As an opening multiplication table question: add up the 100 numbers that you’d find in a typical classroom-style 10 by 10 multiplication table. How did you do it?
I commonly see the wrong answer of 5050, which students get by adding up the numbers 1 through 100, or by assuming that the general method of “how many numbers, times average of first and last” will work for any collection of numbers.
The successful solvers use some more flexible strategies. I see them summing the first row to get 55, then summing the second row to get 110, and so on. Take the big problem and break it into 10 easier subproblems. Each row is easy to sum because it actually is an arithmetic sequence, so 10 times the average of first and last works pretty quickly for each row, and now you have 10 numbers to sum.
But wait! Those 10 numbers are themselves an arithmetic progression, so they’re easy to add up, too. That should make students wonder if there’s a reason for that! If it doesn’t, that seems like an important problem solving skill to learn — take it seriously when people tell you to reflect on the problem and solution after you get the answer! (I find it ironic that after the Wikipedia article on How to Solve It mentions that the first of Polya’s four phases is neglected, it proceeds to shortchange the the fourth phase. Polya recommends several useful subcategories to help you “look back” productively.) I encourage students to give a guess at the beginning, and then to compare their guess to their final result and think about how they could guess better next time. I also recommend that they imagine the sentence they would like to send back in time to themselves at the start of the problem, as a way of getting them to summarize the key step that they had the hardest time seeing.
At some point through this process, someone eventually figures out that your rows are 1 times the first row, 2 times the first row, 3 times the first row, and so on. The distributive property strikes again!
What really opens up the thinking, though, is when you look at the multiplication table (and the distributive property) visually or geometrically instead of numerically. Avery wrote a short blog post that mentioned this way of drawing the multiplication table after attending our summer math teachers’ circle workshop in Berkeley. My friend David Millar brought his graphic design skills to work and now we have some beautiful versions of this multiplication table that you can use in the classroom!
There’s a lot more problem-solving fun to be had with the multiplication table, though, but we’ll have to save some of those questions for future posts. In the meantime, how does the multiplication table relate to the area and perimeter questions from Dan Meyer’s blog? (Does it relate to this more interesting question, too? Maybe not as strongly.) For some of the future multiplication table posts, you can get a preview with this Julia Robinson Mathematics Festival activity and especially the T-shirt design relating Pascal’s triangle to the multiplication table.