As an author of Sketchpad activities, I like to think that I can pose good problems for students to solve. But as I visit elementary classrooms and watch students use Sketchpad, I realize that a large part of the enjoyment they derive from using our software comes from creating their own problems and sharing them with classmates.
Not only do students feel a sense of personal ownership from problem posing, but they also learn a lot of mathematics in the process. Making a problem that is challenging but approachable requires at least as much, if not more, mathematical understanding than solving it. I’m reminded of this when I think about a factoring activity I designed several years ago.
Factor Puzzles is a number game that injects logical reasoning into the study of factors. The Web Sketchpad model below (and here) shows four letters, a, b, c, and d, each of which has been assigned a secret numerical value by the software. Students pick any two letters, drag them to the right across the vertical line, and note their product. By picking other pairs of letters, students learn more information about their hidden values. For example, if students discover that b × d = 30, a × b = 15, a × d = 18, and b × c = 10, they can reason that a = 3, b = 5, c = 2, and d = 6.
Students play the game multiple times, with Sketchpad generating new random values for the four letters. Page 1 of the websketch limits the values to numbers between 1 and 9. Pressing the arrow in the lower-right corner of the model takes you to a second page where the random values lie between 1 and 14.
While students enjoyed solving these challenges, what they really wanted to do was create factor puzzles for each other. As with most of our Sketchpad activities for young learners, we include a “make-your-own” version of the puzzle that allowed students to work in pairs and choose their own values for a, b, c, and d. You’ll find this version on page 3 of the websketch above.
In one third-grade classroom, Zach and Kayla were partners. As Kayla looked away, Zach pondered what values to assign a through d. Not content with the “wimpy” numbers that Sketchpad had chosen, Zach approached the problem with gusto. He entered values likea = 511, b = 296, c = 632, and d = 784. When Zach was done, he hid the numbers and told Kayla, with a devilish grin, that the puzzle was ready to solve.
Kayla, of course, stood no chance.
Dragging a and b across the divider, she discovered that a × b = 151,256. The other products were equally daunting. If nothing else, I thought, Zach and Kayla had learned something valuable about products and factors: While it’s straightforward to multiply two numbers together and obtain a large product, it’s much more difficult to pull large numbers apart into their factors.
I expected Zach and Kayla to call a truce and revert to manageable numbers, but Kayla said she could craft a puzzle with large products that would not be difficult to solve. She asked all of us to turn away while she entered new values for the four letters. With the puzzle ready, Zach dragged pairs of letters across the divider and found that a × b = 60,000, b × c = 80,000, and c × d = 200,000.
Kayla had kept the products large, but cleverly found a way to make the problem tractable. Zach noticed that each product ended with four zeroes, a strong indicator that the values of a through d were multiples of 100. By ignoring the zeroes, Zach was able to reduce the problem to a simpler, related version: a × b = 6, b × c = 8, and c × d = 20. Taking the solution a = 3, b = 2, c = 4, and d = 5, Zach appended two zeroes to each number to obtain the answer to Kayla’s original problem: a = 300, b = 200, c = 400, and d = 500.
Needless to say, Kayla and Zach’s exchange left me impressed. Sketchpad knew nothing about what made for a good or bad puzzle, so it happily allowed the third graders to enter any numbers they wanted for the values of a, b, c, and d. In the process, both Kayla and Zach used their knowledge of small products to reason about much larger products.
As I continue to share our Dynamic Number activities with students, I remain amazed at their ability to take the leap from being problem solvers to problem posers. So watch out, curriculum developers—your students might create better problems than you!