Fall Math Notes I: The Area Table

This fall, I’ve been teaching a math workshop for grade school kids (grades 3-6) in my town. Once a week for an hour and a half, covering the typical topics for kids this age: multiplication, division, place value, fractions. We’ve wrapped up now (last week was the last class), so I wanted to note down a few impressions before I forget it all.

One thing that worked pretty well was identifying multiplication and area. I started one of the first classes with the following exercise: take a grid (say 10 by 10). Pick a box in the grid, count the number of boxes that are above and/or to the left of that box, and write that number in your box. (Alternately, draw a rectangle extending from the top left of the grid to the box you picked, and count the number of boxes in that rectangle.) For example, if you pick the box in the 4th row (counting from the top) and 5th column (counting from the left) of your grid, then the boxes above and to the left are marked in green in the picture below, and there are 20 of them:

You don’t need to do anything more here than count boxes, but of course the point is that 20 boxes = 4 rows × 5 columns and also that 20 is the area of the green rectangle (each 1 by 1 box has an area of 1 square unit, so 20 boxes is 20 square units of area).

I had the kids repeat this for every box in the grid. The number you write down in each box is the product of the corresponding row and column, and you end up with the good old multiplication table. I liked how this worked out for a few reasons:

1. It was a way for the kids to figure out the multiplication table themselves, and get it right. No memorization, and little required in the way of prerequisites or arithmetic skills. (When you’re working with kids with different backgrounds, that’s a very important advantage.) The kids were pretty enthusiastic about doing it.
2. It was a big enough table that counting boxes over and over got tedious, so the kids started to look for shortcuts. They noticed right away that the numbers in each row increase by the index of that row (e.g., 5th row = counting by 5’s). That helped them fill the table out pretty quickly. It also gave us an excuse to talk about why that worked (each time you take a step to the right in the 5th row, you add 5 boxes).
3. When we were done, we didn’t just have a multiplication table, we had good geometric intuition to go with it. Meaning, we knew how to think of the table as a family of overlapping rectangles, and every number in the table as an area! (I almost wanted to call the thing the area table instead of the multiplication table, but decided I shouldn’t saddle the kids with made up terminology.) Then we could find more patterns in the table, and try to explain those patterns geometrically. For example, a 5 × 5 square has one more box than a 6 × 4 rectangle (25 = 24 + 1), a 7 × 7 square has one more box than an 8 × 6 rectangle (49 = 48 + 1), etc., and we could explain that by moving boxes around. For another example, if you just stick to the nested squares, the number of boxes you need to add to each square to get the next square grows in a linear way (4 = 1 + 3, 9 = 4 + 5, 16 = 9 + 7, 25 = 16 + 9, 36 = 25 + 11, and so on), and you can see and count the extra boxes explicitly. There are hints here of algebra (x2 – 1 = (x + 1) (x – 1)) and even calculus (derivative of x2 is 2x), and you can get at them just by counting and moving boxes.

More to come (Division! Fractions! Place value!). You can’t wait, can you? Meanwhile, happy Thanksgiving.