Michael Pershan’s geometry problem part 2 + an extension

[note – sorry, no editing whatsoever on this one, had to run out the door for the evening . . . ]

Yesterday we talked through a geometry problem that Michael Person posted on twitter:

Michael Pershan’s geometry problem

Today he posted another one:

My older son had an after school activity today, so I talked through this problem with my younger son first. He looked at it in two different ways.

First he looked at the entire rectangle and subtracted away the areas that were not shaded. The arithmetic gave him a little difficulty, but he was able to work out the area:

Next I challenged him to find an alternate approach. This time he thought about shifting the top triangle to the left one unit. This approach is a nice little challenge for a younger kid.

When my older son got home I’d planned on going through a problem from an old AMC 8 that gave him a little trouble this morning (partially because of time). I hadn’t looked at the problem, though, and when I did look at it I got a nice surprise – it was a problem we could approach “in your head” just like the two problem from Pershan.

Here’s the problem on Art of Problem Solving’s site:

Problem 25 from the 1999 AMC 8

and here’s my older son talking through the problem this afternoon:

Even if was just a coincidence, I’m happy to see how the ideas you use to talk about beginning problems are also really useful in problems that seem much more complicated. Math is like that – the basic ideas can be really (and surprisingly) powerful.

Michael Pershan’s Geoemtry problem

Yesterday Michael Pershan posted this geometry question:

The question interested me for two reasons. First it is always neat to hear how kids think out loud. The challenge if this problem is to solve it in your head. Second, by coincidence, we’d just talked about Pick’s theorem last week, and this would be a good chance to review the idea (even though the number of grid points is pretty large).

Here’s my older son’s initial reaction – he sees two different ways to approach the problem:

Next I asked him to approach the problem using the ideas in Pick’s theorem. The interesting thing to me here was how he counting the various grid points, and the little bit of difficulty that he had counting these points made me happy that we looked at the problem this way.

Next up, my younger son. We had to run to an even program that he’s in on Monday’s so I only looked at the problem one way with him. Interestingly, though, his first idea was to approach the problem via Pick’s theorem, though we ended up talking about a more traditional geometric approach.

So, a fun question from Michael Pershan. It is always nice to hear the ideas that are happening in their head rather than just crunching out the numbers on paper.