FRACTIONSA Conceptual Unit for

3rd Grade

2010

by Kathy Beach, Dennis Powell, Heather Rader, Leah Workman and Kathie Young

for the North Thurston Public Schools

FractionsA Conceptual Unit for 3rd Grade

3.3 Core Content: Fraction Concepts

Students learn about fractions and how they are used. Students deepen their understanding of fractions by comparing and ordering fractions and by representing them in different ways. With a solid knowledge of fractions as numbers, students are prepared to be successful when they add, subtract, multiply, and divide fractions to solve problems in later grades.

Performance ExpectationsStudents are expected to:

3.3.A Represent fractions that have denominators of 2, 3, 4, 5, 6, 8, 9, 10 and 12 as parts of a whole, parts of a set, and points on the number line.

3.3.B Compare and order fractions that have denominators of 2, 3, 4, 5, 6, 8, 9, 10 and 12.

3.3.C Represent and identify equivalent fractions with denominators of 2, 3, 4, 5, 6, 8, 9, 10, 12.

3.3.D Solve single- and multi-step word problems involving comparison of fractions and verify the solutions.

Big Ideas Essential Question

Fractions are numbers that represent equal parts of a whole

What are fractions?

The relationship between the numerator and denominator determine the relative size of a fraction.

How do you compare the size of fractions?

Fractions can be translated into different forms and still be equivalent.

What are equivalent fractions?

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Table of Contents

Key Vocabulary 4

Daily Math Review 5-8

Mental Math 9

Problem Solving Tasks 10-11

Big Ideas 12

Unit Scope and Sequence 13-14

Number Line Lessons & Printables 15-18

Reflection 19

Big Ideas & Vocabulary Printables 20-26

References 27

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FRACTIONS

Compare & Order

Vocabulary

Understanding Fractional Symbols

& Conventions

NumeratorDenominator

Equivalency

Represent Fractions

Parts of a Whole

Parts of a Set

Points on aNumber Line

3 rd grade Fraction Vocabulary

Compare: To look for similarities and differences.

Denominator *: The bottom number in a fraction. The number of parts the whole is divided into.

Equal : Having the same amount or value.

Fraction: Any equal part of a group number or whole.

Greater than: More than. (Shows relationship between numbers.)

Less Than: Not as many as. (Shows relationship between numbers.)

Numerator *: The top number in a fraction. The number of parts out of the whole that we are talking about.

Order: Arrange according the size, amount or value.

Represent: Show, draw a picture or other visual image.

*Model to develop concept

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Daily Math Review(Step 1 of Five Easy Steps)

Daily Math Review establishes a deliberate progression of mathematical concepts and computational skills that increase in difficulty throughout the school year.

This component of the “Five Easy Steps” program is a simple system for reviewing basic computational skills on a daily basis. Students take the first 20 minutes of math class to

complete a set of five problems. These problems should:

Represent the specific standards for that grade level

Provide practice in several standards or strands

Match the conceptual focus of the current instructional unit or set of lessons

Reinforce prior learning and retention of previously taught concepts and skills

Provide daily practice for the computation sections on district and state assessments

Adapted from Five Easy Steps to a Balanced Math Program

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Daily Math Review Example

Name ___________

Number Sense

Put these numbers in order from least to greatest.

42, 13, 101, 2, 31, 78

____, ____, ____, ____, ____, ____

Patterns

Complete the pattern.

1004, 1006, _____, _____, 1012

Subtraction

4302 - 2148

Division

Jamie had a bag of 28 skittles to share equally with her friends at lunch. There were 4 friends, including Jamie. How many skittles should each friend get?

Current Unit

What fraction of this shape is shaded?

____________________________

Challenge Box

How many different designs can you make that are ¾ red and ¼ yellow?

Draw as many designs as you can.

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Daily Math Review Example

Name ___________

Number Sense

I am a 3 digit number I have a 3 in my tens place. The number in my ones place is 2

more than the number in my tens place.

The number in my hundreds place is the sum of my ones and tens place.

WHAT NUMBER AM I?

_____________

Patterns

Complete the pattern box.

2 4 8 10

8 16 24

Subtraction

6032 - 3281

Multiplication

Brent was cleaning his room. He has 8 toy shelves, so he put 6 toys on each shelf. How many toys did Brent put on all of the shelves?

Current Unit

What fraction of the hearts are shaded?

____________________________

Challenge Box

I picked up a handful of M&M’s. 1/3 of them were red. What might my picture

look like?

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Daily Math Review Example

Name ___________

Number Sense

Write a number with an 8 in the tens place.

____________

Write a number with an 8 in the ones place

___________

Write a number with an 8 in the hundreds place.

___________

Patterns

What is the rule for this box?

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21Subtraction

3002 - 281

Division

Each desk in Jamal’s group at school has 4 legs. He counted 16 desk legs. How many desks are in Jamal’s group?

Current UnitShade 2/9 of the circles.

Challenge Box

Aunt Sallie said that when she was ½ the age she is now, she could touch her toes. How old could she have been when she could touch her toes? How old could she be right now?

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Mental Math

“Students need regular opportunities to develop effective computational strategies that are based on number sense. Helping students use number strategies that they find comfortable and accurate is

an effective way to develop number sense. Students need daily mental practice to develop and retain strong number sense and effective computational skills.” (Christensen, 23)

Mental Math

*Fractions are numbers that represent equal parts of a whole.

-The whole is divided into 5 parts.

a. What fraction names 1 of the 5 parts?

b. What fraction names 4 of the 5 parts?

*The relationship between the numerator and denominator determines the relative size of the fraction.

-A pizza is cut into 8 equal parts.

a. What fraction names the whole?

b. Which is larger 3/8 or 5/8 ?

*Fractions can be translated different forms and still be equivalent/equal.

a. Bob has 2/4 of a candy bar. His brother has an equal amount. What fraction of a candy bar could he have?

b. Name a fraction that is equal to 1/4.

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Problem Solving(Step 2 of Five Easy Steps)

This step provides both a structure for problem-solving activities related to the current conceptual unit focus and a general problem-solving rubric or scoring guide that is used throughout the year to assess student work.

The National Council of Teachers of Mathematics (2000) recommends that students: Communicate their thinking to peers, teachers, and others Analyze and evaluate their math thinking and strategies Use the language of mathematics

Students’ Role Solve a problem independently first, recording work on a data sheet. Share data sheet and solutions with others in a cooperative group/pair. Each cooperative group/pair designs on chart paper a group data sheet that includes

various ways to solve the problem. Math Congress: groups share their solutions with the whole class. Write Up: each student writes an explanation for how to solve the problem.

Teacher’s Role Emphasize mathematical reasoning and verification and use of math vocabulary. Model the process you want students eventually to be able to do on their own. Provide individual or small-group assistance as needed. Don’t rush this process—time invested now pays off later.

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Sample Problem Solving Tasks3.3.A Fractions are numbers that represent equal parts of a whole.

Task: Pizza Party

Part A: Jose and Jane are sharing a pizza.

A) How many parts will the pizza be cut into?

B) How much of the pizza will each person get?

C) Name the part of the pizza each person gets?

Part B: If Jose and Jane each invite a friend to share the pizza with them.

A) How many parts will the pizza be cut into?

B) How much of the pizza will each person get?

C) Name the part of the pizza each person gets?

3.3.B The relationship between the numerator and the denominator determines the relative size of a fraction.

Task 2: I need Chocolate

Part A: Suzy loves chocolate and always wants the most she can get. A giant chocolate bar has 12 equal parts. She has a choice of 1/4, 1/6 and 1/12 of the candy bar, which should she choose to get the most chocolate? Show how you found your answer.

Part B: Now Suzy can choose to have 1/3, 2/3, or 3/3. Which amount will give her the most chocolate? Show your thinking with pictures, words and numbers.

3.3.C Fractions can be translated into different forms and still be equivalent.

Task 3: Brownies Anyone?

Part A: At one party there was a pan of brownies cut into 6 equal parts. Six children were at the party. What fraction of the brownies did each child get? Show with pictures, words and numbers.

Part B: At a second party there was an identical pan of brownies cut into 12 equal parts. Six children were at this party also. What fraction of the brownies did each child get? Show with pictures, words and numbers.

Part C: At the second party did the children get more, less or the same amount of brownies as the first party?

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Developing Fraction Concepts

Big Ideas

1. Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as one. On the number line, the distance from 0 to 1 is the unit.

2. Fractional parts have special names that tell how many parts of that size are needed to make the whole. For example, thirds require three parts to make a whole.

3. The more fractional parts (that) are used to make a whole, the smaller the parts. For example, eights are smaller than fifths.

4. The denominator of a fraction indicates by what number the whole has been divided in order to produce the type of part under consideration. Thus, the denominator is a divisor. In practical terms, the denominator names the kind of fractional part that is under consideration. The numerator of a fraction counts or tells how many of the fractional parts (of the type indicated by the denominator) are under consideration. Therefore, the numerator is a multiplier – it indicates a multiple of the given fractional part.

5. Two equivalent fractions are two ways of describing the same amount by using different-sized fractional parts. For example, in the fraction 6/8, if the eighths share is taken in twos, then each pair of eights is a fourth. The six-eights then can be seen to be three-fourths.

*John Van de Walle Professional Mathematics Series, Teaching Student –Centered Mathematics Grades 3 – 5, Volume 2.

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Scope and Sequence

Trailblazers Unit 11, Lesson 3, “Pizza Problems” (Student Guide, Pages 155-156) (See URG for teacher direction.)

Students can work in small groups to solve problems about sharing pizza. Answers can be found and represented by drawing pictures or paper circles can be used to represent the pizza and to find the fraction each person gets.

To develop the idea of fair shares model a pizza cut into unequal portions and discuss whether students would want to share pizza cut into parts like this, etc.

Trailblazers Unit 11, Lesson 3 Homework (Page 157)

Other problems should also be introduced and solved to continue the development of this concept. Options are included in Chapter 5, Developing Fraction Concepts in Van de Walle’s Teaching Student-Centered Mathematics, Grades 3-5. Marilyn Burns also has many lessons that can be used.

Length and Area models are also commonly used to model fractions.

Fraction strips are teacher or student made representations of Cuisenaire Rods. They provide great flexibility in comparing fractions. See Chapter 5, Developing Fraction Concepts in Van de Walle’s Teaching Student-Centered Mathematics, Grades 3-5. (Page 135.)

Math Trailblazers Unit 11, Lesson 2, “Geoboard Fractions” (Pages 152 – 154) Math Trailblazers Unit 17 Lesson1, “What’s 1?” (SG Pages 274 – 276) This

lesson uses pattern blocks to compare fractions

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Fraction instruction should first focus on a more conceptual approach. The first goal of developing an understanding of fractions should be the idea of fractional parts of the whole or the parts that result when the whole has been divided into equal sized portions.

Most students understand the ideas of fair shares. The goal is to expand this understanding to include fractional parts of all sizes. Sharing tasks are a good way for students to begin to understand fractional parts.

Math Trailblazers Unit 17, Lesson 2, “Folding Fractions” (SG Pages 277 - 285) and the Discovery Assignment Book pages that go with it. Students find and name equivalent fractions through paper folding activities.

Set models are another important way to understand fractions. It is often hard for students to understand that a collection of items can stand for “1” and the separate items are the fractional parts. It does impart real world uses for fraction.

Math Trailblazers Unit 11, Lesson 1, “Kid Fractions” (Page150 – 151) This lesson explores the set model using students as parts of the set. It provides experience for understanding how the size of the whole affects the size of the fraction. It is a good example of the use of numerator and denominator.

Number lines are a more sophisticated model and can be difficult for students to understand.

See attached lesson ideas for teaching fractions using a number line.

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Using a Number Line to Understand Fractions1. Each student will get a copy of the Double Number Line.

They need to cut it out along the heavy dark lines and fold it in half lengthwise.

2. Allow students to pair – share the mathematical observations they can make about the number lines. Share out with the class.

3. Give each student a paper clip. Using the side marked 0 – 1, ask them to slide the paper clip down to the point they think is ½. Then have them turn the paper over to see how close they got.

4. Allow students time to experiment with strategies to get closer, or as close as possible, to ½.

5. Discuss the other marks on the number line that haven’t been labeled with fractions yet. How would they label them? Give them time to pair – share or share at their table group. Call on volunteers to share with the class.

6. After discussion, sketch or use your projector to show the number line. Work with what the students shared to label each mark. Discuss their reasoning and what strategies they used to figure this out.

7. Have the students label each mark with their fractions.

8. Ask them to turn back to the 0 – 1 side and try to slide their paper clip to ¾ of the way along the line. Then check the other side to see how close they got.

9. Repeat with some of the following fractions. (Differentiate to meet the needs of your students.) 1/8 7/8 3/8 1/4 +1/4 1/8 + 1/8 1/2 + 1/8 1/4 + 1/8

Enrichment

Pose story problems and ask them to use their number lines to find the answer. Example: I ran ½ mile. Then I took a rest and ran another ¼ mile. How far did I run in all?

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Number line Fractions: Run a half-class set on cardstock. Cut the sheets in half.

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Name __________________________________________ Date_______________________

Independent PracticeUsing Rulers

You will find, mark and label the following measurements on the pictures of the rulers on this paper. There is an example done for you.

Example: 4 inches

a. 3 inches

b. 2 inches

c. 3 inches

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43

83

87

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85

d. 1 inches

e. 5 inches

f. 3 inches

g. 6 inches

h. 1 inches

Teacher Reflection18

Taken from “Five Easy Steps to a Balanced Math Program”

1. What worked? What didn’t? _____________________________________________________________

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2. What will I do differently next time? _______________________________________________________

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3. What student work samples do I have? _____________________________________________________

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4. What suggestions can I provide for other teachers who may use this assessment? __________________

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I will understand that fractions show

equal parts of a whole.

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I will understand that fractions show

equal parts of a group.

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I will understand that fractions show

equal parts of a number line.

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I will compare the size of fractions,

and put fractions in order.

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I will understand that fractions can

be in different forms and still be

equivalent.24

Fraction Vocabulary

comparedenominator

equalfraction

greater thanless than

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numeratororder

represent

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References

Ainsworth, L. & Christinson, J. (2006) Five easy steps to a balanced math program for upper elementary grades. Englewood: Lead & Learn Press.

Burns, M. (2000). About teaching mathematics. Sausalito: Math Solutions Publications.

Van de Walle, J. A., & Lovin, L. H. (Eds.). (2006). Teaching student-centered mathematics grades3-8. Boston: Pearson Education.

Wagreich, P., & Et. al. (1997). Trailblazers. Dubuque: Kendall Hunt.

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