Area, Perimeter, and Fence Posts

A couple of years ago we stumbled on a pretty neat section of Khan Academy that talks about basic counting techniques:

I remember being surprised at how difficult it was for me to explain the concepts in this section to my older son.    Of course, I struggle to explain lots of math topics, but the struggle talking through these problems really stuck with me.  We even went outside to our deck and actually counted fence posts to try to help make the problems more real.

This week we ran across another “fence post” problem playing around with some old AMC 8 exams on the Art of Problem Solving site:

This problem gave me the idea for our Family Math topic for this weekend – area, perimeter, and fence posts.

As we often do, we started out with snap cubes.   After some informal talk about a few basic geometry concepts, we made a little shape out of 4 snap cubes and talked about area and perimeter.  I was happy that we were able to find  two completely different ways of calculating the perimeter of the shape:

Next we talked about fence posts.  Watching the video just now, I’m not thrilled with how well I explained the fence post idea, but hopefully the example with the line was helpful.  The main thing I was trying to show is that counting the fence posts isn’t always super easy.  With a straight line you get one answer and with a closed “loop” you get a different answer.    Not necessarily intuitive, but there is a nice relationship with the perimeter (at least for the simple shapes we are looking at today).

and having dealt with the cat distraction,  the dog gets into the act:

finally, can we finish up with out another distraction . . . . :

So, despite the distractions, this was a pretty fun project.  I remember the difficulty walking through problems like this a few years ago, and I can see that both kids remain challenged by this type of problem.  Hopefully playing around with the snap cubes and the Penrose tiles help them get a better feel for each of the three geometry concepts we were talking through today.