Multiplication and Division of Fractions: Thinking More Deeply

Nadine Bezuk and Steve KlassSession 322 CMC-S 2005

Today’s Session Welcome and introductions Meanings for operations

• How do we help children model and reason about the operations

• Multiplication and division with whole numbers• Multiplication and division with fractions

Models for multiplication of fractions• Set, Array, Area, Measurement

Models for division of fractions• Set, Array, Area, Measurement

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 aren’t 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.

Types of Models for Considering Fractions

Area/region• Fraction circles, pattern blocks, paper folding,

geoboards, fraction bars, fraction strips/kits Length/linear

• Number lines, rulers, (fraction bars, fraction strips/kits)

Set/discrete• Chips,counters, painted beans

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?

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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?

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 much does each group get?• How does this work?

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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 that’s left?

13

Dealing With Remainders

11

2÷ 1

3

?

13

13

13

13

1

12 1 0

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 that’s left?

12

13

Models for Reasoning About Division

Area/Measurement (batches of cookies)

Linear/Measurement (ribbon - external units)

How do you deal with the remainder?

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

11

2÷ 1

3

Multiplication of FractionsConsider:

How do you think a child might solve each ofthese?

What kinds of reasoning and/or models might they use to make sense of each of these problems?

23

×34

34

×23

An Eighth Grade Problem?

George has of a pie. He ate of that.How much pie did he eat?

17% of 13-year-olds answered correctly, though 60% could correctly calculate the product .

(NAEP, 1983, p. 26)

34

35

78

×32

Another ContextAt one school 3/4 of all eighth graders went to one game. Two-thirds of those who went to the game traveled by car. What part of all the eighth graders traveled by car to the game?

12% chose to multiply, while about 55% decided to subtract and about 8% to divide! (Sowder, 1988)

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

23

×34

=12

34

×23

=12

Contexts for Multiplication Finding part of a part (a reason why

multiplication doesn’t always make things “bigger”)

Pizza (pepperoni on )

Brownies ( is frosted, of the that part has pecans)

Lawn ( is mowed, of that is raked)

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of34

34

34

23

23

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

23

×34

=12

Questions?

References

National Assessment of Educational Progress. (1983) The third national mathematics assessment: Results, trends and issues. Report No. 13-MA-01. Denver: Education Commission of the States.

Sowder, L. (1988) Concept-driven strategies for solving story problems in mathematics. Final Tech. Rep, Center for Research in Mathematics and Science Education, San Diego State University, San Diego, CA (ERIC Document Reproduction Service No. ED 290 629)

Contact Us

nbezuk@mail.sdsu.edusklass@projects.sdsu.edu

http://pdc.sdsu.edu