fractions: getting the whole picture

Fractions: Getting the

Whole Picture

Fraction Hot Topic WorkshopNovember 1, 2012

Complete “Fractions of Words” sheet

Fraction UnderstandingsWhat misconceptions do

students have about fractions? Why are fractions so difficult for

students to understand?What are the important fraction

concepts that students need to understand?

Fractions Researchers have concluded

that fractions is a complex topic and causes more trouble for elementary and middle school students than any other area of mathematics.

Why do you think this is the case?What makes fractions difficult for

students?

Reasons for Difficulties in Learning

Fractions Material is being taught:

too abstractly too procedurally outside meaningful contexts through rote memorization of procedures more attention on algorithms and less

attention on number sense and reasoning without connections with limited models

EOG Weight Distribution

Fractionscan makeSENSE!

What is a FRACTION?

Visualizing Fractions In your mind, picture three quarters.

What image did you create?

Create a different picture of three quarters.

What image did you create?

Visualizing Fractions What image did you create?

three quarters

Fraction Understandings

What early experiences do students have with fractions? There’s a quarter moon tonight. You can have half of my cookie. It’s a quarter past one. The recipe calls for two-thirds cup of sugar. The dishwasher is less than half full. I earned half a dollar.

Fractions in the Real World

Find examples of fractions in the real world

Illustrate or take pictures of examples

Fraction Scavenger Hunt

Fractions Decimals Percents

Meanings of Fractions

Values Fractions are rational numbers that can be

counted and ordered They can represent parts of a region or a set

or a point on the number line Operators

One can find a fractional part of a value A fraction can represent a division problem

Ratios A comparison of two quantities

Fractions are NUMBERS

Name an amount (quantity), a part of a specified whole

Name a point on a number line An infinite amount of fractions exists

between any two whole numbers Can be counted

Partitioning Partitioning is KEY to understanding and

generalizing concepts related to fractions

Partitioning in Grades 1 & 2First Grade Geometry1.G.3 Partition circles and rectangles into

two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Partitioning in Grades 1 & 2

Second Grade Geometry2.G.2 Partition a rectangle into rows and

columns of same-size squares and count to find the total number of them.

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc, and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Second Grade Ms. Nim gave her students a picture of a

rectangle. Then she asked them to shade in one half of the rectangle. Which one shows one half?

Second Grade Which pictures show one

half of the shape shaded?

Partitioning in Grade 3

Third Grade Geometry3.G.2 Partition shapes into parts with equal

areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as ¼ of the area of the shape.

Relationship of a Fraction to its Whole

Fractions are defined in relation to a whole Need to understand what the fraction is “of”

The whole can be One object

A collection of multiple objects

A quantity

Models & Representations

• Area/Region Models

• Linear/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

From Models to Symbols Top Number (numerator)

The counting number. It tells how many shares or parts of a certain size are being counted. Numerator -- Latin word meaning number

Bottom Number (denominator) Tells fractional part being counted If a 4, counting fourths. If a 6, counting sixths… The number of equal parts into which the whole is

partitioned; parts or shares of the wholeDenominator -- Latin word meaning namer

How many of what type of parts

how many

Unit Fraction The amount formed by 1 of the parts when

a whole is divided into b equal parts; 1/b of a whole

Two Fifths What does the

denominator represent? 1 whole object is split

into 5 equal parts Each part is ⅕ of 1

whole object

What does the numerator represent? 2 parts of 1 whole

object, where the size of each part is ⅕

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

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

Unit FractionsThe 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

Unit Fractions Decompose the following fractions in as

many ways as you can

Unit Fraction Counting

Fractional Parts Counting Display pie-piece – tell what fraction this

represents of the whole and count as a class

Example: each pie-piece is one-third

What is another way we can say eight-thirds?

Problem SolvingSome girls were sharing some

bananas so that each person got the same amount. Each girl got ¼ of a banana. How many bananas and how many girls could there have been?

How would students solve this problem? What models/representations would they

Fractions Third Grade

Halves, thirds, fourths, sixths, eighths Fourth Grade

Halves, thirds, fourths, fifths, sixths, eighths, tenths, twelfths, hundredths

Fifth Grade No limits specified

Naming Fractions Students need many opportunities to model

and name non-unit fractions This process begins with regional

representations and is extended to include number lines

The advantage of regional representations is their familiar contexts (pizza slices, sections of apples, cake etc.) for students.

Area/Region Models Region is cut into smaller parts Examples: pattern blocks, grid

paper, geoboards, circles, rectangles, triangles

REGIONS as Models for Fractions

Identify the Thirds Which of these regions is divided into

thirds? Why or why not?

Identify the Fourths

Area ModelsGeoboard Models

Divide the geoboard into half Make halves on the geoboard and record all

of your ideas on dot paperHow do you know the two parts are halves?How do you know the two parts are equal?

Repeat with fourths and eighthsHow will the size of each piece change? Extension: Combine fractions to create a design

Fractional Parts of a Whole (Region)

How many ways can you show fourths on a geoboard?

Fractional Parts of a Whole (Region) Does region show fourths of the square? How do you know?

Fractions and Geoboards

Videos:Annenberg Learner - Learner Express

http://www.learner.org/vod/vod_window.html?pid=905http://www.learner.org/series/modules/express/pages/ccmathmod_07.html

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.

Name the unit fractions that equal one whole hexagon

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

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

Caution

Be sure to identify the whole or whole unit. The unit is not consistent and may change.

Want half of a candy bar?

Identifying Fractional Parts

What part is red? Blue? Green? Yellow?

Area Models Pattern Blocks

Extensions: Create a pattern block design. Assign one

block the value of 1 and find the value of your entire picture.

Change the whole. How does this change the value of your design?

Challenge students to create a design with a predetermined value. (Ex. – If the hexagon is 1, create a design with a value of 24 1/3.)

Create the whole knowing a part…

If the blue rhombus is ¼, build the whole.

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

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.

Fraction Area

Name each piece as a fraction.

What are the mathematical understandings involved in solving this problem?

Mrs. Frances drew a picture on the board.

When she asked her students what fraction it represents. Emily said that the picture represents two-sixths. Raj said that the picture represents two-thirds. Alejandra said that the picture represents 2.

www.illustrativemathematics.org

Illustrative Mathematicshttp://illustrativemathematics.org/

Caution

Be careful about using limited models with students. It is important to have students model fractions with manipulatives but also draw a fraction representation.

Linear Models Linear region is cut into smaller

parts; lengths are compared Examples: fraction tiles, paper

strips, cuisenaire rods, number lines, rulers

0 1 2 3 4

Number Lines

Fractions on a Number Line

Develop understanding of fractions as numbers.3.NF.2 Understand a fraction as a number on the

number line; represent fractions on a number line diagram.a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

Develop understanding of fractions as numbers.3.NF.2 Understand a fraction as a number on the

number line; represent fractions on a number line diagram.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

On a line segmented into fourths, show that three 1/4’s equals 3/4

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)

Number Line – A Linear Model Differs from other models

An identified length represents the unit Can represent iteration of the unit Can simultaneously illustrate subdivisions of

all iterated units

There is no visual separation between consecutive units

Model is continuous

Unit Fractions on a Number Line

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

Number lines show relative magnitude of fractions.

Where would you place 1/3?

0 a b c d e 1

What fraction would be at point a? How far apart are a and b?

Placing Fractions on a Number Line

In turn, place your fraction on the number line 0 1 2

Explain your reasoning for placement

How can a Number Line be a useful tool for making sense of fractions?

Parallel number lines support students in identifying equivalent fractions

Equivalence – Third Grade

Develop understanding of fractions as numbers.3.NF.3 Explain equivalence of fractions in special cases, and

compare fractions by reasoning about their size.a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., ½ = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Equivalence – Fourth Grade

Extend understanding of fraction equivalence and ordering.

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Equivalent FractionsThe Concept:

Two fractions are equivalent if they are representations for the same amount or quantity – if they are the same number.

The Method:Students should EXPLORE equivalent fractions rather than depending on procedures, techniques or algorithms.

Intuitive methods are always best at first.Van de Walle and Lovin, Teaching Student-Centered Mathematics, Grades 3-5

Equivalent Fractions

Explanation 1 Divide the 4 equal parts into 5 small, equal

parts (20); 3 parts subdivided into 5 equal parts becomes 15 parts out of 20.

Every fraction equal to infinitely many other fractions.

Compare the numbers and the size of the parts.

Explanation 2

Every fraction equal to infinitely many other fractions.

Multiply A/B by 1 in the form of N/N

What algebraic property does this represent?

NCTM Illuminations

http://illuminations.nctm.org/

Simplest Form What fractions are in simplest form? How do you know?

Simplest Form There is no whole

number other than 1 that divides both the numerator and the denominator evenly

Comparing – Third Grade

Develop understanding of fractions as numbers.3.NF.3 Explain equivalence of fractions in special

cases, and compare fractions by reasoning about their size.d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Comparing – Fourth Grade

Extend understanding of fraction equivalence and ordering.

4.NF.1 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Comparing Fractions Using the fraction cards, sort the fractions using

the benchmarks 0, 1/2, and 1

0 ½ 1

Common Student Errors

• Students see fractions as two separate whole numbers 2/3 and 3/5 = 5/8

Students think 3/8 is bigger than 2/5 because 8 is greater than 5

Using region models, students often count shaded pieces as whole numbers

Students may see 3/6 and 1/2 as equivalent but they do not understand that 3/6 and 1/2 are identical numbers

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 Game: Getting Closer to 1

First player names a fraction that is close to one (but does not go over one)

Next player names a fraction that is even closer to one and explains why they think it is closer

Continue for several rounds trying to get closer to 1

Variations:Try naming fractions closer to 0 or 1/2

Comparing Fractions Comparing Fractions Worksheet

Compare each set of fractions. Decide which fraction is greater without using algorithms, cross multiplying, or common denominators.

Explain your reasoning.

What strategies did you use to solve each problem and figure out which fraction was greater?

Which fraction is larger?

Comparing FractionsStrategies: Like denominators – When denominators are the

same, the fraction with the bigger numerator is more because there are more of the same-sized parts.

Like numerators – When numerators are the same, the fraction with the smaller denominator is more because the size of the parts is larger.

More or less than 0, 1/2, 1 – Fractions can be compared by determining whether they are more than or less than a benchmark number.

Distance from 0, 1/2, 1 – Fractions can be compared by finding the distance from a benchmark number and then comparing that distance.

Let’s Try It

Caution Be cautious of

teaching rules or algorithms for comparing two fractions. They require no thought about the size of the fractions. Students must develop number sense about fraction size.

Fraction Games Close to 1

Take turns rolling a number cube Decide where you will place the number

(numerator, denominator, or Throw Away box) Object: Try to get the fraction closest to 1

Throw Away

Close to 1Throw Away

Throw Away

Partner 1

Partner 2

Fraction Games Close to 1

Variations: Roll 2 number cubes and add the numbers

together before placing them in the boxes Roll 2 number cubes and multiply the numbers

together before placing them in the boxes Change the target number to 0 or ½

Throw Away

Caution A fraction tells us only

about the relationship between the part and the whole. It does not say anything about the size of the whole or the size of the parts. Comparisons with any model can be made only if both fractions are parts of the same whole.

Set Models Describe your table group with 3 – 4

fraction statements One-half of the group is male One-half of the group is wearing a hat

Set Models Whole is the set of objects and

subsets of the whole make up fractional parts

Examples: counters, teddy bear counters, candy, people

Set Model

Fractional Parts of a Whole (Set)

Take 24 chips.Divide them into thirds if you can.

How many in one third? two thirds?

Divide them into fourths if you can. How many in one fourth? three fourths?

Divide them into fifths if you can. Why can’t you divide them into fifths?

Into what other fractional parts can 24 be divided?

Fractional Parts of a Whole (Set)

Caution: When partitioning sets, children frequently confuse the number of counters in a share with the number of shares. Example: Say: Divide 12 counters into fourths:

(The child correctly makes four equal groups.)

Say: Show me three fourths of 12.

(Some children who correctly divided the set into four equal groups above will now regroup the 12 chips into three groups of

four.)

If you know a fractional part, can you make the

whole?

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

OPERATIONS WITH FRACTIONS

Adding/Subtracting Build fractions from unit fractions by applying and

extending previous understandings of operations on whole numbers.

4.NF.3 Understand a fraction a/b with a>1 as a sum of fractions 1/b.a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8, 2 1/8 = 1 + 1 + 1/8 or 8/8 + 8/8 + 1/8.

Adding/Subtracting Build fractions from unit fractions by applying and

extending previous understandings of operations on whole numbers.

4.NF.3 Understand a fraction a/b with a>1 as a sum of fractions 1/b.c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problems.

Addition and Subtraction• If students have a solid understanding of fraction

concepts, they should be able to add and subtract fractions with like denominators right away.

(e.g., 2/5 + 4/5, or 4 7/8 – 2 3/8)

• Understanding that the numerator is the count and the denominator is what is counted makes adding and subtracting like fractions the same process as adding whole numbers. Two fifths + four fifths is the same problem as adding two apples and four apples.

• If children have problems with adding and subtracting like fractions, they need further work with fraction concept development before moving to adding and subtracting unlike denominators.

Adding/Subtracting Use equivalent fractions as a strategy to add and

subtract fractions.5.NF.1 Add and subtract fractions with unlike denominators

(including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or different for fractions with like denominators.

5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Adding & Subtracting Fractions Begin with simple contextual tasks.

Jack and Jill ordered two identical-sized pizzas, one cheese and one pepperoni. Jack ate 4/6 of a pizza and Jill ate 1/3 of a pizza. How much pizza did they eat together?

Explore with models. Connect the meaning of fraction computation

with whole-number computation. Have students estimate answers and check

for reasonableness of solutions. Develop strategies using estimation and

informal methods.

Addition and SubtractionBegin with informal exploration

Paul and his brother were eating the same kind of candy bar. Paul had 3/4 of his candy bar. His brother still had 7/8 of a candy bar. How much candy did the two boys 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.

Addition and SubtractionMarie and Glen ordered two identical-sized

pizzas, one pepperoni and one veggie. Marie ate 5/6 of a pizza and Glen ate ½ of a pizza. How much pizza did they eat altogether?

What model would you use for the whole? How would you solve this problem without a

traditional algorithm?

How could you solve these problems without finding a common denominator?

3 1 1 1 2 1

4 8 2 8 3 2

1 3 2 3

2 4 3 4

1 1+ +

Fraction Addition

Addition and Subtraction

Build on informal explorations and invented strategies to develop a method for addition and subtraction

Use estimation strategies

These approaches help students see that the common-denominator approach – finding a common “family” – is meaningful

Consider:2 5

4 8Use models.Key question: “How can we change this to a

problem with the parts the same? (like “adding apples and apples”). In this case fourths can be changed to eighths.

Main idea: 2/4 + 5/8 is the same problem as 4/8 + 5/8

Consider:2 1

Focus attention on rewriting the problem in a form that is “like adding apples and apples” so that the parts of both the fractions are the same.

The new form is the same problem as the old form.

Demonstrate with models. (CD Fractions)

Common Denominator

As students work with modeling and rewriting problems to make them easy, they will come to understand that the process of getting a common denominator is actually one of finding a way to change the statement of the problem without changing the problem itself.

From Van de Walle and Lovin, Teaching Student-Centered Mathematics, Grades 3-5

Using Visual ModelsFraction Tracks uses the number line as a visual

model.

http://illuminations.nctm.org

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

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

1 ¾ x

n 1 ½

Multiplying Fractions

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. In general, n x (a/b) = (n x a)/b.)

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.

Multiplying Unit Fractions

Understand a fraction a/b as a multiple of 1/b

is the product of 5 x ( )

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 of5

Multiple Solution StrategiesSolve 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 two whole numbers does your answer lie?

Multiplication with Fractions

Laura had $240. She spent 5/8 of it. How money did she have left?

Eli had 18 cars in his toy car collection. Two-thirds of the cars are blue. How many blue cars does Eli have?

Solve each problem. How are these problems different from the “roast beef” problem?

Multiplying/Dividing Apply and extend previous understandings of

multiplication and division to multiply and divide fractions.

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret ¾ as the result of dividing 3 by 4, noting that ¾ multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share the size ¾. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get?

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd).

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

5.NF.7b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

The meaning of each operation on fractions is the same as the meaning for the operations on whole numbers

For multiplication of fractions, it is useful to recall that the denominator is a divisor. This will allow students to find parts of the other factor.

For division, it is useful to think of the operation as partition and measurement

Fraction Operations

Ramseur Elementary School asked the fifth grade students to help the art teacher design some tile murals for the new art room. The first mural is going to have ¾ of the design as red tiles and ½ of those will have flowers on them. How many tiles will have flowers on them?

How would you model and solve this problem?

Multiplying FractionsHow would you model and solve without an

algorithm?

Raj had 2/3 of his bedroom left to paint. After lunch, he painted 4/5 of what was left. How much of the whole room did Raj paint after lunch?

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

Multiplication of Fractions

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Olivia was sharing a pitcher of lemonade with her sister. Olivia drank 3/5 of the pitcher; then her sister drank

3/4 of what was left. How much of the pitcher of lemonade did her sister drink?

Modeling the Process3 3

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?

Multiplying Fractions You have 6/8 of a pizza left. If you give 1/6 of

the leftover pizza to a friend, how much of a whole pizza will be left for your brother?

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?

Challenges for Students…

Research shows that young students have little trouble multiplying fractions; they can easily multiply the top numbers and bottoms numbers, but they have great difficulty interpreting the meaning of the solution

They interpret whole number multiplication as repeated addition, but have no way to interpret multiplication of fractions

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?

Fractions and Brownies You bake two pans of brownies. If 1/8 of a

pan equals 1 serving, how many servings did you make? Justify your response.

Fractions and Brownies What are the big understandings that are

useful in being able to solve this problem?2÷1/8=16

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?

Words to Symbols 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?

Is it easier for you to think about coffee beans, lengths of ribbon, or symbols?

Context gives meaning to the symbols for numbers, operations, and their relationships

Students need the ability to

decontextualize and contextualize

Importance of Sense-Making

Knowing THAT an algorithm works is not the same thing as knowing WHY an algorithm works (or does not work)

Research states that context gives meaning to the symbols for numbers, operations, and their relationships

Division of Fractions

5 ÷ ⅓ = ?

04/20/23 • page 155

5 ÷ ⅓ =

04/20/23 • page 156

5 ÷ ⅓ = 1 2 3

1311 1210 14 15

4 5 6 7 8 9

During the carnival, Ms. Garcia notices that there 5 bags of balloons. She wants to give 1⁄2 a bag of balloons to some of the volunteers. How many volunteers can she give 1⁄2 a bag to?

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 BrowniesYou have 1/3 of a pan of brownies left after last night’s party. If you and four 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 problem Solve the problem using models and share

your method with table partners

⅓ ÷ 5 =

How is 1/3 ÷ 5 different?

• Use the relationship between multiplication and division to explain that (1/3) ÷ 5 = 1/15 because (1/15) × 5 = 1/3.

• Create a story context.04/20/23 • page 162

During the carnival, Sheila found ½ of a pizza. She wants to share it with her four friends. How much of a whole pizza does each friend get?

The popcorn booth serves tubs that weigh ¾ of a pound. I have a box that weighs 6 pounds. How many tubs can I make from the box?

The face-painting booth has ¼ pint of paint. There are five fifth graders who want their faces painted. How much paint should we use for each child?

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

Fraction Hot Topic Resources Fourth & Fifth Grade Teachers

Lessons for Extending Fractions by Marilyn Burns

Extending Children’s Mathematics by Empson & Levi

Fourth Grade Lessons for Introducing Fractions by

Marilyn Burns Fifth Grade

Lessons for Multiplying and Dividing Fractions by Marilyn Burns

fractions: getting the whole picture

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