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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Fair sharing folktale.
In my writing about fractions, I call situations where some quantity is shared equally among some number of people “equal sharing.” Many other people call these same kinds of situations “fair sharing.” But, as Elizabeth Fennema once pointed out to me, “fair” does not always mean “equal.”
Just a couple of days ago, I ran across a folktale called “A Fair Division” in the October 2003 issue of Highlights for Children. It shows very cleverly why “fair” is not always “equal” and it makes a good math problem too. Here’s my paraphrase of it (or you can view a jpg of the entire text, minus the hints the author gave for figuring it out. Yes, that’s right. I took out the hints!).
Two farmers, Ram and Shyam were eating chapatis. Ram had 3 pieces of the flat, round bread and Shyam had 5. A traveller who looked hungry and tired rode up to the two men. Ram and Shyam decided to share their chapatis with him. The 3 men stacked the 8 chapatis (like pancakes) and cut the stack into 3 equal parts. They shared the pieces equally and ate until nothing was left. The traveller, who was a nobleman, was so grateful that he gave the two farmers 8 gold coins for his share of the food.
After the traveller left, Ram and Shyam wondered how they should share the 8 gold coins. Ram said that there were 8 coins and only 2 people, so each person should get an equal share of 4 coins. “But that’s not fair,” said Shyam, “since I had 5 chapatis to begin with.” Ram could see his point, but he didn’t really want to give 5 of the coins to Shyam. So he suggested they go see Maulvi, who was very wise. Shyam agreed.
Ram and Shyam told the whole story to Maulvi. After thinking for a long time, he said that the fairest way to share the coins was to give Shyam 7 coins and Ram only 1 coin. Both men were surprised. But when they asked Maulvi to explain his reasoning, they were satisfied that it was a fair division of the 8 coins.
What do you think Maulvi’ reasoning could have been? What makes it fair?
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IF you can solve this logic puzzle, THEN…
Here’s a link to a discussion of a logic puzzle, among other things, on none-other-than-RLC’s blog. (He’s my husband; some of the Agents in question are my children.) We all found the logic puzzle fun.
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Try this!
My two sons know that because I am a math educator, I like to give them problems to solve every once in a while. Last year, when they were first and third graders, I drew a picture of a brownie that had been cut with a slice removed and asked them to decide how much had been eaten.
The darkened part shows what part of this brownie Jackie ate. How much of the brownie was eaten?
The son who was in third grade said it was an impossible amount, because the pieces were not all the same size and it therefore could not be “one out of three.” The first grader said the portion was half of a half. How surprising that the third grader, who had been in school longer, gave a non-sensical response!
So I am curious about what other children would do. I have heard other children describe that piece of brownie as “one third,” but I don’t know how widespread this kind of response is. Similarly, I’ve seen other children describe it as half of a half, and I wonder how common this more intuitive response is.
If you’re curious too, give this question to your own students and see how they respond. How many children say the amount is one third? half of a half? one fourth? an impossible amount?
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