?

When your student is able to distinguish between (and solve) problems of Type 1, 2, and 3 readily, you can

teach them how to solve more general word problems involving multiplication and division. Tell them to think

of a container (like a box or pot) or a carrier (like a car or a boat) as a set, and the things contained or carried

as objects in the set.

EXAMPLE: Ten people need to cross a river. A boat can hold two people. How many boats are needed to

take everyone across?

Hint: Think of the boats as sets (or boxes) and the people as objects placed in the sets. This is a problem of

Type 2, as discussed in above, ie. you know the total number of objects and the number of objects in

each set.

Draw 10 lines to represent 10 people:

Put boxes around every 2 lines (each

box represents a boat):

Count the number of boats: 5 boats

This approach also works for things that have parts (think of the things that have parts as sets, and the parts

as objects in the set).

Example: A cat has 2 eyes. How many eyes are there on 5 cats?

Hint: Note that the first statement is really saying that “Each cat has 2 eyes,” so cats are the sets and eyes

are being divided into sets. Use boxes to represent each cat and lines in each box to represent the eyes.

This is a problem of Type 1, as discussed above, i.e. you know the number of sets and the number of

objects in each set.

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Draw 5 boxes to represent the 5 cats:

Draw 2 lines in each box

(representing the eyes):

The number of lines gives you the answer: 10 eyes

This approach also works for things that have a value or a price (think of the thing with value as a set or box,

and the price or value as the objects in the set, i.e. you can think of dollars or cents as lines which you can

place inside the box representing the thing you are buying).

Example: A piece of gum costs 5 cents. You have 15 cents. How many pieces of gum can you buy?

Hint: Again, this is a problem of Type 2.

Draw 15 lines to represent 15 cents:

Put boxes around every 5 lines (each

box represents a piece of gum):

Count the number of boxes: 3 boxes (or pieces of gum)

Once you give your students enough practice with this type of problem, they should eventually see that they

simply have to divide 15 by 5 to find the answer.

Activity: Students could model their solutions to questions 5 a) and b) with counters. It is important,

however, that students also be able to solve the problems by drawing a sketch with dots or lines.

Extension: Tell your students that you met someone from Mars last weekend, and they told you that

there are 3 dulgs on each flut. If you count 15 dulgs, how many fluts are there? Explain the problem-solving

strategy of replacing unknown words with words that are commonly used. For example, replace the object in

the problem (dulgs) with students, and replace the set in the problem (flut) with bench. So, if there are 3

students on each bench and you count 15 students, how many benches are there? It wouldn’t make sense to

replace the object (dulgs) with benches and the set (flut) with students, would it? [If there are 3 benches on

each student and you count 15 benches, how many students are there?] A good strategy for replacing words

is to replace the object and the set and then invert the replacement with the same two words. Only one of the

two versions of the problem with the replacement words should make sense.

Students may wish to create their own science fiction word problems for their classmates. Encourage them

to use words from another language, if they speak another language.

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NS3-66 Remainders Goal: Students will divide with remainders using pictures.

Prior Knowledge Required: Relationships between division and multiplication, addition, skip

counting, number lines

Vocabulary: remainder R, quotient, divisor

Draw:

6 ÷ 3 = 2

7 ÷ 3 = 2 Remainder 1


9 ÷ 3 = 3


Ask your students if they know what the word “remainder” means. Instead of responding with a definition,

encourage them to only say the answers for the following problems. This will allow those students who don’t

immediately see the pattern a chance to detect it.

7 ÷ 2 = 3 Remainder _____

11 ÷ 3 = 3 Remainder _____

12 ÷ 5 = 2 Remainder _____

14 ÷ 5 = 2 Remainder _____

Challenge volunteers to find the remainder by drawing a picture on the board. This way, students who do

not yet see the pattern can see more and more examples of the rule being applied.


SAMPLE PROBLEMS:

9 ÷ 2 7 ÷ 3 11 ÷ 3 15 ÷ 4

15 ÷ 6 12 ÷ 4 11 ÷ 2 18 ÷ 5

What does “remainder” mean? Why are some dots left over? Why aren’t they included in the circles? What

rule is being followed in the illustrations? [The same number of dots is placed in each circle, the remaining

dots are left uncircled]. If there are fewer uncircled dots than circles then we can’t put one more in each

circle and still have the same number in each circle, so we have to leave them uncircled. If there are no

dots left over, what does the remainder equal? [Zero.]

Introduce your students to the word “quotient”: Remind your students that when subtracting two numbers,

the answer is called the difference. ASK: When you add two numbers, what is the answer called? In

7 + 4 = 11, what is 11 called? (The sum). When you multiply two numbers, what is the answer called? In

2 × 5 = 10, what is 10 called? (The product). When you divide two numbers, does anyone know what the

answer is called? There is a special word for it. If no-one suggests it, tell them that when you write

10 ÷ 2 = 5, the 5 is called the quotient.

Have your students determine the quotient and the remainder for the following statements.

a) 17 ÷ 3 = _____ Remainder _____ b) 23 ÷ 4 = _____ Remainder _____

c) 11 ÷ 3 = _____ Remainder _____

Write “2 friends want to share 7 apples.” What are the sets? [Friends.] What are the objects being divided?

[Apples.] How many circles need to be drawn to model this problem? How many dots need to be drawn?

Draw 2 circles and 7 dots.

To divide 7 apples between 2 friends,

place 1 dot (apple) in each circle.

Can another dot be placed in each

circle? Are there at least 2 dots

left over? So is there enough to put

one more in each circle? Repeat this

line of instruction until the diagram

looks like this:


How many apples will each friend receive? Explain. [There are 3 dots in each circle.] How many apples will

be left over? Explain. [Placing 1 more dot in either of the circles will make the compared amount of dots in

both circles unequal.]

Repeat this exercise with “5 friends want to share 18 apples.” Emphasize that the process of division and

placing apples (dots) into sets (circles) continues as long as there are at least 5 apples left to share. Count

the number of apples remaining after each round of division to ensure that at least 5 apples remain.

Have your students illustrate each of the following division statements with a picture, and then determine

the quotients and remainders.

Number in each circle

a) 11 ÷ 5 = _____ Remainder _____ Number left over

b) 18 ÷ 4 = _____ Remainder _____

c) 20 ÷ 3 = _____ Remainder

d) 22 ÷ 5 = _____ Remainder

e) 11 ÷ 2 = _____ Remainder

f) 8 ÷ 5 = _____ Remainder

g) 19 ÷ 4 = _____ Remainder

Explain to students that the word “Remainder” is sometimes written just as “R.” For example,

11 ÷ 5 = 2 R 1.

Teach students that there are many ways to think about division. To find the answer to 14 ÷ 3, students might use

any of the following methods.

a) Forming equal groups (of size 3) using a picture of 14 objects: b) Sharing 14 things (candies for instance), three apiece, among friends: c) Adding threes repeatedly. Stop before you reach 14:

There are 4 groups of 3 with 2 objects left over

3 6 9 12

You stop here, one more three would take you beyond 14 to 15. (You could show this with a number line).

You can share with 4 friends. Two are left over.


So 3 goes into 12 (and 14!) four times.

The remainder of 14 ÷ 3 is the difference between 14 and the number you reached when you were

counting up (12). So the remainder is 2.

d) Guessing and multiplying by 3 until you get close to 14:

e) Subtracting 3 from 14 repeatedly until you get close to 0 (this is not the most practical method). You can

subtract 3 four times from 14 before you reach 2 (the remainder).

Activity: Students could model their solutions to all of the questions on this worksheet with counters.

Extensions:

1. 22 kids go on a picnic. Hot dogs come in packs of 8. Buns come in packs of 12. How many packs of

buns and hot dogs should the children take if each kid wants one hot dog and bun? Will there be any

buns or hotdogs left over?

2. Which number is greater, the divisor (the number by which another is to be divided) or the

remainder? Will this always be true? Have your students examine their illustrations to help explain.

Emphasize that the divisor is equal to the number of circles (sets), and the remainder is equal to the

number of dots left over. We stop putting dots in circles only when the number left over is smaller

than the number of circles; otherwise, we would continue putting the dots in the circles. See the

journal section below.

Which of the following division statements is correctly illustrated? Can one more dot be placed

into each circle or not? Correct the two wrong statements.

15 ÷ 3 = 4 Remainder 3 17 ÷ 4 = 3 Remainder 5 19 ÷ 4 = 4 Remainder 3

Without illustration, identify the incorrect division statements and correct them.

a) 16 ÷ 5 = 2 Remainder 6 b) 11 ÷ 2 = 4 Remainder 3 c) 19 ÷ 6 = 3 Remainder 1

3. Explain how a diagram can illustrate a division statement with a remainder and a multiplication

statement with addition.


3 × 4 + 2 = 14

4 × 3 = 12 5 × 3 = 15

This is too high, so 3 divides into 14 four times (with 2 remainder).


Ask students to write a division statement with a remainder and a multiplication statement

with addition for each of the following illustrations.

4. Compare mathematical division to normal sharing. Often if we share 5 things (say, marbles) among 2

people as equally as possible, we give 3 to one person and 2 to the other person. But in mathematics, if

we divide 5 objects between 2 sets, 2 objects are placed in each set and the leftover object is

designated as a remainder. Teach them that we can still use division to solve this type of real-life

problem; we just have to be careful in how we interpret the remainder. Have students compare the

answers to the real-life problem and to the mathematical problem:

a) 2 people share 5 marbles (groups of 2 and 3; 5 ÷ 2 = 2 R 1)

b) 2 people share 7 marbles (groups of 3 and 4; 7 ÷ 2 = 3 R 1)

c) 2 people share 9 marbles (groups of 4 and 5; 9 ÷ 2 = 4 R 1)

ASK: If 19 ÷ 2 = 9 R 1, how many marbles would each person get if 2 people shared 19 marbles?

Emphasize that we can use the mathematical definition of sharing as equally as possible even when the

answer isn’t exactly what we’re looking for. We just have to know how to adapt it to what we need.

5. Find the mystery number. I am between 22 and 38. I am a multiple of 5. When I am divided by 7 the

remainder is 2.

6. Have your students demonstrate two different ways of dividing…

a) 7 counters so that the remainder equals 1.

b) 17 counters so that the remainder equals 1.

Journal

The remainder is always smaller than the divisor because…


NS3-67 Multiplication and Division and NS3-68 Multiplication and Division (Review)

Goal: Students will consolidate their learning on multiplication and division done so far.

Vocabulary: fact family, multiple, division, remainder, twice as many

Tell students that you have 13 apples and you have divided them into equal groups so that there is

one left over:

ASK: What division statement does this picture show? Then challenge students to find another way to divide

the apples so that there is only 1 left over. (Students might divide the apples into 2, 3 or 4 circles.) Take up

the various solutions.

Draw a 2 × 3 array on the board:

ASK: What multiplication sentence does this show? Challenge students to find another array that shows a

similar multiplication sentence but uses twice as many dots. (Review the phrase “twice as many” if needed.)

ASK: What addition sentence does this show:

ANSWER: 3 + 4 = 7. To guide students, ASK: How many dots are on the left side of the vertical line? How

many are on the right side? How many are there altogether?

Repeat with several similar examples. Then tell students that you are going to show them a more

complicated picture. This time the left side and the right side both show multiplication statements. Can they

tell which ones?


The left side shows 3 × 2 and the right side shows 4 × 2. ASK: What does the whole picture show? What

multiplication statement is shown by all 7 circles? (7 × 2) Emphasize that this picture shows that seven sets

of 2 equals three sets of 2 plus four sets of 2. Do they think that 7 sets of 3 will equal three sets of 3 plus four

sets of three? Have students draw the picture to show this. Then have students change their picture, one

step at a time to show that:

a) seven sets of 3 equals two sets of 3 plus five sets of 3 (move the vertical line one place left – erase the

vertical line that exists and add a new one)

b) eight sets of 3 equals two sets of 3 plus six sets of 3 (add another circle to the right of the line)

c) ten sets of 3 equals four sets of 3 plus six sets of 3 (add two more circles to the left of the line)

d) ten sets of 4 equals four sets of 4 plus six sets of 4 (add one more dot in each circle)

e) ten sets of four equal three sets of 4 plus one set of 4 plus six sets of 4 (add another vertical line after the

third circle; do not erase the vertical line that exists)

Have students practise doing puzzles similar to question 10 on the worksheet.

Use the numbers 3, 4 and 6 once each to make the sentences true:

× + = 18 × + = 22

× + = 27 × – = 21

÷ + = 6 × – = 6

× ÷ = 2 + – = 7

Bonus: Have students make up a similar puzzle for a partner to solve. They can use any operation, but if

they use multiplication or division, it must only be the first operation used; addition and subtraction can be

either first or second. (This condition avoids potential problems with the order of operations, which they are

not required to know at this point.)

Bonus: Make 0 using the numbers 2, 3 and 6 once each in as many ways as you can.

(EXAMPLES: 2 × 3 – 6 = 0, 3 × 2 – 6 = 0, 6 ÷ 3 – 2, 6 ÷ 2 – 3) Note: Students should not be taught the order

of operations at this point. That 6 – 2 × 3 is also 0 should not be discussed.

The activity and extensions 5-8 below are designed to satisfy the Atlantic Curriculum.

Activity: This activity is best done after completing the extensions below. Play a game with your students

to see who (you or your class) can come up with the most multiplication or division questions that you could

solve in one or two steps just from knowing 5 × 6 = 30. Sample questions:

• 6 × 5 = 30

• 5 × 7 = 5 × 6 + 5 = 30 + 5 = 35

• 6 × 6 = 5 × 6 + 6 = 30 + 6 = 36

• 30 ÷ 5 = 6

• 30 ÷ 6 = 5


• 60 ÷ 10 = 6

• 90 ÷ 15= 6

• 120 ÷ 20 = 6

• 60 ÷ 5= 12

• 5 × 12 = 60

• 10 × 6 = 60

• 4 × 6 = 30 – 6 = 24

Extensions:

1. Ask students to write a full fact family for a division problem (including 2 addition statements).

For example: The fact family for 14 ÷ 2 = 7 is 14 ÷ 7 = 2, 7 × 2 = 14, 2 × 7 = 14,

2 + 2 + 2 + 2 + 2 + 2 + 2 = 14, and 7 + 7 = 14.

2. Ask your students to complete the following story problems with their own numbers and solve the

problems.

a) Janice had _____ apples. She shared them equally with _____ friends. How many did each

person get?

b) Tim had _____ boxes. He placed _____ watermelons in each box. How many watermelons did he

have altogether?

3. The BLMs “Always, Sometimes, or Never True (Numbers)” and “Define a Number” will help students

sharpen their understanding of numbers.

4. (Adapted from the Atlantic Curriculum) Show students how 5 sets of 3 can be broken down into subsets

in various ways:

• 4 sets of 3 and 1 set of 3

• 3 sets of 3 and 2 sets of 3

• 5 sets of 2 and 5 sets of 1


Challenge students to write multiplication and division statements that relate to these pictures. For example,

we might write:

• 4 × 3 + 1 × 3 = 5 × 3 or 12 ÷ 3 + 3 ÷ 3 = 15 ÷ 3

• 3 × 3 + 2 × 3 = 5 × 3 or 9 ÷ 3 + 6 ÷ 3 = 15 ÷ 3

• 5 × 2 + 5 × 1 = 5 × 3 or 10 ÷ 5 + 5 ÷ 5 = 15 ÷ 5

5. Ask students what happens to the product when you double one of the numbers in a

multiplication statement:

2 × 3 4 × 3 or 2 × 3 2 × 6

ANSWER: The product doubles.

What happens to the product when you multiply one of the factors by 3? (The product is multiplied by 3.)

Have students investigate what this means for division. For example, 2 × 3 = 6, so 2 × 6 = 12 becomes, in

terms of division: 6 ÷ 3 = 2, so 12 ÷ 6 = 2. Note that doubling both terms of the division statement will keep

the answer the same. Ask students how they could use doubling both terms to find 45 ÷ 5.

6. Remind students that, from Extension 5, the quotient remains the same when both the dividend and the

divisor are multiplied by 2 (or when both are multiplied by 3, or both multiplied by 4, and so on). Ask

students to investigate what happens to the quotient when the dividend is multiplied by 2 and the divisor

remains the same. (The quotient doubles.) What happens to the quotient when the divisor is multiplied

by 2 and the dividend remains the same? (The quotient is divided in half.) Tell students that 21 ÷ 7 = 3.

ASK: What is 42 ÷ 7? 84 ÷ 7? 168 ÷ 3? How does knowing 60 ÷ 6 help you to know 30 ÷ 6?

7. Teach students how to estimate products. Draw a number line as follows:

3 × 0 3 × 10

0 3 6 9 12 15 18 21 24 27 30

ASK: Is 3 × 7 closer to 3 × 0 or 3 × 10? Have a volunteer circle 3 × 7. Have volunteers circle 3 × 1,

3 × 9, 3 × 5. Then draw on the board the following number line:

4 × 0 4 × 10

0 40


ASK: Is 4 × 8 closer to 4 × 0 or 4 × 10? Is 4 × 8 closer to 0 or 40? How much closer? Have a volunteer mark

where 4 × 8 occurs with an X. Then have another volunteer mark where they think 10, 20 and 30 occur.

Which two multiples of ten is 4 × 8 between? Which multiple of ten is 4 × 8 closest to? If you were to

estimate 4 × 8 to the nearest ten, which multiple of ten would you pick?

4 × 0 4 × 10

0 10 20 30 40

Have students use the same method and the following number line to estimate 3 × 17:

3 × 10 3 × 20

30 60

Where are 40 and 50?

8. Teach students how to estimate quotients. For example, draw the following number line:

20 40 60 80

5 10 15 20

Tell your students that the bottom numbers were all obtained from the top numbers in the same way.

Can they see how? (divide by 4) Bring to the students’ attention that the numbers on the bottom, just like

the numbers on the top, go up by a fixed amount. When the numbers on the top skip count by 20, and

you divide by 4 to get the numbers on the bottom, then the numbers on the bottom skip count by

20 ÷ 4 = 5. ASK: What do they think 35 ÷ 4 will be close to? Have students mark where 35 is on the top

and then use that to mark where they think 35 ÷ 4 will be on the bottom. What is the closest whole

number to 35 ÷ 4?


NS3-69 Patterns Made with Repeated Addition Goal: Students will use their calculator to discover patterns made from repeated addition.

Prior Knowledge Required: Addition, patterns

Give each student a calculator. If available, photocopy a calculator onto an overhead transparency, so that

students can see all the buttons. Tell students to use their calculator to find 7 + 2. Have a volunteer show on

the overhead the buttons they pressed to find the answer (7, +, 2, =). Then have all students press the “=”

button again. What number showed up? Why? What operation did the calculator do to get from 9 to 11?

Again, have students press the “=” button. What number does the calculator show? Challenge students to

predict the next number the calculator will show when they press the “=” button, and then the next number.

Explain to students that pressing 7, +, 2, =, =, =, =, =, results in: 7 + 2 + 2 + 2 + 2 + 2.

Challenge students to find, by pressing “=” the correct number of times:

a) 3 + 2 + 2

b) 5 + 3 + 3 + 3

c) 3 + 5 + 5 + 5

d) 8 + 2 + 2 + 2 + 2 + 2 + 2

e) 7 + 3 + 3 + 3 + 3

f) 8 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9

Then show students how to record each step of the process:

8 8 + 9 = 17 8 + 9 + 9 = 26 8 + 9 + 9 + 9 = 35, and so on.

Write the sequence of results, all in a row:

8 17 26 35 44 53 62 71 80 89 98 107

Have students record the patterns of ones digits only:

8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7.

Can your students predict what the next ones digit will be? Have them explain their answer.

Have students record the patterns in the number of tens:

0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10.

Tell your students that you showed this pattern to someone who said that the numbers always increase by 1.

Tell them to look closely at the sequence and to tell you if that is correct. (No, the 8 is repeated) Have

students extend the pattern on their calculators to find out when the next repetition is. ASK: How many

numbers occur before the first repetition? How many numbers come before the second repetition? How

many numbers do they think will occur before the next repetition? Have students write what they think the

next ten terms of the sequence (of the number of tens) will be and then to check their answer using the

calculator.

Activity: Review the Skip Counting Machine activity from section NS3-13: Counting by 5s and 25s.

Have students build a skip counting by 9s machine, starting from any number. Students may also enjoy

building a skip counting by 11s machine.


NS3-70 Counting by Dollars and Coins Goal: Students will use dollar and cent notation for money amounts that involve either only cents or only

dollars. Students will skip count to add money amounts that consist of a single type of coin or dollar bill.

Review the Canadian coins: pennies, nickels, dimes and quarters, and then introduce loonies and toonies.

Give students play coins. Ask them if the amount of money is shown on the coins. Does every coin have the

amount written on it? Is the value of the coin printed on both sides or just one? Ask them if there is a word

that shows the number is talking about money and not how many trees, cats, or houses. Tell them there are

two different words used to show money and challenge them to find both of them (they are written on the

coins). Ask them how cents and dollars are like units in measurement. Remind them that if something is 5

paper clips long, it is shorter than if it is 5 notebooks long. Tell them that if something is worth 3 cents, it is

worth less than if it is worth 3 dollars. A dollar is worth a hundred cents, so if Isobel has 2 dollars and Soren

has 30 cents, Isobel has more money even though 2 is less than 30. If Isobel is 2 m tall and Soren is 30 cm

tall, who is taller?

Write on the board:

“1 dollar = 100 cents and 2 dollars = 200 cents.” Have students look at the amount on their coins and

arrange them in order from least value to most value, keeping in mind that a dollar is worth a hundred cents.

Then have them arrange the coins in order from smallest to largest in size. Which coin is out of place?

To introduce the symbols $ and ¢, tell students that in math we use the words plus, minus, and equals a lot

and ask if there is a symbol we use instead of writing out the words all the time. Write on the board the words

“plus,” “minus,” and “equals” and have volunteers write the symbols used to show those words. Ask them if

there are words we use a lot when talking about money. Ask if they think those words should have a symbol

for them. Ask if anyone knows what the symbols are. Then write on the board, “$1 = 100¢ and $2 = 200¢.”

Teach students that when we count money in cents, we write the cent sign after the amount of cents, but

when we count money in dollars, we write the dollar sign before the amount of dollars. Have students

practice by using the correct notation for the following money amounts (do not include money amounts that

involve both dollars and cents).

a) five cents b) twelve dollars c) 9 cents d) 9 dollars e) 83 dollars f) 46 cents

Draw a circle and write “5¢” inside. ASK: Which coin does this represent? Repeat for several Canadian coin

values (EXAMPLES: 1¢, $1, 10¢, $2, 25¢)

Review skip counting. Then, have students skip count to find the total amount of money:

a) 5¢ 5¢ 5¢ 5¢

b) 10¢ 10¢ 10¢ 10¢


c) 25¢ 25¢ 25¢ 25¢

d) 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢

e) $1 $1 $1 $1 $1 $1 $1

f) $2 $2 $2 $2 $2

To guide students, have them write the subtotal over each coin. For example:

5¢ 10¢ 15¢ 20¢

5¢ 5¢ 5¢ 5¢

Then teach students to skip count without writing the subtotals. Students should count how many coins there

are and then skip count on their fingers to find the total amount of money.

Present a problem: Jane emptied her piggy bank. She asked her elder brother John to help her count the

coins. He suggested she stack the coins of the same value in different stacks and count each stack

separately. Jane has:

19 pennies 8 nickels 7 dimes

3 quarters 6 loonies 3 toonies

How much money is in each stack? Ask students to write the amount in dollar notation when counting

loonies and toonies and to write the amount in cent notation when counting pennies, nickels, dimes and

quarters. Give them more practice questions, such as:

9 pennies 14 nickels 13 dimes

6 quarters 5 loonies 9 toonies

Then ask students how much money they have if they have 6 nickels. ASK: What do I skip count by? How do

I know when to stop? Repeat with several examples of a single type of coin. (EXAMPLES: 3 quarters, 5

dimes, 7 loonies, 4 nickels, 3 toonies).

Then tell students that they are only allowed to use one type of coin. ASK: How can I make 8 cents? (Use 8

pennies.) How can I make 10 cents? (Use 10 pennies or 2 nickels or 1 dime.) How many I make 20 cents?

25 cents? 50 cents? 3 dollars? 4 dollars? Students need only find 1 solution. Bonus: Faster students should

be challenged to find several solutions or even all solutions.

Then write the amount in words that you want the students to make, again using only 1 type of coin.

(EXAMPLES: eight cents, eighty cents, five dollars, forty-five cents, fifty cents) Bonus: Students can try to

make these amounts using more than one denomination. For example, five dollars can be made with a

toonie and three loonies or two toonies and a loonie, or twelve quarters and a toonie, and so on.


Activities:

Heads and Tails Give students play money coins – one of each: penny, nickel, dime, quarter, loonie, and

toonie. Ask them to turn all the coins so that the heads side is up. Then tell them to take a white sheet of

paper and fold it in half so that the fold line separates the top to the bottom. They then unfold the paper and

place all the coins under the top half of the page and they rub a pencil over the paper so that they can see

the coin images. When this is done, they turn the coins around so that the tails side is showing, rearrange

the coins, place the coins under the bottom half of their sheet, and rub the pencil over the paper again. They

should then match each heads side with the tail of the same coin.

Pick the Right Coin Give students a bag of coins including pennies, nickels, dimes, and quarters. Ask

students to try to pick out a dime without looking. Repeat with picking a penny, a nickel and a quarter. What

characteristics are they looking for? What is the easiest coin to pick out? Why is it the easiest? What is the

hardest coin to pick out?


NS3-71 Dollars and Cent Notation Goal: Students will express monetary values less than a dollar in dollar and cent notation.

Prior Knowledge Required: Place value to tenths and hundredths

Canadian coins

Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, tenths, notation, hundredths

Have students show how to make various amounts of money using dimes and pennies (EXAMPLES:

35¢ = dimes and pennies, 22¢, 25¢, 30¢, 36¢, 41¢)

Explain that there are two standard ways of representing money. The first is cent notation: you simply write

the number of cents or pennies that you have, followed by the ¢ sign. In dollar notation, “twenty-five cents”

is written $0.25. The first number to the right of the decimal represents the number of dimes and the next

digit to the right represents the number of pennies.

25¢ = $0.25

2 dimes 5 pennies

Have students write both the cent notation and the dollar (decimal) notation for each amount shown above.

Have students write the total value of a collection of coins both in dollar and cent notation (begin with only

one type of coin, then two types, then three types and finally four types of coins).

a) 25¢, 25¢ b) 10¢, 10¢, 10¢ c) 10¢, 10¢, 10¢, 5¢, 5¢, 5¢

d) 10¢, 10¢, 1¢, 1¢, 1¢, 1¢, 1¢, 1¢ e) 10¢, 10¢, 5¢, 1¢, 1¢, 1¢ f) 25¢, 25¢, 10¢, 10¢, 1¢

g) 25¢, 5¢, 5¢, 5¢, 1¢ h) 25¢, 25¢, 25¢, 10¢, 1¢, 1¢ i) 25¢, 10¢, 10¢, 5¢, 1¢, 1¢

j) 25¢, 10¢, 10¢, 10¢, 5¢, 1¢, 1¢, 1¢.

Then tell students that you have 7 nickels and 4 pennies. Show this on the board in a random arrangement:

5¢, 1¢, 1¢, 5¢, 5¢, 5¢, 5¢, 1¢, 5¢, 5¢, 1¢

Ask your students to write an addition sentence to show the total amount of money. Remind them that when

they have a long sequence of numbers to add, they should keep track as they go along by putting the total at

each stage in squares above the numbers:

6 7 12 17 22 27 28 33 38 39

5 + 1 + 1 + 5 + 5 + 5 + 5 + 1 + 5 + 5 + 1


Then show the coins on the board, putting the nickels first. Tell them you want to add the nickels first and ask

what you should skip count by. Demonstrate doing this until you get to the pennies (5, 10, 15, 20, 25, 30, 35)

by crossing out the fives as you count them and then ask: Should I still skip count by 5s? Now what do I

count by? Demonstrate continuing the count (36, 37, 38, 39) by crossing out the pennies as you count them.

ASK: Did we get the same total both ways?

Repeat with pennies and dimes, then dimes, nickels and pennies and then quarters, dimes, nickels and

pennies, all arranged randomly and then in order. ASK: What if they didn’t have nickels or dimes or quarters,

and they only had pennies. How would that affect their counting? (It would make it a lot slower to count how

much money they have and it would make carrying the money around a lot heavier.)

Then introduce dollar and cent notation for money amounts that are at least a dollar:

100¢ = $1.00, 200¢ = $2.00.

Ask students how they would show, in dollar notation, the following various amounts: 300¢, 700¢, 900¢,

500¢. Bonus: 1200¢, 1000¢, 3000¢.

Then have students total the following amounts and then write the dollar notation for the total:

a) $1, $1, $1, $1, $1 b) $2, $2, $2, $2 c) $2, $2, $1, $1, $1

d) $5, $1, $1 e) $5, $5, $2, $2, $1 f) $10, $5, $5, $2, $2, $2, $1

Then tell students that 300¢ is written as $3.00 in dollar notation and 47¢ is written as $0.47 in dollar

notation. Challenge students to guess how 347¢ is written in dollar notation. (ANSWER: $3.47)

$3.47

dollars dimes pennies

Have students (volunteers at first and then individually) show how to write the following cent amounts in

dollar notation: 321¢, 21¢, 320¢, 301¢, 478¢, 408¢, 78¢, 470¢, 603¢, 57¢, 430¢, 541¢)

Activities:

1. Ask students to pretend that there is a vending machine which only takes loonies, dimes and pennies.

Have them make amounts using only these coins (EXAMPLE: 453¢, 278¢, 102¢, etc).

2. A Game for Two: The Change Machine

One player makes an amount using nickels and quarters. The other (“the machine”) has to change the

amount into loonies, dimes and pennies.

Extensions:

1. Discuss the difference and similarities between the dollar and cent notation and the metre and

centimetre notation. Tell students that 134 cm can equally be written as 1m 34 cm or as 1.34 m. But

134¢ can be written as $1.34 (not 1.34 $) but not as 1 $ 34 ¢.


2. Have students write in symbols to make the following sentences correct (do the first one for them):

135¢ = $1 + 35¢

246 = 2 + 46

887 = 8 + 87

432 = 32 + 4

Remind students that the dollar sign goes to the left and the cents sign to the right of the number, but

you always say 3 dollars, not dollars 3. If students are comfortable with cm and m, have them fill in the

same number sentences with those units to make the sentences true and then discuss the comparison.

Then give only the number of cents (a 3-digit number) and have students break it up into dollars and

cents.


NS3-72 Counting and Changing Units and NS3-73 Converting Between Dollar and Cent

Notation

Goal: Students will convert between dollar and cent notation.

Prior Knowledge Required: Understanding of dollar and cent notation

Knowledge of place value

Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, notation, convert

Changing from dollar to cent notation is easy—all you need to do is to remove the decimal point and dollar

sign, and then add the ¢ sign to the right of the number.

Changing from cent notation to dollar notation is a little trickier. Make sure students know that, when an

amount in cent notation has no tens digit (or a zero in the tens place), the corresponding amount in dollar

notation must have a zero in the dimes place.

EXAMPLE: 6¢ is written $.06 or $0.06, not $0.60.

0 dimes 6 pennies 6 dimes 0 pennies

Make a chart like the one shown below on the board and ask volunteers to help you fill it in.

Amount Amount in Cents Whole Dollars Dimes Pennies Amount in Dollars

3 toonies 800 8 0 0 $8.00

2 loonies

2 quarters

5 dimes

2 nickels

4 quarters

2 pennies

Students could also practice skip counting by quarters, toonies and other coins (both in dollar and in cent

notation).

$2.00, $4.00, ____, ____ $0.25, $0.50, ____, ____, ____, ____

200¢, 400¢, ____, ____ 25¢, 50¢, ____, ____, ____, ____


Bonus: What coin is being used for skip counting?

$1.00, ____, ____, ____, $1.20

$2.00, ____, ____, ____, $3.00

You may wish to give your students some play money and to let them practice in pairs—each player picks 2

coins and then both students individually write the total amount of the 4 coins in dollar and cent notation.

Partners then compare answers.

Assessment

1. Write the total amount in dollar and cent notation:

a)

Total amount = _____ ¢ = $ ______

b)

Total amount = _____ ¢ = $ ______

2. Convert between dollar and cent notations:

$ .24 = ___¢, _____=82¢, _____ = 6¢, $12.03 = _____, $120.30 = ______

Bonus:

3. Change these numbers to dollar notation: 273258¢, 1234567890¢.

4. Change $245.56 to cent notation. Try these amounts: $76.34, $12.03, $120.30, $123.52, $3789.49.

Challenge students to make various money amounts using exactly two coins (EXAMPLES: 6¢, 10¢, 11¢,

15¢, 20¢, 30¢, 35¢, 50¢, Bonus: $2, $3, $1.05, $2.25)

Then challenge students to make various money amounts using exactly three coins (EXAMPLES: 7¢, 12¢,

15¢, 27¢, 30¢, 31¢, 40¢, 51¢, 75¢, Bonus: $3, $4, $5, $6, $2.10, $1.10, $1.15, $1.11)

Challenge students to make various money amounts using exactly four coins (EXAMPLES: 8¢, 13¢, 22¢,

30¢, 40¢, 45¢, 60¢, 65¢, 70¢, 76¢, 80¢, 81¢, 85¢, Bonus: $4, $5, $6, $7, $8, $1.40, $2.10, $3.25, $4.30,

$3.50, $5.10)

25¢ 10¢ $2 5¢ 1¢ 1¢

25¢ $1 10¢ 5¢ 1¢


Activities:

1. Guess the hidden coins Player 1 hides several coins: for example, a loonie, a toonie and a dime or

three loonies and a dime or a loonie, a toonie and two nickels for $3.10. Player 1 tells her partner how

many coins she has and the value of the coins in dollar notation. The partner has to guess which coins

she has. There may be more than one possibility for the answer. The player should try to use money and

coin amounts that have more than one possibility to increase the chances of an incorrect guess.

Students might discuss whether it is a good strategy for this game to use the least number of coins. (It is

not, because the least number of coins is always unique and quite easy to guess.) Challenge students to

change the rules of the game so that it is better to use the least number of coins. Then play the game.

2. Money Matching Memory Game. (See the BLM “Money Matching Memory Game”) Students play this

game in pairs. Each student takes a turn flipping over two cards. If the cards match, that player gets to

keep the pair and takes another turn. Students will have to remember which cards are placed where and

also match up identical amounts written in dollar and cent notation.

3. Each pair should have the BLM “Adding or Trading Game” as a game board and each player should

have a different token to use as their playing piece. They will also need a die to know how many pieces

to move forward. When they roll, they move forward the correct number of squares and receive the coin

shown on the board. When both players are at the end of the board (not necessarily by the exact amount

shown on the die), they count up their money – the player with the most amount of money wins.

Variation: The player with the fewest coins wins; players may trade from a shared bank for equal value

coins at the end of the game to try to have fewer coins.

4. Trading Game Have students work in pairs with different goals. Give each player 10 pennies, 4 nickels,

and 1 dime. Player One’s goal is to get 10 coins and Player Two’s goal is to get 20 coins. They must only

trade for equivalent values. To ensure this is always the case, have students add their money at the

beginning and periodically. They should always have 40¢. Note that both players will achieve their goals

at the same time, so they should cooperate.

Variation: Player One’s goal: 18 coins; Player Two’s goal: 12 coins.

Variation: Give each player 5 pennies, 5 dimes, 4 nickels, and 1 quarter. Player 1 tries to get 20 coins –

the other person will then try to get 10 coins. Students should repeatedly check that they have a total of

$1 or 100¢.

Variation: Give each player 2 loonies, 3 quarters, 10 dimes and 7 nickels, so that each player has 22

coins. Player 1 aims for 11 coins while Player 2 aims for 33 coins. Students should repeatedly check that

they have a total of $4.10.

Extensions:

1. Provide the BLM “Smallest Number of Coins Chart.”

2. Give students the BLM “Dimes, Pennies, and Base Ten Materials” to show them the relationship

between dimes and base ten blocks, and pennies and ones blocks.

3. (Atlantic Curriculum A5.11) Mary won $5000 in a contest. If she wants all her prize money in $10 bills,

how many would she receive?

4. (Atlantic Curriculum A4.4) Pretend that you won three thousand dollars. How many hundred dollar bills

would that be?

5. (Atlantic Curriculum A4.5) Martin said the car cost thirty-four hundred dollars, while Sam said he thought

it cost over three thousand dollars. Are they disagreeing? Explain.


NS3-74 Canadian Bills and Coins Goal: Students will identify Canadian coins by name and denomination, and express their values in dollar

and cent notation. They will identify correct forms of writing amounts of Canadian currency.

Prior Knowledge Required: Dollar and cent notation

Vocabulary: penny, toonie, nickel, dollar notation, dime, cent notation, quarter, currency, loonie

Remind students that when writing in dollar notation, the number of full dollars is written to the left of the

decimal point. There is no limit to how many place value columns this can extend to. Demonstrate by writing

$2.00, $22.00, $222.00 and $2222222222222.00 on the board.

However, there are only two place value columns to the right of the decimal place. Remind students that, if

you have a number of cents which is only a single digit (say 3), in dollar notation a zero is placed in the

dimes column.

Ask the class to invent some ways to write money amounts in incorrect notation. Allow them to come up

with a wide variety of ideas and welcome silly answers. Something that looks like this: 54$, is incorrect. Add

several examples yourself, for instance:

2.89$, $26.989, $67¢, ¢45, ¢576, 37.58¢, ¢67.89, $12.34¢, $1 35

Review the names and values of Canadian coins and bills. Discuss the images depicted on the coins. Point

out that the animal on the quarter is a caribou (not a moose) and that the dime shows the Bluenose, a type

of sailing ship called a schooner.

Discuss with students the relationships between various coins and bills.

Ask students how many…

a) dimes are in a loonie, toonie, or a 5 dollar bill.

b) nickels are in a loonie or toonie.

c) quarters are in a loonie, toonie, 5 or 10 dollar bill.

d) toonies are in a 10 or 20 dollar bill.

Students could use play money as manipulatives for the above problems.


Activities:

1. As a class or in small groups, have students visit the Canadian Currency Museum website and their

exhibit on Canada’s Coins. This has great information about the history and symbolism of Canada’s

coins. You could have students write a short report about one of the symbols on a Canadian coin and

why they are important enough to appear on a Canadian coin.

http://www.bankofcanada.ca/currencymuseum/eng/learning/canadascoins.php

2. Cross-curricular connection: Students could design coins that represent important symbols of the

First Nation peoples in Upper Canada in the years around 1800. Ask them to explain what image they

chose and why they think the symbol could be important.

3. (Atlantic Curriculum A4) Have students play “Race For a Loonie.” Ask each student to repeatedly

toss a die and count out pennies on a mat. Ten pennies are exchanged for a dime, and ten dimes for

a loonie. You may wish to have the students toss the die onto a plate to prevent the die from going

too far.

4. (Atlantic Curriculum A4.2) Have the students use a mat with sections marked off for $1, 10¢ and 1¢.

Ask them to toss two dice, find the sum, and place the total on the mat. Have them exchange 10

pennies (or 10 counters in the 1¢ section) for one dime (or one counter in the 10¢ section) and

continue until they have reached a dollar.

Extensions:

1. The Royal Canadian Mint has great resources available on their website. Their Currency Timeline might

be of great interest to the students. This outlines the history of Canadian settlement and development

and all the varieties of currency that have been used from the early 16th century to the present. Use this

as a resource and have students research other kinds of coins (denominations, forms, images, etc.),

that have existed in Canada’s past.

www.mint.ca/teach

Project Ideas

Choose a coin and find the following information on the mint.ca/teach web-site.

• What are the different images that have appeared on this coin and what did they

commemorate?

• Who drew the design for the coin?

• When was the coin introduced?

• Were there changes in the metal used to make the coin? When and why were these changes

made?

2. If you have any students who have lived in or travelled to other countries, have them bring in samples

of the other currency as a ‘show and tell’ for the class. Discuss the different images and shapes of the

coins and bills.


NS3-75 Adding Money Goal: Students will be able to add money amounts using traditional algorithms.

Prior Knowledge Required: Adding three-digit numbers

Adding using regrouping

Converting between dollar and cent notation

Familiarity with Canadian currency

Vocabulary: penny, quarter, nickel, loonie, dime, toonie, cent, adding, dollar , regrouping

Review adding 2-digit numbers. Demonstrate the steps: line up the numbers correctly, then add the digits in

each column starting from the right. Start with some examples that would not require regrouping, and then

move on to some which do (EXAMPLES: 15 + 23, 62 + 23, 38 + 46).

Use volunteers to pass to 3-digit addition questions, first without regrouping, then with regrouping.

SAMPLE QUESTIONS:

545 + 123, 123 + 345, then 132 + 259, 234 + 556, 578 + 789, 346 + 397.

Tell your students that you want to add $5.45 + $1.23. ASK: How many cents are in $5.45? (545¢) How

many cents are in $1.23? (123¢) How many cents is that altogether? (668¢) What is that in dollar notation?

($6.68) Show on the board:

$5.45 545¢

+ $1.23 123¢

$6.68 668¢

When adding money, the difference is in the lining up—the decimal point is lined up over the decimal point.

ASK: Are the one dollars lined up over the one dollars? The ten dollars over the ten dollars? The dimes

over the dimes? The pennies over the pennies? Tell them that if the decimal point is lined up, all the other

digits must be lined up correctly too, since the decimal point is between the ones and the dimes. Students

can model regrouping of terms using play money: for instance, in $2.33 + $2.74 they will have to group 10

dimes as a dollar.

Students should complete a number of problems in their notebooks. Some SAMPLE PROBLEMS:

a) $5.08 + $1.51 b) $3.13 + $2.98 c) $1.74 + $5.22

d) $3.95 + $4.28 e) $1.79 + $2.83

Ask volunteers to help you solve several word problems, such as: Julie spent $4.98 for a T-shirt and $3.78

for a sandwich. How much did she spend in total?


Students should also practice finding the value of sets of coins and bills and writing the amount in dollar

notation, such as:

There are 19 pennies, 23 nickels and 7 quarters in Jane’s piggybank. How much money does

she have?

Helen has a five-dollar bill, 1 toonie and 9 quarters in her pocket. How much money does she have?

Randy paid 2 toonies, a loonie, 5 quarters and 7 dimes for a parrot. How much did his parrot cost?

A mango fruit costs 69¢. I have a toonie. How many mangos can I buy?

If I add a dime, will it suffice for another one?

Have students do the following questions individually.

1. Add:

a) $7.25 b) $3.89 c) $5.08 d) $5.37 e) $2.86 + $5.17

+$1.64 +$6.23 +$3.87 +$2.79

2. Sheila saved 2 toonies, 4 dimes and 8 pennies from babysitting. Her brother Noah saved 4 loonies and

6 quarters from mowing a lawn.

a) Who has saved more money?

b) They want to share money to buy a present for their mother. How much money do

they have together?

c) They’ve chosen a teapot for $9.99. Do they have enough money?

Bring in fliers from local businesses. Ask students to select gifts to buy for a friend or relative as a

birthday gift. They must choose at least two items. They have a $10.00 budget. What is the total

cost of their gifts?

Extension: Fill in the missing information in the story problem and then solve the problem.

a) Betty bought _____ pairs of shoes for _____ each. How much did she spend?

b) Una bought ____ apples for _____ each. How much did she spend?

c) Bertrand bought _____ brooms for _____ each. How much did he spend?

d) Blake bought _____ comic books for _____ each. How much did he spend?

Bonus: Have students make up their own story problem and have a partner solve it.


NS3-76 Subtracting Money Goal: Students will subtract money amounts using traditional algorithms

Prior Knowledge Required: Subtraction of 3-digit numbers

Subtracting using regrouping

Converting between dollar and cent notation

Familiarity with Canadian currency

Vocabulary: penny, toonie, nickel, cent, dime, dollar, quarter, subtracting, loonie, regrouping

Review 2-digit and 3-digit subtraction. Start with some examples that would not require regrouping: 45 − 23,

78 − 67, 234 − 123, 678 − 354.

Show some examples on the board of numbers lined up correctly or incorrectly and have students decide

which ones are done correctly.

Demonstrate the steps: line up the numbers correctly, subtract the digits in each column starting from the

right. Move onto questions that require regrouping, EXAMPLE: 86 − 27, 567 − 38, 782 − 127, 673 − 185,

467 − 369.

Tell your students that you want to subtract $0.38 from $5.67. ASK: How many cents are in $5.67? (567)

How many cents are in $0.38? (38) What is 567 – 38?

567¢

– 38¢

529¢

ASK: What is 529¢ in dollar notation? ($5.29) Show on the board:

567¢ $5.67

– 38¢ – $0.38

529¢ $5.29

When subtracting money, the difference is in the lining up—the decimal point is lined up over the decimal

point. ASK: Are the one dollars lined up over the one dollars? The dimes over the dimes? The pennies over

the pennies? Remind them that if the decimal point is lined up, all the other digits must be lined up correctly

too, since the decimal point is between the ones and the dimes. Students can model regrouping of terms

using play money: for instance, in $2.74 – $2.36 they will have to group a dime as ten pennies.


Students should complete a number of problems in their notebooks. For EXAMPLE:

Subtract:

a) $8.89 b) $9.00 c) $4.00 d) $8.37 e) $8.47 - $0.62

−$1.64 −$7.23 −$3.87 −$5.79

Bonus: $28.14 – 17.43

ASK: Alyson went to a grocery store with $10.00. She would buy buns for $1.69, ice cream for $3.99 and

tomatoes for $2.50. Does she have enough money? If yes, how much change will she get?

Next, teach your students the following fast way of subtracting from powers of 10 (numbers such as 10,

100, 1 000 and so on) to help them avoid regrouping:

For example, you can subtract any money amount from a dollar by taking the amount away from 99¢ and

then adding one cent to the result.

1.00 .99 + one cent .99 + .01

− .57 = −.57 = −.57

.42 + .01 = .43 = 43¢

As another example, you can subtract any money amount from $10.00 by taking the amount away from

$9.99 and adding one cent to the result.

10.00 9.99 + .01

− 8.63 = − 8.63

1.36 + .01 = 1.37

NOTE: If students know how to subtract any one-digit number from 9, then they can easily perform the

subtractions shown above mentally. To reinforce this skill have students play the Modified Go Fish game

(in the MENTAL MATH section) using 9 as the target number.

Literature Connection

Alexander, Who Used to Be Rich Last Sunday. By Judith Viorst.

Last Sunday, Alexander’s grandparents gave him a dollar—and he was rich. There were so many things

that he could do with all of that money!

He could buy as much gum as he wanted, or even a walkie-talkie, if he saved enough. But somehow the

money began to disappear…

A great activity would be stopping to calculate how much money Alexander is left with every time he ends

up spending money.

Students could even write their own stories about Alexander creating a subtraction problem of their own.


Extensions:

1. a) Ella paid for a bottle of water with $2 and received 35¢ in change. How much did the water cost?

b) Jordan paid for some spring rolls with $5 and received 47¢ in change. How much did his meal cost?

c) Paige paid for a shirt with $10 and received 68¢ in change. How much did the shirt cost?

d) Geoff paid for a hockey stick with $10 and received 54¢ in change. How much did the hockey

stick cost?

2. Fill in the missing information in the story problem and then solve the problem.

a) Kyle spent $4.90 for a notebook and pencils. He bought 5 pencils for ______. How much did the

notebook cost?

b) Sally spent $6.50 for a bottle of juice and 3 apples. The apples cost ______How much did the

juice cost?

c) Clarke spent $9.70 for 2 novels and a dictionary. The ______ cost ______. How much did the

dictionary cost?

d) Mary spent $16.40 for 2 movie tickets and a small bag of popcorn. ___________________. How

much did her popcorn cost?

3. (Atlantic Curriculum B4.1) Count back the change for $5.00, if the bill totalled $3.59.

4. (Atlantic Curriculum B4.3) Write a subtraction problem that includes $1.40 and 16¢. Solve the problem.


NS3-77 Estimating Goal: Students will use rounding to estimate money amounts.

Prior Knowledge Required: Rounding

Canadian bills and coins

adding and subtracting money

dollars and cents

word problems

Vocabulary: rounding, estimating, estimate, about

Begin with a demonstration. Bring in a handful of change. Put it down on a table at the front of the class.

Tell your students that you would like to buy a magazine that costs $2.50. Do they think that you have

enough money? How could they find out?

Review rounding. Remind the class that rounding involves changing numbers to an amount that is close to

the original, but which is easier to use in mental calculations. As an example, ask what is easier to add:

8 + 9 or 10 + 10.

Explain to the class that they will be doing two kinds of rounding—rounding to the nearest 10¢ and rounding

to the nearest dollar. To round to the nearest 10¢ look at the digit in the pennies column. If the pennies digit

is less than 5, you round down. ASK: For which pennies digits would you round down? 7? 0? 3? 4? 9?

(round down for 0, 1, 2, 3 or 4) If the pennies digit is 5 or more, you round up. For which ones digits would

you round up? (5, 6, 7, 8 and 9) For example 33¢ would round down to 30¢, but 38¢ would round up to 40¢.

Use volunteers to round: 39¢, 56¢, 52¢, 75¢, 60¢, 44¢.

Model rounding to the nearest dollar. In this case, the number to look at is in the tenths place (the dimes

place). If an amount has 50¢ or more, round up. If it has less than 50¢, round down. So, $1.54 would round

up to $2.00. $1.45 would round down to $1.00. Use volunteers to round: $1.39, $2.56, $3.50, $4.75, $0.60,

$0.49.

Give several problems to show how rounding can be used for estimation:

Make an estimate and then find the exact amount:

a) Dana has $5.27. Tor has $2.38. How much more money does Dana have than Tor?

b) Mary has $3.74. Sheryl has $5.33. How much money do they have altogether?

a) Jason has saved $9.95. Does he have enough money to buy a book for $4.96 and a binder for $5.99?

Why is rounding not helpful here?

Ask students to explain why rounding to the nearest dollar isn’t helpful for the following question:

“Millicent has $2.15. Richard has $1.97. About how much more money does Millicent have than Richard?”

(Both round to $2. Rounding to the nearest 10 cents helps.)


Make sure students understand that they can use rounding to check the reasonableness of answers. Ask

students to explain how they know that $2.52 + $3.95 = $9.47 can’t be correct.

Have students solve the following problems individually.

1. Make an estimate and then find the exact amount: Molina has $7.89. Vinijaa has $5.77. How much

more money does Molina have?

2. Benjamin spent $2.94 on pop, $4.85 on vegetables and dip and $2.15 on bagels. About how much did

he spend altogether?

Activities:

1. Ask students to estimate the total value of a particular denomination (EXAMPLE: dimes, loonies, etc.)

that would be needed to cover their desk or a book. Students could use play money to test their

predictions. Students could also trace an outline of their hand and predict the value of dimes that would

cover their hand and then test their prediction.

2. Place a handful of play money bills and coins on a table and cover up the amount after students have

had a chance to look at it 5 or 10 seconds. Ask students to estimate the total amount. Alternatively, you

might ask students to estimate how much extra money would be needed to make a particular amount

(say $5 or $10).

1. Pairs of students practice estimating by using play money coins. Player 1 places ten to fifteen play

money coins on a table. Player 2 estimates the amount of money before counting the coins.

Extensions:

1. What is the best way to round when you are adding two numbers: to round both to the nearest dollar, or

to round one up and one down?

Explore which method gives the best answer for the following amounts:

$2.56 + $3.68 $4.55 + $4.57 $6.61 + $1.05

Students might notice that, when two numbers have cent values that are both close to 50¢ and that are

both greater than 50¢ or both less than 50¢, rounding one number up and one down gives a better

result than the standard rounding technique. For instance, rounding $2.57 and $3.54 to the nearest

dollar gives an estimated sum of $7.00 ($2.57 + $3.54 ≈ $3.00 + $4.00), whereas rounding one

number up and the other down gives an estimated sum of $6.00, which is closer to the actual total.

2. (Adapted from Atlantic Curriculum B10)

a) Popsicles cost 10¢ each. How many popsicles could you buy with a loonie? (ANSWER: 10)

b) Popsicles cost 20¢ each. How many popsicles could you buy with a loonie? (ANSWER: 5)

c) Popsicles cost 14¢ each. About how many popsicles could you buy with a loonie? Will the answer

be closer to 5 or 10? If I want to buy a popsicle for myself and five others, will a loonie be enough?

d) If erasers are on sale for 19¢, how many would you estimate you could buy with a loonie?

e) If I have a loonie, can I buy 3 party favours that cost 29¢ each?

f) If I have a loonie, can I buy 5 packages of stickers that cost 21¢ each?


NS3-78 Equal Parts Goal: Students will verbally express fractions given as diagrams or numerical notation.

Prior Knowledge Required: The ability to count

Fraction of an area

Ordinal numbers

Vocabulary: part, numerator, whole, denominator, fraction, area

Draw:

Ask your students how many circles are shaded.

Draw then ask them again how many circles are shaded.

Explain that the whole circle is no longer shaded. Ask your students if they know the word for a number that

is not a whole number, but is only part of a whole number. [Fraction.]

Explain that a fraction has a top and bottom number, then ask your students if they know what the numbers

represent. Explain that the shaded fraction of the circle is written as 34 (pronounce this as “three over four” for

now). What does the 3 represent? What does the 4 represent? Draw more fractions and ask students who

understand the significance of the numbers to identify the fractions without explaining it to the rest of the

class.

14

34

24

24

Draw examples with different denominators. Ask: What does the top number of the fraction represent? [The

number of shaded parts.] What does the bottom number of the fraction represent? [The number of parts in a

whole.]

If some students say that the bottom number is counting the number of parts altogether, tell them that from

what they’ve seen so far, that’s a good answer, but later we will see improper fractions where we have more

than one whole pie, so if each pie has 4 pieces and we have 2 pies, there are 8 pieces altogether, but we still

write the bottom number as 4. For example, if you bought 2 pies and 3 pieces were eaten from each pie, you

could say that 64 of a pie was eaten. (You could also say that

68 of what you bought was eaten, but that would

change the whole to 2 pies instead of 1 pie). Explain that fractions aren’t generally pronounced as they are.

Ask your students if they know the expressions for “three over four.” [Three-fourths or three-quarters.] Then

ask if they know the expression for 12 . Draw a half-shaded circle to encourage answers.


Draw several diagrams and have students express the fraction for the shaded parts of each whole by

writing the top and the bottom numbers.

Gradually increase the number of parts in each whole and shaded parts.

Ask them to explain their methods (skip counting or multiplication, for example) for determining the total

number of squares. At first, shade parts in an orderly way to facilitate a count. Then shade parts randomly,

but never exceed more than 20 shaded parts, and start with a small number of parts.

Ask students if they know which number—the top or bottom—is called the numerator. [Top.]

ASK: Does anyone know what the bottom number is called? [The denominator.] Which number—the

numerator or denominator—expresses the amount of equal parts in the whole?

Have students shade the correct number of parts to illustrate the following fractions.

16

35

47

Extension: A sport played by witches and wizards on brooms regulates that the players must fly higher

than 5 m above the ground over certain parts of the field (shown as shaded). Over what fraction of the field

must the players fly higher than 5 m?


NS3-79 Models of Fractions Goal: Students will name fractions given as diagrams or in numerical notation. Students will understand

that parts of a whole must be equal to determine a fraction that one or more parts represent.

Prior Knowledge Required: Expressing fractions

Fraction of an area

Fraction of a length

Vocabulary: numerator, denominator, fraction, part, whole

Illustrate these fractions — 12 ,

13 ,

14 ,

15 ,

16 ,

17 ,

18 — with circles.

Ask students if they know the proper way to pronounce these fractions. Remind them of ordinal numbers.

Say: If Sally is first in line, Tom is second in line and Rita is in line behind Tom, in what place is Rita? If Bilal

is in line behind Rita, in what place is Bilal? Continue to the eighth position. Explain that most of the ordinal

numbers—except for first and second—are also used for fractions. No one refers to half of a pie as one-

second of a pie, but we do say one-third, one-fourth, one-fifth, and so on.

Then ask your students to pronounce these fractions: 1

11 , 124 ,

113 ,

119 ,

1100 ,

192 .

If your students are comfortable with ordinal numbers up to a hundred, 1

92 could lead to some confusion,

since “first” and “second” are usually unused when dealing with fractions. In this case, the fraction is

expressed as “one ninety-second of a whole.” Explain that fractions with a numerator larger than one are

expressed the same way, with the numerator followed by the ordinal number. For example, 311 is expressed

as “three elevenths”. The ordinal number is pluralized when the numerator is greater than one, i.e., one

eleventh, two elevenths, three elevenths, and so on. Some ESL students might find it helpful to contrast this

with how we say 200 = two hundred, not two hundreds.

Ask your students to pronounce these fractions: 314 ,

295 ,

17100 ,

9495 ,

6183 ,

4151 ,

3052 .

Include fractions on spelling tests by writing the numeric fraction on the board.

Ask your students if they have ever been given a fraction of something (like food) instead of the whole, and

gather their responses. Bring a banana (or some easily broken piece of food) to class. Break it in two very

unequal pieces. Say: This is one of two pieces. Is this half the banana? Why not? Emphasize that the parts

have to be equal for either of the two pieces to be a half. Then draw the following rectangle.


Ask your students if they think the rectangle is divided in half. Explain that the fraction 12 not only expresses

one of two parts, but it more specifically expresses one of two equal parts. Draw numerous examples of this

fraction, some that are equal and some that are not, and ask volunteers to mark the diagrams as correct or

incorrect.

�� x x �

Ask: Which diagram illustrates one fourth? What’s wrong with the other diagram? Isn’t one of the four pieces

still shaded?

Then draw the following diagram on the board:

Ask volunteers to come to the board to show 34 in different ways. Then challenge a volunteer to draw the

circle themselves and to show 34 in yet a different way. Have students practice individually drawing a circle

divided into fourths in their notebooks and to find as many ways of showing 24 as they can.

Then show students the four ways of showing 14 :

Challenge students to find another way of showing 14 in a circle. Draw the following pictures on the board to

help them:

Have volunteers draw their own circle divided into fourths to show yet another way of modelling 1/4. You

could help them get started by drawing one line for them:

Then challenge students to individually show in their notebooks many ways of dividing a pie into halves, and

then into thirds (they should find a pie drawn in the workbook that is already divided into thirds to help them)

and then into eighths.


Bonus: Cut the following pies into thirds:

Activity: After students have completed exercise 6 and are familiar with the words “half,” “third,” “fourth,”

and so on, ask them to identify fractions (i.e. of pizzas, of objects in the classroom such as the blackboard,

the carpet, their desk, their pencil, and of diagrams) using phrases such as “one half,” “two thirds,” etc.

Extension: Have students ask their French teacher if ordinal numbers are used for fractions in French,

and have them tell you the answer next class.


NS3-80 Fractions of a Region or Length Goal: Students will understand that parts of a whole must be equal to determine an entire measurement,

when given only a fraction of the entire measurement.

Prior Knowledge Required: Expressing fractions

Fraction of an area

Fraction of a length

Vocabulary: numerator, denominator, fraction, part, whole

Explain that it’s not just shapes like circles and squares and triangles that can be divided into fractions, but

anything that can be divided into equal parts. Draw a line and ask if a line can be divided into equal parts.

Ask a volunteer to guess where the line would be divided in half. Then ask the class to suggest a way of

checking how close the volunteer’s guess is. Have a volunteer measure the length of each part. Is one part

longer? How much longer? Challenge students to discover a way to check that the two halves are equal

without using a ruler, only a pencil and paper. [On a separate sheet of paper, mark the length of one side of

the divided line. Compare that length with the other side of the divided line by sliding the paper over. Are

they the same length?]

Have students draw lines in their notebooks and then ask a partner to guess where the line would be

divided in half. They can then check their partner’s work.

ASK: What fraction of this line is double?

Say: The double line is one part of the line. How many parts, equal to that one, are in the whole linne,

including the double line? [5, so the double line is 15 of the whole line.]

Mark the length of the double line on a separate sheet of paper. Compare that length to the entire line to

determine how many of those lengths make up the whole line. Repeat with more examples.

Then ask students to express the fraction of shaded squares in each of the following rectangles.


Have them compare the top and bottom rows of rectangles.

ASK: Are the same fraction of the rectangles shaded in both rows? Explain. If you were given the

rectangles without square divisions, how would you determine the shaded fraction? What could be used to

mark the parts of the rectangle? What if you didn’t have a ruler? Have them work as partners to solve the

problem. Suggest that they mark the length of one square unit on a separate sheet of paper, and then use

that length to mark additional square units.

Prepare several strips of paper with one unit shaded, and have students determine the shaded fraction

without using a pencil or ruler. Only allow them to fold the paper. For instance they could fold the following

pieces of paper into 3, 4 and 5 equal parts respectively.

Draw a shaded square and ask students to extend it so the shaded part becomes half the size

of the extended rectangle.

Repeat this exercise for squares that are one-third and one-quarter the size of extended rectangles. ASK:

How many equal parts are needed? [Three for one-third, four for one-quarter.]

How many parts do you already have? [1.] So how many more equal parts are needed?

[Two for one-third, three for one-quarter.]

Activities:

After students have completed Question 3 from worksheet NS3-80, ask them to use a ruler to draw

rectangles in which:

a) 13 of the area is shaded b)

34 of the area is shaded

Extensions

1. On grid paper draw a rectangle with a width of 2 boxes and a length of 3 boxes. Shade 13 of the boxes.

2. On grid paper draw a rectangle with a width of 2 boxes and a length of 5 boxes. Shade 15 of the boxes.

3. a) Sketch a pie and cut it into fourths. How can it be cut into eighths?

b) Sketch a pie and cut it into thirds. How can it be cut into sixths?

4. Ask students to identify several fractions they see in the classroom and name them (e.g. about one third

of the chalk board is covered with writing, the room is about 4 fifths wide as it is long).


NS3-81 Equals Parts of a Set Goal: Students will understand that fractions can represent equal parts of a set.

Prior Knowledge Required: Fractions as equal parts of a whole

Review equal parts of a whole. Tell your students that the whole for a fraction might not be a shape like a

circle or square. Tell them that the whole can be anything that can be divided into equal parts. Brainstorm

with the class other things that the whole might be: a line, an angle, a container, apples, oranges, amounts

of flour for a recipe. Tell them that the whole could even be a group of people. For example, the grade 3

students in this class is a whole set and I can ask questions like: what fraction of students in this class are

girls? What fraction of students in this class are eight years old? What fraction of students wear glasses?

What do I need to know to find the fraction of students who are girls? (The total number of students and the

number of girls). Which number do I put on top: the total number of students or the number of girls? (the

number of girls). Does anyone know what the top number is called? (the numerator) Does anyone know

what the bottom number is called? (the denominator) What number is the denominator? (the total number of

students). What fraction of students in this class are girls? (Ensure that they say the correct name for the

fraction, for example: “eleven twentieths” instead of “eleven over twenty”) Tell them that the girls and boys

don’t have to be the same size; they are still equal parts of a set. Ask students to answer: What fraction of

their family is older than 8? Younger than 8? Female? Male? Some of these fractions, for some students,

will have numerator 0, and this should be pointed out. Avoid asking questions that

will lead them to fractions with a denominator of 0 (For example, the question “What fraction of your siblings

are male?” will lead some students to say 0/0).

Then draw pictures of shapes with two attributes changing:

a)

ASK: What fraction of these shapes are shaded?

What fraction are circles?

What fraction of the circles are shaded?

b)

ASK: What fraction of these shapes are shaded? What fraction are unshaded? What fraction are

squares? What fraction are triangles? Bonus: What fraction of the triangles are shaded? What fraction

of the squares are shaded? What fraction of the squares are not shaded?

Have students write fraction statements in their notebooks for similar pictures.

Then tell your students that you have five squares and circles. Some are shaded and some are not. Have

students draw shapes that fit the puzzles:


a) 25 of the shapes are squares.

25 of the shapes are shaded. One circle is shaded.

SOLUTION:

b) 35 of the shapes are squares.

25 of the shapes are shaded. No circle is shaded.

c) 35 of the shapes are squares.

35 of the shapes are shaded.

13 of the squares are shaded.

Ask some word problems:

A basketball team played 5 games and won 2 of them. What fraction of the games

did the team win?

A basketball team won 3 games and lost 1 game. How many games did they play altogether? What

fraction of their games did they win?

Bonus: A basketball team won 4 games, lost 1 game and tied 2 games. How many games did

they play? What fraction of their games did they win?

Activities:

1. On a geoboard, have students enclose a given

fraction of the pegs with an elastic.

For instance, 1025 .

2. (Adapted from Atlantic Curriculum Grade 4) Have the students “shake and spill” a number of two-

coloured counters and ask them to name the fraction that represents the red counters.

Extensions:

1. There are 8 figures in total. 12 of the figures are squares. The rest of the shapes are triangles.

58 of the figures are shaded. One triangle is shaded. How many squares are shaded?

2. a) Complete the following sentences (by writing a fraction in the first blank):

ii) _____ of the children in my family are _____ ii) _____ of the children in my class are _____

b) Make up your own sentence like the ones above in a).

3. Then draw the following picture.

ASK: How many pieces are shaded? (1) How many pieces are there altogether? (4)

Are one quarter of the pieces shaded? (yes) Is 14 of the shape shaded? (No, because

the four pieces do not have equal area.) Note that, just like boys and girls don’t have

to be the same size to be equal parts of a set, the pieces don’t have to be the same

size to be equal parts of a set.


NS3-82 Parts and Wholes Goal: Students will compare fractions with the same denominator. Students will compare fractions with

the same numerator. Students will divide shapes into equal parts to determine what fraction of the shapes

are shaded.

Prior Knowledge Required: Fractions as area

Parts of a whole must be equal to determine a fraction that one or

more parts represent.

Comparing fractions with the same denominator.

Draw on the board:

Have students name the fractions shaded and then have them say which circle has more shaded. ASK: Which is

more: one fourth of the circle or three fourths of the circle? Then draw different shapes on the board:

14

34

14

34

Have volunteers show the fractions on the board by shading and ASK: Is three quarters of something

always more than one quarter of the same thing? Is three quarters of a metre longer or shorter than a

quarter of a metre? Is three quarters of a dollar more money or less money than a quarter of a dollar? Is

three fourths of an orange more or less than one fourth of the orange? If possible, bring in an actual orange,

or use a paper circle, cut into quarters. Put one quarter aside and count the remainder: one quarter, two

quarters, three quarters. ASK: Which is more: one quarter or three quarters? Is three fourths of the class

more or less people than one fourth of the class? If three fourths of the class have brown eyes and one

quarter of the class have blue eyes, do more people have brown eyes or blue eyes?

Tell students that if you consider fractions of the same whole—no matter what whole you’re referring to—

three quarters of the whole is always more than one quarter of that whole, so mathematicians say that the

fraction 34 is greater than the fraction

14 . Ask students if they remember what symbol goes in between:

34

14 (< or >).

Remind them that the inequality sign is like the mouth of a hungry person who wants to eat more of the

pasta but has to choose between three quarters of it or one quarter of it. The sign opens toward the bigger

number:

34

14 or

14

34 .


Ensure that students understand that three of anything is always more than one of them. ASK: Are three

fifths more than one fifth?

Count as you colour in the fifths: one fifth, two fifths, three fifths.

ASK: Are three eighths more than one eighth? (Draw a picture of two pies, one with one eighth shaded and

the other with three eighths shaded). Which is more: five eighths or seven eighths? Five apples or seven

apples? $5 or $7? If Sally gets one sixth and Tony gets two sixths, who gets more? If Sally gets three sixths

and Tony gets two sixths, who gets more?

Have volunteers circle the largest fraction: 35 ,

25 ;

17 ,

47 ;

36 ,

56 ;

58 ,

38 .

Have students solve the following problems individually.

19 or

29 ?

112 or

212 ?

153 or

253 ?

1100 or

2100 ?

1807 or

2807 ?

29 or

59 ?

311 or

411 ?

911 or

811 ?

3587 or

4387 ?

91102 or

52102 ?

Bonus: 7 432

25 401 or 869

25 401 ? 52 645

4 567 341 or 54 154

4 567 341 ?

Comparing fractions with the same numerator.

Draw on the board:

12

13

14

15

Have a volunteer colour the first part of each strip of paper and then ask students which fraction shows the

most: 12 ,

13 ,

14 or

15 . Ask: Do you think one sixth of this fraction strip will be more or less than one fifth of it?

Will one eighth be more or less than one tenth? Then colour the second fifth and ASK: Which is more: two

fifths or one half. ASK: Two is more than one; why aren’t two fifths more than one half? (The fifth-sized

pieces are smaller than the half-sized pieces. It’s like saying two pencils are longer than one desk because

2 is more than 1 – show the two pencils next to your desk).


Then draw two circles the same size on the board:

Have volunteers shade one part of each circle. ASK: What fractions are represented? Which fraction

represents more? Repeat with different shapes:

12

14

12

14

12

14

Ask: Is one half of something always more than one quarter of the same thing? Is half a metre longer or

shorter than a quarter of a metre? Is half an hour more or less time than a quarter of an hour? Is half a dollar

more money or less money than a quarter of a dollar? Is half an orange more or less than a fourth of the

orange? Is half the class more or less than a quarter of the class? If half the class has brown eyes and a

quarter of the class has green eyes, do more people have brown eyes or green eyes?

Tell students that no matter what quantity you have, half of the quantity is always more than a fourth of it, so

mathematicians say that the fraction 12 is greater than the fraction

14 . Ask students if they remember what

symbol goes in between: 12

14 (< or >). What symbol goes in between now?

14

12

Tell your students that you are going to try to trick them with this next question so they will have to listen

carefully. Then ASK: Is half a minute longer or shorter than a quarter of an hour? Is half a centimetre longer

or shorter than a quarter of a metre? Is half of Stick A longer or shorter than a quarter of Stick B?

Stick A:

Stick B:

Ask: Is a half always bigger than a quarter?

Allow everyone who wishes to attempt to articulate an answer. Summarize by saying: A half of something is

always more than a quarter of the same thing. But if we compare different things, a half of something might

very well be less than a quarter of something else. When mathematicians say that 12 >

14 , they mean that a

half of something is always more than a quarter of the same thing; it doesn’t matter what you take as your

whole, as long as it’s the same whole for both fractions.

Draw the following strips on the board:

Ask students to name the fractions and then to tell you which is more.


Have students draw the same fractions in their notebooks but with circles instead of strips.

Is 34 still more than

38 ? (yes, as long as the circles are the same size)

Bonus:

Show the same fractions using a line of length 8 cm.

Ask students: If you cut the same strip into more and more pieces of the same size, what happens to the

size of each piece?

Draw the following picture on the board to help them:

1 big piece

2 pieces in one whole



Many pieces in one whole

Ask: Do you think that 1 third of a pie is more or less pie than 1 fifth of the same pie? Would you rather have

one piece when it’s cut into 3 pieces or 5 pieces? Which way will you get more? Ask a volunteer to show how

we write that mathematically (13 >

15 ).

Do you think 2 thirds of a pie is more or less than 2 fifths of the same pie? Would you rather have two pieces

when the pie is cut into 3 pieces or 5 pieces? Which way will get you more? Ask a volunteer to show how we

write that mathematically (23 >

25 ).

If you get 7 pieces, would you rather the pie be cut into 20 pieces or 30? Which way will get you more pie?

How do we write that mathematically? (720 >

730 ).

Ask the students to answer individually: Which is greater?

a) 12 or

17 b)

13 or

14 ? c)

19 or

16 ? d) Bonus:

1132 or

1147

e) 37 or

38 f)

419 or

415 g)

822 or

825 h) Bonus:

132234 or

132198

SAY: Two fractions have the same numerator and different denominators. How can you tell which fraction is

bigger? Why? Summarize by saying that the same number of pieces gives more when the pieces are

bigger. The numerator tells you the number of pieces, so when the numerator is the same, you just look at

the denominator.

The bigger the denominator, the more pieces you have to share between and the smaller the portion you

get. So bigger denominators give smaller fractions when the numerators are the same.


SAY: If two fractions have the same denominator and different numerators, how can you tell which fraction

is bigger? Why? Summarize by saying that if the denominators are the same, the size of the pieces are the

same. So just as 2 pieces of the same size are more than 1 piece of that size, 84 pieces of the same size

are more than 76 pieces of that size.

Emphasize that students can’t do this sort of comparison if the denominators are not the same. ASK: Would

you rather 2 fifths of a pie or 1 half? Draw the following picture to help them:

Tell your students that when the denominators and numerators of the fractions are different, they will have

to compare the fractions by drawing a picture or by using other methods that they will learn in later grades.

The same fraction of the same thing are always equal.

Draw:

ASK: What fraction of the first square is shaded? What fraction of the second square is shaded? Are the

squares the same size? Are the shaded parts the same size?

Repeat for various pairs of shapes:

Challenge students to find various ways of dividing these shapes into quarters.

EXAMPLES:

Dividing shapes into equal parts.

Draw on the board the shaded strips from before:


ASK: Is the same amount shaded on each strip? Is the same fraction of the whole strip shaded in each

case? How do you know? Then draw two hexagons as follows:

ASK: Is the same amount shaded on each hexagon? What fraction of each hexagon is shaded? Then

challenge students to find the fraction shaded by drawing their own lines to divide the shapes into

equal parts:

Give your students pattern blocks. Ask them to make a rhombus from the triangles. How many triangles do

they need? What fraction of a rhombus is a triangle? Challenge them to find:

a) What fraction of a hexagon is the rhombus? The trapezoid? The triangle?

b) What fraction of a trapezoid is the triangle?

Activities:

1. Give each group of 3 students 3 large congruent shapes, but cut differently into the same number of

equal parts. EXAMPLE:

Have students shade one part of each shape and then cover, in a single layer, the shaded part with as

many small counters as they can. Did each quarter of the shape require the same number of counters, at

least approximately?

2. Have students toss several coins (or two-colour counters). What fraction of the coins came up heads?

What fraction of the coins came up tails? What do the two numerators add to? (the denominator) Why do

they think that happened? Will it always happen?

3. Ask students to enclose a fraction of the pegs on a geoboard and then to write the fraction of pegs that

are not enclosed. Have students investigate with many examples, so that they see that the two

numerators always add up to the denominator. For extra practice with fractions that add to 1, use the

2-page BLM “Fractions That Add to 1.”


Extensions:

1. a) What fraction of a tens block is a ones block?

b) What fraction of a tens block is 3 ones blocks?

c) What fraction of a hundreds block is a tens block?

d) What fraction of a hundreds block is 4 tens blocks?

e) What fraction of a hundreds block is 32 ones blocks?

f) What fraction of a hundreds block is 3 tens blocks and 2 ones blocks?

2. On a geoboard, show 3 different ways to divide the area of the board into 2 equal parts.

EXAMPLES:

3. Give each student a set of pattern blocks. Ask them to identify the whole of a figure

given a part.

a) If the pattern block triangle is 16 of a pattern block, what is the whole?

Answer: The hexagon.

b) If the pattern block triangle is 13 of a pattern block, what is the whole?

Answer: The trapezoid.

c) If the pattern block triangle is 12 of a pattern block, what is the whole?

Answer: The rhombus.

d) If the rhombus is 16 of a set of same-shaped pattern blocks, what is the whole?

Answer: 2 hexagons or 6 rhombuses or 12 triangles or 4 trapezoids.

2. Draw each shape below onto cm grid paper so that each square takes up a 4 cm by 4 cm square. Have

students find the area of each shaded piece:

Bonus:

To help students count half squares and whole squares, see ME3-30.

5. The pattern block triangle represents 14 . What might the whole be?


6. (From Atlantic Curriculum A3.9) This shape is 12 of a larger one. What could the larger one look like?

How many different possibilities can you find?

a) b) c) d)

NOTE: If you prefer, you could assign these shapes on grid paper (rather than dot paper or a

geoboard).

7. Students can construct a figure using the pattern block shapes and then determine what fraction of the

shapes is covered by the pattern block triangle.

8. What fraction of the figure is covered by…

a) The shaded triangle

b) The small square

9. This extension is best done after Activity 2 above. Compare the following fractions by comparing how

much of a whole pie is left if the following amounts are eaten: 34 or

45 . Emphasize that the fraction with

a bigger piece left-over is the smaller fraction.

10. Write the following fractions in order from least to greatest. Explain how you found the order. 13 ,

12 ,

23 ,

34 ,

18

HINT: Use the ideas from Extension 9 above.

11. Fold a strip of paper 3 times to create eighths. Write the following fractions over top of the

corresponding folds:

18

38

12

78 Bonus:

14

12. Give each student three strips of paper. Ask them to fold the strips to divide one strip into halves, one

into quarters, one into eighths. Use the strips to find a fraction between

a) 38 and

58 (one answer is

12 ) b)

14 and

24 (one answer is

38 ) c)

58 and

78 (one answer is

38 )


13. Have students fold a strip of paper (the same length as they folded in EXTENSION 12 into thirds by

guessing and checking. Students should number their guesses.

EXAMPLE:

Try folding here too short, so try a little further

1st guess

2nd

guess

Is 13 a good answer for any part of Extension 12? How about

23 ?

14. Why is 23 greater than

25 ? Explain.

15. Why is it easy to compare 25 and

212 ? Explain.


NS3-83 Sharing and Fractions Goal: Students will find fractions of numbers. Students will see examples of equivalent fractions.

Prior Knowledge Required: Fractions as area

Fractions of a set

Finding fractions of whole numbers.

Brainstorm the types of things students can find fractions of (circles, squares, pies, pizzas, groups of

people, angles, hours, minutes, years, lengths, areas, capacities, apples).

Brainstorm some types of situations in which it wouldn’t make sense to talk about fractions. For example:

Can you say 3 12 people went skiing? I folded the sheet of paper 4

14 times?

Explain to your students that it makes sense to talk about fractions of almost anything, even people and

folds of paper, if the context is right: EXAMPLE: Half of her is covered in blue paint; half the fold is covered

in ink. Then teach them that they can take fractions of numbers as well. ASK: If I have 6 hats and keep half

for myself and give half to a friend, how many do I keep? If I have 6 apples and half of them are red, how

many are red? If I have a pie cut into 6 pieces and half the pieces are eaten, how many are eaten? If I have

a rope 6 metres long and I cut it in half, how long is each piece? Tell your students that no matter what you

have 6 of, half is always 3. Tell them that mathematicians express this by saying that the number 3 is half of

the number 6.

Tell your students that they can find 12 of 6 by drawing two circles. Put one dot in each circle until you have

placed 6 dots.

2 dots 4 dots 6 dots

Now, one half of the dots are in the first circle and one half of the dots are in the second circle. So 12 of 6

is 3. Have students use this method to find 12 of a) 10 b) 8 c) 14

Then have students draw 3 circles to find 13 of a) 6 b) 12 c) 9 d) 15 e) 3 f) 18.

Then have students write a division statement for each picture a) to f) above. (6 ÷ 3 = 2 and 12 ÷ 3 = 4 and

so on)


Then have students draw a picture and write a division statement for each fraction of a number:

a) 12 of 8 b)

13 of 12 c)

14 of 8 d)

12 of 10

e) 15 of 10 f)

15 of 20 g)

14 of 12 h)

15 of 15

Write a division statement to find the fraction of the number without using pictures.

a) 12 of 22 b)

14 of 84 c)

15 of 100 d)

110 of 70

Teach your students to see the connection between the fact that 6 is 3 twos and the fact that 13 of 6 is 2. The

exercise below will help with this:

Complete the number statement using the words “twos”, “threes”, “fours” or “fives”. Then draw a picture

and complete the fraction statements. (The first one is done for you.)

Number Statement Picture Fraction Statement

a) 6 = 3 twos 13 of 6 = 2

23 of 6 = 4

b) 12 = 4 ________ 14 of 12 =

24 of 12 =

34 of 12 =

c) 15 = 3 ________ 13 of 15 =

23 of 15 =

Equivalent fractions

Tell your students that sometimes two fractions that look different can mean the same thing. Show the

following pictures:


ASK: What fraction does each picture show? (12 ,

24 ,

48 and

36 ) ASK: Is the same amount shaded in each

picture? Ask students to find other fractions that mean the same as 12 . ASK: If a pie had 10 pieces, how

many would be half? What fraction with denominator 10 means the same thing as 12 ?

Have students find fractions that mean the same as 12 and have given denominators:

12 = 6

12 = 12

12 = 8

12 = 10

12 = 20

12 = 14

Then have students find fractions that mean the same as 12 and have given numerators:

12 =

2

12 =

5

12 =

3

12 =

4

12 =

8

12 =

50

Finally, mix up questions of both types:

12 =

6

12 = 6

12 =

10

12 = 10

12 = 16

12 =

16

Activities:

1. Have students compare using fraction strips 12 and

24 ,

34 and

68 ,

14 and

28 ,

12 and

48 ,

24 and

48 .

2. Students can find fractions of a whole number using the following method: Find 12 of 12.

STEP 1 – Make a model of 12 using 12 yellow counters

STEP 2 – Replace yellow counters one at a time with red counters until an equal number of the

counters are red and yellow.

Ask students to use this method to find 12 of 6, 8, 10, 14, etc.

3. Students can find 12 of 6 by drawing rows of dots. Put 2 dots in each row until you have placed 6 dots.


There are 3 dots in each column, so 3 is 12 of 6. Have students find

12 of 10, 12, 8 and 16 using this

method. Students might also find 13 of various numbers by drawing rows of dots with 3 dots in each row.


Extensions:

1. How many months are in:

a) 12 year? b)

13 year? c)

14 years?

2. How many minutes are in:

a) 13 of an hour? b)

12 of an hour? c)

14 of an hour?

3. How many hours are in:

a) 12 of a day? b)

13 of a day? c)

14 of a day?

4. Give your students counters to model the following problems.

a) 2 is 13 of a set. How many are in the set? b) 3 is

14 of a set. How many are in the set?

5. Show students how to find 23 of 6 dots.

STEP 1. Find 13 of 6 dots.

3 dots 6 dots

So 13 of 6 is 2.

STEP 2. Multiply by 2.

Each circle has 13 of 6 dots, so 2 circles have

23 of 6 dots.

23 of 6 is 4. Have students use this method

to find:

a) 23 of 9 b)

34 of 12 c)

23 of 12 d)

25 of 10 e)

35 of 15

6. Have students investigate.

a) 34 of 4 b)

45 of 5 c)

37 of 7 Bonus: What is

13215 of 215?


7. (Atlantic Curriculum A3.4) Tell the student that Lee and Teddy bought their mother a gift for Christmas

which cost $20. Lee paid 34 of the cost, and Teddy paid the balance. Ask: How much did each pay?

Provide coloured counters to help him/her solve the problem.

8. (Adapted from Atlantic Curriculum A3.5)

a) Pair each student with a partner to solve this problem: Eight-year-old Samantha, whose birthday is

January 25th, said, “I can’t wait until I’m 8 and 1112 .” Ask: Why was she excited?

b) Natalia said that she will turn 8 and 13 on Christmas day. When is her birthday?

9. (Atlantic Curriculum A3.6) Ask the student to tell why, whenever you see a model of 13 , there is always a

model of 23 associated with it.

10. (From Atlantic Curriculum A3.8) You have 8 coins. Half of them are pennies. More than 18 of them

are quarters. The others are nickels. Use coins to represent the situation. How much money might

you have?


NS3-84 Comparing Fractions Goal: Students will decide when a picture shows more than half or less than half, about a half, or

about a third or about a fourth.

Prior Knowledge Required: Comparing fractions

Using models to represent fractions

Vocabulary: less than, more than, half, close to

Ask students to tell you whether the shaded fraction is more than half or less than half and how they can tell:

Then repeat the exercise, but this time have a volunteer extend one of the border lines of the shaded region

to show half. Then give several similar problems for students to do individually, and tell them to imagine

where the line would be extended.

Ask students to tell you whether the shaded fraction is less than 13 , more than

13 but less than

23 or more than

23 and have students explain how they can tell:

Then repeat the exercise, but this time have a volunteer use one of the border lines of the shaded region to

show the pie cut into thirds. Then give several similar problems for students to do individually, and tell them

to imagine where the lines would be to divide the pies into thirds.

Repeat with fractions that show less than a quarter, between one and two quarters, between two and three

quarters and more than three quarters.

Draw some fraction strips on the board:

12

13

14


Give students fraction strips with a part shaded. EXAMPLES:

Have students estimate what fraction of the strips are shaded. Challenge the students to find a way to check

their answer by folding the fraction strips.

Then ask: Which figure shows about 13 ?

Then show on a strip of paper the following circles:

Ask students to estimate what fraction of the circles are shaded. ASK: How can we check by folding the

paper? Demonstrate folding the paper and then circling the 4 groups of 3 circles. ASK: How many circles are

shaded? (3) How many groups of 3 circles are there altogether? What fraction of the circles are shaded?

(one group out of four groups are shaded, so one fourth or one quarter of the circles are shaded. Show

students how they can do this without folding.

ASK: How many circles are shaded? (2) Demonstrate circling groups of 2 circles until you have circled all the

circles. ASK: How many groups of 2 circles are there? (5) What fraction of the circles is shaded? (one group

out of five groups are shaded, so one fifth of the circles is shaded. Give students several such problems to

practise with.

Activities:

1. Oranges or Apples

Bring oranges or apples into the class and cut some in fourths and some in half and some in eighths. Ask

students to decide which is more between different fractions: for example, one half or three eighths.

Encourage them to put three eighths together so that they can see that it is not quite half.

2. Mark off 13 and

23 of a clean plastic cylinder with scotch tape. Partially fill the cylinder with water and turn

the cylinder so students can’t see the tape marks. Ask them whether the cylinder is closer to 13 or

23 full.

Then turn the cylinder so that students can see the tape marks. Did they guess correctly? Repeat with

different amounts of liquid


Extension:

Ordering Fraction Strips

Use the BLM “Shaded Fraction Strips” and give each student three fraction strips to compare. They can tape

the fraction strips to a coloured piece of paper in order from smallest fraction to largest fraction and write

their conclusions on the same paper. If they are familiar with the < and > symbols for “less than” and “greater

than” students can write their answers in the form 38 <

12 <

23 . Otherwise, they can write sentences: “

38 is less

than 12 ” and “

12 is less than

23 .”

Literature/Cross-Curricular Connections:

Apple Fractions by J. Pallotta Different apples are used to teach kids all about fractions. Students will learn

to divide apples into halves, thirds, fourths, and more. See the above activity.


NS3-85 Fractions Greater than One Goal: Students will model fractions greater than 1 by using pictures.

Prior Knowledge Required: comparing fractions, using models to represent fractions

Vocabulary: less than, more than

Tell students that beads come in packs of 4. Ask volunteers to show, by modelling beads with circles:

a) one whole pack, b) one half of a pack, c) 2 packs, d) one pack and another half of a pack.

Tell students that, rather than saying one pack and then another half of a pack, we describe this as one and

a half packs. Ask a volunteer to draw what they think two and a half packs will look like.

Then tell students that beads come in packs of 6. Ask students to show individually in their notebooks: a) one

half of a pack, b) one and a half packs, c) two packs, d) two and a half packs, e) three and a half packs.

Then tell students that beads come in packs of 3. Ask students to show individually in their notebooks: a) one

third of a pack, b) two thirds of a pack, c) one and a third packs, d) two and a third packs, e) four and two

thirds packs.

NOTE: You might also demonstrate what it means to have “one and a half” or “two and a half” of something

using an area model: for instance, one and a half pizzas.

Extension: Teach students that one whole can be written as a fraction in many different ways.

Have students name the fractions shaded. Tell them that they are all one whole and write 1 = 44 and 1 =

66 .

Then have a student volunteer to fill in the blanks: 9 7

Then repeat with larger numbers and have students fill in the denominator (give them only the numerator).

Repeat the above exercises with fractions that show two wholes. Ensure that students understand that to

find the numerator, they double the denominator, or to find the denominator, they take half of the numerator.

Gradually increase the denominators to make them more difficult to double: 3, 7, 23, 34, 36, 52, 47, 74, 78.

1 = 1 =


NS3-86 Puzzles and Problems Goal: students will consolidate and apply their learning about fractions.

Prior Knowledge Required: fractions as equal parts of an area or set

fractions that look different but mean the same amount

comparing fractions

fractions greater than one

fractions of numbers

Vocabulary: less than, more than

This worksheet is review. It can be used as an assessment.


NS3-87 Decimal Tenths Goal: Students will learn the notation for decimal tenths. Students will add decimal tenths with sum up

to 1.0. Students will recognize and be able to write 10 tenths as 1.0.

Prior Knowledge Required: Fractions Thinking of different items as a whole (EXAMPLE: a pie, a hundreds block)

Place value

0 as a place holder

Vocabulary: decimal, decimal tenth, decimal point

Draw the following pictures on the board and ask students to show the fraction

110 in each picture:

Tell students that mathematicians invented decimals as another way to write tenths: One tenth ( 1

10 ) is

written as 0.1 or just .1. Two tenths ( 2

10 ) is written as 0.2 or just .2. Ask a volunteer to write 310 in decimal

notation. (.3 or 0.3) Ask if there is another way to write it. (0.3 or .3) Then have students write the following

fractions as decimals:

a) 710 b)

810 c)

910 d)

510 e)

610 f)

410

In their notebooks, have students rewrite each addition statement using decimal notation:

a) 310 +

110 =

410 b)

210 +

510 =

710 c)

210 +

310 =

510 d)

410 +

210 =

610

Bonus:

Include subtraction problems such as:

a) 710 –

310 =

410 b)

910 –

410 =

510 c)

310 –

110 =

210 d)

610 –

310 =

310


Draw on the board:

ASK: What fraction does this show? (410 ) What decimal does this show? (0.4 or .4)

Repeat with the following pictures:

Have students write the fractions and decimals for similar pictures independently, in their notebooks.

Then ask students to convert the following decimals to fractions, and to draw models in

their notebooks:

a) 0.3 b) .8 c) .9 d) 0.2

Demonstrate the first one for them:

0.3 = 3

10

Have students write addition statements, using fractions and decimals, for each picture:

+ = + = + =

Draw on the board:

0 110

210

310

410

510

610

710

810

910 1

Have students count out loud with you from 0 to 1 by tenths: zero, one tenth, two tenths, … nine

tenths, one.

Then have a volunteer write the equivalent decimal for 1

10 on top of the number line:

0.1

0 1

10 210

310

410

510

610

810

910 1

Continue in random order until all the equivalent decimals have been added to the number line.


Then have students write, in their notebooks, the equivalent decimals and fractions for the spots marked on

these number lines:

a) b)

0 1 0 1

c)

0 1

d)

0 A 1 B C 2 D 3

Have volunteers mark the location of the following numbers on the number line with an X and the

corresponding letter.

A. 0.7 B. 2 710 C. 1.40 D.

810 E. 1

910

0 1 2 3

Invite any students who don’t volunteer to participate. Help them with prompts and questions such as: Is

the number more than 1 or less than 1? How do you know? Is the number between 1 and 2 or between

2 and 3? How do you know?

Tell your students that there are 2 different ways of saying the number 1.4. We can say “one decimal four” or

“one and four tenths”. Both are correct.

Have students write the following numbers as decimals:

a) four tenths b) one and six tenths c) three and one tenth d) two and five tenths

Have students write the following decimals as words:

a) 1.2 b) 2.1 c) 3.4 d) 7.3 e) 9.1 f) 2.9

Have students place the following fractions on the number line from 0 to 3:

A. three tenths B. two and five tenths C. one and seven tenths

D. one decimal two E. two decimal eight

Which of the numbers (from A, B, C, D and E above) are less than 1? Which are more than 1 and less

than 2? Which are more than 2? Is 2.3 more or less than 1.8? How do you know? Which two whole numbers

is 2.3 between? Which two whole numbers is 1.8 between?


Teach students to count forwards and backwards by decimal tenths using dimes. ASK: How many dimes

make up a dollar? (10) What fraction of a dollar is a dime? (one tenth) Tell students that we can write the

dime as .1 dollars, since .1 is just another way of writing 110 . ASK: What fraction of a dollar are 2 dimes?

How would we write that using decimal notation? What fraction of a dollar are…

a) 7 dimes? b) 3 dimes? c) 9 dimes? d) 6 dimes? e) 10 dimes?

Have students write all the fractions as decimals.

ASK: How many dimes are in … a) $0.70 b) $0.20 c) $0.60 d) $0.30

ASK: How many tenths of a dollar are in … a) $0.70? b) $0.20 c) $0.60 d) $0.80 e) $0.90

Activity: (From Atlantic Curriculum A8)

Decimal War. Prepare a deck of cards with numbers such as 0.1, 0.2, …, 0.9, 1.0, 1.1, …,

1.9, 2.0, 2.1, …, 2.9 for each pair of students. Each student gets half the deck. They both turn over one

card at a time. The student with the card showing the greater number keeps both cards. Play continues

until someone has all the cards. Variation: Give each pair two identical sets of cards so that ties are

possible.

You might choose to have students play the same game with cards numbered 1 through 29 instead of 0.1

through 2.9 and have them compare the two games. Notice that 1.1 is greater than 0.9 precisely because

11 (tenths) is greater than 9 (tenths).

Extensions:

1. Put the following sequence on the board: .1, .3, .5, _____ and have students extend the pattern.

2. If your students are familiar with equivalent fractions, have them convert each fraction to an equivalent

fraction with denominator 10 and then to a decimal:

a) 25 b)

12 c)

45

3. If your students are familiar with equivalent fractions, have them rewrite each addition or subtraction

statement using decimal notation by first changing all fractions to an equivalent fraction with

denominator 10:

a) 12 +

15 =

710 b)

12 +

25 =

910 c)

35 –

12 =

110 d)

12 –

15 =

310

4. Teach students numbers with two decimal places and where to find the tenths.

ASK: How many dimes are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35

ASK: How many tenths of a dollar are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35

ASK: Where do you see the number of tenths in each number:

a) 0.78 b) 0.93 c) 0.21 d) 0.35


5. (From Atlantic Curriculum A8)

a) Identify the decimals and then put them in order:

A.

B.

C.

D.

b) Model the decimals and then write and model a decimal that is between the two decimals:

3.4 and 3.8 2.8 and 3.1 1.4 and 3.9 0.5 and 1.2 1.9 and 3.2

c) Complete the pattern: 2.9, _____, 3.1, 3.2, _____, 3.4

6. (From Atlantic Curriculum A7)

a) Which number is larger: 2.9 or 4.2? How do you know?

b) Which number is larger: 6.2 or 40? How do you know?

c) Which number is 0.2 more than 0.4? Use a number line or ten frame to help you.

d) Which number is 0.2 less than 1? Use a number line or ten frame to help you.

e) Teach students that decimals are equivalent to fractions with denominator 10 and just as they can

take a fraction of a set, they can take a decimal of a set. Ask students to circle about 0.4 (or 410 ) of

the dots.


NS3-88 Word Problems (Warm Up) Goal: Students will decide when to use addition, subtraction, multiplication or division in word problems

given in point-form notation.

Prior Knowledge Required: addition, subtraction, multiplication, division

Ask students whether the underlined words make them think of adding or subtracting:

1) Three more joined.

2) Two children left to go skipping.

3) There are five altogether.

4) She took five away.

5) There are seven in total.

6) How many are leftover?

7) How many cookies are left?

8) How many altogether?

9) How many are not red?

10) How many more apples than oranges are there?

11) How many fewer days are in a week than in a month?

12) How many days are in a month and a week altogether?

13) How much longer is a school bus than a car?

Have students write down the important words in each question and then the symbol (+ or –) that it makes

you think of:

a) How many apples were sold altogether?

b) How many more red apples than green apples were sold?

c) How many apples were not red?

d) How many stickers were collected altogether?

e) How many stickers were not from Canada?

f) How much longer is a ruler than my pencil?

g) How long are my ruler and pencil when placed end to end?

Tell students that you went to a zoo and saw 14 spotted animals and 9 striped animals and 6 plain animals.

ASK:

a) How many animals are there altogether? 14 + 9 + 6 = 29

b) How many animals are not plain? (29 – 6 = 23)


c) How many spotted and striped animals are there? (Have students discuss two different solutions:

14 + 9 = 23, or 29 – 6 = 23)

d) How many animals are not striped? Discuss the different solutions.

e) How many more spotted animals are there than plain animals?

f) How many fewer plain animals are there than striped animals?

g) Make up your own question and have a partner solve it, then check the partner’s solution.

Tell students that you went to a different zoo and saw 12 spotted animals and 15 striped animals and 34

animals altogether. ASK:

a) How many spotted and striped animals are there altogether?

b) How many more striped animals are there than spotted animals?

c) How many plain animals are there? (I.e. How many animals are there that are NOT spotted or striped?)

d) How many animals are not spotted? (Discuss the two solutions here.)

e) How many animals are striped or plain?

Then ask students to decide between addition and multiplication and how they know:

a) There are 4 pages in each chapter. There are 5 chapters. How many pages are there altogether?

b) There are 7 pages in Chapter 1, 4 pages in Chapter 2 and 8 pages in Chapter 3. How many pages are

there altogether?

c) There are three bookshelves. The shelves have 9, 4, and 6 books each. How many books are there

altogether?

d) There are three bookshelves. There are 6 books on each shelf. How many books are there altogether?

Ask students to tell you how they know whether to add or multiply. Emphasize that when the same number is

on each shelf or in each page, or on each whatever, then they know to multiply (they could add too, but

multiplication is less work).

Have students solve the following multiplication word problems:

a) There are 5 bookshelves and 3 books on each shelf. How many books are there altogether?

b) There are 100 cm in each metre. How many cm are in 3 metres?

c) There are 3 medals given in each event. How many medals are given in 7 events?

d) Bonus: There are 52 weeks in each year. If Jenny turned 8 today, how many weeks until she turns 10?

Remind students how to decide between multiplication and division (see NS3-65):


Have students decide which two pieces of information they are given (between the number of sets, the

number of objects in each set and the total number of objects) and which piece they need to determine.


a) 30 boxes. 6 shelves. How many boxes on each shelf?

6 × _________ = 30

number of sets × number of objects in each set = total number of objects

b) 30 books. 6 books on each shelf. How many shelves?

_______ × 6 = 30


c) 3 books on each shelf. 5 shelves. How many books?

5 × 3 = _______


When they are given the total number, they should divide. When they have to find the total number, they

should multiply. Give students more practice with this skill.


NS3-89 Word Problems and NS3-90 Planning a Party and NS3-91 Additional Problems

Goal: Students will solve word problems involving addition, subtraction, multiplication and division.

Prior Knowledge Required: addition, subtraction, multiplication and division, word problems

Vocabulary: sum, difference

Most of these worksheets provide review and extra practice.

For Question 5 on NS3-89, remind students how to list in an organized way the pairs of numbers that sum to

a given number. They could then look at all their pairs to see if there is one pair with the correct difference.

Extension: After all students have completed Question 6 on NS3-89, discuss the various solutions. For

example, some students might multiply 79¢ by 5 to find the total and then subtract the total from $5.00. Other

students might calculate the change from $1 for each pack (21¢) and then multiply by 5. This works because

the total number of dollars is the same as the number of packs.


NS3-92 Charts Goal: Students will use charts to find possibilities.

Prior Knowledge Required: addition, subtraction, multiplication, division

Tell students that a restaurant has orange juice or apple juice to drink and eggs or pancakes or French toast

for breakfast to eat. If we want to choose one of each, what are the possible choices? Have students

volunteer possibilities. Then tell them that you want an organized way to make sure that you don’t miss any

choices. Write down orange juice on the board and ask how many choices to eat you can have if you pick

orange juice to drink. What are those choices? Write on the board:

Orange juice, eggs

Orange juice, pancakes

Orange juice, French toast

Apple juice,

Apple juice,

Apple juice,

ASK: Why did I write down apple juice three times? Have a volunteer finish the chart. ASK: How many

choices are there altogether? Did we find all of them? How do you know? Could I have organized my chart

differently? Write on the board:

Eggs,

Eggs,

Pancakes,

Pancakes,

French toast,

French toast,

ASK: How am I organizing my chart now? Why did I write each choice twice instead of 3 times? Then have a

volunteer finish the chart.

Repeat with several similar examples, always keeping two choices for one option and three choices for

the other.

Then show students a dart board where they get 2 points for hitting the board but 5 points for hitting the

centre of the board:


Ask students to make a chart to show all the scores they could get by throwing the dart twice. Assume that

they will never miss the board, so they never get 0 points (if they miss the board, they get another throw).

Help get them started:

1st

dart 2nd

dart Total Score

2

2

ASK: Why did I write 2 twice?

Have students individually finish the chart in their notebooks?

Bonus:

Assume that they can get a score of 0 by missing the board. Help them get started:

ASK: Why did I write each first score three times?

1st

dart 2nd

dart Total Score

0

0

0

2

2

2

5

5

5


NS3-93 Arrangements and Combinations and NS3-94 Arrangements and Combinations

(Advanced)

Goal: Students will learn problem-solving.

Ask: I want to make a 2-digit number that uses the digits 1 and 2 each once. How many different numbers

can I make? (2) What are they? 12 and 21. What 2-digit numbers can I make using the digits 3 and 5 each

once? 4 and 7? 2 and 9?

Ask students to make 2-digit numbers:

a) an even number using the digits 3 and 4.

b) An odd number using the digits 5 and 6.

c) A number greater than 40 using the digits 1 and 7.

d) A number less than 30 whose tens digit is four times smaller than its ones digit.

e) A number divisible by 5 using the digits 3 and 5.

f) A number divisible by 5 using the digits 0 and 6.

Ask students to find a 3-digit number that uses each of the digits 1, 2 and 3 once. Write down all their

answers and do not stop until they found all of them. Encourage them to write the 6 different numbers in an

organized list. Repeat with the digits 2, 5 and 7 and then with the digits 3, 4, and 8 and then with the digits 4,

6 and 7. Ask them how they are organizing their list (for example, to write all the numbers using digits 1,2

and 3 each once, they can start with numbers having hundreds digit 1, then numbers with hundreds digit 2

and then numbers with hundreds digit 3; there will be 2 of each) Ask how does the organization make it easy

to know whether they have all of them or not?

Ask students to make 3-digit numbers:

a) An even number using the digits 3, 4 and 5

b) An odd number using the digits 4, 5 and 6.

c) A number greater than 400 using the digits 2, 3, and 6.

d) An even number greater than 400 using the digits 3, 4 and 5.

e) An odd number less than 200 using the digits 1, 3 and 4.

f) An odd number divisible by 5 using the digits 0, 3 and 5.

Draw on the board:


Show students how to arrange these circles in a row in 2 different ways: or

Then draw a third colour and have them find different ways to arrange the circles in a row. Have

volunteers come up to show the different ways. After all 6 ways are shown, tell your students that it is a good

idea to reflect back and ask if there is an organized way to find all 6 ways. If we want to make sure we found

all of them and didn’t leave out any, how could we start? (start by deciding what colour the first one is) Let’s

say the first one is white. What can the second one be? (solid or striped) Have a volunteer show the 2

possibilities that start with the white circle. Have another volunteer show the 2 possibilities that start with the

solid circle and another volunteer show the 2 possibilities that start with the striped circle.


NS3-95 Guess and Check Goal: Students will learn how to improve previous guesses to get closer to an answer.

Bring in a large strip of paper and ask a volunteer to guess where half is. Ask the class: Should the next

guess be further or not as far? How can they tell? Repeat until half is found. (NOTE: One way of deciding if

the guess is too far or not far enough is to directly compare the two halves either by folding the paper or

tracing one part of the line and placing it over the other part.

Then write on the board: 23 + __ __ = 58. Have students guess an answer that they think is close to the

correct answer. Students might say 30 or 35 or 40, for example. Then ASK: Is the guess too high or too low?

Should I guess a larger number or a smaller number next? Continue until students have found the correct

answer.

Then write on the board: 47 + __ __ = 74. Repeat the process. ASK: How am I improving my guess each

time? Would guessing and checking be a good strategy if I didn’t improve my guess each time? What if I just

randomly tried a bunch of numbers and hoped to eventually get it right—is that a good strategy?

Ask students if they know of any games that use the guess and check strategy by improving the guess each

time? (warm and cold, the person tells you whether you are looking close or far away from the object and

you adjust your search accordingly; guessing a number between 1 and 20 that the other person is thinking of

and they tell you whether you are too high or too low) Which games do they know that use guessing without

improving their guess each time? (Hide and seek unless you are given clues such as noise) Which game

usually takes longer to find the correct answer? Games like guess a number between 1 and 20 or games like

hide and seek? Encourage students to share times when hide and seek took a long time precisely because

guessing the possibilities didn’t help with eliminating any option except the one they just checked.

Then write on the board: 2 ___ + ___ 4 = 65. ASK: How is this problem different from: 47 + __ __ = 74?

Emphasize that now they are guessing digits of different numbers instead of guessing the whole number.

Tell students that it might seem a bit overwhelming to try to guess two different numbers at the same time.

One way they can do this is to guess something for one of the numbers and then change the other number

repeatedly until you get close to the answer. For example: 20 + ____ 4 is about 65. We can try 1 and see

that 20 + 14 is 34, which is too low, so our next guess should be higher. Continue in this way until students

are confident they found the closest number they can: 20 + 44 = 64, is about 65.

Then have students adjust the first number until they get 65: 21 + 44 = 65. Repeat this with several examples

where no regrouping is required in the adding of 2-digit numbers. (EXAMPLES: 3 __ + __ 4 = 89,

4 __ + __2 = 59) Bonus: 2 __ 4 + __ 3 ___ = 796.

Then write on the board: 3 __ + __ 7 = 85 Again have students guess a number to fill in: 30 + __7 is about

85. (30 + 57 = 87 and 30 + 47 = 77) Since 87 is closer to 85, students might think that 5 is the best digit to

use. However, they are not allowed to change the 7 to a 5, so they would have to reduce the 30. This would

require changing the 3 to a 2, which they are also not allowed to do. Emphasize that 30 is the smallest the

first number can be, so whatever digit we put in front of the 7 cannot make the answer become more than


85. Since 87 is too high, the second number must be 47. Once we found the second number, the problem

becomes 3 __ + 47 = 85. This can again be done by guessing and checking. Give students practice with

more examples of this sort (EXAMPLES: 1__ + __8 = 81, 3__ + __ 9 = 56.)

Then tell students that you are looking for two numbers that add to 10 and have a difference of 2. Ask

students for some numbers they know that have a difference of 2. Is there a way of listing numbers with

difference 2 so that their sums always increase? (1 and 3, 2 and 4, 3 and 5, and so on). So we can guess

a pair and check to see if the sum is too high or too low. We then know whether to go further in the list

or earlier in the list. For example, if we try 5 and 7, we know that the sum is too high, so try 4 and 6.

We’re done.

Have students find two numbers with difference two that add to:

a) 16 b) 24 c) 22 d) 48 e) 56

Bonus: Find two numbers that add to 234 and have a difference of 2. Students might notice the pattern that

the answer is always very near to half of the given number. For example 7 and 9 are close to 8, which is half

of 16.

Ask the class to list in order the first ten pairs of numbers with difference 3: (1 and 4, 2 and 5, 3 and 6, … ,

10 and 13). Tell students that you are looking for a pair of numbers with difference 3 that add to 37. Have a

volunteer guess a number they think will be close to the right number. For example 15 + 18 is 33, which is

too low. Then try 16 + 19 = 35, again too low. Then try 17 + 20 = 37; done.

Repeat for various such examples.

Cross-Curricular Connection: If you have a piano available and a tape recorder, record yourself playing

several piano notes with enough time in between so that rewinding between notes is easy. Take your

students to a room with a piano. Play your first recorded note once and tell your students that you want to

guess which note this was. Demonstrate guessing a note on the piano right in the middle. ASK: Should my

next guess be higher or lower? Where should my next guess be – to the right or to the left? Rewind and play

the note again. If the next guess should be higher, ASK: A lot higher or a little higher? How about here?

Repeat several times until they guess it correctly. If your students have trouble listening to musical notes, put

tape on certain keys (say 10 keys apart) and tell your students that the note you played is one of the taped

keys. This will make it easier to guess and check. As your students become more comfortable, you can

gradually move the tape pieces closer together.

Extensions:

1. Find two numbers that:

a) Have a difference of 9 and add to 9;

b) Have a difference of 9 and multiply to 10

2. Refer back to finding pairs of numbers that add to 10 and have a difference of 2. Ask students if, instead

of starting with numbers that differ by 2, could they have started by listing numbers that add to 10. What

is an ordered way of listing numbers that add to 10? (Start with the first number 1 and increase the first

number by 1 each time: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5) Then notice that the differences

decrease so if you start by guessing 2 and 8, your next guess needs to be further in the list.


NS3-96 Puzzles Goal: students will improve their problem-solving ability.

Prior Knowledge Required: adding two or three 1-digit numbers, trading for coins of equal value.

Write the following numbers on the board: 1, 2, 3, 4, 5, 6. Tell your students that they are only allowed to use

each digit once for each question below. There may be more than one solution for each part.

a) + = 4 (ANSWER: 1, 3)

b) + = 6 (ANSWER: (1, 5 or 2, 4)

Ask students to place numbers from 1 to 6, at most once each, so that any pair of numbers joined by a

straight line add to 6:

c) + = 8 (ANSWER: 2, 6 or 3, 5)

Repeat the puzzle above with numbers joined by a straight line adding to 8 instead of 6.

d) + = 7 (ANSWER: 1, 2, 4)

e) + = 8 (ANSWER: 1, 2, 5 or 1, 3, 4)

Take up both answers for part e) with the class. Then tell the students to solve the following problem

using the numbers from 1 to 5 each once so that both of the diagonal sums equal 8:


ASK: Which number is in both sums? Where does that number need to be placed? How do you know? Can

any other number be at the top? Why not?

Then challenge your students to use the numbers from 1 to 5 each once so that both lines of 3 numbers

add to 8:

f) + + = 9 (ANSWER: 1, 2, 6 or 1, 3, 5 or 2, 3, 4)

Challenge students to find all possible solutions, then have them solve the following problem using the

numbers from 1 to 6 each once so that all of the edge sums equal 9:

ASK: Which numbers need to be in the corners? Why?

e) Repeat part d) with + + = 10 (ANSWER: 1, 3, 6 or 1, 4, 5 or 2, 3, 5)

Now tell students that you have 3 dimes, 5 nickels and 5 pennies. ASK: How can this be evenly split

among 2 people. Draw the coins or write their letters (D, D, D, N, N, N, N, N, P, P P, P, P) and 2 circles

to divide them into. Divide the dimes first, using nickels if necessary:

D, N, N D, D D, N, N D, D

N, N N, P, P

P, P, P

N, N, N, P, P, P, P, P


Adding or Trading Game _________________________________________________2

Always, Sometimes, or Never True (Numbers) ________________________________3

Define a Number _______________________________________________________4

Dimes, Pennies, and Base Ten Materials _____________________________________5

Fractions That Add to 1 __________________________________________________6

Money Matching Memory Game ___________________________________________8

Number Lines to Twenty _________________________________________________9

Shaded Fraction Strips __________________________________________________10

Smallest Number of Coins Chart __________________________________________12

NS3 Part 2: BLM List

Adding or Trading Game

END 1¢ 5¢ 1¢

1¢ 5¢ 1¢ 5¢10¢ 5¢

1¢ 5¢ 1¢25¢ 10¢ 1¢

START 5¢ 10¢ 1¢ 10¢25¢ 1¢ 1¢

1¢

1¢

10¢5¢

1¢

1¢

10¢ 25¢

1¢

1¢

10¢1¢10¢25¢



Always, Sometimes or Never True (Numbers)

Choose a statement from the chart above and say whether it is always true, sometimes true, or never true. Give reasons for your answer.

1. What statement did you choose? Statement Letter

This statement is…

Always True Sometimes True Never True

Explain:

2. Choose a statement that is sometimes true, and reword it so that it is always true.

What statement did you choose? Statement Letter

Your reworded statement:

3. Repeat the exercise with another statement.

AA 4-digit number is greater than

a 3-digit number.

BThe product of two multiples

of 5 is odd.

CIf you multiply a 2-digit number by a 1-digit number, the answer

will be a 2-digit number.

DIf you multiply a number by zero,

the answer will be zero.

EWhen you subtract a 1-digit

number from a multiple of 100 you will have to regroup.

FThe product of two

even numbers is even.

GWhen you divide, the remainder

is less than the number you divide by.

HThe product of two numbers

is greater than the sum.

IIf you have two fractions, the

one with the smaller denominator

is the larger fraction.

JMultiples of 8 end in

even numbers.

KTenths are larger than

hundredths.

L10 thousands is the same

as 10 thousandths.

MMultiples of 5 are divisible by 2.

NThe product of 0

and a number is 0.

OA number that ends in an

even number is divisible by 2.


Define a Number

1. Name a number that statement D applies to:

2. Name a number that statement C and O apply to:

3. Name three numbers that statements N and A apply to: , ,

4. a) Name a number that statements B, D, G and O apply to:

b) Name a number that statements D, L and O apply to:

5. a) Which statements apply to both the number 22 and the number 30?

b) Which statements apply to both the number 12 and the number 32?

6. Can you find four numbers that statement Q applies to? , , ,

AThe number is even.

BThe number is odd.

CYou can count to the

number by 4s.

DYou can count to the

number by 3s.

EYou can count to the

number by 25s.

FYou can count to the

number by 100s.

GIf you multiplied the

number by 5, the product would be larger than 100.

HThe number has 3 digits.

IThe ones digit is one less than the tens digit. The ones digit is

5.

JThe number has 3 or more

digits.

KThe sum of its digits

is greater than 9.

LThe number has 2 digits

and the ones digit is greater than the tens digit.

MThe number is less than 40.

NIf you rolled two dice and

added the numbers together, you could get the number.

OThe number is less than 25.

PThe ones digit of this number

is divisible by 3.

QYou can get this number by

multiplying another number by itself (EXAMPLE: 9 = 3 × 3).

RThe ones digit of this number

is more than the tens digit.

Each statement describes at least one whole number between 1 and 100.



Dimes, Pennies, and Base Ten Materials

Show each amount using tens blocks and ones blocks.

Look at the pictures.

Fill in the blanks.

The number of dimes is equal to the number of ___________

blocks.

The number of pennies is equal to the number of ___________

blocks.

tens or ones

tens or ones


Fractions That Add to 1

What fraction is shaded?

What fraction is not shaded?

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

shaded

not shaded

1

4

3

4



Colour in the rest to make 1 whole.

How much did you colour?

+ = 1 whole 1

4

3

4

+ = 1 whole 1

3

+ = 1 whole 1

6

+ = 1 whole 1

9

+ = 1 whole 1

5

Fractions That Add to 1 (continued)


Money Matching Memory Game

$0.75 75¢ $7.50

750¢ 20¢ $0.20

200¢ $2 $1

1¢ 100¢ $0.01

$2.02 $2.20 22¢

202¢ 220¢ $0.22



Nu

mb

er L

ines

to

Tw

enty

10

11

12

13

14

15

16

17

18

19

20

98

76

54

32

10

10

11

12

13

14

15

16

17

18

19

20

98

76

54

32

10

10

11

12

13

14

15

16

17

18

19

20

98

76

54

32

10

10

11

12

13

14

15

16

17

18

19

20

98

76

54

32

10


Shaded Fraction Strips



Shaded Fraction Strips (continued)


Smallest Number of Coins Chart

Use the smallest number of coins to make each amount.

HINT: Use as many quarters as you can first, then dimes, then nickels, and then pennies.

10¢25¢ 5¢ 1¢ TOTAL

8¢

35¢

30¢

26¢

52¢

61¢

71¢

12¢ 1 2 3




ME3-18 Analogue Clock Faces Goal: Students will order the numbers on clock faces.

Prior Knowledge Required: Familiarity with clocks

Understanding of ordinal numbers

Vocabulary: clock, analogue, quarters, sequence, before, after, first, second, third

ASK: What is a clock? How do we use clocks? What does a clock measure? Record the students’ ideas.

Draw a circle on the board or on chart paper with boxes where the numbers on a clock would be. (See

below for initial example.) Ask students to tell how many numbers are usually on a clock’s face. Show

students exactly what part of the clock is referred to as its face. Explain that the clock face is divided into

four parts called quarters. The easiest way to draw a clock face is to first put the numbers 12, 3, 6, and 9 in

the proper places. Pointing at the box where the 12 should be, ask students to tell which number fits into

that spot. Have a volunteer come to the board and write in the 12. Repeat with the numbers 3, 6, and 9, in

any order.

Next, show a similar clock face where the 12 and the 6 have been added, but the 3 and the 9 are missing.

Have another volunteer insert the missing numbers.

Finally, remove all four “key” numbers and encourage a volunteer to use the preceding clocks and the clock

in the classroom (if there is one) to fill in all four missing numbers.

Now ask students what numbers come before and after the 3. Write those numbers on the clock face.

Repeat for the 6 and the 9. Ask students what comes before the 12. Fill in the 11. Now discuss with

students what number usually comes after the number 12. Most should say 13. Some will say 1, based on

their prior knowledge of clocks. Explain that the clock face shows only 12 hours and then starts again. You

may wish to explain at this point that a day consists of 24 hours and that it is divided into two periods called

a.m. and p.m. (a.m. starts at midnight and ends at noon, while p.m. starts at noon and ends at midnight).

Before moving on to the next lesson, ensure all students understand that a clock tells time sequentially—

one hour follows another.

12

9 3


Activities:

1. Brainstorm with students everything they know about time. Record all information on a chart.

2. Working in small groups, have students search through magazines to find and cut out pictures of

watches, clocks, timers, etc. They can use the pictures to create a collage. Remind students to give

their collage a title. Students who are ready to read and write the time could write the time shown on

each clock underneath the picture.

3. The BLM “Analogue Clock Faces” shows the clock face divided into quarters, with 15-minute intervals

noted for students who need an extra visual support.

4. http://www.learningplanet.com/act/tw/index.asp?contentid=410

In this interactive online activity, students can fill in the missing numbers on the clock face.

Extensions:

1. Students can research the origins of the clock and why it measures time in 12-hour increments.

2. Challenge students to order times on the half-hour! Show clock faces with these times: 2:30, 4:30, 1:30,

12:30.

3. Challenge students to order times on the hour and the half-hour. Clock faces might show 12:00, 1:30,

5:30, 3:00, and 11:30.

Journal:

Describe a clock (what it looks like and how it works). Draw a picture to go along with your explanation.


ME3-19 Hands on an Analogue Clock Goal: Students will identify the hour and the minute hands on an analogue clock.

Prior Knowledge Required: Familiarity with a clock face

Familiarity with the hands on a clock

Vocabulary: clock face, clock hands, hour, minute

Show students a clock and draw their attention to the two hands. ASK: How are the hands the same?

How are they different? Do you know what each hand is called? You can use a graphic organizer, such

as a Venn diagram, to record students’ answers. Once all ideas have been recorded, summarize key

points for the class. Make sure students understand the key difference—the minute hand is longer than the

hour hand.

Next, show the hands at different positions on the clock, and have students identify which hand is pointing

where. Ask questions such as: Which hand is pointing at 12, minute or hour? Which hand is pointing at 3?

Now put the minute hand at the quarter hours only (12, 3, 6, and 9) and ASK: What number is the minute

hand pointing at? Once students are comfortable with these basics, show the minute hand at different five-

minute intervals. Keep the hour hand at 12 while you change the position of the minute hand.

Activity: Students can make their own clock. Each student will need a paper plate, some sturdy paper

and a paper fastener.

The plate is the face of your clock. Draw or paint the numbers on the plate. Cut out a large and a small

arrow for the hands. Use a paper fastener to affix them to the centre of the clock. The clock can be used to

show different times during this and the next lessons.

Extensions: Project:

Which non-digital clocks do not have hands? (Examples for students to investigate: hourglass, sundial,

water clock) Choose one of the clocks without hands. How does it show time? Where was it used and why?

Journal:

The hour and the minute hands are similar and different because…


ME3-20 Telling Time The Hour Hand Goal: Students will identify hours on an analogue clock.



Ability to differentiate the two hands on a clock


Review the previous lesson. Draw two clock faces—one with the hour hand pointing at 2, and one with the

hour hand pointing at 3 (leave off the minute hand). Ask students which comes first (assuming they are

focusing only on a 12-hour period), two o’clock or three o’clock. Repeat with six o’clock and five o’clock.

Continue with more examples until students can readily identify which time comes first. Finish with twelve

o’clock and one o’clock.

Next, draw three clock faces without the minute hand, showing three, two, and one o’clock, and have

volunteers determine which came first. Ask them to order the times using 1st, 2

nd, and 3

rd. Repeat with more

sets of three times.

Remind your students that the hour hand shows the hour. Ask your students to tell you what hour it is in the

clocks you have drawn. Write the answers in two forms: “The hour is 3” and “3:00.”

ASK: How do people answer the question, What time is it? Record the answers students have heard on the

board in 2 columns—times “past the hour” and times “to the hour”

EXAMPLES:

Ten minutes after three Twenty minutes before two

Quarter past five Five to ten

Half past one Quarter to eight

Eight thirty-five

…and so on

Start with the times “past the hour.” ASK: The time is twenty minutes after three. Which hour has just

passed? (three o’clock) So the hour is 3. Is it 3:00? No, it’s after three, so the minutes are different. Where

is the hour hand pointing? Explain that the hour hand is not pointing directly at 3; it has already started

moving towards 4. Draw an analogue clock without the minute hand and with the hour hand pointing

between 3 and 4. Draw several more clock faces without the minute hand and with the hour hand pointing

between hours. Ask your students to write the hour for each time in two ways: Hour:___ and ___: .

Repeat the exercise, but draw the minute hand as well. Students do not need to write the minutes yet.


As a challenge, you might draw several clock faces without the hands, give several times in verbal form

(e.g., five past ten) and in digital form (e.g., 4:15), and ask your students to draw the hour hand for the

times. Students can also use the clocks they created during the activity in ME3-19.

Extension: The time is ten minutes before two. Is it already two o’clock? Which hour has passed? What

is the hour?

Write the hour for these times:

Five minutes to ten

Twenty minutes before two

Fifteen before twelve

Five minutes to nine


ME3-21 Telling Time Five-Minute Intervals Goal: Students will identify minutes (in five-minute intervals) on an analogue clock.




Skip counting by 5s


Review skip counting by 5s. Draw a close-up of part of a clock face on the board so that students can see

the minutes clearly. Let students count the number of minutes between each pair of numbers.

Remind your students that when a minute hand moves from one number to the next, five minutes have

passed. Ask your students where the minute hand points at the round hour, that is, when it is exactly three

o’clock or four o’clock (or any o’clock). Draw a minute hand pointing at 1 and ASK: How many numbers did

the minute hand move from the round hour? How many minutes have passed since the round hour? Write

“5” outside the clock face, beside the 1, and continue moving the hand and skip counting by 5s until you

reach 60. Explain to your students that there are 60 minutes in one hour, so when one hour ends and a new

hour begins, we start back at 0 minutes.

Draw several clocks with only a minute hand pointing at different numbers. Ask your students to skip count

by 5s around the clock, writing down the minutes as they go, until they reach the position of the hand. Ask

your students to tell you how many minutes have passed since the round hour on each clock.

Draw several clocks with both hands and ask your students to tell how many minutes past the hour it is, and to write that down in this form: :45.

Activity: Each student will need two dice and a clock face with moveable hands (e.g., the play clock

made in the activity in ME3-19). Each student rolls the dice, adds the results, and points the minute hand

towards the sum. For example, if a student rolls 3 and 4, he should set the minute hand pointing at 7,

i.e. 35 minutes. Then the student writes down the minutes past the hour given by the minute hand.


ME3-22 Telling Time Putting It Together! Goal: Students will tell time (in five-minute intervals) on an analogue clock.




Skip counting by 5s


Review the previous two lessons. Tell your students that today they will tell both the hour and the minutes,

like grown-ups. Explain that, to tell time, we look at the hour hand first and then the minute hand. Draw

many clock faces showing various times (five-minute intervals only) and ask students to say and to write the

time. The should write the time in both digital form (using numbers) and verbal form (using words).

Encourage your students to say the time in various ways, such as “five past three,” “ten minutes after

twelve,” and “four and twenty-five minutes”.

Assessment:

What time is it? Write each time in digital and in verbal form: What time is it? Draw the hands on an analogue clock and write these times in verbal form: 12:50 2:45 1:05

Activities:

1. Each student will need two dice and a clock face with moveable hands (e.g., the play clock made in the

activity in ME3-19). The student rolls the dice, adds the results, and sets the hour hand pointing at the

sum. The student rolls the dice again, adds the results, and sets the minute hand pointing at the sum.

Then the student writes down the time. For example, if a student rolls 7 at the first roll and 9 at the

second roll, the clock should be set at 7:45.

2. Each pair of students will need a die and a clock face with moveable hands (e.g., the play clock made in

the activity in ME3-19). Player 1 rolls the die three times. He or she adds the results of the first two rolls

and writes them down as the hour, then multiplies the result of the third roll by 10 and chooses whether

or not to add 5 for the minutes. (If the third roll is 4, the minutes could be :40 or :45. If the third roll is 6,

the player should write :00 or :05 instead of :60 or :65.) Player 2 has to set the play clock to this time.

12

6

9 3

1

2

8

7

4

5

11

10

12

6

9 3

1

2

8

7

4

5

11

10

12

6

9 3

1

2

8

7

4

5

11

10

12

6

9 3

1

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11

10


ME3-23 Digital Clock Faces Goal: Students will tell time (in five-minute intervals) using both digital and analogue clocks.




Skip counting by 5s

Vocabulary: clock face, clock hands, hour, minute, digital, analogue

Show students an analogue and a digital clock (perhaps an alarm clock). ASK: How are these clocks

similar? How are they different?

Next, review the two ways students have learned to write time. Have volunteers write various times using

numbers (e.g., 12:00) and words (e.g., twelve o’clock). Sample times: 12:30, 12:15, 3:45, 2:50, 6:10.

Ask students to look at the digital clock face again, and to compare it to the times they’ve written. What

does the time on the digital clock face look like? (The time on the digital clock faces matches one of the

ways students have learned to write time.)

Show one o’clock on an analogue clock. Ask a volunteer to show what the time would look like on a digital

clock. Repeat with several times, to give students opportunities for practice. Then do the reverse: state and

write digital times and have students show the times on the analogue clock.

Activities:

1. Draw the following table and have students fill in the blanks.

2. On the BLM “Time: Digital Clock Faces,” students are asked to match the digital time to the

analogue clock.

3. Use the BLM “Time Memory Game” to play a game. Directions: Arrange the cards face down in a

rectangular array. Players take turns turning over pairs of cards. If the times on the cards match, the

player lays the pair aside. If the times do not match, the cards are turned face down again. The player

with the largest number of pairs wins.

1 hour ago now 1 hour later

3:15

7:45

11:30

1:45

8:00


Extensions:

1. Ask students to calculate how much time passed between the given times (assume the times are both

a.m. or both p.m.).

a) 7:20 and 7:25 b) 10:30 and 10:45 c) 8:20 and 8:40

d) 1:10 and 1:30 e) 3:45 and 3:50 f) 2:00 and 2:20

g) 3:15 and 3:55 h) 11:05 and 11:40 i) 10:15 and 10:45

HARDER:

j) 5:40 and 6:05 k) 6:45 and 7:20 l) 8:30 and 9:20

2. At various times during the day, ask your students to record the time on the classroom analogue clock

digitally. At the end of the day, ask them to calculate the amount of time that passed between each

reading.


Telling Time: How to tell time on digital and analogue clocks by J. Older

This book introduces the how and why of analogue and digital clocks. Read this as an introduction to time

and to the relationship between digital to analogue.

Journal:

List the differences and similarities between how we record digital and analogue times.


ME3-24 Timelines Goal: Students will find elapsed time using number lines.

Prior Knowledge Required: Thorough understanding of the concept of time

Ability to differentiate between long periods of time and short

periods of time

Understanding of hours and minutes

Ability to write the time

Knowledge of the number line

Vocabulary: passed, elapsed, hours, minutes

Draw this number line on the board:

Ask students to tell what they know about number lines and how they have used them in the past. (Adding,

subtracting, skip counting, and measuring are some of the possible answers.)

Draw a leap starting at nine o’clock and ending at ten o’clock:

ASK: How many hours have passed? How can you tell? Add another leap from ten o’clock to eleven o’clock

and repeat the question. Continue until students are comfortable using the number line to count the number

of hours that have passed.

Next, using the same number line, draw a leap from eleven o’clock to noon. Ask students how many hours

have passed. Add a leap to one o’clock. Repeat the question. Point out that although the numbers are not

sequential, the hours are. (The number 1 does not follow the number 12, but one o’clock comes after twelve

o’clock.) Students should focus on the number of leaps to help them determine how much time has passed.

Draw a new number line using half-hour increments. Ask how much time has passed with this leap:

Students should reply a half-hour or thirty minutes. Repeat the exercise, adding to the number of leaps

incrementally. Then use a different starting time, such as 10:30, and repeat the exercise to ensure that

students are using the leaps to help them count the half-hour increments that have passed.

9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00

9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00

9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 1:00


3:50 3:55 4:00 6:00 8:00 9:00 9:05 7:00 5:00

Finally, repeat the exercises with five-minute increments. Let your students tell how much time passed from

9:15 to 9:40, from 9:45 to 10:05, from 9:55 to 10:25, and so on.

Assess the understanding before proceeding to more complicated number lines. Do not continue until all

students are able to find elapsed time by counting forward using an equally divided number line.

Assessment:

Sindi played hockey from 9:45 to 10:20. How long did she play?

Draw this number line:

Tell students you want to calculate how much time passed between 3:50 and 9:05. Ask your students to

count by 5s until they get to the hour. How much time elapsed? (10 minutes) Draw a leap from 3:50 to 4:00

and label it “10 min.” Ask a volunteer to count by hours from 4:00 to 9:00. Draw another leap from 4:00 to

9:00 and label it “5 hours.” Ask another volunteer to draw a third leap from 9:00 to 9:05 and to label the arrow

“5 min.” Ask a volunteer to add up the labels to get the total time that passed from 3:50 to 9:05. Students will

need more practice with problems of this type.

SAMPLE PROBLEMS:

• Rita worked from 8:00 to 4:05. How long did she work?

• Sam’s birthday party started at 12:45 and ended at 3:10. How long was his birthday party?

Make sure all students are able to find longer period of elapsed time using number lines with leaps of

different length before proceeding to the next material.

Show 7:15 on a play clock or a clock drawn on the board. SAY: This is the time Eve wakes up. She eats

breakfast at 7:40. How much time passes between the time Eve wakes up and the time she eats breakfast?

Turn the hand slowly and ask a volunteer to count the minutes on the clock by 5s. Give your students several

practice questions, such as:

Eve brushes her teeth at 7:30 and gets to school at 8:15. How much time elapsed?

Eve arrived at school at 8:15. The math lesson started at 9:05. How much time elapsed?

Let your students solve such problems by counting by 5s or by drawing number lines marked off in five-

minute intervals. For harder questions involving times with two different hours (e.g., 4:15 and 5:35), suggest

that students use number lines. Invite volunteers to present their solutions.

9:40 9:45 9:50 9:55 10:00 10:05 10:10 10:15 10:20 10:25 10:30


SAMPLE PROBLEMS:

• Eve put a cake in the oven at 7:55. The cake should bake for 25 minutes. When should she take the

cake out?

• The art lesson starts at 1:30 and lasts for 55 minutes. When does it end?

• The TV show is on from 8:15 till 8:55. How long is the show?

Assessment: Cyril has to catch a school bus at 8:25. He woke up at 7:40. How much time does he have before the

bus leaves?

Bonus:

A witch is cooking a potion. The potion turns purple at 3:45. Twenty minutes after it turns purple, the witch

has to add snake heads. When should the witch add the snake heads? The snake heads should stay in the

potion for 35 minutes. Then the witch should stir the potion quickly 7 times clockwise, remove the heads,

and let the potion boil for 35 minutes more. After this, the potion will be ready. When should the witch take

the cauldron from the fire?

Activity:

http://www.shodor.org/interactivate/activities/ElapsedTime/

This website illustrates skip counting in “See” mode and lets students check their own skip counting in

“Guess” mode.

Extension:

Ask students to calculate how much time passed between the given times.

a) 7:25 a.m. and 11:25 a.m. b) 10:05 a.m. and 11:45 a.m. c) 4:20 p.m. and 8:50 p.m.

d) 3:10 p.m. and 8:30 p.m. e) 1:45 a.m. and 7:50 a.m. f) 2:00 p.m. and 8:20 p.m.

g) 9:00 p.m. and 1:30 a.m. h) 11:15 a.m. and 3:15 p.m. i) 11:00 p.m. and 3:35 a.m.


ME3-25 Intervals of Time Goal: Students will express time intervals in different units.



periods of time



Vocabulary: passed, hours, decade, century, centuries

Ask your students to suppose that a relative is coming to visit in two weeks and two days. How many days is

this? Students might reason as follows: A week is 7 days long, so in 2 weeks there are 7 + 7 = 14 days. The

relative is coming in two weeks and two days; since 14 + 2 = 16, the relative will arrive in 16 days. Give your

students more time intervals to express in days, such as 3 weeks and 4 days, 4 weeks and 1 day, and 6

weeks and 5 days.

Suggest that your students use models, such as dots grouped in 7s, to find out how many weeks are in 19

days. Students can draw 19 dots and circle sets of 7 dots to see that there are 2 full sets (2 weeks) and 5

dots (5 days) left over.

Ask your students to convert the following time intervals into weeks and days:

14 days 17 days 21 days 24 days 29 days 27 days

ASK: How many minutes are in one hour? In two hours? Three hours? Record the numbers in a table. Write

two time intervals and ask your students to tell which one is longer. Start with simple pairs of intervals, such

as 20 minutes and two hours, 40 minutes and one hour, 65 minutes and one hour, 60 minutes and two

hours. Continue with harder pairs, such as 90 minutes and one hour, 90 minutes and two hours, 130

minutes and two hours. Ask your students to explain how they know which interval is longer. Explain to your

students that when they compare two measurements in different units, it is convenient to convert both

measurements into smaller units. For example, to compare 145 minutes to 2 hours and 10 minutes,

students should convert the second time interval to minutes: 2 hours + 10 minutes = 120 minutes + 10

minutes = 130 minutes). As a final series of challenges, ask your students to compare more such pairs of

intervals.

SAMPLE INTERVALS:

1 hour and 40 minutes and 2 hours

3 hours and 50 minutes and 225 minutes

three periods of 45 minutes and 2 hours


Explain to your students that longer periods of time are measured not only in years, but also in decades and

centuries. Write the singular and plural forms of “decade” and “century” on the board. Explain that a decade

is 10 years long, and a century is 100 years long. Ask your students to give an example of a period of time

that is measured in decades or in centuries (e.g., age of a country, age of a tree, time since the Middle

Ages). List several periods of time and ask your students to write them in centuries and/or decades.

SAMPLE PROBLEMS:

50 years = _____ decades 110 years = _____ decades

90 decades = _____ years = _____ centuries 1 century = _____ decades

300 years = _____ centuries = _____ decades 210 years = _____ centuries + _____

decades

Assessment:

Convert:

7 weeks = _____ days

31 days = _____ weeks and _____ days

50 decades = _____ years = _____ centuries

2 centuries = _____ decades = _____ years

500 years = _____ centuries = _____ decades


ME3-26 Estimating Time Intervals Goal: Students will estimate various time intervals.



periods of time



Vocabulary: decade, century (centuries), hour, minute, second, week, month

Ask your students to name some units people use to measure time. Record the answers on the board. Ask

students which units they would use to measure these intervals of time:

• the length of Christmas holidays

• the length of summer holidays

• the age of a person

• the length of a music lesson

• the length of a movie

• the age of a baby that is not yet one month old

Next, ask students if they know what unit of measurement is used to count an amount of time smaller than a

minute. Students may say seconds, nanoseconds, milliseconds, etc. Accept all correct answers. Explain

that seconds are a very short amount of time. There are sixty seconds in one minute and sixty minutes in

one hour. Draw a long line on the board and tell student that it represents one hour. Ask a volunteer to draw

a line that represents one minute, and ask another volunteer to draw a line that represents one second. The

final drawing may look something like this:

An hour _____________________________________________________

A minute ____

A second _

Ask a volunteer to clap 5 times, to do 5 sit-ups, and to run around a desk. Measure the time each activity

takes with a stopwatch, to help students develop a better understanding of how long one second is.

Divide your students into 7 groups. Give each group a sheet of paper, and assign each group a unit for

measuring time from the vocabulary for this lesson. Ask each group to list 10 time intervals that would

normally be measured in their unit. To decide in which order the lists will be shared with the class, ask a

volunteer to order the units of time from least to greatest.

Ask your students to estimate the time it takes to do various daily activities, such as sleeping, washing,

brushing teeth, eating, studying, walking a dog, etc. To help students estimate the time, you might ask

questions like: When do you go to bed? When do you get up? How much time elapsed?


Activities:

1. Ask students to estimate how long it will take to…

a) write 100 words

b) walk up a flight of stairs in the school

c) read a chapter of a book

d) count to 50

e) clap your hands 30 times

f) sing the national anthem

g) do 20 jumping jacks

(Students should check some of their estimates using a clock with a second hand.)

2. Students can use the BLM “Time: How Long is a Minute?” to record activities that can be done in one

minute. Students will need a stopwatch to time themselves doing various activities.

Extensions:

1. How many…

a) seconds are in a minute?

b) minutes are in an hour?

c) hours are in a day?

d) days are in a week?

e) weeks are in a month?

f) years are in a decade?

2. How many…

a) seconds are in 3 minutes?

b) minutes are in 2 hours?

3. If a ones block (in base ten materials) represents 1 year, which block would represent…

a) a decade?

b) a century?


ME3-27 Cumulative Review

This worksheet is a review worksheet for time and money. Review the basic concepts of money before

assigning it.


ME3-28 Area Goal: Students will measure area in non-standard units.

Prior Knowledge Required: Concept of a surface

Concepts of size (big and small)

Spatial sense

Vocabulary: surface, area, big, small

ASK: What is a surface? How would you measure the size of a surface? Record and discuss students’

ideas.

Introduce the term “area” and explain that to measure the area of a surface, you can cover it with same-

sized units, such as squares, and count them. Draw a rectangle on the board and affix some squares to it,

so that many gaps are left. Ask your students if this is a good way to cover the shape. Repeat with

overlapping squares. Is this better? Repeat with squares that cover much more space than the shape itself.

Invite volunteers to affix the squares in the right way (close to but not overlapping one another and only on

the shape).

Draw this shape on the board or on graph chart paper.

ASK: What is the area of this shape in squares? (2)

Next, draw this shape and ask what its area is.

Ask students which shape has the larger area and have them explain how they know.

Put more and more complex shapes on the board, and challenge students to find their areas. Here are

some examples (see the worksheet for more):

For the more complex shapes, students will have to keep track of the squares they have counted. Ask

students how they can keep track. They may use the reading pattern, count in rows, check boxes off as

they count and tally, etc. The more methodical they are about the process, the easier it will be to find the

area of the more complex shapes.


Repeat the sequence of exercises above for shapes made up of triangles. For each shape, ASK: What is

the area of this shape in triangles?

Draw a shape that is not divided into squares or triangles and ask your students to estimate how many

squares might cover the shape. Let them check and record the actual area. To help your students make

good estimates, draw a shape that can be covered by squares or triangles, ask your students to estimate

the area, then affix the squares or triangles one by one and ask your students after each shape is added if

they would like to revise their estimate. Continue until the whole shape is covered.

Draw two different shapes with the same area.

Invite a student to measure the area of one of the shapes with squares. Then ask your students to estimate

and then measure the area of the second shape. Record the measurements and the estimate. Students will

see that the shapes have the same area. Now provide your students with a different measurement unit,

such as a right-angled triangle that has half the area of the square. Ask a volunteer to measure the area of

the first shape with triangles. Record the area, so that your students have an opportunity to notice the

pattern (the area in triangles is twice the area in squares). ASK: What will be the area of the second shape

in triangles? Let your students check their predictions.

Activities:

1. Guide students to name or find objects in the classroom that they can use to illustrate an understanding

of area. EXAMPLES: “The blackboard has a big area.” “The cover of this book has a smaller area

than...” “The wall has a bigger area than...”

2. Students can choose a unit of measurement and prove that one object has a bigger area than another.

They can record this in their journals. Encourage them to use some form of graphic organizer, such as

a T-table with the headings “big area” and “small area,” to record their results.

Extensions:

1. Order the two shapes in QUESTION 8 on the worksheet according to their area in triangles. Does the

order change when you measure their area in rhombuses?

2. Have students draw a square with a perimeter of 12 cubes. Ask them to find the length of each side of

the square and then the area. Have them predict whether the area of a square with perimeter 12 cubes

would be the same as the area of a rectangle with perimeter 12 cubes. Have them check their

predictions.

Journal:

Area is….


ME3-29 Area in Square Centimetres Goal: Students will measure the area of shapes on grids in square centimetres.

Prior Knowledge Required: Concept of a surface

Area

Concepts of size (big and small)

Spatial sense

Measuring and drawing lines with a ruler

Perimeter

Vocabulary: surface, area, big, small, centimetre, square centimetre, centimetres squared (cm2),

2-dimensional, perimeter, rectangle

ASK: What units do we use to measure length? Accept all correct answers. Draw a shape on the board and

ask students what units they would use to measure its length—metres, centimetres, or kilometres. Would

they measure the length in linking cubes? Suppose they have to explain how large the rectangle is to a

person that has never seen linking cubes. Would they still use linking cubes or would they prefer a more

standard unit? Ask your students if they think that there might be a standard unit for area as well.

Explain to your students that area is often measured in units called “centimetres squared” or cm2. Show

students an example of a square centimetre, that is, a square whose sides are all 1 cm long.

Draw several rectangles and other shapes (EXAMPLE: L-shape, E-shape) on the board and subdivide them

into equal squares (or draw the shapes on a grid on the board). Label the side length of one square “1 cm”

Ask volunteers to count the number of squares in each shape and write the area in cm2.

Then draw several more rectangles and mark their sides at regular intervals, as shown below.

Ask volunteers to divide the rectangles into squares by joining the marks using a metre stick. Ask more

volunteers to calculate the area of these rectangles.

Ask students to draw their own shapes on grid paper and to find the area and perimeter for each one.

Activities:

1. Students could try to make as many shapes as possible with area 6 units (or squares) on a geoboard.

For a challenge, students could try making shapes with half squares. For an extra challenge, require

that the shapes have at least one line of symmetry. For instance, the shapes below have area 6 units

and a single line of symmetry.


2. Students work in pairs. One student draws a shape on grid paper, and the other calculates the area and

the perimeter.

Extension: Sketch the shape below (at left) on centimetre grid paper. What is its area in cm2? (16) Now

calculate the area using a different unit: a 2 cm × 2 cm square (see below right). What is its area in these

units? What happens to your measurement of area when you double the side lengths of the square you are

measuring with? (The area measurement decreases by a factor of 4.) If the area of a shape is 20 cm2, what

would its area be in 2 cm × 2 cm squares?

2 cm × 2 cm

square unit


ME3-30 Half Squares Goal: Students will find the area of shapes built with whole squares and half squares (i.e., triangles).

Prior Knowledge Required: Division by 2

Area

Vocabulary: area

Before the lesson, cut a square into two triangles by cutting across the diagonal. Now you have 2 congruent

right triangles whose base and height are the same length. Hold up the 2 triangles and show students that

they are congruent. Then demonstrate how you can put the triangles together to form a square. SAY: If I

know that the base and height of the triangles is 2 cm, what is the area of the square? (2 × 2 = 4 cm2) What

part of the square does each triangle cover? (half) What area does it have? (2 cm2)

Draw this shape on the board:

Tell students the square has area 1 cm2. ASK: What is the area of the whole shape? How did you figure

that out? Invite students to draw another shape with the same area.

Draw several shapes made up of whole squares and half squares (i.e., triangles) on a grid and ask students

to calculate the shapes’ area in terms of whole squares. Then draw a shape with 3 half squares and ASK:

What is the area of this shape? (one and a half squares) Ask students to draw shapes using both whole

squares and half squares. They should swap their drawings with a partner and calculate the area of their

partner’s shapes. Students can then check each others’ area calculations.

Draw several shapes made with even an number of half squares, say four. Ask your students to find the

area in triangles. ASK: If you have a shape built from four half squares, what is its area in whole squares?

What did you do to calculate the area in whole squares? (divided the number of triangles by 2) What is the

area of a shape made of 10 half squares? 200 half squares? 100 half squares and 100 full squares?

Assessment Calculate the area of the shape.

Height 2

Base 2


Bonus:

1. Calculate the area of the shape.

2. Write your name on grid paper using only squares and half squares. What is its total area? Compare

your name with a partner’s. Whose name has the greater area? Who in the class has the name with the

greatest area? The least area?

Activities:

1. Students use geoboards or dot paper (see the BLM “Dot Paper”) to create shapes with a given area

using whole squares. Then students make shapes that include a particular number of half squares.

(EXAMPLE: Build a shape that has an area of 8 squares and contains 8 half squares.)

2. Draw a 6 × 6 quilt pattern using whole squares and half squares. Use shading or colour to create a

design on your quilt. Calculate the area of the shaded or coloured squares. VARIATION: Use different

colours and calculate the area covered by each colour separately.

Extension: If it takes Sandra two seconds to colour one (triangle or half square). How long will it

take her to colour the whole shape from Bonus 1?


ME3-31 Puzzles and Problems ME3-32 Investigating Units of Area These are review worksheets that can be complemented with the following activities and extension.

Activities:

1. Students can make up their own problems like those in QUESTIONS 2 and 3 on worksheet ME3-31.

Partners can use graph paper or a geoboard to solve each other’s problems.

2. A rectangle has area 6 cm2 and perimeter 10 cm. What is the length of the rectangle?

Extension:

History of Measurement: An Investigation

Weights and measures are some of the earliest tools invented by humans.

Length: The first measurements were based on lengths of parts of the human body (for example, the width

of a thumb) or distances between them (for example, the distance between the end of an outstretched arm

and the chin). The Egyptian cubit was based on the length of the arm from elbow to outstretched finger tip.

Roman soldiers measured marching distances by counting paces, the distance from the heel of one foot to

the heel of the same foot when it next touched the ground.

Capacity: To compare the capacities of containers such as gourds or clay or metal vessels, they were filled

with plant seeds which were then counted.

Mass: The first weights were seeds and stones. The Egyptians and the Greeks used a wheat seed as the

smallest unit of weight. Today's carat (used to measure the mass of gems) is based on the weight of a

carob seed, used by the Arabs. The Babylonians compared the weight of an object with a set of special

stones and used different stones for weighing different things.

Standard Units

People and seeds come in different sizes. For measurements to be useful, everyone

needs to be using exactly the same unit. The units need to be standard. Often, kings

or queens decided what a standard for a measurement would be. These standards

spread through trade and commerce.

Metric System

About 200 years ago, the French created the metric system. This system of

measurement is now used throughout the world. It includes measurements for

length, capacity, volume, and weight. The metric system is based on 10s. To

change a measurement from one unit to another, you just move the decimal point.

For example, 300 centimetres = 3.00 metres.

U N

I T

S


1. With a partner, choose a part of the body (for example, length from elbow to fingertip, length of foot) as

a non-standard unit of measure for length. Write down a description for your measurement unit and

choose a name for your unit (e.g., the “stretch”: as far as you can stretch your thumb and index finger).

Choose 5 items to measure using your unit. Each partner will estimate the length of each item in their

unit of measurement, measure each item, and record the measurement (e.g., 4 stretches). Partners

should then compare their measurements.

2. Repeat the activity in Question 1 for capacity (for example, you could measure the capacity of a

container using ones blocks or linking cubes).

3. Repeat the activity in Question 1 for mass (for example, you could measure the mass of an object using

pennies or unsharpened pencils).

Class Projects:

1. Students can each measure a single length (for example, the width of the classroom) using a non-

standard measure, such as a cubit, then compare their measurements. They can then repeat the

activity using metres and centimetres.

2. Students can bring in stones of 3 approximate sizes—large (about 3 cm wide), medium (about 1 cm

wide), and small (3–5 mm wide)—and use them to measure the mass of various items. They can

construct a simple scale using a coat hanger with small foil pie plates hung from each end.


Analogue Clock Faces ___________________________________________________2

Dot Paper _____________________________________________________________3

Time Memory Game _____________________________________________________4

Time: Digital Clock Faces _________________________________________________5

Time: How Long is a Minute? ______________________________________________6

ME3 Part 2: BLM List

1st quarter

2nd quarter

4th quarter

3rd quarter

Analogue Clock FacesThe hour and minute hands m

ove in th

is dire

ction

.:00

12

6

11

5

1

48

7

9

10 2

:45

:30

:153



Dot Paper


Time Memory Game

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Time: Digital Clock Faces

A digital clock face looks like this . It’s a quarter past 3.

It is exactly the same as on an analogue clock.

12

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1011

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7

8

Match the analogue clock faces to the digital times.

12

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Time: How Long is a Minute?What can you do in a minute? Have your teacher time you.

How many times can you write your name?

How many jumping jacks can you do?

What number can you count to? Can you write the alphabet?

How many sit-ups can you do? How many times can you blink?




A Note about Terms in Probability

Terms and definitions for methods of representing data (bar graphs, etc.) as well as concepts in

probability have been included on the worksheets. We give a general summary of the concepts underlying

probability below.

Outcomes and Events

A simple action (such as “rolling a die,” “flipping a coin,” or “spinning a spinner”) has various possible results.

These results are called the outcomes of the action. If you flip a coin, the outcomes are: “You flip a head,”

and “You flip a tail.” If you identify a particular outcome or set of outcomes of an action (such as “rolling a

six,” “rolling an even number,” “tossing a head,” or “spinning red”) you identify an event.

In probability theory, the terms outcome and event have very precise meanings. But in elementary texts, the

term outcome is occasionally misused.

On the spinner below there are three possible outcomes:

The outcomes on the spinner are:

1. The pointer lands in the blue region.

2. The pointer lands in the red region.

3. The pointer lands in the green region.

Some textbooks will identify the outcomes as:

1. You spin blue.

2. You spin red.

3. You spin green.

Identifying outcomes by colour only can cause confusion when two or more regions of a spinner are painted

the same colour. Here is a spinner with four coloured regions:

R B

• G

R B

• B B


There are 4 outcomes of spinning the spinner:

1. The pointer lands in the blue region at top right.

2. The pointer lands in the blue region at bottom right.

3. The pointer lands in the blue region at bottom left.

4. The pointer lands in the red region.

This is clearly not the same as saying that the outcomes are:

1. You spin red.

2. You spin blue.

“You spin red” and “You spin blue” are events, not outcomes. To assess the relative likelihood of spinning

red or blue, students must recognize that the pointer may land in four distinct regions of the spinner (so that

there are four possible outcomes). In three of the four outcomes, the spinner lands in a blue region. Hence,

the event “You spin blue” is more likely than the event “You spin red.”

The outcomes for Question 1 e) on Worksheet PDM3-15 are “The spinner lands in region 1,” “The spinner

lands in region 2,” and “The spinner lands in region 3.” A student may write something more concise, such as

“You spin a 1,” “You spin a 2,” “You spin a 3.” Accept these answers, as long as your student knows that

when different regions of a spinner have the same number or colour, each region must be counted as a

distinct outcome. (To avoid confusion, we only use the term outcome on worksheets when the regions of the

spinner are uniquely coloured or labelled. When the same colour or label appears more than once on the

spinner, we use phrases like “ways of spinning red” instead of “outcomes.”)


PDM3-15 Outcomes

Goals: Students will identify the possible outcomes of various events.

Prior Knowledge Required: Experience in playing with a spinner and rolling a die.

Vocabulary: outcome

Tell students that today they will start learning how to predict the future! Hold up a die and ask students to

predict what will happen when you roll it. Can it land on a vertex? On an edge? No, the die will land on one of

its sides. Ask students to predict which number you will roll. Then roll the die (more than once, if necessary)

to show that the prediction about landing on a side works, but the number students picked does not

necessary come. Explain that the possible results of rolling the die are called outcomes. To predict the future

students must learn to identify which outcomes of various actions are more likely to happen and which are

not. But first, students must learn to identify outcomes correctly.

Hold up a coin and ASK: What are the possible outcomes of tossing a coin? How many outcomes are there?

Show a spinner and a set of marbles. What are the possible outcomes of spinning the spinner or picking a

marble with your eyes closed? Ask students to identify the possible outcomes of a soccer game. How many

outcomes are there? (3 outcomes: team A wins, team B wins, a draw)

ASK: You have to make a spinner with 5 possible outcomes. How would you do this? Invite volunteers to

draw possible spinners. Then draw the spinner below, shade each region with a different colour, and ASK:

How many outcomes are there for this spinner?

Are all the outcomes bound to come equally, or is there an outcome that might happen more often than the

other ones? Which colour are you most likely to spin? Draw the second below and ASK: How many

outcomes does the second spinner have? (4) Will the pointer ever be in the grey region? (no, never)

Activity: Your students will need a spinner like the first spinner above, with all regions coloured

differently. Students spin the spinner 10 times and tally the results. Which colour occurs the greatest number

of times?


PDM3-16 Even Chances Goals: Students will identify situations where the chances of an event are even.

Prior Knowledge Required: Outcome, Half of a number, Visual representations of fractions

Vocabulary: outcome, even chances, draw

Review with your students how they can find half of a number, half of a pie, half of a set of objects. You

might wish to use Questions 1–5 on worksheet PDM3-16 for this review.

Draw six squares on the board. Invite a volunteer to circle half of the squares. Ask your students to shade

less than half of the squares red. Draw a different set of six squares and ask your students if they can shade

a different number of squares blue, so that the number of blue squares will be still less than half of six.

ASK: What is half of eight? I need a number that is less than half of eight—which numbers fit this

description? Repeat these questions for different even numbers. Then draw even numbers of shapes and

ask students to find several ways to shade more than half of the shapes. Have volunteers help you solve

several problems like the following:

� I have 10 marbles. Half of them are red. How many marbles are red?

� I have 12 marbles and 6 of them are green. How many of my marbles are green: half, less than half, or

more than half?

� I had 14 marbles. I lost 6 of them. Did I lose more than half, less than half, or exactly half?

Questions 6–9 on worksheet PDM3-16 can be used to assess the understanding of the concept.

Hold up a coin. ASK: What are the possible outcomes of flipping this coin? How many outcomes are there?

Which is more likely—flipping a head or a tail? Explain that the chances are the same—you have even

chances of flipping a head or a tail. Write the term even chances on the board and explain that the chances

of an event are even when the event happens in exactly half of the outcomes. Flipping a tail is 1 out of 2

possible outcomes, and 1 is half of 2. ASK: How many outcomes are there when you roll a die? (6) How

many outcomes are numbers that are more than 3? (3) Since half of the outcomes are numbers greater than

3, you have even chances of rolling a number greater than 3 (and even chances of rolling a number that is

3 or less).

SAY: We have 8 marbles in a box. I take out 1 marble (without looking!). How many outcomes are possible?

(8, regardless of the colour of the marbles) What is half of 8? I would like to have even chances of taking out

a green marble. How many marbles should be green? (exactly 4) Does it matter what colour the other

marbles are? (No, provided they’re not green.) Invite a volunteer to draw a collection of 8 marbles (or to

create a collection using actual marbles, if available) that gives even chances of drawing a green marble.

If the collection uses only 2 colours, ask another volunteer to make a collection that uses at least 3 colours

but still gives even chances of drawing a green marble.


Draw a spinner:

ASK: Which part of the spinner is shaded green? What are the possible outcomes for this spinner? How

many outcomes are there? Which part (or fraction) of the outcomes is “The pointer lands in a green region”?

Are the chances of spinning green even?

Repeat this exercise with the spinner below. Emphasize, if necessary, that this spinner has 4 possible

outcomes because it has 4 different regions, but 2 of the 4—half—are green.

Draw 3 spinners:

Ask students to identify the outcomes for each spinner. ASK: In which spinners do you have even chances of

spinning red? (the one on the left and the one on the right) Draw more spinners on the board and ask

students to identify the spinners where you have even chances of spinning red.

Ask students to describe an event with even chances for rolling a die. (Possible answers: roll a number that

is 3 or less; roll an even number; roll 2, 3, or 5.) Ask students to describe another event with even chances.

Encourage students to think of examples that do not involve rolling dice, spinning spinners, or drawing

marbles. (EXAMPLE: Several pairs of boots are in a closet. I pick a boot in the dark. It is either a left boot or

right boot. So “I pick a left boot” has even chances.)

Assessment:

1. Circle the spinners where you have even chances of spinning red.

G R • Y

G R

• R B

G R

•

Y G

• G B

B

R •

G

R

• Y

B

G R

• B

G R

• R

B R

B

G R • Y

G R •

R

B


2. Draw a collection of 6 marbles that gives even chances of picking a green marble.

3. Draw a collection of 8 marbles of at least 3 colours that gives even chances of picking a yellow marble.

Activity: Divide students into 3 groups. Let each group make one of the 3 spinners below, spin it 24

times, and record the results. Then have groups make a bar graph of the results.

ASK: Did your group spin red in more than half, less than half, or exactly half of the spins? Is this what

you expected? Discuss as a class.

Extension: Complete each statement by writing “more than half” or “less than half.”

a) 2 is ___________________ of 5

b) 3 is ___________________ of 7

c) 6 is ___________________ of 13 d) 7 is ___________________ of 11

e) 11 is ___________________ of 15 f) 5 is ___________________ of 11

G R

• R B

G R

• B R

G R

• B

R B


PDM3-17 Even, Likely and Unlikely Goals: Students will describe the chances of events as even, likely, and unlikely.

Prior Knowledge Required: Outcome, Half of a number, Visual representations of fractions

Vocabulary: outcome, even chances

Have students make some predictions: ask them to tell you if the following events are likely or unlikely.

� The sun will rise tomorrow.

� The teacher will give the answers to the test before giving the test.

� An alien will walk into the class in the next minute.

� We will have lunch in half an hour.

� It will rain tomorrow.

� It will snow in June.

Invite students to name some events and have other students tell if the events are likely or unlikely. You

could ask students to compare the likelihood of events. For example, it is unlikely to snow in June, but it is

more unlikely that an alien will walk into the class!

Draw the spinner below and ask students if it is likely that the spinner will point to red. Conduct an

experiment with 12 spins. Use a tally chart to record the results. Explain to your students that

mathematicians call an event likely if it is expected to happen more than half the time and unlikely if it

is expected to happen less than half the time. From the last lesson, they know that an event with even

chances is expected to happen exactly half the time. Write all three terms on the board. Using the tally chart,

describe the chances of spinning red, blue, and green in these terms.

Describe several events (see Examples below) and ask students to count the total number of outcomes and

the number of outcomes that suit the event, and to tell whether the event is likely, unlikely, or has even

chances. Remember: If the event matches more than half the outcomes, then that event is likely—chances

are it will happen more than half the time. If the event matches less than half the outcomes, it is unlikely.

EXAMPLES:

There are 4 pairs of boots in a dark closet—2 black, 2 brown. Events:

� Pull out a right boot � Pull out a black boot

� Pull out a brown left boot � Pull out a boot that is either

� Pull out a boot that is not left brown black or right brown

G R

• B


A model (draw 8 shoes and colour them) could be helpful for students who have trouble counting outcomes.

Events: Spin green. Events: Spin blue.

Spin purple. Spin purple.

Assessment:

1. Are the chances likely, unlikely, or even?

a) Pull a black sock from a box with 6 green socks and 4 black socks.

b) Pull a penny from a pocket with 5 pennies, 4 dimes, and 1 nickel.

2. Harold rolls a die. Give an example of a likely event and an unlikely event for Harold. Both events should

be possible.

P

B

P Y

P

P P

B P G

GB G

B

B


PDM3-18 Describing Probability Goals: Students will describe the likelihood of events.

Prior Knowledge Required: Outcome, Half of a number, Likely, Unlikely,

Vocabulary: outcome, even chances, likely, unlikely, certain, impossible

Review the meaning of the terms likely and unlikely with your students. Write the terms on the board. Ask

your students which word people would use to describe an event like meeting a live dinosaur in the street.

Can that happen at all? Add the word impossible to the list. Ask your students which words describe an

event that will definitely happen, like rolling a number less than 7 on a die. Add the word certain to the list.

Ask students to describe the following events as likely, unlikely, certain, or impossible:

� It will rain on September 12

� It will snow on July 15

� There will be a math test before the end of the month

� You will grow wings

� Pull a green sock from a drawer with 10 green socks and 2 red socks

� Pull a $5 bill from a wallet with three $20 bills and one $5 bill

� Roll a number greater than 0 on a die

� Meet a green panther

NOTE: Although students might use the word impossible to describe the likelihood of meeting a dinosaur,

this event is not necessarily impossible (scientists might find a way to clone dinosaurs in the future). The only

events that are strictly impossible are events that are contradictory—like rolling a number greater than 6 on a

regular die. You might discuss this distinction with students.

Ask students to give examples of various events and explain whether they are likely, unlikely, certain, or

impossible. Encourage students to think of events using marbles, dice, money, and other objects, as well

events from daily life, such as meeting a tiger or an astronaut on the way to school.

Draw the spinners below on the board and ask your students to describe the following events as likely,

unlikely, having even chances, certain, or impossible for each spinner:

� Spinning green

� Spinning red

� Spinning blue

� Spinning yellow

G R

• B

G R

• • B G Y

• B

G R

• B R


Ask students to draw a spinner to match this description:

1. You are likely to spin yellow.

2. You are unlikely to spin green.

3. It is impossible to spin blue.

Show students a collection of marbles or coloured counters:

ASK: What are your chances of picking green? Which colour are you most likely to pick? Which colour is less

likely to be picked, yellow or red? So which colour is least likely to be picked?

Assessment:

Draw a collection of marbles of at least three colours so that:

1. You are likely to draw a green marble.

2. You are unlikely to draw a yellow marble.

3. It is impossible to draw a purple marble.

Bonus:

Design 3 spinners with different number of regions that fit this description:

1. You are likely to spin yellow.

2. You are unlikely to spin green.

3. It is impossible to spin blue.

Activity: Have students flip a coin 10 times and keep a tally of the number of heads. It is likely that not

every student flipped heads exactly half the time. Ask students to identify the result that was furthest from the

expected number of 5 heads. Then add up the total number of heads from all the tallies and the total number

of tosses (10 × number of students in class). The overall proportion of heads should be closer to half of the

total number of tosses. Explain to students that the more trials you conduct (i.e., the more times you repeat

an experiment) the more closely the actual outcome will match the expected outcome for an event.

Extension: Invent or describe a game where a certain player’s chance of winning is very close to certain.

What are the chances of the other player(s) winning?

R B B G G G G G Y Y

Y


PDM3-19 Describing Probability (Advanced) Goals: Students will describe and compare the likelihood of events using vocabulary words.

Prior Knowledge Required: Outcome, Half of a number, Likely, Unlikely

Vocabulary: outcome, even chances, likely, unlikely, certain, impossible, most likely, very unlikely

Review the previous lesson.

Draw the following collection of marbles and a probability line on the board:

ASK: How many marbles are in the collection? How many marbles of each colour do we have? Which colour

are you most likely to pick? Is your chance of picking green even, likely or unlikely? Which colour is less

likely to be picked: yellow or red? Why? Which colour is least likely to be picked? Ask your students to mark

on the line the probability of picking…

� a green marble � a blue marble � a yellow marble

� a red marble � a pink marble � a marble of any colour

Ask your students to give examples of various mathematical or real-life events. For each one, ASK: Where

on the probability line would you put this event? Why?

R B B G G G G G Y G R B

impossible certain even unlikely likely


PDM3-20 Fair Games Goals: Students will use the concept of even chances in games.

Prior Knowledge Required: Outcome, Half of a number, Even chances

Vocabulary: outcome, even chances, likely, unlikely, certain, impossible, most likely, very unlikely,

fair game

SAY: I would like to play a game with you. The rules of the game are simple. I will spin a spinner. If I get red,

I win; if I get blue, the class wins. Ask students if they agree to play by these rules. Now show them the

spinner. Do they still want to play? Why not?

Write the term fair game on the board. Ask students to explain what they think this term might mean.

Encourage students to use math vocabulary in their explanations. Point out that in a fair game, both players

have equal chances, or are equally likely, to win.

Give the rules for another game: There are 6 marbles in a box. If I draw a red marble, I win; if I draw a blue

marble, the class wins. To make the game fair, how many blue marbles should be in the box? How many red

marbles?

Vary the game: If I draw a red marble, I win, but if I draw any other colour, the class wins. If 2 of the 6

marbles are red, who has more chances of winning, me or the class? What if 5 marbles are red? What

should be in the box to make the game fair? Encourage the class to think of more than one solution.

Assessment: Two players are spinning this spinner. Invent rules of play to ensure that the game is fair.

R

• B

R • B

W


Activities:

1. Pair up students and give each pair a container with 2 red counters and 4 blue counters inside. Either

student can pick counters from the container (no peeking!). Player 1 wins if the counter drawn is red and

Player 2 wins if the counter drawn is blue. Ask students to explain whether the game is fair or not. Have

students play the game 20 times (replacing the counter each time) and keep a tally of who wins each

time. Ask students if the results are what they expected.

2. Ask pairs to keep track of who wins and who loses in 20 repetitions of the following game: Players take

turns rolling a die. Player 1 wins if the number rolled is a 1 or a 6; Player 2 wins otherwise. ASK: Is this

game fair? Are the results what you expected?

Extensions:

1. How are the games in Activities 1 and 2 similar? Is the first game less fair than the second? (From the

point of view of probability, the two games are identical. In both games, Player 2 has 4 out of 6 chances

of winning: the probability of Player 2 winning is 46 .)

2. Introduce the concept of equal likelihood.

Draw this spinner on the board:

ASK: What is more likely, to spin red or to spin green? Green or blue? Red or something else? What is

more likely to happen, rolling 2 or 3 on a die? These pairs of events are equally likely to happen.

SAY: A game is fair if both players are equally likely to win. Is the following game fair?

Players spin the spinner above. If they spin green, Player 1 wins. If they spin blue, Player 2 wins. If they

spin red, it is a draw. (The game is fair, since both players have equal chances of winning—the

probability of winning is 14 for both.)

G R

• B


PDM3-21 Experiments and Expectations Goals: Students will compare theoretical and experimental probability.

Prior Knowledge Required: Outcome, Half of a number, Even chances

Vocabulary: outcome, even chances, fair game

Draw a spinner on the board:

ASK: Which part of the spinner is coloured red? How can you describe the chances of spinning red? (even

chances) How many outcomes are there for spinning this spinner? (4) Which part of the outcomes is

“spinning red”? (2 out of 4, or half) If I spin the spinner 4 times, how many times would I expect to get red?

What if I spin it 10 times? 20 times? Give your students spinners like the one shown, and ask them to spin

the spinner 10 times and to tally the results. Did everybody get 5 reds? (No) What was the most common

number of red spins? What was the smallest number of red spins? The largest number?

Pool the results for the whole class. Compare the class results with the individual results using graphs or

other visual representations. For example, individuals could make bar graphs of their results (showing the

number of times they spun red, blue, and green) and you could put all the results together in a class bar

graph. Discuss with your students the difference between the class results and the individual results. ASK:

Which results are nearer to the prediction? Which data do you think is more reliable: the individual data or

the group data? Why? Point out that the more results you have, the closer you get to the expected outcome.

The number of reds spun by the class will be closer to half of the total (the expected outcome) than many of

the individual results.

Repeat with several more experiments, such as:

� Flip the coin 10 times. How many times will you get a head?

� Draw a marble from a box with 6 marbles, 3 blue and 3 green. Return the marble to the box. Repeat

12 times. How many times do you expect to draw a blue marble?

� Draw a marble from a box with 6 marbles: 3 blue, 1 yellow, and 2 green. Return the marble to the box.

Repeat 12 times. How many times do you expect to draw a blue marble?

Ask your students to design an experiment where they would expect to get a certain result 10 times in 20

repetitions (in other words, the result has even chances).

Extension: See BLM “Shape Spinner.”

G R

• B

R


PDM3-22 Cumulative Review

PDM3-22 is a review worksheet for Probability and Data Management, parts 1 and 2.

Extension: Write the numbers from 1 to 10 on ten cards (one number per card). If you select 6 cards at

random, ask students to think about the probability of any one or more of the following events:

� The sum of the numbers will be greater than 60.

� All the numbers will be even.

� Two numbers will be neighbours.

� No numbers will be neighbours.

� The sum of the numbers will be less than 12.

� The sum of the numbers will be greater than 25.

Ask students to predict whether each event is certain, impossible, likely, or unlikely. Invite individuals, pairs,

or small groups to conduct 10 experiments to check their predictions. Combine the individual results to get

class results. How do the class results compare to the individual results? How do the class results help

students to confirm (or revise) their predictions? Can students explain why some of the events are certain or

impossible?

Workbook 3 - Probability & Data Management, Part 2 1BLACKLINE MASTERS

Shape Spinner _________________________________________________________2

PDM3 Part 2: BLM List

Shape Spinner

You will be asked to spin

the spinner until it lands on

the same shape 10 times.

Predict which shape this

will be:

My Prediction My Partner’s Prediction

Create a spinner like the one below.

Test your predictions. Take turns spinning with your partner.

The first player with 10 tallies in one column wins!

Record the data in these charts.

My Chart My Partner’s Chart




G3-18 Introduction to Coordinate Systems Goal: Students will identify columns and rows of coordinate systems given their numbers.


Horizontal

Vertical

Vocabulary: coordinate system, row, column

To illustrate the concept of a coordinate system, you can start with the following card trick.

1. First, deal out nine cards—face up—in the arrangement shown below:

Row 3

Row 2

Row 1

Column 1 Column 2 Column 3

2. Next, ask your students to select a card in the array and then tell you only what column it’s in (they

shouldn’t identify the card!).

3. Gather up the cards, with the three cards in the column your students selected on the top of the deck.

4. Deal the cards face up in another 3 × 3 array making sure the top three cards of the deck end up in the

top row of the array.

5. Ask your students to tell you what column their card is in now. The top card in that column (i.e., the card

in row 3) is their card, which you can now identify!


6. Repeat the trick several times and ask your students to try to figure out how it works. You might give

them hints by telling them to watch how you place the cards, or even by repeating the trick with a

2 × 2 array.

When your students understand how the trick works, you can ask the following questions:

• Would there be any point to the trick if the person picking the card told the person performing the trick

both the row number and the column number of the card they had selected? Clearly there would be no

trick if the performer knew both numbers. Two pieces of information are enough to unambiguously

identify a position in an array or graph. This is why graphs are such an efficient means of representation;

two numbers can identify any location in two-dimensional space (in other words, on a flat sheet of

paper). This discovery, made over 300 years ago by the French mathematician René Descartes,

was one of the simplest and most revolutionary steps in the history of mathematics and science.

His idea of representing position using numbers underlies virtually all modern mathematics, science,

and technology.

• Will the trick work with a larger array? Have students try the trick with a 4 × 4 array. They should see that

as long as the array is square (with an equal number of rows and columns), the trick works for any

number of cards. Ask your student to explain why this is so and why the trick doesn't work if the array

isn't square (for instance, try it with 2 columns and 6 rows).

• Will the trick work if the person picking the card tells the performer which row the card is in rather than

which column? What does the performer have to do differently in this case? Have your students show

you how the new trick would be performed. The fact that the trick works equally well in both cases

illustrates a very deep principle of invariance in mathematics. In a square array, there is no real

difference between the rows and columns. In fact, if you rotate the array by a quarter turn, the rows

become columns and vice versa. More generally, once you fix an origin in space, it doesn't matter how

you set up your grid (the lines representing the rows and columns). In all cases you need only two

numbers to identify a position.

• How many numbers would be required to represent the position of an object relative to an origin in three-

dimensional space? (The answer is three. Think of the origin as being situated on a plane, or a flat piece

of paper, that has a grid or graph on it. You need two numbers to tell you how to travel from the origin

along the grid lines on the plane to situate yourself directly above or below the object, and one more

number to tell you how far you have to travel up or down from the plane to reach the object.)

Now draw a 3 × 3 array and number the columns and rows:

1 2 3

3

2

1

C R O W

L U

M N


The diagram next to the array illustrates the orientation of columns (horizontal) and rows (vertical). Point out

a row and a column in the array, and stress that we order rows from bottom to top, and columns from right to

left. Ask several volunteers to locate the third column, second row, etc. Then ask your students to complete

the worksheet for this lesson.

Assessment:

Join the dots in the given column and row:

a) Column 3, Row 2 b) Column 1, Row 3 c) Column 3, Row 3

Bonus:

Which letters of the alphabet can be written on a grid and described in terms of rows and columns only?

(See Question 4 on the worksheet.) Which numbers can be written this way?

Extension:

The card trick above can be modified for non-square arrays if one makes an extra rearrangement. Deal out

an array of 3 columns and 9 rows. Have a student select a card and tell you what column it’s in. Re-deal the

cards so that all nine cards from the chosen column land in the top three rows of the new array. Ask the

student to tell you what column their card is in now, and re-deal the top three cards in that column into the

top row of a new array. Once the student tells you what column their card is in, you can identify the card—it

will be the top card in that column.

This version of the trick illustrates a powerful general principle in science and mathematics: when you are

looking for a solution to a problem, it is often possible to eliminate a great many possibilities by asking a well-

formulated question. In the card trick, one is able to single out one of 27 possibilities by asking only three

questions. Repeat the trick, asking your students how many possibilities are eliminated by the first question

(18), by the second question (6), and by the third (2).


G3-19 Coordinate Systems Goal: Students will locate a point in an array given its column and row numbers.

Prior Knowledge Required: Horizontal

Vertical

Columns and rows in coordinate systems

Vocabulary: coordinate system, row, column

Review the previous lesson. Explain to your students that today you will give them a point in the array and

they will have to identify the column and the row. Draw several arrays of dots on the board, join the points in

a column and a row in each array, and ask the students to identify the highlighted row and column. NOTE:

You can create bonus questions using larger arrays, i.e., arrays with more rows and columns.

Draw more arrays and circle a dot in each. Ask your students to identify the column and the row for each dot.

Students that have trouble identifying the column and the row should first highlight them. Then reverse the

task—give the column and the row and ask students to find the dot.

Assessment:

1. Circle the dot where the two lines meet:

a) Column 2, Row 3 b) Column 3, Row 1 c) Column 1, Row 1

2. Identify the proper column and row for the circled dot:

a) b) c) d)

Column ____

Row ____

Column ____

Row ____

Column ____

Row ____

Column ____

Row ____

Activity:

Ball Game The students are the points in a coordinate system. Ask each student to say which column and

row they are in. Give one of the students a ball and identify a point on the array with a column number and a

row number. The student with the ball has to toss it to the student at the given point. Students can continue

tossing the ball around by calling out coordinates rather than names.


G3-20 Introduction to Slides (or Translations) Goal: Students will slide a dot on a grid.


Distinguish between right and left

Vocabulary: slide, translation

For this lesson, a magnetic board with a grid on it (or an overhead projector with a grid drawn on a

transparent slide) would be helpful. Let your students practice sliding a dot, in the form of a small circular

magnet, right and left, then up and down. Students should be able to identify how far a dot slid in a particular

direction (how many squares, or units) and also be able to slide a dot a given distance (e.g., 2 units up, 5

units left, 4 units right).

If any students have difficulty distinguishing between right and left, write the letters L and R on the left and

right sides of the board or overhead. You can also spend some time teaching them to distinguish between

left and right:

• Have students raise their right hand, then their left. (Show them that their left index finger and thumb

makes a correct L, whereas their right index finger and thumb makes a backwards L.)

• Draw two figures on the board and have students identify the one that is on the right and the one that is

on the left. Repeat.

After students can slide a dot in a given direction, show them how to slide a dot in a combination of

directions. EXAMPLE: Slide the dot 4 units up then 1 unit left. Slide the dot 3 units right then 3 units down.

Assessment:

Slide the dot:

a) 3 units right; 3 units up b) 6 units left; 3 units down c) 7 units left; 2 units up

Activities:

1. Ball Game The students are points on the grid, and you give directions such as “The ball slides 3 units

to the right.” The student with the ball has to toss it to the right “point” on the grid.

2. In the schoolyard, draw a grid on the ground. Ask your students to move a certain number of units,

in various combinations of directions, by hopping from point to point in the grid.


G3-21 Slides Goal: Students will slide a shape on a grid.

Prior Knowledge Required: Slide a dot on a grid


Vocabulary: slide, translation

Tell your students the following story. You might use two actual figures to demonstrate the movements in

the story.

Suppose you have a pair of two-dimensional figures and you wish to place one of the figures on top of the

other. But the figures are very heavy—and very hot—sheets of metal. You need to program a robot to move

the sheets, and to write the program you have to divide the process into very simple steps. It is always

possible to move a figure into any position in space using some combination of the following three

movements:

1. You may slide the figure in a straight line (without allowing it to turn):

SLIDE or TRANSLATION

2. You may turn the figure around some fixed point (usually on the figure):

TURN or ROTATION

3. You may flip the figure over:

FLIP or REFLECTION


Two figures are congruent if the figures can be made to coincide by some sequence of flips, slides, and

turns. For instance, the figures in the picture below can be brought into alignment by rotating the right hand

figure counter-clockwise a quarter turn around the highlighted point (also a vertex), then sliding it to the left.

It is not always possible to align two figures using only slides and turns. To align the figures below you must,

at some point, flip one of the figures:

One way to flip a figure is to reflect the figure through a line that passes through an edge or a vertex of

the figure.

Tell your students that today you are going to teach them about slides. Show students the following picture

and ask them how far the rectangle slid to the right. Ask for several answers and record them on the board.

You may even call for a vote.

Students might say the shape moved anywhere between 1 and 7 units right. Take a rectangular block and

perform the actual slide, counting the units with the students. The correct answer is 4.

Show another picture:

This figure has a dot on its corner. How much did it slide? This time it is easier to describe the slide—just use

the benchmark dot on the corner. Check with the block.

Show a third picture:


Is this a slide? No, this is a slide together with a rotation. You cannot slide this block from one position to the

other without turning it.

Activity: Give your students a set of pattern blocks or Pentamino pieces and ask them to trace a shape

on dot paper so that at least one of the corners of the shape touches a dot (use BLMs if needed).

Ask students to slide the shape a given number of units in one or more directions. After the slide, trace the

pattern block again.


G3-22 Slides (Advanced) Goal: Students will slide a shape on a grid and describe the slide.



Vocabulary: slide, translation, translation arrow

Draw a shape on a grid on the board and perform a slide, say 3 units right and 2 units up. Draw a translation

arrow as shown on the worksheet. Ask your students if they can describe the slide you’ve made. If they have

trouble describing the slide, suggest that they look at how the vertex of the figure moved (as shown by the

transition arrow). To help students describe the slide, you might tell them that the grid lines represent streets

and they have to explain to a truck driver how to get from the location at the tail of the arrow to the location at

the tip of the arrow. The arrow shows the direction as the crow flies, but the truck has to follow the streets.

Make sure your students know that a slide is also called a translation. Ensure that students also understand

that a shape and its image under a translation are congruent.

Extensions

1. Slide the figures however you want, and then describe the slide.

2. Describe a move made by a chess knight as a slide. Describe some typical moves of other chess pieces,

such as a pawn or a rook (castle).


G3-23 Slides on a Grid

Goal: Students will describe slides on a grid.



Vocabulary: slide, translation, translation arrow

Draw a hockey rink on a magnetic grid (or use the overhead projector) and place several “players” on the

intersections of grid lines, as shown below. You might use a small magnet to show the passes of the puck.

Ask your students to describe the following passes:

From Player 1 to Player 2: ____ units up

From Player 2 to Player 3: ____ units _________

From Player 3 to Player 4: ____ unit _________ and ___ units down

From Player 4 to Player 5 ____ units _________ and ___ units _________

From Player 5 to Goalie: ____ units _________ and ___ units _________

Pass the puck between other players to provide your students with more practice. Suggest that your students

move the players and describe more passes. Students can also answer questions like these:

• Player 3 passes the puck 5 units right and 3 units down. Who receives the pass?

• Player 5 wants to pass the puck to Player 4. How many units left and how many units down should the

puck go?

1

2 3

4

5 G


G3-24 Coordinate Systems and Maps Goal: Students will describe and perform slides on a grid.

Prior Knowledge Required: Slides

Coordinate systems

Vocabulary: slide, translation, translation arrow, row, column, coordinate system, coordinates

Assign a letter to each row of desks in your class and a number to each column. Ask your students to give

the coordinates of their desks. Then invite students to write a short message to a classmate and to include

the recipient’s “address” in coordinates. A volunteer “postman” then delivers the letters. The postman has to

describe how the letter moved (two to the front and one to the left, for example). Reverse the roles: the

postman delivers the letter and the addressee has to say which slide the letter performed. Let several

students perform the role of postman.


G3-25 Mapping Exercise Goal: Students will use coordinates to describe and find points on a map.

Prior Knowledge Required: Slides

Coordinate systems

Vocabulary: slide, translation, translation arrow, row, column, coordinate system, coordinates

Place a transparency with a map of Saskatchewan on the overhead projector (see the BLM “Map of

Saskatchewan”). Ask volunteers to find the cities on the map and to answer these questions:

• What are the coordinates of Saskatoon?

• What are the coordinates of Regina?

• What are the coordinates of Uranium City?

• What are the coordinates of Prince Albert?

• What can you find in the square A4? D5? D1?

Ask more questions of this sort.

Activity:

Memory Game

Students will need a grid and 1 to 4 small objects (such as play money of different values or beads of

different colours). The objects are placed on intersections of the grid. Player 1 slides one of the objects while

Player 2’s back is turned, and Player 2 then has to guess which object was moved and how (describe the

slide). Students will have to study the board and memorize the original coordinates of the objects. They can

then compare the coordinates after the objects were moved with the coordinates before the objects were

moved to determine exactly which object moved and how.


G3-26 Flips Goal: Students will reflect simple shapes through vertical and horizontal lines parallel to their sides.

Prior Knowledge Required: Symmetry

Vocabulary: symmetry, line of symmetry, horizontal, vertical

Draw a scalene right-angled triangle and a line parallel to one of the legs of the triangle on a transparency

and project them on the board. Invite a volunteer to trace the shape and the line on the board. Flip the

transparency so that the line coincides with the line in the drawing and the projected triangle is a reflection of

the triangle in the drawing. Explain to your students that the movement that you performed is called a flip or

reflection of the triangle over the line. Ask your students if they remember a name that refers to these two

triangles together (congruent, symmetrical). What is the line called? (line of symmetry)

Give your students an assortment of Pentamino pieces. Ask them to trace each piece on grid paper (see

BLM section), draw a mirror line through a side of the piece, flip the shape over the line, and then trace the

flipped piece. As a challenge, ask them to draw the mirror line parallel to, but not touching, one of the sides

and repeat the exercise. Students should use the squares of the grid to determine the right distance between

the mirror line and the flipped shape.

Activities:

1. Students could create their own shape and trace the shape before and after a flip.

2. Find-a-Flip Game

Divide your students into groups of 2-5 players each. Each group will need two copies of the BLM

“Find-a-Flip Game.” Let the students cut out the cards. Players shuffle the deck, deal out 4 cards for

each player, and lay a card face up on the table. If a player has a card that is a reflection of the card on

the table over one of its sides, he or she can lay the card on the table and pick a new card from the deck:

Players take turns placing cards on the table until they have a square:


The player that placed the last card in the square obtains 1 point.

If a player does not have a card that is a reflection of one of the cards on the table, the player picks cards

from the deck until he or she can either add a card to the shape on the table or create a square of

shapes from his or her own cards. A player that does not have any more cards in hand picks 4 cards

from the deck. If a player runs out of cards and there are no more cards in the deck, the player exits the

game and obtains 2 points. The player with the greatest number of points wins.


G3-27 Reflections Goal: Students will reflect simple shapes on a grid through vertical and horizontal lines.


Vocabulary: symmetry, line of symmetry, horizontal, vertical

Give your students an assortment of Pentamino pieces. Ask them to trace each piece on grid paper, draw a

mirror line through a side of the piece, and then draw the reflection of the piece in the mirror line. Students

could check if they have drawn the image correctly by flipping the Pentamino piece over the mirror line and

seeing if it matches the image. Remind your students that a flip is also called a reflection. Students should

notice that each vertex on the original shape is the same distance from the mirror line as the corresponding

vertex of the image. Let your students practise reflecting shapes with partners: one student draws a shape of

no more than 10 squares and chooses the mirror line; the partner has to reflect the shape over the given

mirror line.

ADVANCED GAME: One student draws a shape of no more than 10 squares and its reflection in a mirror

line, but misplaces one of the squares in the reflection. The partner has to correct the mistake.

Extension:

Students could try to copy and reflect a shape in a slant line.

EXAMPLES:


G3-28 Flips and Slides Goal: Students will distinguish between flips (reflections) and slides (translations).


Reflections

Slides

Vocabulary: symmetry, line of symmetry, horizontal, vertical, slide, reflection, flip, translation

Draw a parallelogram on a grid on the board. Ask your students which transformations they learnt to do.

Slide the parallelogram several units right and ASK: What did I do with the shape? Ask your students to

describe the slide. Reflect the initial parallelogram over a horizontal line and ask them to describe the

transformation. Repeat with several more shapes, including a shape with a horizontal line of symmetry, such

as a square or rectangle (so that the horizontal reflection does not change the shape). Discuss with your

students which transformations could take this shape onto its image. Check with a cut-out of the shape that

both a slide and a reflection can bring the shape onto its image.

Activity:

Divide your students into groups of 2–5 players each. Each group will need two copies of the BLM “Find-a-

Flip Game.” Let the students cut out the cards. Players shuffle the deck, deal out 4 cards for each player,

and lay a card face up on the table. If a player has a card that is a slide or a reflection of the card on the table

over one of its sides, he or she adds the card to the table (and picks a new card from the deck):

or or or

Players take turns placing cards until they have a square or rectangle of area 4 (or more) cards. The player

that placed the last card in the rectangle obtains a number of points equal to the area of the rectangle. Check

with the students which rectangles are possible (They are: 4 × 1, 2 × 2, 2 × 3, 2 × 4. The rectangle 5 × 1

seems possible but isn’t because the previous player will have claimed the 4 × 1 rectangle.) A player is not

allowed to place a card so that there will not be enough cards to make a rectangle. (There are only 8 cards

with shapes that can go into the same rectangle.)

This is not a rectangle.

This shape is not allowed – you will not have enough cards to make a rectangle


If a player does not have a card that is a reflection or a slide of one of the cards on the table, the player picks

cards from the deck until he or she can either add a card to the shape on the table or make a rectangle from

the shapes on his or her own cards. A player that does not have any more cards picks 4 cards from the deck.

If there are no more cards left in the deck, the player exits the game and obtains 2 points. The player with the

greatest number of points wins.


G3-29 Turns Goal: Students will describe rotations that are multiples of a quarter turn.

Prior Knowledge Required: Fractions: 12 ,

14 ,

34

Clockwise, counter-clockwise

Vocabulary: rotation, clockwise, counter-clockwise

Review the division of the analogue clock face into quarters. If a large clock is available, put the hands of the

clock so that they are perpendicular and ask your students to identify the angle between them. Move the

hands into various positions, such as 3:00, 9:00, or 3:30, but also 1:20 or 10:38. Review the meaning of the

terms clockwise and counter-clockwise using the clock and also by drawing arrows on the board. Ask

volunteers to rotate the minute hand, clockwise and counter-clockwise, a full turn, a half turn, and a quarter

turn. You might also ask your students to be the clocks: each student stands with an arm outstretched and

turns clockwise (CW) or counter-clockwise (CCW) according to your instructions.

Review with your students these fractions of a circle: 14 ,

12 ,

34 . Let them both name shaded fractions of a

circle and shade fractions of a circle.

Sample fractions to name:

Sample fractions to shade:

Shade: 14

12

34

34

14

12

Draw pairs of arrows on the board, such as those shown below, and ask your students to shade the part

between the arrows. Ask them to lay a pencil on the “start” arrow and to turn it to the “finish” arrow. How

much has the pencil turned?


___ turn CCW ___ turn CW ___ turn CW ___ turn CCW

Students who have trouble deciding which part of the circle is shaded might extend the hands of the clock to

the other side of the central dot to divide the circle into quarters. Repeat, but this time let your students

decide whether the turn was clockwise or counter-clockwise.

Bonus:

How much has the arrow turned?

___ turn CW ___ turn CCW ___ turn CCW ___ turn CW

start

finish start finish

start

finish start

finish

start

finish

start

finish

start finish

start

finish


G3-30 Rotations Goal: Students will describe and perform rotations that are multiples of a quarter turn.


14 ,

34



Review the previous lesson with the students.

Draw several clocks on the board as shown below and ask your students to tell you how far and in which

direction each hand moved from start to finish:

12 turn CCW _______________ _______________ _______________

Then draw examples with only one arrow and ask students to turn the arrow:

a) 12 turn CCW b)

14 turn CCW c)

14 turn CW d)

34 turn CCW

Assessment

1. Describe the rotation of the arrow:

2. Show the position of the arrow after each turn:

a) 14 turn CW b)

34 turn CW c)

34 turn CCW d)

14 turn CW


G3-31 Rotations (Advanced) Goal: Students will describe and perform rotations that are multiples of a quarter turn.


14 ,

34



Review the previous lesson by drawing several arrows or clock hands. To help your students visualize the

effect of a rotation on a shape, have them make a small flag (as in Question 1 on the worksheet) by taping a

triangular piece of paper to a straw. Ask students to rotate the flag and trace its image after the rotation.

Students could also cut out other shapes used on the worksheet and trace the images of these shapes after

a rotation. Then ask your students to trace or draw a figure, decide on a rotation, and draw the new shape

without a prop. Students could also practise rotating pattern blocks or Pentamino shapes (around vertices of

the shapes) on a grid.

Assessment: Draw the shape after each turn:

a) 14 turn CW b)

12 turn CW c)

14 turn CCW d)

34 turn CW

Activities:

1. NOTE: This activity was adapted from the Atlantic curriculum.

Trace a pattern block trapezoid on a sheet of paper. Rotate the block around one of the vertices a half-

turn clockwise and trace it again. Repeat with a half-turn counter-clockwise. What do you notice? Repeat

with a different shape (not necessarily a pattern block).

2. Play another version of the Find-a-Flip game. Each group of 2–4 students will need two copies of any

three rows of shapes on the BLM “Find-a-Flip Game.” Let your students play the game as in Activity 2 of

G3-26, but this time each card placed should be a quarter turn rotation of the adjacent cards.

Extension: Using pattern blocks or cardboard polygons, trace a figure on a sheet of paper. Then choose

a vertex and rotate the shape around the vertex a half turn. Trace the figure again. Would you get the same

result if you had reflected the figure? (Right-angled trapezoids are particularly interesting in this case.)


G3-32 Flips, Slides and Turns Goal: Students will distinguish between flips (reflections), slides (translations), and turns (rotations).


14 ,

34


Reflections. rotations, and slides

Vocabulary: rotation, turn, clockwise, counter-clockwise, reflection, flip, slide, translation

Show your students several pairs of shapes and ask if one shape was moved onto the other by a flip, a turn,

or a slide. Start with non-symmetric shapes, such as an L-shape and a right-angled trapezoid, and continue

to shapes that have a center of symmetry or a line of symmetry, such as parallelograms and kites.

Encourage students to give multiple answers. For example, these two pairs of shapes could be moved by a

slide as well as by a reflection (kite) or a rotation (parallelogram).

Repeat with shapes that have more than one line of symmetry, such as a rectangle and a rhombus. Invite

volunteers to trace the shapes on tracing paper and to check the answers.

As a challenge, present the following pair of figures:

Ask your students if the second shape was obtained by rotating, reflecting, or sliding the first shape. Let your

students try all three moves. Students should see that none of the three moves produces the second shape.

Give your students a cut-out of the shape and ask a volunteer to check if the figures are congruent. (ASK:

Maybe the figures are different? Maybe this is not the same figure?) Ask the students to follow the

movements of the volunteer closely, so that they can spot which transformation or transformations are

performed. Repeat several times if needed, doing one movement at a time, until all the students understand

that the figure has to be both rotated and reflected.


Invalid placement – this shape is a clockwise turn of the shape

on the left, but it cannot be obtained from the shape above it by

a rotation, slide, or reflection over the common side.

Good placement – this shape is a counter-clockwise turn of the

shape on the left and a reflection of the shape above over the

common side.

Activities:

1. NOTE: This activity was adapted from the Atlantic curriculum. Students will need grid paper

and tracing paper.

Draw a rectangle on a piece of grid paper. Trace the rectangle on the tracing paper. Keep the tracing

paper on the grid paper, choose a vertex of the rectangle, and press a pencil point firmly to this point.

Rotate the shape on the tracing paper around the fixed point a quarter turn clockwise. Mark the vertices

of the rotated shape using a sharp pencil. Remove the tracing paper and join the vertices of the image.

Compare the shapes. Could you move the first rectangle onto the second rectangle by a slide? A

reflection?

Repeat this activity using a half turn, a quarter turn counter-clockwise, and three quarters of a turn

clockwise. Compare the images. What do you notice?

Repeat the activity with a square, a rhombus, and a parallelogram.

2. Ask students to create their own shape and move it from the start box to the finish box in Question 2 on

the worksheet. Students should describe the flips, slides, and turns they used to move their shape.

3. Play another version of the Find-a-Flip Game. Each group of 2–4 students will need two copies of any

three rows of shapes on the BLM “Find-a-Flip Game.” Let your students play the game as in the Activity

of G3-28, but this time each card placed could be a reflection, a quarter turn, or a slide of the shapes on

the adjacent cards.

Students should name the transformation they used.


G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids.

Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons

Vocabulary: edge, vertex, vertices, face, pyramid, skeleton, base, triangular, rectangular, pentagonal,

hexagonal

Start with a riddle: “You have 6 toothpicks. Make 4 triangles with them. The toothpicks must touch each other

only at the ends.” Let your students try to solve the riddle using toothpicks and modelling clay to hold the

toothpicks together at the vertices of the triangles. The answer, of course, is the triangular pyramid. You

might give your students the hint that the solution is three-dimensional.

Sketch a rectangular pyramid on the board and shade the base. Ask volunteers to mark the edges and

the vertices (count them and make a tally chart). Write the words “base,” “edge,” “vertex,” and “vertices” on

the board.

Give your students modelling clay and toothpicks. Show them how to make a pyramid. Start with a base,

then add an edge to each vertex of the base and join the edges at a point. The students should make

triangular, square, and pentagonal pyramids. Then let them fill in the chart and answer the questions on the

worksheet. After finishing the worksheet, they may check their prediction for the hexagonal pyramid by

making one.

Tell your students that the shapes they have built are called “skeletons” of pyramids. You might write the

following “equation” on the board: Skeleton = Edges + Vertices. As animal skeletons are covered with flesh

and skin, the skeleton of a pyramid can be covered with paper, glass, or other substances and will have

faces. Show a pyramid (with faces) and write the word “faces” on the board as well.

Assessment: Add a row to the chart on the worksheet for a pyramid with a heptagonal (7-sided) base,

and fill it in.

Activity: Build skeletons of pyramids using marshmallows and toothpicks or straws.

Extensions

1. How many faces, edges, and vertices would a pyramid with a 10-sided base have?

2. Ask your students to bring to class pyramids or pictures of pyramids (e.g., Egypt, Mexico, Japan,

entrance to Louvre in Paris) that they can find at home. You can use these pyramids in lesson G3-38:

Edges, Vertices and Faces.

3. PROJECT: Ask your students to learn about a pyramidal structure and give a presentation about it—

what was the structure used for, when and where was it built, why does it have the pyramidal form?


G3-34 Building Prisms Goal: Students will build skeletons of prisms and describe properties of prisms.

Prior Knowledge Required: Polygons: triangles, quadrilaterals, rectangles, pentagons, hexagons

Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, triangular, rectangular,

pentagonal, hexagonal

Give each student several pattern block triangles and ask them to place the triangles on the table one on top

of the other, aligning the sides. Ask your students what shape the stack has. What does this shape look like

from above? (triangle) What does it look like from the side? (rectangle) Explain that mathematicians call this

3-D shape a triangular prism. Let your students build prisms from other pattern blocks. Point out the faces,

the edges, and the vertices. Explain that the shape that was used to build the prism (triangle, for example) is

called the base. Sketch a prism on the board and shade the bases. Ask volunteers to mark the edges and

the vertices (count them and make a tally chart). Write the words “base,” “faces,” “edges,” “vertex,” and

“vertices” on the board.

Give your students modelling clay and toothpicks. Show them how to make a prism. First make two copies of

the base, and then join each vertex on one base to a vertex on the other base with an edge. Students should

make triangular prisms, pentagonal prisms, and a cube. Let them fill in the chart and answer the questions

on the worksheet. After finishing the worksheet, they may check their prediction for the hexagonal prism by

making one.

ASK: What have you built? (skeletons of prisms) What do we call the “bones” of your skeletons? (edges)

What do the skeletons need to become prisms? (faces)

Assessment: Add a row to the chart on the worksheet for the prism with a heptagonal (7-sided) base,

and fill it in.


G3-35 Edges, Vertices and Faces Goal: Students will identify vertices, edges (including hidden edges), and faces (side, back, front, top, and

bottom) in the drawings of 3-D shapes.

Prior Knowledge Required: Count edges of polygons

Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape

Remind your students that lots of 3-D shapes in the world around us are either pyramids or prisms. As an

example, you might show them a photo of the pyramids in Egypt.

Hold up a 3-D shape and draw a picture of the shape on the board. Write “3-D shape” next to your drawing.

Ask volunteers to identify the edges, the faces, and the vertices on the shape itself and on the drawing, and

write the terms “edge,” “face,” and “vertex” on the board. Remind your students that the plural of “vertex” is

“vertices.”

Your students will need the skeletons of the cubes they made during the last two lessons. Give each student

2 squares made of paper and 4 squares made of transparent material. Ask them to add

• the paper (non-transparent) squares as the bottom face and the back face;

• the transparent squares as the top, front, and side faces.

It is a good idea to show the students how to add faces on a larger model before they work on their own

models. Add the faces one at a time, emphasizing the position and name of each one.

ASK: Which edges of the cube do you see only through the transparent paper. If the transparent faces were

made of paper, would you see these edges? (no) The edges that would be invisible if all the faces were non-

transparent are called the “hidden edges.” On a two-dimensional drawing of a cube these hidden edges are

marked with dotted lines.

Extension: Ask students to hold or place their cubes in various positions and to look at them from

different angles (on the table, on the floor seen from above, slightly above their heads, and so on.) Ask

students to describe what the faces look like when seen from different angles (they look like a square, a

parallelogram, etc.). The outline of the shape itself can look like a square, a rectangle, a hexagon, a

trapezoid, and a rhombus.


G3-36 Pyramid Nets Goal: Students will build pyramids from nets.


Polygons: triangle, square, pentagon, hexagon


Show your students nets for hexagonal and square pyramids. ASK: What do these drawings have in

common? What is different? Let your students count the faces in the net. What shape do they have? How

many shapes of each kind? If your students do not mention that each net has one face that is different from

the others, point that out and ask what this face is called. (the base) Ask your students what the net for a

triangular pyramid might look like.

Ask students to complete the worksheet for this lesson. They can use the nets on the worksheet to answer

Question 1. Alternatively, you can give them copies of pyramid nets (see BLM section) to cut and fold. Let

your students count the edges, the vertices, and the faces of the shapes they make.

Draw a net for a triangular pyramid on the board and ask your students to count the number of edges on the

net. Is it the same as the number of edges in the 3-D pyramid? Let the students explain. (There are nine

edges on the net and only six edges in the shape. The six edges on the outside of the net are glued in pairs,

so they produce three edges in the 3-D shape. The three internal edges of the net together with the other

three produce six edges in the pyramid.) Repeat with a square pyramid. As a challenge, draw a net of a

pentagonal pyramid and ask your students to tell from the net how many edges are in the 3-D shape.

Extensions:

1. ASK: What happens if you cut one of the side faces in a pyramid net and try to glue it at some other

place? Students might actually cut off one of the triangles and reattach it to another edge. You might

draw the following examples on the board. Will the net still fold into a pyramid if the face is glued in this

position?

no yes no

2. Describe the nets for different shapes. Describe the shape of each face and count the number of faces of

a given shape. Draw a freehand sketch of all the faces that make up a particular 3-D shape.

For example, the parts of a square prism are and the net is .


G3-37 Prism Nets Goal: Students will build prisms from nets.




Repeat the previous lesson for prism nets.

Activities:

1. After your students have constructed the pyramids and prisms from the nets on worksheets G3-36 and

G3-37, or the BLM section, ask them to sketch the nets from memory. They can test their nets by

making the shapes. You might also ask students to sketch and test nets for pentagonal and hexagonal

pyramids and prisms.

2. Give students square or pentagonal pyramids and ask them to trace the faces on a piece of paper, so

that they create a net. Ask them to cut out the nets they have drawn. Let them cut off faces of the net and

reattach the faces at different places. Will the new net fold into the same pyramid? Which edges are

places where you would want to re-glue the faces and which are not? Repeat this exercise with a prism.

This activity is important because it lets students explore various ways to create nets for the same solid

rather than memorizing a single net shape.


G3-38 Prisms and Pyramids Goal: Students will compare prisms and pyramids.



Prisms

Pyramids

Vocabulary: edge, vertex, vertices, face, skeleton, base, 3-D shape, prism, pyramid, triangular, square,

rectangular, pentagonal, hexagonal

Divide your students into groups. Give each group several 3-D shapes, so that each group has some

rectangular and triangular pyramids, rectangular and triangular prisms, and a cube. Ask your students to

count the faces of the shapes. If some students are having trouble keeping track of the number of faces, they

might mark each face with a chalk dot or a small sticker. Ask your students to count the edges and vertices

on the 3-D shapes as well (They might shade edges with chalk and mark vertices with stickers.) Ask them to

write the results of their count in the table on the worksheet (see Question 2).

Draw a pentagonal pyramid and a triangular prism on the board and let volunteers count the edges, faces,

and vertices of these figures. Ask them to mark the edges and circle the vertices as they count.

Review the difference between a skeleton and a 3-D shape. Show your students a pentagon made of

toothpicks and modelling clay. SAY: I want to create a skeleton of a pyramid. What do I need to add to this

shape to turn it into a skeleton? How many more vertices (balls of clay) and edges (toothpicks) do I need?

Invite a volunteer to add the vertex and the edges. Show your students another pentagon as above and ask

what is needed to turn it into a prism. Ask the students to compare the skeletons. What do they have in

common? What is different? What was different in the way they built the skeletons?

Activities:

1. Show your students an example of a cone and a cylinder. Explain that a cone has one curved surface

and one flat surface, while a cylinder has two flat surfaces and one curved surface. Ask students to find

as many examples of pyramids, prisms, cones, and cylinders in the classroom as they can.

2. Ask your students to bring to class objects that are prisms, pyramids, cubes, cylinders, cones. Create a

collection of such shapes for future use. EXAMPLES: paper cylinders, glasses, boxes (sometimes you

can find boxes or tins in the shape of cylinders or hexagonal prisms), juice or milk cartons (these are

pentagonal prisms).

3. From the Atlantic curriculum: Put a variety of prisms and pyramids in a bag. Have the students, using

only their sense of touch, describe the shape and name it before bringing it out of the bag to check.


4. Review the notion of congruency. Then give each student a pentagonal pyramid, a triangular prism, and

a cube and ask them to do the following:

a) Place each shape—base down—on a piece of paper and trace the base. (That way you can verify

that each student knows how to find the base.)

b) Write the name of the figure beside the base and indicate whether the figure has one or two bases.

c) If all faces of the figure are congruent, indicate this.

Extension:

Ask your students to add the number of faces and vertices of a cube and subtract the number of edges

(6 faces + 8 vertices – 12 edges). The result is 2. What happens if you do that to another solid? (The result

will again be 2. This fact is known as Euler’s formula and was discovered by the great Swiss mathematician

Leonard Euler in the 18th century.)

Let your students construct shapes from Polydrons. They should make both regular 3-D shapes, like prisms

and pyramids, and irregular “shapes,” like animals and buildings. Count the faces, edges, and vertices.

Check if Euler’s formula holds.


G3-39 Drawing Pyramids and Prisms Goal: Students will draw skeletons of prisms and pyramids.



Prisms

Pyramids


Show your students how they can draw a picture of a cube on dot paper.

STEP 1 Draw a 2 × 2 square that will become the front face.

STEP 2 Draw another 2 × 2 square so that the centre of the first square is a corner of the second square.

STEP 3 Join the vertices with lines as shown.


Follow the same three steps to draw the second diagram below on the board and ASK: What is the

difference between this cube and the previous one (the 2 × 2 × 2 cube)? How are Steps 1 and 2 performed

differently? Draw the remaining two diagrams below. ASK: What is the difference between the three shapes?

How do we see that in the drawing? How is Step 2 performed differently in each drawing?


ASK: What would you do in Step 2 to draw a very long rectangular prism? Invite a volunteer to draw it. If the

drawing does not look good (i.e., if edges overlap), suggest to students that the corner of the back face not

sit on the diagonal:

rather than

Suggest that your students use the same method to draw other prisms, such as triangular or pentagonal

prisms. In this case students should draw bases in STEPS 1 and 2. EXAMPLES:

ASK: What did you draw, prisms or skeletons? As a challenge, students might change some of the lines in

the finished skeletons to dotted lines to show hidden edges and faces.

Review with your students the fact that a pyramid has one base and a point opposite to it. Show your

students this method for drawing a triangular pyramid: draw a base, choose a point, and join the vertices of

the base to the point. Let your students practise drawing various prisms and pyramids.

Activity: After students have learned to sketch pyramids and prisms, ask them to draw a picture that

includes some buildings that have those shapes. For instance, they might draw a scene from Ancient Egypt

(after consulting picture books or non-fiction books for historical details for their drawing).


Extension: Show your students how to draw a skeleton of a prism standing on a base rather than on one

of the rectangular faces. To see the distortion of the base in this position, suggest that your students hold a

pattern block horizontally, slightly below eye level. They should see that the polygon in the base appears

shorter and wider from this angle than when you look at it head on. To draw a prism standing on a base,

draw the base wider than it is when viewed head on, then draw the second (top) base directly above the

bottom base and join the vertices to produce the side faces.

EXAMPLE:


G3-40 Properties of Pyramids and Prisms Goal: Students will compare prisms and pyramids systematically.

Prior Knowledge Required: Faces, edges, vertices of 3-D shapes


Prisms

Pyramids


rectangular, pentagonal, hexagonal, cone, cylinder

Show your students how you can make a cone and a cylinder from a piece of paper. ASK: Where have you

seen this shape before? (ice cream cone, clown hat, etc) Does a cone remind you of some other geometric

shape that we have studied? (a pyramid) If you have an octagonal pyramid (or a pyramid with more than 8

sides in the base), show it to the students and ASK: What do these shapes have in common? Repeat with a

cylinder and an octagonal prism.

Give your students pentagonal pyramids and prisms or have them make these shapes using the nets from

the BLM. Ask your students to draw the following chart in their notebooks and to fill it in, using the shapes.

Property Pentagonal

Prism

Pentagonal

Pyramid Same? Different?

Number of faces

Shape of base

Number of bases

Number of faces

that are not bases

Shape of faces

that are not bases

Number of

edges

Number of

vertices


When students are finished, invite a volunteer to use the information in the table to write a paragraph

comparing the two shapes. Repeat with another pair of shapes, such as a rectangular prism and a

pentagonal prism.

Draw several shapes on the board and ask your students which 3-D shape they make.

SAMPLES:

This would be a good time for students to try the Activity.

Assessment:

1. a) Make a property chart for rectangular and triangular prisms.

b) Use the chart to write a paragraph comparing rectangular and triangular prisms.

2. Who am I?

a) I have only rectangular faces.

b) I have 8 faces, and 6 of them are rectangles.

c) I am a prism with 9 edges and 5 faces.

d) I have one circular base.

Activity:

A Game in Pairs Your students might need a table of the number of faces, edges, and vertices in prisms and pyramids (see G3-33 and G3-34, or G3-38). One player gives a description of a shape; the other has to name the shape.

ADVANCED: The player gives only two of the three possible pieces of information: number of edges,

number of vertices, number of faces.

EXAMPLES:

12 edges and 6 faces (a rectangular prism)

12 edges and 7 faces (a hexagonal pyramid)

Extension: Have students complete the BLM “Word Search Puzzle (3-D Shapes).”


G3-41 Sorting 3-D Shapes Goal: Students will sort prisms and pyramids according to their properties.

Prior Knowledge Required: Venn diagrams

Prisms and pyramids


rectangular, pentagonal, hexagonal

Give individual students or small groups a deck of shape cards and a deck of property cards. These cards

are in the BLM section of the guide. (If you have enough 3-D shapes in your classroom, students can use

actual shapes instead of the cards.) Let them play the following games:


Each student flips over a property card and then sorts the shapes into two piles according to whether a

shape on a card has the property or not. Students get a point for each card that is in the correct pile. (If you

prefer, you could choose a single property for the class and have everyone sort the shapes using that

property.)

Once students have mastered this sorting game they can play the next game.

3-D Venn Diagram Game

Give each student a copy of the BLM “Venn Diagram” (or have students create their own Venn diagram on a

sheet of construction paper or Bristol board). Ask students to choose two property cards and place one

beside each circle of the Venn diagram. Students should then sort their shape cards into the Venn diagram.

Give 1 point for each shape that is placed in the correct region of the Venn diagram.

Assessment:

Sort the shapes below into a Venn diagram according to these properties:

1. One or more rectangular faces

2. One or more triangular faces

A B C D E F

Extension: Draw a Venn diagram to sort the shapes on the worksheet according to these properties:

1. Pyramid

2. One or more triangular faces

What do you notice about your Venn diagram? Explain why part of one of the circles is empty.


G3-42 Classifying Shapes and Making Patterns Goal: Students will review the entire Geometry unit and make patterns with transformations.

Prior Knowledge Required: Venn diagrams

Polygons

Slides, reflections, and rotations

Attributes

Extending patterns

Vocabulary: edge, vertex, vertices, triangle, square, rectangle, pentagon, hexagon, slide, turn, flip,

translation, rotation, reflection

This is a review worksheet.

Activity: Students will need a spinner as shown and 8 cards (4 copies of one shape and 4 copies

of the flipped shape) from the BLM “Find-a-Flip Game.” Player 1 spins the spinner so that Player 2 does

not see the result. Player 1 builds a pattern with the cards using the transformation shown by the spinner

and shows the pattern to his or her partner. Player 2 has to determine which transformation was used to

make the pattern.

Extension:

Use the shapes from Question 1 on worksheet G3-42 for this problem:

Each chart identifies shapes with given attributes. Can you figure out what the two attributes are?

a) b)

Shapes with Attribute 1 Shapes with Attribute 2

B, D

A, D

Shapes with Attribute 1

Shapes with Attribute 2

F, C

F, E

Flip Turn

• Slide


G3-43 Geometry in the World Goal: Students will see applications of geometry in real life.

Prior Knowledge Required: Slides, reflections, and rotations

Symmetry

3-D shapes

Vocabulary: edge, vertex, vertices, face, triangle, square, rectangle, pentagon, hexagon, octagon, slide,

turn, flip, translation, rotation, reflection, prism, pyramid

The worksheet G3-43: Geometry in the World is an extension and review worksheet. Students can also

complete one or more of these cross-curricular projects:

Symmetry

1. Flags and Coats of Arms of Canadian provinces/cities—Which ones have lines of symmetry? Which

ones have more than one line of symmetry? (None!)

2. Flags of the world—Make a list of countries with flags that have two lines of symmetry.

3. Coats of Arms of Soccer/Baseball/Hockey clubs—Which ones have lines of symmetry? Which ones have

more than one line of symmetry?

4. Cultural Diversity: Alphabets—Find letters in a non-Latin alphabet (e.g., Greek, Arabic, Hindi, Hebrew,

Korean) that have lines of symmetry. Are there any letters or symbols that have more than one line of

symmetry?

5. Snowflakes—Make several designs of snowflakes. How many lines of symmetry do your

snowflakes have?

Geometry in Everyday Life

1. Bee hives and hexagons—Research why bees build hexagonal shapes in the hive. Why don’t they use

rectangular or triangular shapes?

2. Geometrical floor patterns—Use reflections and rotations to create a pattern design.

3. Traffic signs—Which geometric shapes are used in various traffic signs?

4. A robot is used to draw letters. The robot understands the following commands:

• Draw a line from the current position to the point ___ units up/down and ___ units right/left.

• Move ___ units up/down and ___ units right/left without drawing a line.

Give the robot directions to draw various letters of the alphabet. Can you make the robot

write your name?


Geometry and History

1. Ancient Egypt—Which geometric shapes were used in ancient Egyptian buildings?

2. Ancient Egypt: The Pyramid of Khufu—How many right angles can you find in this pyramid? Find more

interesting facts about this or other Egyptian pyramids.

3. Ancient Maya—Which geometric shapes were used in ancient Mayan temples?


G3-44 Problems and Puzzles

The worksheet G3-44: Problems and Puzzles is a review worksheet and may be used for extra practice.


3-D Shape Sorting Game _________________________________________________2

Dot Paper _____________________________________________________________6

Find-a-Flip Game _______________________________________________________7

Grid Paper _____________________________________________________________8

Map of Saskatchewan ___________________________________________________9

Nets for 3-D Shapes ____________________________________________________10

Pattern Blocks _________________________________________________________13

Pentamino Pieces ______________________________________________________14

Venn Diagram _________________________________________________________15

Word Search Puzzle (3-D Shapes) _________________________________________16

G3 Part 2: BLM List






Four or more

triangular-

shaped faces

Square-

shaped base

Fewer than

six faces

More than

four faces

Triangular-

shaped base

Two or more

square-

shaped faces




Ten or more

edges

Four or more

vertices

Pyramids

Six or fewer

vertices

Exactly

twelve edges

Prisms



Dot Paper



Find-a-Flip Game


Grid Paper (1 cm)



Map of Saskatchewan




SquarePyramid

Nets for 3-D Shapes

Tri

an

gu

lar

Pyra

mid


Nets for 3-D Shapes (continued)

Cube

Tri

angula

rP

rism



Nets for 3-D Shapes (continued)

Pentagonal Pyramid

Pentagonal

Prism

Pattern Blocks

Triangles

Squares

Rhombuses

Trapezoids

Hexagons


Pentamino Pieces



Venn Diagram


Word Search Puzzle (3-D Shapes)

WORDS TO SEARCH:

base pyramid

edge rectangular

face skeleton

hexagonal triangular

net vertex

pentagonal vertices

prism

r s k e l e t o n o

a e e s a b g e m a

l i v l n o t d r r

u p v a o p r a e n

g e e n g a t r p g

n v r o a l d n y i

a e t g t m s i r p

t r i a n g u l a r

c t c x e c a f m g

e e e e p e a m i u

r x s h l c n p d l



Teacher's Guide: Workbook 3 - JUMPMath - Common Drive

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