Let's look at how three upper elementary students approached the following problem:

First, think about the problem and imagine approaches that students with minimal prior experience with division and ratios might take. Then, read the teacher's questions and the students' statements. As you reflect on these conversations, consider the general usefulness of the students' methods in other problem-solving situations.

Teacher: What did you find out about this problem?
Teresa: Group 3 paid a lot.

Teacher: Can you explain what you mean by "paid a lot"?
Teresa: Look at Group 2: 12 and 12 is 24, but 4 and 4 is only 8. It would have been cheaper for Group 3 to buy two 12-packs.

Teacher: So what does that tell us about each can that Group 3 bought?
Teresa: I think their cans cost more.

Teacher: Why is that?
Teresa: They add up to $10, not $8, so each one costs a little more.

Teacher: OK. Evan, what about Group 1's cans? What did you find out?
Evan: First I thought maybe they cost 50 cents each -- but it's too much. I added 50 cents six times and got $3.

Teacher: How can we find out how much each can costs for Group 1?
Evan: Six cans for $1.80 is like a sharing-division problem. Each one has an equal part of the cost, and they should add up to $1.80.

Teacher: Tell us more about what you are thinking. If we had a picture of each of the six cans, what would you do?
Evan: They each need the same price. I could share out the $1.80, I think. I don't know how to start, because there's only one dollar.

Teacher: Shall we try 10 cents each at first? Will that make $1.80?
Evan: No, count by tens; it's only 60 cents. (They continue, trying 20 cents each, then 25 cents, then 30 cents.)

Teacher: Can we use division to check to make sure that 30 cents for each can is correct?
Evan: I think so; since it's a sharing-division problem, we can do $1.80 divided by 6. (Evan writes the long-division problem and decides that $0.30 is the correct written answer.)

Teacher: Who agrees? (The others in the group nod in agreement.) I think you might also divide in your head, use a calculator, or even use manipulatives to divide. Teresa, what could we divide to find out how much each can costs, to find out the unit cost, for Group 2's cans?
Teresa: I think you could divide, like Evan did. Divide the money amount by how many cans.

Teacher: OK. Kelly, can you help us decide whether Group 3 or Group 1 paid more for each can?
Kelly: You can divide the price by how many cans to see how much they paid for one can. For Group 3, it's 10 dollars divided by 24 cans.

Teacher: What's that on the calculator?
Kelly: It's a long number, 0.4166666.

Teacher: Yes, sometimes when you share out -- that is, distribute or partition evenly -- the answer doesn't come out exactly even. With money problems, we usually care about rounding to the nearest cent. Who can underline the part of this decimal that shows the dimes and cents? (Teresa underlines the 4 and 1.)

The numbers behind the cents' place show that there is part of a cent more for each can. I see that this decimal is closer to 42 cents than to 41 cents. We'll learn more about decimals like these sometime soon.

Take a minute to make a final decision about which group paid more for each can of soda. Write a statement about how to find the cost of one can in your journal.

  Reflect on the problem