I did my best during this experience to let the kids figure out as much for themselves as they could. It is a pleasure to watch them turn into little scientists when there was something about which they are really curious. Through the course of this investigation, they had to conjecture, test, and discover several things that those of you reading this probably take for granted:

- horizontal motion can be considered independently from vertical motion
- angle of incidence = angle of return if no spin on the ball
- the distance from a person to a wall is the shortest distance
- the distance from a person to a wall is perpendicular to the wall
- a diagram helps make sense of relationships between distances
- a neat and labeled diagram makes it easier to discuss
- the easiest way to measure distances in a room with a square grid tile floor is by counting floor tiles
- triangles that have the same angles are the same shape
- triangles that have the same shape have proportional sides

Without further ado:

Once they had a good diagram, they had to find a way to partition the wall in the same proportion as d:d'. All of this discussion was in terms of number of floor tiles. They marked the nice-acting ratios first:

2:1 marked 2/3 of the way

1:2 marked 1/3 of the way

1:3 marked 1/4 of the way

I had them do a bunch of these just to give them practice with some tangible fractional amounts.

After we had marked several of these and proved they worked, the kids were honestly kind of over it. I wanted to talk about any distance, and we eventually got here: a = (d/ (d+d'))w. But that last part, I felt like I was just doing it for them. That was ok, though.

**The Teacher Has Her Fun**

I was also interested in modeling the path of the ball with an absolute value function. Here is my derivation.

So we have y = c*abs(x - a) + d, where a is given above in terms of d, d', and w, and since this thing has to go through the origin, c = -d/a. I don't know what I'd do with this, but I was trained as an engineer and I couldn't help myself. Thanks for reading! I hope you enjoyed my little adventure.