Dividing fractions
Consider these two division problems:
1. 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?
2. 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?
Which one do you think is more difficult for children to solve?
To help you decide, 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 #2), 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 #1), 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.
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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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“This one blew our minds.”
Linda Jaslow has been working on fractions in a first-grade classroom using equal sharing problems. She sent this description of one girl’s insight about how to generate equivalent fractions:
We also had an amazing revelation by a first grader on equivalent fractions. The class had done a great deal of playing with the fractions and were pretty able in terms of equivalence. But this first grader blew our minds. The problem was 8 [people] sharing 14 [of something, e.g., pancakes]. So we had the different responses but one little girl said that 14/8 was equivalent to 7/4 because 2/8 was equal to 1/4 so you could double the 7/4 and get 14/8 or you could half 14/8 to get 7/4. It was too close to 7/4 x 2/2 idea to be an accident…. I am not sure how far you would take this with a first grader but it felt like a teachable moment to explore. Curious…would you have just let that big idea slide or done something with it?
I wonder how she arrived at that conclusion. I think she might be reasoning by analogy — a form of reasoning not specific to mathematics — rather than from mathematical principles — such as multiplication by 1 = 2/2. I’m also curious whether she can relate her ideas back to the sharing situation (e.g., a child in a group of 8 people sharing 14 pancakes gets as much pancake as a child in a group of 4 people sharing 7 pancakes) and to what kinds of other fraction relationships she can extend this reasoning. My guess is it would be harder for her to reason about equivalence relationships that involved tripling or “thirding” the numerators and denominators, and so on. But all of this doesn’t make her insight any less fascinating or powerful in the context of learning fractions. And yes, I’d definitely want to explore it!
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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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