Why I love the Art of Problem Solving books – part 2

Had two really wonderful experiences talking through problems in our math books tonight. My older son is studying in Art of Problem Solving’s Introduction to Geometry book and my younger son is studying in their Prealgebra book.

The current topic for my older son is 3D geometry and the specific topic today was pyramids. An exercise that gave my son a little trouble today is an absolutely brilliant problem about pyramids. Here’s our walk through it tonight:

The current topic for my younger son is a little introductory geometry and the specific topic is circles. The problem we talked through tonight was the last example in the circle section – I love that this problem is just right at the edge of his reach right now. His ideas are really wonderful and open the door for some really fun follow up conversations tomorrow morning:

I’m sure there are other great math books for kids out there, but I love being able to crack open these books and see wonderful example after wonderful example after wonderful example. I love that the variation in problem difficulty, too. The problems always seem to keep the kids engaged and excited about math. It is so much fun to have the opportunity to go through these books with them.

Using the area of a circle formula for a little algebra review

My younger son and I started talking about circles this morning. Even before the first example problem in our Prealgebra book there’s a sketch of the proof that the area of a circle is \pi r^2. Seeing that proof, I closed the book and decided to walk through it with my son for today’s project.

To start I asked him what he already knew about circles – turns out that he’s heard a lot about circles. He even thinks to find the area of a circle by chopping it up into pieces:

In the next part we got a little sidetracked by some algebra. That’s fine. Instead of putting all of the focus on the geometry proof, I decided to take a little detour to try to make the algebra a bit more clear. I suspect that my son’s algebra mistakes here are pretty common mistakes. Hopefully we’ll get through some of these difficulties with a bit more practice.

Now back to the geometry. We chop up the circle into 16 pieces now and try to rearrange the shapes into a (near) rectangle. Our drawings probably aren’t exactly right, but at least you get the idea that the shape is starting to look more and more like a rectangle. With the extra algebra practice from the last video, the formula for the area of a circle now falls into place.

Maybe the next project is comparing the famous “proof” that \pi = 4 and seeing if he can see where things are going wrong. For today, though, a nice little proof and hopefully some valuable algebra practice. Fun morning.

Henri Picciotto says in two sentences what I couldn’t figure out how to say in two weeks

There’s been quite a conversation on the math ed corner of twitter about Michael Pershan’s thoughts about hints. Henri Picciotto wrote a great piece that I just saw tonight.

The two big takeaways for me from that piece are:

“Improvisation is absolutely required in every facet of teaching . . . .”


“All the hints Anna lists may be useful, depending on the situation.”

That second quote finally got me unstuck.

After both reading and watching some of Michael’s thoughts, my mind had drifted to a description of a Bobby Fischer / Tigran Petrosian game from 1971. Specifically the commentary from Graham Burgess here:

Graham Burgess on Fischer / Petrosian Game 7

The quote I was thinking about relates to Fischer’s 22nd move. I’ll add emphasis on the parts that I thought were relevant to Michael’s hints:

“This is one of the most talked-about moves in chess history. It looks extremely unnatural to exchange off the strong, beautifully-placed knight for Black’s bad, awkward bishop. Yet it wins the game quickly and efficiently. Is there something wrong with the principles that would lead many players not even to consider the move? Not really. Nine times out of ten (if not more frequently) it would be wrong to exchange a good knight for a bad bishop. The problem is if a useful general principle takes on the status of a hard-and-fast rule, rather than it always being governed by the proviso, “unless the specifics of the position demand another move.”

So, thanks to Henri, I now see the connection better. If you get too caught up in general principles, you risk failing to improvise when you need to (or at least making it much harder to improvise). You also risk the possibility of dismissing hints that may well be useful in different situations.

I’m glad that I can now stop thinking about descriptions of chess games from 40+ years ago – ha!

Also, thanks for writing the wonderful Zome Geometry book, Henri. I use it with my kids all the time 🙂

Walking the boys through a pretty challenging problem from Solve My Maths

Saw this problem through a re-tweet from John Golden:

This is a pretty challenging geometry problem. I asked me older son about it and his first observation was about the inscribed and circumscribed circle of triangle BCD. I really didn’t expect him to have that idea, so I thought it might be fun to try to walk through the problem with him.

As I thought about how to do that, I decided to have my younger son join in as well since we are studying angles right now.

I did a lot of the heavy lifting here, but it was still a really fun project.

First up, showing the kids the problem. The main thing here was to have my older son give his idea about the circumscribed and inscribed circles:

[a quick note on the thumbnails. I seem to be having the same thumbnail troubles that started yesterday. Sorry if some of the video previews are all grey. The videos are uploaded and published and should play fine. I don’t know why the thumbnails are getting goofed up. ]

Before diving into the calculations, we built some pentagons / pentagrams out of our Zometool set. Our construction allowed us to explore some similar triangles that will help us solve the problem:

Next we moved to the whiteboard to look more carefully at some of the angles in the problem. I really enjoyed the conversation with my younger son here and I think he was able to get a little bit better of an understanding about angles in triangles from this part of the project.

Next we moved on to discuss some of the similar triangles that we saw in our Zometool set. The relationship between these triangles will help us determine the (surprising) relationship between the long blue pieces and the medium blue pieces. This video has a quadratic formula calculation that has fun result:

The next part is included only for completeness. Here we walk through the long calculation to arrive at the solution to the problem. Well, a special case of the solution anyway. I think that my older son isn’t quite ready to tackle calculations like this one, but I was hoping that seeing how to work through something like this would be instructive for him. One nice surprise was that my younger son seemed very interested in every step.

Finally, instead of spending more time on the calculation from the last video, we just go to Wolfram Alpha to finish the job. This last step allows the boys to see the number giving the final relationship without seeing another messing square root calculation:

So, a problem that is obviously over both kid’s heads at this point, but nonetheless has a couple of fun parts that are accessible to them. Finding the golden ratio relationship in the blue struts was a pretty fun surprise for them, too. Fun little exercise.