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- Multiplication and Division of Fractions

Thinking More Deeply - Nadine Bezuk and Steve Klass
- Session W106 GSDMC 2006

Todays Session

- Welcome and introductions
- Meanings for division and multiplication
- How do we help children model and reason about

the operations? - Division and multiplication with whole numbers
- Division and multiplication with fractions
- Models for division of fractions
- Area, linear
- Models for multiplication of fractions
- Area, linear
- Contexts for division of fractions
- Questions

What Students Need to Know Well Before Operating

With Fractions

- Meaning of the denominator (number of equal-sized

pieces into which the whole has been cut) - Meaning of the numerator (how many pieces are

being considered) - The more pieces a whole is divided into, the

smaller the size of the pieces - Fractions arent just between zero and one, they

live between all the numbers on the number line - A fraction can have many different names
- Understand the meanings for operations for whole

numbers.

Solving a Division Problem With Fractions

- How would you solve ?
- How would you solve ?
- How might a fifth or sixth grader solve these

problems and what answers might you expect? - How can pictures or models be used to solve these

problems?

What Does Elliot Know?

- What does Elliot understand?
- What concepts is he struggling with?
- How could we help him understand how to model and

reason about the problem?

What Do Children Need to Know in Order to

Understand Division With Fractions?

What Does Elliot Know?

- What does Elliot understand?
- What concepts is he struggling with?
- How could we help him understand how to model and

reason about the problem?

Reasoning About Division

- Whole number meanings for division
- 6 2 3
- Sharing / partitive
- What does the 2 mean? What does the 3 mean?
- Repeated subtraction / measurement
- Now what does the 2 mean and what does the 3 mean?

Now Consider 6

- What does this mean?
- How can it be modeled?
- What contexts make sense for
- Sharing interpretation
- Repeated subtraction interpretation

Reasoning About Division With Fractions

- Sharing meaning for division
- 1
- One shared by one-third of a group?
- How many in the whole group?
- How does this work?

Reasoning About DivisionWith Fractions

- Repeated subtraction / measurement meaning
- 1
- How many times can one-third be subtracted from

one? - How many one-thirds are contained in one?
- How does this work?
- How might you deal with anything thats left?

Materials for Modeling Division of Fractions

- How would you use these materials to model
- ?
- Paper tape
- Fraction circles
- You could also use
- Pattern blocks
- Fraction Bars / Fraction Strips

Using a Linear Model With a Measurement

Interpretation

How many one-thirds are in one and one-half?

Using an Area Model With a Measurement

Interpretation

- Representation of with fraction

circles.

How Many Thirds?

?

A Context For Division of Fractions

- You have 1 cups of sugar. It takes cup to make

1 batch of cookies. How many batches of cookies

can you make? - How many cups of sugar are left?
- How many batches of cookies could be made with

the sugar thats left?

Multiplication of Fractions

- Consider
- How do you think a child might solve each of

these? - What kinds of reasoning and/or models might they

use to make sense of each of these problems?

Reasoning About Multiplication

- Whole number meanings - U.S. conventions
- 4 x 2 8
- Set - Four groups of two
- Array - Four rows of two columns
- Measurement - Four units by two units
- 2 x 4 8
- Set - Two groups of four
- Array - Two rows of four columns
- Measurement - Two units by four units

Reasoning About Multiplication

- Fraction meanings - U.S. conventions
- Set - Two-thirds of one group of three-fourths
- Array - Two-thirds of a row of three-fourths of

one column - Measurement - Two-thirds of one unit by

three-fourths of one unit - Set - Three-fourths of one group of two-thirds
- Array - Three-fourths rows of two-thirds of one

column - Measurement - Three-fourths of one unit by

two-thirds of one unit

Models for Reasoning About Multiplication

- Area/measurement models (fraction circles)
- Linear/measurement (ribbon)
- Set models (eggs in cartons)

Materials for Modeling Multiplication of Fractions

- How would you use these materials to model

? - Paper tape
- Fraction circles
- You could also use
- Pattern blocks
- Fraction Bars / Fraction Strips
- Paper folding

Using an Area Model with Fraction Circles for

Fraction Multiplication

- How would you use these materials to model

?

Using an Area Model with Fraction Circles for

Fraction Multiplication

- How would you use these materials to model

?

Contexts for Multiplication

- Finding part of a part (a reason why

multiplication doesnt always make things

bigger) - Pizza (pepperoni on )
- Brownies ( is frosted, of the that part has

pecans) - Lawn ( is mowed, of that is raked)

Thinking More Deeply About Multiplication and

Division of Fractions

- Estimating and judging the reasonableness of

answers - Recognizing situations involving multiplication

or division of fractions - Considering and creating other contexts where the

multiplication or division of fractions occurs - Using meaning as a springboard to understand why

invert and multiply works

Questions/Discussion

Contact Usnbezuk_at_mail.sdsu.edusklass_at_projects.s

dsu.eduhttp//pdc.sdsu.edu

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