These pool tables remind me of the book Math Curse, where Mrs. Fibonacci (the main character's math teacher) says, "You know, you can think of almost everything as a math problem." Jacobs made the most interesting math problems out of pool tables...

If you're not done, you probably don't want to read this, as it contains answers. If people want, I could put anything like this in the file section in future, just so no one sees it before they're ready. Let me know.

This is not detailed answers. Bob has posted his detailed answers in the files section.

Here's what I got out of the first two lessons:

As we drew the tables, we were gathering data, and eventually, with Jacobs' help, we could see that, if you reduced the ratio of width to length, odd width to even height ended up in the top left, even width to odd height ended up bottom right (same shapes, length and width reversed), and odd width to odd height ended up top right. We used inductive reasoning for this. What he starts explaining in lesson 3 is that inductive reasoning is not enough to be sure of our results. Is there any way to prove (deductive reasoning) that this result always holds true? My edition didn't have the paper folding exercise, but that sounds like it could maybe lead to a proof.

If you couldn't simplify (reduce) the ratio, the ball went through every square. If you could simplify the ratio, the ball did not go through every square. You can see why if you take a 2 x 4 table, and then make double-size squares in it, which would make it a 1x2 table.The ball goes through each double-size square, which takes it through only 2 of the 4 original squares in that double-sizer.

If you counted the endpoints, the ball bounced as many times as the sum of the simplified ratio. I don't have a good sense of why on that one.

We're supposed to get experience looking for (and finding!) patterns.

If you're not done, you probably don't want to read this, as it contains answers. If people want, I could put anything like this in the file section in future, just so no one sees it before they're ready. Let me know.

This is not detailed answers. Bob has posted his detailed answers in the files section.

Here's what I got out of the first two lessons:

As we drew the tables, we were gathering data, and eventually, with Jacobs' help, we could see that, if you reduced the ratio of width to length, odd width to even height ended up in the top left, even width to odd height ended up bottom right (same shapes, length and width reversed), and odd width to odd height ended up top right. We used inductive reasoning for this. What he starts explaining in lesson 3 is that inductive reasoning is not enough to be sure of our results. Is there any way to prove (deductive reasoning) that this result always holds true? My edition didn't have the paper folding exercise, but that sounds like it could maybe lead to a proof.

If you couldn't simplify (reduce) the ratio, the ball went through every square. If you could simplify the ratio, the ball did not go through every square. You can see why if you take a 2 x 4 table, and then make double-size squares in it, which would make it a 1x2 table.The ball goes through each double-size square, which takes it through only 2 of the 4 original squares in that double-sizer.

If you counted the endpoints, the ball bounced as many times as the sum of the simplified ratio. I don't have a good sense of why on that one.

We're supposed to get experience looking for (and finding!) patterns.

- Notes by Sue VanHattum