Reconceptualizing Mathematics Part 1 Reasoning About Numbers and

Reconceptualizing Mathematics Part 1: Reasoning About Numbers and Quantities Judith Sowder, Larry Sowder, Susan Nickerson CHAPTER 7 – COMPUTING WITH FRACTIONS 1 © 2010 by W. H. Freeman and Company. All rights reserved.

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No doubt computations involving fractions are easier when the denominators are the same. Under the partwhole view of fractions, having a “common denominator” means that in the case of the fractions involved, that they are all cut into equal sized pieces. And since that is the case, then dealing with the numerator is easy under addition and subtraction because the objects are the same size to begin with. 7 -3

However with different denominators, in order to add or subtract it becomes obvious why it is important to make them the same… we need “pieces” that are the same size in order to make the calculation. Can you figure out what size for the denominator works in the case we looked at earlier (listed below)? 3/8 + 1/6 + 2/15 7 -4

Example: 0, 30, 60, 90, 120…. are all common multiples of 6 and 15. But 30 is the least common multiple. 7 -5

DISCUSSION 7 -6

ACTIVITY 7 -7

DISCUSSION 1. Does commutativity of addition hold for fractions? Explain your answer. 2. Does associativity of addition hold for fractions? Explain your answer. 3. Give examples of story problems that involve fractions or mixed numbers and could be answered by subtracting. 7 -8

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EXAMPLE continued…. 7 -11

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In the previous example, recall that 4/5 refers to the whole lawn, but that the 2/3 referred to the mowed part. We need to be careful in describing what each fraction refers to, or its “referent unit. ” The referent unit for the 4/5 is the whole lawn. But the referent unit for the 2/3 is 4/5. Further, the referent unit for the answer, 8/15, is also the whole lawn. Do you see this clearly? 7 -13

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ACTIVITY 7 -15

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7. 2 7 -17

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We can now generalize and say that 1 ÷ 1/b = b. 7 -19

We’ve already agreed then that there are three onethirds in one. But then there would be twice as many in two: 2 ÷ 1/3 = 6. Activity 7 -20

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EXAMPLE 7 -22

ACTIVITY 7 -23

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Consider the following problem: Kathleen had 3/4 of a gallon of milk. She gave each of her cats 1/12 of a gallon to drink. How many cats got milk? Here we are asking the question of how many twelfths are in three quarters. In this case, the referent unit for both the 3/4 and the 1/12 is one whole gallon. But the referent unit for the answer, nine, is the number of cats. 7 -25

DISCUSSION What are the referent units in this case? 7 -26

ACTIVITY 7 -27

EXAMPLE 7 -28

ACTIVITY 7 -29

DISCUSSION 7 -30

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For the problem presented on the previous slide, a teacher should have written 12 1/6 for the first problem, and then 1/6 12 for the second. Although order does not matter in terms of the answer to the computation, it matters a great deal in terms of conceptualizing the expressions involved in the problem. Also, the first problem can be represented using repeated addition, while the second requires fractional part. For elementary school children who think of multiplication only in terms of repeated addition, this can be a stumbling block that needs discussion. 7 -33

Eighth graders in one fairly large-scale testing were given the following problem to solve: Only 12% of the children choose to multiply, while over 50% choose to subtract, and another 8% choose to divide. Misconceiving this problem was largely attributed to the fact that these children wanted to stay away from multiplication because they said “…it always makes bigger. ” 7 -34

Research shows that many elementary school students, and even some adults, believe that “multiplication makes bigger and division makes smaller. ” This misconception leads to confusion when solving problems. Also problematic in these types of situations is knowing to what quantity the different fractions presented are referring. As we’ve seen, not all fractions in a problem are referring to the “whole. ” 7 -35

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