Modeling Multiplication and Division of Fractions

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So what’s new about fractions in Grades 6-8?

SOL 6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

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Thinking About Multiplication

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The expression…

We read it… It means… It looks like…

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Thinking About Multiplication

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The expression…

We read it… It means… It looks like…

2 times 3two groups of three

2 timestwo groups of one-third

times one-half group of one-third

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Making sense of multiplication of fractions using paper folding and area models

• Enhanced Scope and Sequence, 2004, pages 22 - 24

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The Importance of Context

• Builds meaning for operations• Develops understanding of and helps

illustrate the relationships among operations• Allows for a variety of approaches to solving

a problem

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Contexts for Modeling Multiplication of Fractions

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The Andersons had pizza for dinner, and there was one-half of a pizza left over. Their three boys each ate one-third of the leftovers for a late night snack.

How much of the original pizza did each boy get for snack?

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One-third of one-half of a pizza is equal to one-sixth of a pizza.

Which meaning of multiplication does this model fit?

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• Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race.

• What portion of the race had Andrea run when she stopped for water?

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Another Context for Multiplication of Fractions

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Students need experiences with problems that lend themselves to a linear model.

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Another Context for Multiplication of Fractions

• Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers.

• What fraction of the 24 stickers did she use on the perfect test papers?

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• Problems involving discrete items • may be represented with set models.

What meaning(s) of multiplication does this model fit?

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One-third of one-half of the 24 stickers is of the 24 stickers.61

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What’s the relationship between multiplying and dividing?

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Multiplication and division are inverse relationsOne operation undoes the otherDivision by a number yields the same result as multiplication by its reciprocal (inverse). For example:

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Meanings of Division

For 20 ÷ 5 = 4… Divvy Up (Partitive): “Sally has 20 cookies. How many cookies can she give to each of her five friends, if she gives each friend the same number of cookies?

- Known number of groups, unknown group size Measure Out (Quotitive): “Sally has 20 minutes left on her cell phone plan this month. How many more 5-minute calls can she make this month?

- Known group size, unknown number of groups

Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

Sometimes, Always, Never?

• When we multiply, the product is larger than the number we start with.

• When we divide, the quotient is smaller than the number we start with.

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“I thought times makes it bigger...”

When moving beyond whole numbers to situations involving fractions and mixed numbers as factors, divisors, and dividends, students can easily become confused. Helping them match problems to everyday situations can help them better understand what it means to multiply and divide with fractions. However, repeated addition and array meanings of multiplication, as well as a divvy up meaning of division, no longer make as much sense as they did when describing whole number operations.

Using a Groups-Of interpretation of multiplication and a Measure Out interpretation of division can help:

Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

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“Groups of” and “Measure Out”1/4 x 8: “I have one-fourth of a box of 8 doughnuts.” 8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are there?” 1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what fraction of a gallon of gas do I have in my tank?” 1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans do I have?” 3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½ laps? 3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be easier to describe using divvy up. 2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?” 3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a pie?” 2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you make if you have 2½ cups of oil left?”

Thinking About Division

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The expression…

We read it… It means… It looks like…

20 ÷ 5

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Thinking About Division

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The expression…

We read it… It means… It looks like…

20 ÷ 5 20 divided by 5

20 divided into groups of 5;

20 divided into 5 equal groups…

How many 5’s are in 20?

20 divided by

20 divided into groups of …

How many ’s are in 20?

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Thinking About Division

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The expression… We read it… It means… It looks like…

one-half divided by one-third

divided into groups of …

How many ’s are in ?

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Is the quotient more than one or less than one? How do you know?

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Contexts for Division of Fractions

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The Andersons had half of a pizza left after dinner. Their son’s typical serving size is pizza. How many of these servings will he eat if he finishes the pizza?

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pizza divided into pizza servings = 1 servings21

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serving

1 serving

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Another Context for Division of Fractions

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Marcy is baking brownies. Her recipe calls for cup cocoa for each batch of brownies. Once she gets started, Marcy realizes she only has cup cocoa. If Marcy uses all of the cocoa, how many batches of brownies can she bake?

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1 cup

cup

0 cups

21 1 batches2

1

One batch (or cup)31

Two batches (or cup)32

Three batches (or cup)33

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Another Context for Division of Fractions

Mrs. Smith had of a sheet cake left over after her party. She decides to divide the rest of the cake into portions that equal of the original cake.

How many cake portions can Mrs. Smith make from her left-over cake?

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What could it look like?

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What does it look like numerically?

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What is the role of common denominators in dividing fractions?

Ensures division of the same size units Assist with the description of parts of the whole

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What about the traditional algorithm?

•If the traditional “invert and multiply” algorithm is taught, it is important that students have the opportunity to consider why it works.•Representations of a pictorial nature provide a visual for finding the reciprocal amount in a given situation.•The common denominator method is a different, valid algorithm. Again, it is important that students have the opportunity to consider why it works.

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What about the traditional algorithm?

Build understanding: Think about 20 ÷ . How many one-half’s are in 20? How many one-half’s are in each of the 20 individual wholes?

Experiences with fraction divisors having a numerator of one illustrate the fact that within each unit, the divisor can be taken out the reciprocal number of times.

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What about the traditional algorithm?

Later, think about divisors with numerators > 1. Think about 1 ÷ . How many times could we take from 1? We can take it out once, and we’d have left. We

could only take half of another from the remaining

portion. That’s a total of .

In each unit, there are sets of .

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Multiple Representations

Instructional programs from pre-k through grade 12 should enable all students to –•Create and use representations to organize, record and communicate mathematical ideas;•Select, apply, and translate among mathematical representations to solve problems;•Use representations to model and interpret physical, social, and mathematical phenomena.

from Principles and Standards for School Mathematics (NCTM, 2000), p. 67.32

Using multiple representations to express understanding

Given problem

Check your solution

Contextual situation

Solve numerically Solve graphically

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Using multiple representations

to express understanding of

division of fractions