Warm Up: Connor ran in a race on Saturday. After completing 2/3 of the race, he had run 3/4 mile.How long was the whole race?Show your work……..

Sense Making With Fractions Through Common Core State Standards

Steele Creek Elementary – March 27, 2013

Fractions: Teaching and Learning

What makes fractions so hard for students?

• taught too abstractly with limited models

• Students tend not to make connections between models and between models and symbols

• taught with rote memorization of procedures

• not taught in meaningful contexts• more attention to algorithms rather

than to developing number sense and

reasoning

Fraction and Decimal Standards

When you look at the highlighted standards for grades 3-5 you will notice a strong emphasis on the following:

Visual Fraction ModelsLine plot/ number lines

EquationsSymbols

CCSS Standards for Mathematical Practice

2. Reasoning and3. Explaining

4. Modeling with mathematics and5. Using tools

7.Seeing and using structure and8. Generalizing

Adapted from work of William McCallum

Fraction MODELS and REPRESENTATIONS

• Area/Region Models

• Linear or Measurement Models

• Set Models

• Symbols (with meaning)3 7 14 8 2

Models introduced in 3rd grade

Model added in 4th grade

A Fraction Represents…

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts;

Understand a fraction a /b as the quantity

formed by a parts of size 1/b

Unit Fractions

A unit fraction is a proper fraction with a numerator of 1 and a whole number denominator

is the unit fraction that corresponds to or to or to

As there are 3 one-inches in 3 inches, there are 3 one-eighths in

51

52

53

517

83

Unit Fractions

Unit fractions are the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of whole numbers

Unit fractions are formed by partitioning a whole into equal parts and naming fractional parts with unit fractions 1/3 +1/3 = 2/3 1/5 + 1/5 + 1/5 = ?

We can obtain any fraction by combining a sufficient number of unit fractions

1 b

Unit Fractions

The numerator 3 of ¾ shows that 3 is the number you get by combining 3 of the 1/4 ’s together when the whole is divided into 4 equal parts

A fraction such as 5/3 shows combining 5 parts together when the whole is divided into 3 equal parts – best shown on a number line

Fractional Parts of a Whole

If the yellow hexagon represents one whole, how might you partition the whole into equal parts? Name the fractional parts with unit fractions

Fractional Parts of a Whole

Name the unit fractions that equal one whole Hexagon

1/3 1/2

1/3 1/6

1/6 1/6 1/6 1/6

Fractional Parts of a Whole

Two yellow hexagons = 1 whole

• How might you partition the whole into equal parts? Name the unit fraction for one triangle; one hexagon; one trapezoid and one rhombus

Fractional Parts of a Whole

One blue rhombus = 1 whole

• What is the value of the red trapezoid, the green triangle and the yellow hexagon?

• Show and explain your answer

Identifying Fractional Parts of a Whole

What part is red?

14

Create the whole if you know a part…

If the blue rhombus is 2/3, build the whole.

If the red trapezoid is 3/8, build the whole.

If you know a fractional part, can you make the whole?

Set model

If this is two fifths of a set, make the whole set.

If you know a fractional part, can you make the whole?

Make the whole line if this is one third.

Make the whole shape if this is three fourths.

c

c

c c

cc

Fractions Greater than One

How much is shaded?

18

(¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ ) = 15/4 4/4 + 4/4 + 4/4 + ¾ = 15/4

1 + 1 + 1 + ¾ = 3 ¾

4 x 1/4 + 4 x 1/4 + 4 x 1/4 + 3 x 1/4 = 15 x 1/4

NUMBER LINES as Fraction Models

Understanding Number Lines

Number lines represent the order of numbers and their magnitude

Numbers to the right of any given number are greater in value; numbers to the left of any given number are less in value

Once two numbers are marked on the number line, the location of all other numbers is fixed

Shaughnessy (2011)

Fractions on a Number Line

Parallel number lines support students in identifying equivalent fractions

Unit Fractions on a Number Line

Fractions allow for more precise measurement of quantities , including fractional parts greater than 1 whole.

Close to…

• Name a fraction close to 1 but not more than 1.

• Name a fraction that is even closer to 1 than that.

• Why do you believe it is closer?• Name a fraction that is even closer than

the previous fraction.• Again…

Comparing Fractions

Which is greater? Explain.

3 58 or 8

5 59 or 6

2 3 3 or 4

Which is greater

3/8 or 2/6?

Explain how you know..

1/4 or 1/2

Can 1/4 be larger than 1/2?

How could a student justify their answer to this question?

Another way to compare fractions

Draw two squares of equal size.

Partition the first one in half vertically and shade in half of the square

Partition the second one in thirds vertically and shade in one third of the square.

Now, partition the first square into thirds horizontally and the second square in half horizontally.

Comparing ½ and 1/3

How does this model help build understanding of “finding common denominators?”

How will this model help students add and subtract fractions with unlike denominators?

Fraction Addition and Subtraction

Begin with informal exploration

Juan and Tiana were each eating the same kind of candy bar. Juan had 3/4 of his bar left. Tiana still had 2/3 of her bar left. Who had the most candy left? How do you know?

How much candy did the two children have together?

Using nothing other than simple drawings, how would you solve this problem without using an algorithm and finding common denominators?

Try to think of two different methods.

Using Estimation

Estimate the answer to 12/13 + 7/8 A. 1B. 2C. 19D. 21

• Only 24% of 13 year olds answered correctly

• Equal numbers of students chose the other answers

NAEP

Fractions in Balance Problems

Find the missing values.

Figures that are the same size and shape must have the same value.Adapted from Wheatley and Abshire, Developing Mathematical Fluency, Mathematical Learning, 2002

30

1 ¾ x

n 1 ½ n

1 ¾

n

Addition and Subtraction with Mixed Numbers

• A separate algorithm for adding and subtracting mixed numbers is not necessary.

• Include mixed numbers in all fraction addition and subtraction activities.

• Let students solve these problems in ways that make sense to them.

• Students tend to work with the whole numbers first.

3 ½ + 5 ¾ = ?

9 1/3 – 7 2/3 = ?

Fraction Computation

A problem-based number sense approachBegin with simple tasks in contexts.Connect the meaning of fraction computation

with whole-number computation.Develop strategies using estimation and

informal methods.Use models to explore each of the operations.

Adapted from Van de Walle and Lovin, Teaching Student-Centered Mathematics, 2006

Multiplying Unit Fractions

Understand a fraction a/b as a multiple of 1/b is the product of 5 x ( )

= 5 x

45

45

41

41

Multiplying Unit Fractions

Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number

3 sets of is the same as 6 sets of52

51

Multiple Solution Strategies

Solve word problems involving multiplication of a fraction by a whole number

At your table, solve in 2 ways… If each person at a party will eat of a

pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what 2 whole numbers does your answer lie?

85

Multiplication with Fractions

How would you solve?Carolyn has 14 cookies to share with her three

friends. How many cookies will Carolyn and each of her friends get?

A sharing problemDividing by 4 (Charlotte + three friends) is the same

as multiplication by ¼. Or think of the 14 cookies as the whole. How many

in ¼? Cookies are used because they can be subdivided.

Multiplying Fractions: No Subdividing of Unit Parts

How would you model and solve?

Rita used 1/10 of a bottle of vanilla flavoring for a cookie recipe, leaving 9/10 of the bottle. If she then used 2/3 of what was left in a cake recipe, how much of the whole bottle did she use?

Alexander used 2 ½ tubes of red paint in his picture. Each tube holds 4/5 ounce of paint. How many ounces of red paint did Alexander use?

Explain your reasoning.

Multiplying Fractions: Subdividing Unit Parts

How would you model and solve without an algorithm?

Raj had 2/3 of his bedroom left to paint. After lunch, he painted 3/4 of what was left. How much of the whole room did Raj paint after lunch?

Olivia was sharing a jar of lemonade with her sister. Olivia drank 2/5 of the jar; then her sister drank 2/3 of what was left. How much of the jar of lemonade did her sister drink?

The type of model can impact students’ understanding of their solution.

Modeling the Process

3 35 4

means “3/5 of a set of ¾”Make ¾, then take 3/5 of it.

Why does extending the lines (the dotted part) help?

x

Consider

How many fifths are in two wholes? How would you begin to think about this

question? Create at least two representations to

show your solution

What operation is represented by this problem?

Look for patterns in the next slides.

Connecting What We Know

Consider 1 ÷ ½ = ? To determine how many of the unit

fractions of the Divisor (1/2) are in the Dividend (1), think about it as:

How many one-halves are in 1?

How is this the same as thinking of 36 ÷9 as “How many nines are in 36?”

Words to Symbols

Write an equation for each situation:A grocer has 10 pounds of coffee beans.

If he sells the beans in ½ pound bags, how many will he have to sell?

If you have a spool with 6 feet of ribbon, and you need 1 ½ foot long pieces for a craft project , how many can you make?

Building

Understanding

How many one-sixths are in 2? 2÷⅙ = ?

How many one-halves are in 3? 3÷½ = ?

How many one-fifths are in 2? 2÷⅕ = ?

What patterns do you see?How might these patterns help develop

a method for dividing by fractions?

More Brownies

You have 1/3 of a pan of brownies left after last night’s party. If you and three friends share what is left of the brownies, how much of the whole pan of brownies will each of you get to eat?

Write an equation to solve this problemSolve the problem using models and share

your method with table partners

Making Sense of Fractions

We must go beyond how we were taught and teach how we wish we had been

taught.Miriam Leiva, NCTM Addenda Series, Grade 4, p. iv

Resources:

Marilyn Michue Elementary Math Curriculum Resource marilyns.michue@cms.k12.nc.us 980-343-2792

http://www.smarterbalanced.org

http://www.illustrativemathematics.org

K-5 Math Teaching Resources

http://www.parcconline.org