Since Leo is obsessed by mazes and games (real world or computer), and all his recent creative work is some combination of those, I decided that I would try out an idea that I have about thinking of algebra as a maze solving activity.
The general idea, which I’ve only so far tried to execute once, and not all that successfully, is that at the beginning you start out with an equation. Here it’s at the top right: (90+8)/2=7^2:
On the left is the maze, which starts out as just an empty box with just and S for Start, and an E for End (the S is off the top left of the picture). Unfortunately, you’re seeing the page AFTER play, so you’ll have to imagine the above picture with just an empty rectangle on the left (with S and E), and just the starting equation at the top.
The “play” is that at each point is that I give Leo several choices of whether to turn, go straight, left, or right, just like in a maze, but that each of these is an arithmetic operation on the equation. It’s a little hard to see, but the first set of options upon entering the maze from the upper left is to multiple by 2 (2x) or do the square (7^2). Leo choose to do the square, and so we did it. This leads to more choices, and so on. The next step the choices where multiple by 2, and subtract 8. As you can tell from the math on the right, he chose to subtract 8. (Just like in a real maze, backtracking is always an option).
We did manage to get through this one, although I was making it up pretty much as fast as I could as I went along, in order to keep up with him. You can see that after some dead ends and backtracking, we ended up with the right answer, which we checked by plugging it back in at the top.
So after we did that, sort of semi-satisfactory algebra maze, Leo decided he was going to make one for me. Here’s his:
It’s actually pretty clever in that he has the idea that you start at the outside of the spiral and the problems get harder until you get to the center, at which point you level up — entirely his design! Here’s my semi-faked making of mistakes (in green) so that he would have to correct me:
I got to make the next level for him. Note that I went all out, with squares and cubes and even cube roots and some algebra.
To my surprise, he solved what I thought was pretty tricky algebra immediately. I wrote: X=5 and Y=X*X^2 and Y? (asking for Y). Straight away he said: “That’s just X cubed”, so it’s 125!
I let him play REAL computer games for that impressive feat of reasoning! (On the ride to his break camp, he asked me to explain what a paradox is. I told him I’d explain it tonight. This ought to be interesting! 🙂 )
Leo did an algebra maze spontaneously this morning, but it was a different (simpler) spin on the idea:
In this version, when you hit a dead end, you get stuck until you solve the algebra problem in that location. He drew the maze and made me put problems into the dead ends. They started out easy, and got more complex. Leo was actually able to do nearly up to the last few, although I had to talk him through this one:
And this one he a simply didn’t understand, even after I talked him through it:
I guess that’s not too surprising that someone under 6 wouldn’t understand how to compute negative roots. (Although when I broke it into srqt(25)*sqrt(-1) he right away got that these were 5 and i respectively!)