Parker - Unit Plan

The unit which I chose to plan is for a 7th

grade mathematics class. The unit is titled

Ratios, Proportions, and Relationships with Rational Numbers. The purpose of this unit is three-

fold. First, the unit addresses a number of state standards for 7th

grade math which is essential to

any quality unit of instruction. Secondly, the unit teaches and reinforces knowledge and skills

which are absolutely essential to future academic success in mathematics. Finally, this unit

incorporates content which is thematically tied together. The theme which is reinforced or

introduced in this lesson (dependent upon students’ background), is relationship. There are many

important themes present in math which it would enrich students’ education to understand, and

relationship is one such theme.

My learning goals for this unit are taken directly from the Iowa Core state standards for

7th

grade math. Math, as a subject-area, is unique in that the state standards often lend themselves

very directly to learning objectives. The standards which this unit addresses are no exception.

They are as follows:

a) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas

and other quantities measured in like or different units. For example, if a person walks

1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per

hour, equivalently 2 miles per hour. (7.RP.A.1)

b) Recognize and represent proportional relationships between quantities. (7.RP.A.2)

c) Apply and extend previous understandings of operations with fractions to add, subtract,

multiply, and divide rational numbers. (7.NS.A)

d) Use properties of operations to generate equivalent expressions. (7.EE.A)

This unit addresses each of these objectives to varying degrees. The unit addresses

objectives ‘a’ and ‘b’ in their entirety; however, I do not spend a great deal of time on these

topics. These are objectives which will crop up throughout the semester, as unit rates and

proportionality are applicable to a number of other chapters and sections. So, in this unit, I

addressed the objectives, but left some unchartered territory to be explored in later units.

The unit focuses heavily on objectives ‘c’ and ‘d’. These objectives are very closely tied

together, as both relate back to the basic mathematical operations of addition, subtraction,

multiplication, and division. Understandings about rational numbers, what they are and how they

are useful, will be increasingly important as students progress in mathematics, thus I felt these

objectives deserved a great deal of time and attention. Likewise, the ability to apply basic

operations to new situations is incredibly important. Students will use these operations all

throughout their time in math and the flexibility to apply these skills, even when the material

looks different, is invaluable.

As I looked out upon the formation of this unit and imagined how it might look and how

it might progress, I accepted the challenge to break away from the lecture-assign-test, format of

traditional math classes. This rather idealistic goal was modified as the unit began to actually

take form. That said, this unit uses lecture as the primary mode of instruction, however, an

exploration activity is used at the beginning of the unit as an opportunity for students to discover

some of the pieces that will be formally introduced later, through lecture. Many of the concepts

covered in this unit are basic, however they have a vast number of applications and though the

concepts do not change, the application can be difficult in new situations. Discovery-based

instructional strategies are heavily relied on in this unit, especially for those students who

progress more quickly than others. There are numerous opportunities during this unit for

enrichment activities which will very much leave these students to their own devices as they

navigate the newly-learned concepts. This is not to say my support will not be available, because

it most certainly will.

Additionally, discussion will be utilized as an instructional strategy through the daily

Q&A sessions over homework assignments. While this will look very different from discussion

in most other content areas, these sessions will be conducted such that students will need to

communicate back and forth in order to reach a solution. This will not be a time for me to simply

dole out answers. Informal discussion opportunities will also be incorporated into lectures.

This unit does not rely heavily on pre-assessment. The primary role of pre-assessment in

this unit is to gain insight into the students as individuals. This is appropriate, as this is the first

unit of the term and the students will be new to me, and I to them. Formative assessment drives

this unit, particularly through board work, homework assignments, and self-assessments. This

assessment strategy is ideal because it allows for a continuous flow of information about student

progress, in both formal and informal ways. Summative assessment strategies utilized include: a

quiz, a unit test, and an authentic peer teaching activity. The quiz and test are ideal for assessing

the more objective pieces which math is typically associated with. However, the teaching activity

works well to assess the more implicit understanding goals tied to personal preference and

intuition in math.

The largest challenge I anticipate with this unit, is the rigorous pace. I intentionally built

in days to slow things down and allow students a chance to catch their breath. However, if there

are students who struggle severely to comprehend the material or possibly are just slower than

the rest of the class, the pace could be a bit fast for them. I anticipate that students who are not

able to keep up will come from one of two camps. One being, students who simply choose not to

do the work out of laziness, in which case, they will inevitably suffer the consequences of their

choices. The other students I anticipate are special education students, who will likely have an

aid or associate with them either in the classroom or as extra support outside of the classroom.

That being said, on the days that I offer time for make-up work and remediation my focus for

those days is to work particularly with special needs students or other students who are putting in

the effort, but simply falling behind.

This raises another possible challenge. On those days when I am differentiating in so

many ways and offering different opportunities and activities for different students, I fear that I

may be pulled in too many directions and not be able to focus enough time and energy on any

one group or individual. I plan to combat this in two ways. First, the more I plan ahead and

prepare for those days, the better. The more I can have things laid out and ready to go, so as to

enable more students to be self-sufficient, the better. Additionally, I will need to prioritize by

student and I will do this by informally evaluating them in terms of effort. I simply will not allow

a student who is working their tail off but not getting it, to lack the support they need because I

am burning time helping another student, who chose to be lazy, to avoid the consequences of

their actions.

I have mentioned a number of ways which I plan to differentiate for students all along the

spectrum from gifted to special needs, but I have a few more ideas which I can explain here. I

philosophically do not believe in giving gifted students more work in terms of quantity, instead I

plan to offer them enrichment opportunities grounded in quality. This can be done in math by

offering more difficult problems or problems which necessitate extended understanding and

discovery in order to solve. Tutoring opportunities may also be given to gifted students, if there

are good circumstances for this arrangement. For special needs students, besides offering them

more of my time and re-teaching opportunities, I have no issue shortening assignments for these

students. In math, repetition and practice are very important; however there is a point for every

student where they start to experience diminishing marginal returns. So, for a special education

student who takes longer to solve problems for whatever reason, the fifteenth problem may

literally be more harm than good for them, whereas that point may be the 45th

problem for

another student. In assignments, it is most important that students are receiving exposure to the

necessary applications, not that they have enormous, long assignments to complete.

In summary, this lesson and its objectives are grounded in the Iowa Core state standards

for 7th

grade mathematic. It relies primarily on lecture, student-discovery activities, and

discussion as instructional strategies. In addition, this lesson will be continually informed by a

variety of formative assessments and summative assessment will be in the form of a quiz, test,

and peer-teaching activity. While the pace may be challenging, there are numerous ways to

adjust the pacing and/or to help students to adapt and catch up. I certainly hope to find use for

this lesson in my future teaching, as I believe it would be very effective in teaching students

about Ratios, Proportions, and Relationships with Rational Numbers.

Unit Title: Ratios, Proportions, and Relationships with Rational Numbers. Grade level: 7

th Grade

Length of unit: 2-3 weeks (beginning of term)

Stage 1 – Desired Results

Meaning

Enduring Understandings: student will understand that…

1. Recognizing, interpreting and manipulating mathematical

relationships will enable us to make sense of math. (a)

2. Practicing math develops intuition which helps us to recognize

relationships. (b)

3. When quantities are related we can create expressions which

represent the relationship mathematically. (a, b)

4. Rational numbers are essential to establishing and

understanding relationships. (c)

5. Mathematical operations add flexibility which enables

mathematicians to find solutions. (d)

Essential Questions:

How do mathematicians solve proportions?

What does it mean to be related mathematically?

What is proportionality and how is it relational?

What does equality mean in math?

Are unequal relationships useful?

How is the number 1 useful in math?

Knowledge & Skills Acquisition

Learning Goals: (standards from the Iowa Core for 7th

grade mathematics)

a) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or

different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles

per hour, equivalently 2 miles per hour. (7.RP.A.1)

b) Recognize and represent proportional relationships between quantities. (7.RP.A.2)

c) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

(7.NS.A)

d) Use properties of operations to generate equivalent expressions. (7.EE.A)

Knowledge: Students will know…

1. Complex fractions can be simplified by using the

multiplicative inverse of the denominator. (U 1)

2. Equality is maintained in expressions by mirrored

operations on either side of the equal sing. (U 1)

3. They can always multiply by 1 w/o changing value and 1

can be written in infinitely many different ways. (U 1)

4. Relationships can look different and are not always easy

to find or establish. (U 2)

5. That practice problems will make math easier and more

beneficial to them. (U 2)

6. What information to look for when trying to establish

relationships. (U 3)

7. Rational Number – Any number which can be written as a

fraction. (U 4)

8. Order of operations must be applied when performing

operations involving rational numbers. (U 4)

9. By applying operations to rational numbers, we can

simplify expressions. (U 4)

10. The basic mathematical operations. (U 5)

11. When to apply basic operations to find solutions. (U 5)

12. Terms: equivalence, LCD, inverse, basic operations, unit

rates, ratio, proportion, rational number, expression, scale

factor.

Skills: Students will be able to…

Organize given information to create complex fractions. (K 1)

Utilize the multiplicative inverse of the denominator of fractions to simplify expressions. (K 1)

Compute unit rates. (K 1)

Interpret conversion charts to achieve like units. (K 1)

Perform algebraic manipulations without changing the value of an expression (i.e.: maintaining equality). (K 2)

Multiply by 1 to simplify expressions. (K 3)

Represent the number 1 in a variety of ways. (K 3)

Recognize and establish relationships using given information.

(K 4)

Apply themselves to sets of practice problems. (K 5)

Organize given information represent relationships through mathematical expressions. (K 6)

Write rational numbers. (K 7)

Manipulate expressions containing rational numbers using basic operations. (K 8,9)

Apply the basic operations of addition, subtraction, multiplication, and division. (K10)

Explain their applications of basic operations when manipulating

equations. (K 11)

Resources/Materials:

Textbook (HOLT Middle School Math Course 3 – Algebra Readiness)

Syllabus

Pre-Assessment: interview sheets and pre-test

Number lines

First in Math license

Computers

White board and markers

Smart board

Lecture notes

Worksheets & problem sets: exploration activity, reciprocals, remedial, practice algebra, practice word problems, group word problems, challenge sets, unit review sets, supplementary individual practice sets, enriched challenge

sets, modeled word problems

Manipulatives

Pie charts and diagrams

Quiz

Teaching Activity: assignment sheet, rubric, and grading sheets

Unit test

Stage 2 – Evidence (Assessment)

Types of assessment: Selected-Response (tests, quizzes); Personal Communication (interview, oral exam, discussion);

Written Response (short constructed response questions, entrance/exit slips, essays); Performance Assessment (role-play,

Simulation, labs, dramatization)

Pre-assessment:

Interview:

o What relationships can you think of in mathematic? (K 4)

o What is your attitude towards assignments in mathematics and why? (K 5)

o How might your attitude be changed?

o What ways will you find to motivate yourself this semester?

Pre-test over operations involving rational numbers (addition, subtraction, multiplication, division, reduction, multiplicative inverse). o Multiplicative inverse (K 1), Algebraic operations to solve for a variable. (K 2, 8, 10, 11), Multiplying by 1. (K 3)

Formative Assessment:

Daily textbook assignments (accuracy)

Practice problems administered as a group, but worked out individually on white boards - These will be used to address questions and misconceptions prior to beginning homework assignments.

Skill sets and modules via computer program (completion) - These will be used to reiterate previous material and to build fact fluency.

Weekly self-assessment (of effort and competency) via exit slip on Thursdays

- Fridays will be dedicated to addressing issues referenced in Thursday’s exit slips, make up work, re-teaching, etc. (mustard days).

Word problem pertaining to proportionality and scale: Students will create a unique word problem which involves scale and/or proportionality. The problem must be presented in such a

way that it could be given to another student to solve. This problem will be assessed for understanding of proportions and students

will have revision opportunities until the problem is such that it is ready to be solved.

Summative Assessment:

Quiz

Unit test covering all content.

Small group teaching experience: Students will be divided into groups of 3 and each student will solve the word problem which was previously created by the other

two.

Unit: Ratios, proportions, and relationships with rational numbers.

Unit Calendar

Week Day 1 Day 2 Day 3 Day 4 Day 5

1

First Day of School

KEY

* SB = smart board * Board Work = all students

work out a problem on the white board. They individually solve the problem, but are free to collaborate. * Individual Practice:

While seated students work individually to solve a given problem. After adequate work time, teacher works the problem on the board and invites questions. * Correct: Students trade papers and correct each others’ assignments. * Patch Up: In no more than

5 minutes, address issues from the bell ringer problems. Self-Assess. Exit Slip:

W/o solving the problems, look at each problem and rate it from 1-10 (10 meaning you know exactly how to solve it).

Rate your level of effort this week from 1-10 (10 being everything you had). Be sure to consider your effort during lecture, board work, on assignments, & during Q&A sessions.

Self introduction & background. Class discussion to establish expectations. Classroom procedure & norms. Syllabus walkthrough. Explanation of SB notes being uploaded for review. Explanation of tutoring times & before/after school hours (info also included in syllabus).

Unit Intro (SB): Understandings Objectives Pre-Assess: Begin individual interviews. Record on interview

sheets. Small group exploration activity (during interviews): Groups of 3 or 4. Worksheet guides

students thru problems with questions leading from one to the next.

Q&A session over upcoming unit. Which understandings

confuse you? What things do you

think you already understand?

Which understandings seem most difficult?

Pre-Assess: Continue interviews (if necessary) Pre-Assess: Administer pre-test.

Evaluate results over-night to inform assignments and speed of progression.

Lecture (SB): Lesson 3.1

Define rational number Simplify (relatively

prime) Simplest form Decimals to fractions Fractions to decimals Practice: Lesson 3-1, 1-16 (evens) Assign: 3-1, 17–50 ( odd)

Correct: 3.1 homework. Q&A over missed problems. Board Work: 3.1, #’s 53, 55, 57 Lecture (SB): Lesson 3.2 Hand out number

lines. Adding rational

numbers (students practice using number lines).

Add & subtract fractions (like & unlike denominators).

Assign: 3.2, 2-17 & 23-38 (odd)

Unit: Ratios, proportions, and relationships with rational numbers.

Unit Calendar

2

Lessons 3.3 - 3.6

Bell Ringer: 3.1 #’s 26, 38, & 46 Patch Up Correct: 3.2 homework. Q&A over homework.

First in Math: Introduce and give an overview. Via projector, orient students. First in Math: Handout to guide students in navigating software. Board Work: Intro 3.3 through sample problems.

Entrance Slip: When you multiply two

numbers together, you get a product. If your product is negative, what must be true of the two numbers? What if your product is positive?

Group discussion over entrance slips.

Lecture (SB): Lesson 3.3

Multiplying fraction and integer.

Multiplying fractions. Multiplying decimals. Reciprocals. Reciprocals worksheet. Assign: 3.3, 1-32 (odd)

Correct: 3.3 homework. Q&A over homework. Lecture (SB): Lesson 3.4

Reiterate reciprocal. Dividing fractions =

multiplication by reciprocal.

Show problems with division sign and as fractions w/in fractions.

Decimal division Using calculators & by

multiplying by 1. Board Work: 2- 5 problems to practice multiplying by 1. Assign: 3.4, 24-47 (odd)

Correct: 3.4 homework. Q&A over homework. Lecture (SB): Lesson 3.6

Solving equations w/ decimals.

Solving equations w/ fractions

Board Work: Lots of practice solving equations w/ rational numbers. Students who are

ready, may move on to assign. 3.6: 14-19 & 27-41 (odd).

Struggling students will move closer together & I will continue practice w/ them.

No assignment due Friday, Self-Assess. Exit Slip: Evaluate responses

from self-assessment to inform Friday’s differentiated instruction.

Based upon self-assessment data, homework scores, informal assessments during board work, and student preferences divide students into groups by the following activities: Independent work:

Students may work on assignment 3.6. When finished, work in fluency thru First in Math or tutor other students.

Make-up work: Students with missing or incomplete assignments may finish them for partial credit. Students w/ D or F assignments may do a remedial assignment to improve their grade up to 70%.

Students who were uncomfortable in specific areas may work on specified practice problems.

Students struggling severely will work in a group w/ the teacher.

Assign: 3.6, 14-19 & 27-41 (odd).

Unit: Ratios, proportions, and relationships with rational numbers.

Unit Calendar

3

Lessons 3.7, 7.1, & 7.2

Bell Ringer: 3.3 #’s 46 & 54, 3.4 #’s 28 & 44, 3.5 Patch up Correct: 3.6 homework. Q&A over homework. Lecture (SB): Lesson 3.7 Solving inequalities w/

decimals. Solving inequalities w/

fractions. TIPS: How are inequalities and equalities similar? How are they different?

Assign: 3.7, 14-24 (odd), 45 & 46.

Correct: 3.7 homework. Q&A over homework First in Math: Practicing operations on rational numbers. Use progress on skills

modules as formative assessment.

Reinforcement activities: Manipulatives to

practice operations of fractions.

Pie charts and bar diagrams to visualize

using multiplicative inverse.

Worksheets to practice solving for variables using mirrored operations.

Lecture: Lesson 7.1

Define ratios and equivalent ratios.

Finding equivalent ratios by multiplying by 1.

Concept: looking for unseen relationships.

Identifying proportions. Individual Practice: Problems will be word problems involving ratios.

These will serve as practice and models for the creation of their own word problems. Quick Write: First draft of student-created word problems. Assign: 7.1, 9-25 (odd), 28, & 31.

Bell Ringer: 3.7 #’s 16 & 36, 3.2 #’s 28 & 40 Correct: 7.1 homework. Q&A over homework Lecture: Lesson 7.2 Recall definition of ratio. Define rate (diff. units). Define unit rate. Relate unit rate to unit

price using business example.

Board Work: Problems from lesson 7.2 Assign: Chapter 3 Mid-Chapter Quiz, 1-33 (odd) Announce quiz tomorrow over 3.1-3.7 Self-Assess. Exit Slip: Evaluate responses

from self-assessment to inform Friday’s differentiated instruction.

Quiz: Sections 3.1-3.7

~10 solution-based problems.

3 short answer concept-based questions.

Correct: Mid-Chapter quiz. Small Groups: In groups of 3 students will work out word problems centered

around rates and proportions. Groups will rotate problems when they finish one. Quick Write: Edit previous quick write using individual feedback given by teacher. Based upon formative assessments divide students into groups by the following activities: Make up work or

corrections. Small group work with

teacher. Challenge problem

sets pertaining to current topics.

Announce review day to be the next day.

Unit: Ratios, proportions, and relationships with rational numbers.

Unit Calendar

4

Review Activities: Stations w/ review sets

from different sections where students may work in groups.

Individual practice exercises and word problems.

Pre-solved word problems on white board for students to analyze and ask questions about.

Quick Write: Final draft of word problem. Assess these overnight

so that if any final changes are necessary, they can be made tomorrow.

Explanation of teaching experience: Hand out assignment

sheet. Hand out and talk about

rubric. Briefly model teaching

experience with 2 students.

Last minute changes to individual word problems (if necessary). Teaching Experience: Group students by 3’s. Students will take turns

administering their word problem.

This can take the entire period if necessary, but monitor closely to be sure students are not cutting corners.

Free Time: Students may use the remainder of class to review, work on First in Math, or work on something for another class (absolutely zero games, YouTube, Facebook, or phones).

Unit Test:

2 or 3 solution-based problems from each section covered (3.1-3.7, 7.1 & 7.2)

3 word problems pertaining to ratios and proportionality.

Quick Write: 2 questions from the pre-unit interview: What relationships can

you think of in mathematic?

What is your attitude towards assignments in mathematics and why?

Next Unit: Very informal introduction (if time).

Small Group Teaching Activity

In groups of three you are asked to solve each others’ word problems which you have been creating over the past week. Each

of you is asked to pre-teach the information that is necessary to solve your problem. The method you use needs to yield the correct

answer, however, it does not necessarily have to be a method which I explicitly taught in class.

To assess your group members’ understanding, each teacher will present their problem for the other members to solve. As

students, when you work out your group members’ problems, you need to use the method which they taught to you. If a group

member is not able to solve a problem using the method, the teacher needs to work with that person individually to develop the

understanding.

You will be assessed in a variety of ways on this assignment. Please read the attached rubric carefully so that you understand

what I am looking for in your group work and how you will be assessed.

Criteria Command

(10 pts.)

Proficiency

(9.25 pts.)

Developing

(7.5 pts.)

Basic

(5 pts.)

Accuracy of solution method

Method yields the

correct answer in all

applicable cases.

Method yields the

correct answer in

most cases.

Method happens to

yield correct answer,

but not in most

cases.

Method does not

yield the correct

answer.

Explanation of/teaching with solution

method

Student explains their

method completely

using examples and

with evidence of

understanding.

Student explains

method well for

particular case with

evidence of

understanding.

Student explanation

makes skips in

logic, but

demonstrates some

understanding.

Student explanation

is incorrect and may

or may not

demonstrate any

understanding.

Practice problem

Problem is creative

and highlights the

pros of using the

particular method

being taught.

Problem

effectively and

appropriately

assesses

knowledge and

ability to utilize the

method.

Problem can be

solved with the

method. Method

may or may not be

most appropriate to

the problem.

Problem does not

test for the method

taught and/or does

not have a solution.

Re-teaching and support

Student patiently

scaffolds others’

learning through pre-

prepared and

intentional strategies.

Student scaffolds

others’ learning by

re-explaining the

method.

Student attempts to

scaffold others’

learning, but a lack

of clarity or

understanding.

Student does not

address issues and/or

has no patience with

others’ struggles.

* This assessment is worth 30 points. If no, or little, re-teaching or support is necessary, you will only be assessed over the top 3

categories. If re-teaching and support are necessary, your score on this area will replace the lowest score of the top 3 categories.