Two Lessons: Frog Leap and Beach Ball
I call it Frog Leap, but you may know this common game as Stepping Stones or Traffic Jam. NRICH has a nice interactive applet.
The challenge is for the 3 boys and 3 girls to switch places in the fewest moves possible. A legal “move” means each person can either jump onto an adjacent empty space or jump over another player onto an empty space. (Two people just switching places is not okay.)
I placed 7 pieces of paper on the floor and asked six kids to come up and act it out. They were stuck a bunch of times, despite a lot of input from the audience.
Once they got the idea of the game, I let them explore it individually using small cubes. When they thought they had found the fewest moves, they wrote down their answer and called me over to check—this, of course, was so no one would shout the answer aloud. If they had the correct answer, I asked them to try it with 8 cubes, 4 on each side.
After about 15 minutes of individual work time, I randomly paired the kids up. I asked them to track the moves—meaning, which cube moved to which position.
The kids already knew that, eventually, we’d want to figure out the fewest moves for any number of people. Two students found the correct equation after testing it with 12 people—six on either side. We tracked the moves together as a class so we could see the cool symmetry in them.
Instead of looking at the total number of people, one could also just work with pairs of people. The two students who figured out the equation regarded each pair as a step number—so step 3 meant 3 pairs of people, or 6 people—and their equation was:
fewest moves = (step number)² + 2(step number)
When we do these types of problems, it's really no big deal to me if the kids don't arrive at the general rule. I think the process itself is more important. But I try to expose them to a lot of problem-solving, and they are getting better at it. And they are persevering.
We're doing simple constructions by hand in geometry. So far, they've learned how to copy a segment and an angle, along with how to bisect a segment and an angle. But nothing too exciting has come out of this yet. I'm now looking for geometric theorems that kids could construct from scratch—so that when we're done, something exciting awaits us!
One of my favorite sites is Math Fun Facts by Professor Francis Su of Harvey Mudd College. I geekishly introduced myself to Dr. Su at a UCLA Math Festival four years ago—in my world, he is a rock star and I'm just a big fan!
Su posts a theorem he calls Pizza Slices:
Take a pizza and pick an arbitrary point in it. Suppose you cut the pizza into 8 slices by cutting at 45-degree angles through that point, and color the alternate pieces red and green.
Surprising theorem: The total area of the red slices and the total area of the green slices will always be the same!
And that’s what my geometry kids did! They needed to construct perpendicular chords and then bisected each of those 90-degree angles to get the 45-degree angles. My kids called them beach balls.
Even though the proof for this theorem requires calculus and polar coordinates, that doesn’t mean we can’t know about it and appreciate the fun fact.
[09/28/12: Joshua Zucker wrote:]
There’s a much simpler proof of the “pizza theorem,” as your beach ball problem is often called. Take a look at Stan Wagon’s dissection proof, shown at this Wikipedia page, for example. No need for calculus or polar coordinates! Just a pair of scissors. Well, and some logic explaining why the pieces that look congruent really are.
[09/29/12: Suzanne Alejandre wrote:]
Fawn, you might find this blog post that I wrote on Traffic Jam interesting:
http://mathforum.org/blogs/suzanne/2011/08/14/whats-in-a-touch/I find it fascinating to watch students move between concrete manipulatives, virtual manipulatives, their bodies as manipulatives, and then paper/pencil representations of the mathematics. I think this particular activity—whether it’s called Leap Frog or Traffic Jam or ?—is a perfect one to help students practice perseverance, as you’ve noted.