Lesson 4 – Not everything is what it seems

The most obvious pattern didn’t work for those regions in the circle (see Circle Problem page). In that context, what is obvious isn’t true. When we look at optical illusions, sometimes we’re sure one line is longer than another, when in reality it isn’t.

This section is meant to tie together the preceding sections on inductive reasoning (finding patterns) with the following sections on deductive reasoning (proving things). Most math texts treat math as purely deductive, when in reality mathematicians do lots of their thinking in inductive mode, searching for patterns.

  • If we see a pattern, how do we know it’s right? We have to prove it.

  • Why would we want to prove something that seems obvious? Sometimes what’s obvious isn’t what’s true.

I’ll answer R’s questions from the study group here. I think they make a good lead-in:

R: In the 1st edition, set 1, #2, all I see is a set of parallel lines. What is the illusion here?
Sue: Is that group of lines a square formation or rectangular? If it’s rectangular, is it taller, or wider? What do you see at first, and what do you get when you measure?

R’s other questions concerned checking the area of various figures. The book assumes you will check the area by multiplying length times width, since all the figures seem to be squares or rectangles. Jacobs is usually good about not making assumptions, but he goofed here, I guess. (R, what you were trying to do was a reasonable thing to try, since you didn’t know what he expected, but it’s terribly hard and almost unworkable, because of the pieces that are not obvious fractions of a square.)

R: In set II, I had trouble checking that the area is 63 units.
Sue: What is the original area (in the book, using length times width), and what is the shaded area that’s thrown out (again, look at lengths and widths in the diagram in the book)? Can you check it this way?

I want to back up here. Does it make total sense to you that we measure area of a rectangle by measuring length times width? If it doesn’t, use graph paper to draw rectangles of various sizes, count the squares to get the area, and then think about how to count the squares quickly. If you count row by row, what does that look like?

- Notes by Sue VanHattum