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” is each person can 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 share the answer aloud. And if they had the correct answer, I asked them to try with 8 cubes, 4 on each side.


After about 15 minutes of individual work time, I randomly paired kids up. I asked them to track the moves, meaning which cube went to which position.


The kids already knew that eventually we would want to figure out the fewest number of moves forany number of people. Two students found the correct equation after testing for 12 people, 6 on either side.

We tracked the moves together as a class so that 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 as:

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 could come out of this. Not until I saw Kate Nowak’s post that helped put the oohs and aahs into ourDay 12 constructions.

To continue with this same idea, I’m now looking for geometric theorems that kids could construct from scratch, that way 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 4 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 bisected each of these 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, it doesn’t mean we can’t know about it and appreciate this 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 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 blogpost that I wrote on Traffic Jam interesting: 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 have noted.

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