I started the club with the broad idea of using graph paper to help understand the relation between measures of linear distance and of area. I expect there to be detours along the way: getting there is half the fun, if not more than half! We will use graph paper to make tables and graphs; we will look for patterns and relationships (e.g. among square and triangular numbers); and I will stress some of my favorite themes in math, such as making mistakes, then learning from, checking for, and correcting them.
We started by gathering and organizing some data. I asked the class to draw a sequence of squares, and then a sequence of right (isosceles) triangles with side length 1, 2, 3, 4, and so on. We counted the number of graph-paper squares inside each figure: for the triangles, we counted both the number of complete squares and the total (counting each little triangle as half of a graph-paper square). We then recorded these numbers in a table, something like the following:
size | square | triangle (complete) | triangle (diagonal) | triangle (total) |
---|---|---|---|---|
1 | 1 | 0 | 1 | 0.5 |
2 | 4 | 1 | 2 | 2.0 |
3 | 9 | 3 | 3 | 4.5 |
4 | 16 | 6 | 4 | 8.0 |
5 | 25 | 10 | 5 | 12.5 |
6 | 36 | 15 | 6 | 18.0 |
7 | 49 | 21 | 7 | 24.5 |
8 | 64 | 28 | 8 | 32.0 |
9 | 81 | 36 | 9 | 40.5 |
10 | 100 | 45 | 10 | 50.0 |
Today we did two things, perhaps not in the following order. We made a bar graph using the first two columns of the table above. On a standard sheet of graph paper (8.5 by 11 inches, four squares to the inch) there is room to graph the first six rows. We also looked for patterns in the table; each of us wrote them down in our own words, with examples. Here is my list:
One more thing: we discussed ways of checking our work. For example, suppose I want to check my bar graph. I could count the 36 boxes that make up the last rectangle that fits on my page. There are two problems with this. First, it takes a long time. Second, if I did make a mistake the first time I did this, I am likely to make the same mistake if I count it a second time. A better way to check would be to take out my ruler: 36 boxes should measure 9 inches. Another way to check, if I have already checked the 25 boxes in the rectangle next door, is to start with 25 at the top of that rectangle, and then to count 11 more: 26, 27, ..., 36 to get to the top of the big rectangle.
Today we talked about looking at patterns from different points of view. We first noticed the patterns from the table of numbers. We made a more complicated bar graph, showing the number of boxes in a square, the number of complete boxes in a triangle, and the total number of boxes in a triangle. Then we talked about how to see the patterns in the bar graph. Finally, we went back to the pictures of triangles and squares and tried to see the patterns there. This is the best way to see why the patterns work.
Using the original pictures, it is easy to see why the total number of boxes in a triangle is half the number of boxes in a square: a diagonal line divides the square into two copies of the triangle. Leonard also noticed a new pattern: the number of boxes in a square starts at 0 and then increases by odd numbers: 0 + 1 = 1, 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, and so on. The question came up of how to predict which odd number to add. For example, if you got to 36 by adding 11 to 25, then you know to add 13; but what if you got to 36 by multiplying 6 by 6? We answered this question when we understood how to see the pattern in the picture: start with a 6-by-6 square and extend the sides to get a 7-by-7 square. It is not hard to see that there are 13 additional boxes, and you can think of the 13 as 13 = 2 * 6 + 1 or 13 = 2 * 7 - 1 or (my favorite) 13 = 6 + 7. We then used this pattern to do some mental arithmetic: if you know that 20 squared is 400, can you figure out 21 squared and 19 squared?
If we were to continue in this direction, I would like to point out the relations between the various patterns: