Investigating fractions operations in Palm Springs.
These problems and teachers’ investigations of them were part of my presentation at the California Mathematics Council’s recent conference in Palm Springs.
1. Jason ate 2/3 of an ice-cream sandwich. He let the rest melt. But he was still hungry, so he ate 5/6 of another ice-cream sandwich. He let the rest melt. Jason’s brother told him he had eaten a lot. Jason didn’t think so. Did Jason eat more or less than 1 whole ice-cream sandwich altogether? How much did Jason eat?
2. Lupita has 4 ice-cream sandwiches. She ate 2/3 of them. How many ice-cream sandwiches did she eat?
3. At a birthday party, 2/3 of a watermelon is left on the table. There are 4 children at the party who want to share this left-over watermelon. They all want the same amount and they want to finish it off. How much can each child have?
4. Okhee has a snowcone machine. It takes 2/3 of a cup of ice to make a snowcone. How many snowcones can Okhee make with 4 cups of ice?
Teachers gave these problems to their students and brought several student-invented solutions to the conference for discussion. We decided that problems #1 (ice cream sandwiches, addition) and #3 (watermelon, division) were harder than problems #2 (ice cream sandwiches, multiplication and #4 (snowcones, division). Some people might find this observation surprising, since #1 is an addition problem. And isn’t it interesting that two division problems vary so much in difficulty?
Why is adding unlike fractions harder than multiplying and dividing fractions?
Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder. It requires knowledge of fraction equivalencies. To add two fractions, you have to know that they must be thought of in terms of like units. We take this for granted when we add whole numbers: 3 + 5 is really 3 ones + 5 ones — but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths.
Why is one division problem hard and the other easy?
Problems #3 (watermelon) and #4 (snowcones) are both division problems, but differ in difficulty. To illustrate, consider one fifth grader’s strategies for each and how the strategies are related to the problem structure.
To figure out how many snowcones Okhee could make (problem #4), he started with the fact that 2/3 of a cup of ice could make 1 snowcone and figured out how many two-thirds-cups were in 4 cups, the total amount of ice Ohkee had.
He added 2/3 twice and got 1 1/3. He then remembered (from a previous problem) that there were 3 one-and-one-thirds in 4, which meant that there were 6 two-thirds in 4. He concluded Okhee could make 6 snowcones. Besides the last bit of relational thinking which is not integral to the approach he took, he treated the division problem primarily in terms of addition of like units (here, thirds). To figure out how much watermelon each of the 4 children would get if they shared 2/3 of a watermelon equally (problem #3), he drew a circle, divided into thirds. Then he blacked out one of the thirds to show the part of the watermelon already gone. Next he decided to cut each of the remaining thirds into 4 parts to share with the 4 children. He knew this would make 8 parts altogether and that each child would get a total of 2 of those parts, so he marked those off:
He confidently reported the answer as “2/8 of 2/3.” (Decide for yourself: Does this make sense?) Yes, but … we really want to know how much of the whole watermelon that is, don’t we? He quickly drew in the missing third divided into 4 parts:
and reinterpreted his answer in terms of the whole watermelon. He concluded 2/8 of 2/3 was “2/12.” See how his unit of reference for the whole shifted, from 2/3 of a watermelon to the entire watermelon? Similarly, the same piece was first interpreted as an eighth of 2/3 then as a twelfth of the whole. The ability to describe a part and its relationship to different unit wholes is a hallmark of more advanced fraction thinking. So although sharing a quantity among 4 children is a fairly straigthforward kind of problem, in the context of part of parts of wholes, interpreting the resulting share for each child as a fractional quantity is not.
Handout
Children’s thinking about fraction operations. Here you’ll find the four problems from above, with a set of progressively more difficult number combinations to insert into each.
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