Revisting a challenging AMC 10 problem with Patty Paper

Earlier this week my older son and I looked at problem #11 from the 2011 AMC 10a:

That earlier project is here:

and the entire 2011 AMC 10 a can be found here:

The 2011 AMC 10a hosted at Art of Problem Solving

Today I wanted to revisit the problem from the perspective of a project rather than as a contest problem. This approach came to mind after watching Po-Shen Loh’s use of a problem from the 2010 International Mathematics Olympiad as fun project during a public lecture at the Museum of Mathematics:

So, inspired by Loh’s approach, I started with a much easier version of the problem – how do we make perpendicular bisectors if we are given a line segment? My younger son (4th grade) makes the perpendicular bisector with a ruler and protractor, and my older son (6th grade) uses some folding ideas:

At the end of the last video I was asking the boys about some special properties of perpendicular bisectors. Upon reflection I don’t know why I assumed they’d notice that the line was equidistant from the two endpoints of the line segment – it isn’t as if this is an obvious property for kids to see. Although I was surprised when we sort of hit a wall at the end of the last video, I really shouldn’t have been.

Instead of just telling them what the property was, I let them play around a little more and eventually my older son noticed the it. We then talked about why the perpendicular bisectors had this property. This 5 minute detour wasn’t really planned, but I didn’t want to just tell them about the property – I wanted them to find it. I really liked how Loh had success with a similar approach in his MoMath lecture. There he showed a simpler version of the IMO problem first – and allowed lots of exploration from the audience – which prepped everyone for thinking about the harder problem.

Now, with a little bit of background on perpendicular bisectors, we looked at the old AMC 10 problem. The background work on perpendicular bisectors helped my younger son approach the problem – it was actually pretty cool to see him take the lead in drawing the region R. After the boys had drawn the region, they had lots of great questions about the picture.

I didn’t go all the way through to finding the area of the region because I didn’t think that would be a great use of time with my 4th grader – after you’ve got the region drawn, finding the area is mostly just calculation.

Finally, we wrapped up the project by approaching the problem using folding techniques. It was nice that this approach revealed additional properties of the pentagon / region R that were not as clearly visible in the last video.

So, I was happy to see the success that Loh had using an IMO problem in a public lecture, and it was fun to mimic his idea on a smaller scale with a contest problem that was also accessible to kids. Contest problems might lend themselves to this approach a little more easily that research problems since, essentially by design, they have short solutions. Still, though I’m excited to think a little bit more about how to use an approach like Loh’s share non-contest math ideas like Larry Guth’s “no rectangles” problem and John Conway’s Surreal Numbers with kids.