Multiplication of Fractions: Thinking More Deeply Steve Klass 48th Annual Fall Conference of the California Mathematics Council - South Palm Springs, CA, Nov. 2, 2007

2 Today’s 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

3 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.

4 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?

5 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 Nadine’s brownie problem?

6 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

7 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.

8 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

9 Models for Reasoning About Multiplication Area/measurement models (e.g. fraction circles) Linear/measurement (e.g paper tape)

10 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

11 Using a Linear Model With Multiplication How much is of ?

12 Using an Area Model with Fraction Circles for Fraction Multiplication How could you use these materials to model

13 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

14 Using a Linear Model With Multiplication How much is of ?

15 Using an Area Model with Fraction Circles for Fraction Multiplication How could you use these materials to model ?

16 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?

17 Mixed Number Multiplication Can you find out what each square is worth? What about partial squares?

18 Making Connections …

19 Try it Yourself How can you use these materials to model ? What contexts can you construct for these two problems?

20 Other Contexts for Multiplication of Fractions 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) Ribbon (you have yd, of the ribbon is used to make a bow)

21 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

22 Questions/Discussion

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