A fun experiment with my older son

We are in the middle of a *slow* move to the Boston area and that has meant a lot of driving back and forth between the NY suburbs and Boston this year. All of that driving has provided lots of time in the car to talk math with the kids. I’ve used some of that time to conduct a fun experiment with my older son which seems to have had good results.

At some point last year I read about a geometry course that was taught – at least partially – in the dark. I can’t remember where I read it, but I think it was in Siobhan Roberts’ King of Infinite Space . (Which is a great book even if I’ve goofed up the reference!).

I didn’t go all the way to teaching in the dark, but the three hour drives allowed us to have long conversations about math problems without pen and paper. During the drive up yesterday we talked for probably 15 to 20 minutes about problem 16 from the 2006 AMC 10a:

Problem 16 from the 2006 AMC 10a

Here’s the problem as it appears in the link above:

2006_AMC10A-16

A circle of radius 1 is tangent to a circle of radius 2. The sides of \triangle ABC are tangent to the circles as shown, and the sides \overline{AB} and \overline{AC} are congruent. What is the area of \triangle ABC?

As you can see, this is a pretty challenging problem all by itself. Talking through it without pen and paper adds to the challenge but the lack of pen and paper also seems to have helped by son develop his geometric insight over the last few months. It was really cool to hear him describe seeing a right triangle to draw in and then hear him describe the similar triangle relationships necessary to solve the problem. Again, it probably took 20 minutes to talk through it, but it seems that this extra work is really helping him improve his geometric insight.

Today we tackled one of the challenge problems from the 3D geometry section of our Introduction to Geometry book. The problem also requires you to make a connection between two similar triangles and I think the practice he’s had visualizing similar problems helps him make that connection here.

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We’ve got quite a few more car trips to go, too, so I’m looking forward to experimenting with these “no paper” solutions over the summer. I’m wondering if it something to try with other areas of math, too, even when we aren’t kn the car. Counting and probability, for example, seems like an area where this approach could be fun since organizing your thoughts in counting problems is so critical.

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