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Multiplication and Division of Fractions: Thinking More Deeply Nadine Bezuk and Steve Klass Session 322 CMC-S 2005

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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

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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.

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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

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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?

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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?

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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 Division With 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?

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Dealing With Remainders ?

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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?

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Models for Reasoning About Division Area/Measurement (batches of cookies) Linear/Measurement (ribbon - external units) How do you deal with the remainder?

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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

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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?

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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)

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Another Context At 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)

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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

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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

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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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Models for Reasoning About Multiplication Area/measurement models (fraction circles) Linear/measurement (ribbon) Set models (eggs in cartons)

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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

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Questions?

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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 )

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