Problem Pairs

I'm happy to share that I will be speaking at the NCTM Annual Meeting this year in a couple of weeks, about a topic that I've been thinking about for a few years now.

	A	B
1	​​Solve for x: 3x = 9	Solve for x: 3x + 1 = 9
2	​​What is the product of 12 × 13 ?	​What is the product of (x + 2)(x + 3)?
3	​Gerald can bike 6 miles in 3 hours. How many miles could he bike in 1 hour at the same rate?	Gerald can bike m miles in h hours. How many miles could he bike in 1 hour at the same rate?
4	Monica is trying to wrap a present in the shape of a box. The box measures 3 inches by 4 inches by 5 inches. How much wrapping paper will she need?	Monica is trying to figure out how many 1 inch cubes she can fit in a box. The box measures 3 inches by 4 inches by 5 inches. How many 1 inch cubes will fit inside it?
5	What is the area of this figure?	What is 2 × 6 + 3 × 3 ?
6	​Randall bought some cartons of milk at the store. If each carton contains the same amount of milk, how much milk did he buy?	Randall bought 3 cartons of milk at the store. If each carton contains 32 ounces of milk, how much milk did he buy?
7	A game controller that is sold to stores for $29.99 is made of materials that cost $1.19 . A worker is paid $8.50 per hour to make game controllers, and can assemble 5 per hour. Write an expression that shows how much profit the company makes from selling 10 game controllers.	A game controller that is sold to stores for $29.99 is made of materials that cost $1.19 . A worker is paid $8.50 per hour to make game controllers, and can assemble 5 per hour. Write an expression that shows how much a worker earns from making 10 game controllers.
8	Melanie has 6 liters of lemonade. She is going to pour it into containers that can each hold 2 liters. How many containers will she need?	​Melanie has 6 liters of lemonade. She is going to pour it into containers that can each hold 1/2 liter. How many containers will she need?

I've thought of these as problem pairs (or perhaps "paired problems"?), and I think they have a lot of potential instructional value. Mostly, the idea is that pairs of problems that are similar-looking can draw a student's attention to the differences between them. Thinking and discussion naturally focus on how the problems are different, which may have a few different purposes:

The problem pair extends student thinking. Sometimes, the first problem in the pair activates students' prior knowledge, while the second problem offers some extension that students have a good shot at figuring out. (See for example #1 or #8 above, which are examples of this.) So often we miss opportunities to activate prior knowledge, because there isn't time; we also sometimes activate at times or in ways that aren't connected to new learning. By pairing a prior skill with a new one, in a way that looks really similar yet is different, I think there's a better chance students will be able to activate and apply what they know. And then the new thing doesn't seem that new at all!

The problem pair draws attention to structure. So much of math is about noticing and taking advantage of structure! These problems may or may not have a clear "order" to them (see example 2 above, which does have an implied order; or 5, which does not). By emphasizing structure, we give students a better chance of seeing the "big picture" in math; we also can help demystify a lot of algebra by helping students see it as generalized arithmetic.

The problem pair draws attention to context. By keeping the numbers the same, and changing the question or context, students are forced to think carefully about what the quantities mean and what mathematics may be necessary to employ. (See 4 above.) In cases like this, there might be an advantage to showing both problems at the same time, which encourages students not to rush to compute, but rather to carefully study each context and make sense of it. This type can also offer opportunities for sociopolitical critique (see 7 above); some examples may also remove numerical values entirely (see 6 above) to focus attention on the context.

Let me know what you think in the comments! And hopefully see you at NCTM. My "Problem Pairs" talk will be on Friday morning, October 27 at 8:00am in the Marriott, Union Station room.