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- Multiplication of Fractions Thinking More Deeply

Steve Klass

48th Annual Fall Conference of the California

Mathematics Council - South Palm Springs, CA,

Nov. 2, 2007

Todays Session

- Welcome and introductions
- What students need to know well before operations

with fractions - Contexts for multiplication of fractions
- Meanings for multiplication
- Models for multiplication of fractions
- Discussion

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.

A Context for Fraction Multiplication

- Nadine is baking brownies. In her family, some

people like their brownies frosted without

walnuts, others like them frosted with walnuts,

and some just like them plain. - So Nadine frosts 3/4 of her batch of brownies

and puts walnuts on 2/3 of the frosted part. - How much of her batch of brownies has both

frosting and walnuts?

Multiplication of Fractions

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

these? - Do both representations mean exactly the same

thing to children? - What kinds of reasoning and/or models might they

use to make sense of each of these problems? - Which one best represents Nadines brownie

problem?

Reasoning About Multiplication

- Whole number meanings - 2 U.S. textbook

conventions - 4 x 2 8
- Set - Four groups of two
- Area - Four rows of two columns

Reasoning About Multiplication

- 2 x 4 8
- Set - Two groups of four
- Area - Two rows of four columns
- When multiplying, each factor refers to something

different. One factor can tell how many groups

there are and the other, how many in each group.

The end result is the same product, but the

representations are quite different.

Reasoning About Multiplication

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

one whole - Area - Two-thirds of a row of three-fourths of

one column - Set - Three-fourths of a group of two-thirds of

one whole - Area - Three-fourths rows of two-thirds of one

column

Models for Reasoning About Multiplication

- Area/measurement models
- (e.g. fraction circles)
- Linear/measurement (e.g paper tape)

Materials for Modeling Multiplication of Fractions

- How could you use these materials to model

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

Using a Linear Model With Multiplication

Using an Area Model with Fraction Circles for

Fraction Multiplication

- How could you use these materials to model

Materials for Modeling Multiplication of Fractions

- How could you use these materials to model

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

Using a Linear Model With Multiplication

Using an Area Model with Fraction Circles for

Fraction Multiplication

- How could you use these materials to model

?

Mixed Number Multiplication

- Using a ruler and card, draw a rectangle that is

by - inches, and find the total number of

square inches. Find your answer first by

counting, then by multiplying. - Compare your answers, are they the same?

Mixed Number Multiplication

- Can you find out what each square is worth?
- What about partial squares?

Making Connections

Try it Yourself

- How can you use these materials to model

? - What contexts can you construct for these two

problems?

Other Contexts for Multiplication of Fractions

- 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) - Ribbon (you have yd , of the ribbon is used

to make a bow)

Thinking More Deeply About Multiplication of

Fractions

- Estimating and judging the reasonableness of

answers - Recognizing situations involving multiplication

of fractions - Considering, creating and representing contexts

where the multiplication of fractions occurs - Making careful number choices

Questions/Discussion

Contact Ussklass_at_projects.sdsu.eduhttp//pdc.s

dsu.edu

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