Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction (Grades 3 – 5) Milligan College, 2015 ITQ Grant Program Dr.

Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction

(Grades 3 – 5)Milligan College, 2015 ITQ Grant Program

Dr. Lyn HowellDr. Angela Hilton-Prillhart

Dr. Jamie Price

June 15-19, 2015

Monday, June 15 8:30 AM – 11:30 AM: Morning Session (Whole Group) 11:30 AM – 1:00 PM: Lunch (on your own) 1:00 PM – 4:00 PM: Accountable Talk/Lesson Planning

Introduction Tuesday, June 16 – Thursday, June 18

8:30 AM – 11:30 AM: Morning Sessions 8:30 AM – 10:00 AM: Math Workshop (Group A/B) 10:00 AM – 11:30 AM: Literary Workshop (Group A/B) 11:30 AM – 1:00 PM: Lunch (on your own) 1:00 PM – 4:00 PM: Lesson Planning/Afternoon math

workshop

Agenda for the Week

Friday, June 19 8:30 AM – 10:30 AM – Post test/Concept Maps 10:30 AM – 11:30 AM – Teacher Group

Presentations 11:30 AM – 1:00 PM – Lunch (on your own) 1:00 PM – 4:00 PM – Teacher Group

Presentations

Agenda for the Week

Keep students at the center.

Be present and engaged.

Monitor air time and share your voice.

Challenge with respect.

Stay solutions oriented.

Risk productive struggle.

Balance urgency and patience.

BE OPEN TO NEW IDEAS!

Workshop Expectations

Some Call It Art

Introduction/Ice Breaker

1. Concrete: At this stage, students are introduced to a new concept with the aid of manipulatives/hands-on work

2. Pictures (Representational): At this stage, students are able to draw pictures to explain their reasoning used to solve a problem. Students may draw pictures to indicate what they would have done with manipulatives.

3. Symbols (Abstract): This is the most abstract stage. Students are able to use symbols (numbers, operation signs, algorithms, etc.) to solve the problem.

Stages for Learning Mathematics

Stages for Learning Mathematics

While the stages for learning should progress in order as students learn a concept, once students reach the symbol (abstract) stage, they should understand the relationship between the symbols and the previous two stages.

Adding It Up Framework(Adding It Up: Helping Children Learn

Mathematics, NRC 2001)

Five Strands of Mathematical Proficiency

Conceptual Understanding—comprehension of mathematical concepts, operations, and relations Students know more than just isolated facts and

methods Students understand why a mathematical idea is

important and the kinds of contexts in which it is useful

Students have organized their knowledge into a coherent whole

Conceptual understanding supports retention; students can reconstruct facts and methods that are forgotten when needed (p. 118)

Five Strands of Mathematical Proficiency

Procedural Fluency – knowledge of procedures, when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently Students need to be efficient and accurate in

performing basic mathematical computations Students need to be able to estimate the result

of a procedure Students use a variety of mental strategies to

solve various problems (p. 121)

Five Strands of Mathematical Proficiency

Strategic Competence – ability to formulate mathematical problems, represent them, and solve them Students must first understand the situation and

determine the key features Generate a mathematical representation of the

problem that captures the key features and ignores irrelevant ones (drawing, equation, graph, etc.)

Students come up with multiple approaches to solving the problem and choose flexibly among various approaches (reasoning, algebraic, guess and check) (p. 124)

Five Strands of Mathematical Proficiency

Adaptive Reasoning – capacity to think logically about the relationships among concepts and situations Adaptive Reasoning is the glue that holds

everything together Includes not only informal explanation and

justification of a solution, but also intuitive and inductive reasoning based on pattern, analogy, and metaphor

Ability to justify one’s work (p. 124)

Five Strands of Mathematical Proficiency

Productive Disposition – refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics If students are to develop in any of the other

strands of proficiency, then they must possess a productive disposition towards mathematics

A productive disposition develops when the other strands do and helps each of them develop (p. 131)

Five Strands of Mathematical Proficiency

Examining Student Work

Consider the student responses to the problem on the back of the Adding It Up Handout.

What mathematical strands of proficiency are shown in each student’s response?

What mathematical strands of proficiency appear to be missing?

Reflecting On Your Practice

Take a few minutes to reflect on your own practice, especially related to teaching fractions.

Within your own practice, do you feel that you help students develop all strands of mathematical proficiency related to understanding fractions?

Are there particular strands that you feel that you develop more than others? Why do you think this is so?

Are there particular strands that you feel you never really develop? Why do you think this is so?

Other comments/thoughts?

Keisha receives her paycheck for the month. She spends 1/6 of it on food. She then spends 3/10 of what remains on her mortgage payment. She spends 3/7 of what is now left for her other bills, and 5/8 of what now remains for entertainment. This leaves her with $300. What was her original monthly take-home pay?

Keisha’s Paycheck

The rectangle represents the paycheck (our whole).One of the six parts is shaded yellow to represent the amount Keisha pays for food.5 equal size pieces remain.

3/10 of remaining is spent on mortgage.But, only 5 equal size pieces remain.What should I do?

food

Cut each piece in half to get 10 white pieces (12 pieces now in the whole).

The yellow (food) section is now divided into two pieces and represents 1/6 or 2/12.

food

Shade 3 of the 10 white pieces. This is rent.7 pieces remain.

food

Rent

Food

Rent

Food

Of the 7 pieces remaining (white pieces), Keisha spends 3/7 on other bills. Shade 3 white pieces pieces to represent other.

What fraction of the whole paycheck is represented by food? What fraction of the whole paycheck is represented by rent?

Food

Rent

Other

4 white pieces remain. But, 5/8 of the remaining is spent on entertainment. What do I do?

Rent

Food

Fun

Other

I need 8 pieces, so each piece is cut in half again.Five are shaded blue. There are 3 white pieces remaining.

How many pieces are now in the whole? What fraction of the whole paycheck is represented by food? Rent? Other? Fun?

$300

Rent

Food

Fun

Other

The 3 white pieces represent $300 left from her paycheck. How much is each white piece?

$300

Rent

Food

Fun

Other

Each piece is $100. How much does Keisha make?

Keisha earns $2400.

Keisha’s Paycheck and Fractions

Consider the Keisha’s Paycheck problem. What concepts related to fractions are

addressed in this single problem?

Keisha’s Paycheck and Mathematical Proficiency

Consider the various solutions presented to the Keisha’s paycheck problem.

What strands of mathematical proficiency are addressed in each solution?

Understanding Fractions

Suppose you have a fraction of the form A . B

We call the value of A the numerator of the fraction and we call the value of B the denominator of the fraction.

What do the numerator and denominator represent in a given fraction problem?

Types of Fraction ModelsTypes of Fraction ModelsModel Example Description

Area/Region

2/5 of the picture is blue

Set

2/5 of the counters are red

Length

The object is 2/5 of a unit long

1 0

Area Model vs. Set Model

What are some similarities to an area model for fractions and a set model for fractions?

What are some differences between these two models?

Which model was represented by the picture solution to the Keisha’s paycheck problem?

Fraction Hexagon Task Handout

Fractions as Parts of Sets

Lesson One from Lessons for Introducing Fractions by Marilyn Burns (Teaching Arithmetic Series)

Homework for Tuesday

For Tuesday, read selections from Chapter 7 of Adding it Up: Helping Children Learn Mathematics Read pp. 231 – 241 (stop at Proportional

Reasoning) Read pp. 246- 247 (section titled Beyond

Whole Numbers) Be ready to have a discussion of the reading in

your math session on Tuesday

Task-Based Lesson Plan

A mathematical task is a problem or set of problems that focuses students’ attention on a particular mathematical idea and/or provides an opportunity to develop or use a particular mathematical habit of mind.

from http://commoncoretools.me

What is a math task?

A high-quality math task has the following characteristics: Aligns with relevant mathematics content

standard(s) Encourages the use of multiple representations Provides opportunities for students to develop

and demonstrate the mathematical practices Involves students in an inquiry-oriented or

exploratory approach

What are the characteristics of a high-quality math task?

Allows entry to the mathematics at a low level (all students can begin the task) but also has a high ceiling (some students can extend the activity to higher-level activities)

Connects previous knowledge to new learning Allows for multiple solution approaches and

strategies Engages students in explaining the meaning of

the result Includes a relevant and interesting contextfrom Putting Essential Understanding of Fractions

into Practice (3-5), p. 8

What are the characteristics of a high-quality math task?

Work with the members of your team to create a task-based lesson plan which satisfies the following: Addresses at least one Math Common Core State

Standard from your grade level related to fractions Addresses multiple Mathematical Practices Problem for the task must relate to a chosen piece

of literature selected by your team Problem for the task must be original work Complete the template provided Be prepared to present your task with the members

of your team on Friday morning or afternoon

Your Task-Based Lesson Plan

Requirements/Expectations