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 Sketchpad number game that injects logical reasoning into the study of factors. The Sketchpad model 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 a vertical line, and Sketchpad displays their product.
In the example below, a student has dragged b and d and learned that b × d is 30. Students can drag letters back and forth across the divider line and display the product of any two letters at a time. By picking various pairs of letters, a student might learn that b × d = 30, a × b = 15, a × d = 18, and b × c = 10 and reason that a = 3, b = 5, c = 2, and d = 6.
A student drags two letters, b and d, across a vertical divider to reveal that b × d equals 30.
Students play the game multiple times, with Sketchpad generating new random values for the four letters. To keep the computations reasonable, I programmed the puzzle so that no letter ever exceeded a value of 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 included 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.
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