Background: I’m wrapping up a unit on modeling with quantities. While proportional reasoning isn’t in the 8th grade standards, my students struggle with it, and I wanted to spend a few days working with units, conversions, and proportional relationships, then introduce scientific notation and do similar problems with very big and very small numbers. Overall, I’m pretty happy with it, and I learned some great stuff along the way.
One topic my students continue to struggle with is deciding which operation to use in a word problem. It’s a classic challenge of middle school math, and there are no easy answers. For too many students, word problems mean “pick out the numbers, look for a key word, if you can’t find one guess the operation and move on as fast as you can”.
Anyway, on Tuesday we were working on solving problems using scientific notation. I found this great task from Illustrative Mathematics: If you stacked all the pennies minted in the last 100 years, how high would that stack be?
I think it’s a great task, but it’s pretty ambitious for where I wanted to get. Instead, I narrowed the focus, and took out all the numerical information.
First question: If I took all of the pennies that the US has created in the last 100 years and stacked them, how high would the stack be? I had students talk to their neighbor, then they shared out answers. Got a range of answers, from the Empire State Building, to the sun, to a bunch of kids who pointed out that it was likely impossible. I wrote these on the board.
Next question: What information do you need to answer this question? I had students write first, then share with a partner. Each class settled on “how many pennies have been created” and “how thick is a penny”. Wrote those two on the board.
Next (and, I think, most important) question: What are you going to do with these two numbers? Gave students some think time, then they told a partner. Each class was unanimous: multiply. The engagement in the room is the key part here. My usual approach is to leave the numerical information in the problem, read it as a class, then ask how to approach it. But then, a bunch of the high kids are already solving ahead of the rest of the class and don’t feel like contributing to the discussion, and deciding on an operation feels like a chore. Doing it in the abstract prevented anyone from rushing ahead, and made deciding what to do a meaningful step on the way to a solution.
Next, give the information, with a slight twist: There have been 500 billion pennies created, and each is about 1.6 x 10^-3 meters thick. Students solve. Share student approaches, comparing students who used scientific notation to students who multiplied in other ways, discuss which method students prefer. Whole thing takes 5-10 minutes, and creates some great student thinking about a meaningful problem.
Anyway, to generalize a broad approach out of this:
Part 1: Choose a meaningful problem that exists in a world that students can estimate and make sense of.
Part 2: Withhold numerical information until students have named what they need, and what they will do once they get it.
Part 3: Give students the information, and let them solve. Consider giving key information with a twist.
Part 4: Compare and make connections between student approaches.
Nothing revolutionary here, but I overlooked for awhile that, while big Three-Act Tasks like everything Dan Meyer and Andrew Stadel have created are awesome, they take lots of time. The basic structure of a 3-Act has a lot of value in helping students create a robust mental model for what mathematical tools represent, and cutting down on the time it takes to complete a task gives students more practice and more different perspectives to build their knowledge. This has a lot to do with what Dan Meyer has been writing recently on developing the question. Take a bit longer to get there, but create a meaningful mathematical experience along the way.
Thinking about the Common Core, the standards name three types of rigor: conceptual understanding, procedural skill and fluency, and application. I think this type of task is critical in developing skill in application. Multiplication is not a set of facts to be memorized. It is rectangular area, an array, repeated addition, scaling, stretching, conversions, proportions, and more, all at the same time. In order for students to understand all of those ideas, they need to 1) care about the problem, 2) think deeply about how to solve it, 3) practice, and 4) have an opportunity to make connections between different approaches to an idea.
That’s my insight for the day. It seems small in hindsight, but I think it will make a big difference as I make it a part of our class routine.