The second half of the summer left me mentally bursting at the seams with mathematics. Working with Dawson during Session 3 (special topics in math) I experienced that kind of “math buzz” that I hadn’t felt in a long time. We started every day with a puzzler: some that required specific mathematical knowledge and computations, and some that were more logic-puzzle-y. Since allowing the students enough time to struggle/persist/think through the puzzlers each day meant that we had plenty of time to just walk around, peek at their work, and think, I often found myself daydreaming up variations on the puzzles – expanding them, changing some elements, generalizing… Lucky for me Dawson always indulges me in my mathematical wanderings and we would often stay during breaks or after work to work through some of the interesting questions we had.

Some of my favorite puzzlers and problems from the summer, then, for posterity: (Since I have the world’s absolute worst memory. Actually I might not even be able to generate this list accurately, at this point, a month later. WE WILL SEE.)

- Tweaking the “Math on the ever slowing drive” puzzler (75 miles away from your destination, you drive 75 mph for the 1st mile, 74 mph for the 2nd mile, 73 mph for the 3rd mile, etc… how long until you arrive?) so that you change speeds every half mile, by .5 mph. Trying to figure out how to write that in sigma notation, given that sigma notation is only going to capture the integer values. Realizing there are a couple of completely different ways to write the sigma notation for this, and it is WEIRD that they are equal to each other, algebraically. I actually still need to think about this one a little more.
- Also weird about the “ever slowing drive” puzzler… the answer to our “tweaked” problem (75 miles away; 75 mph, 74.5 mph, 74 mph, etc…) actually turns out to be the same as “doubling” (not really the right word for it, but sure) the original problem (150 miles away; 150 mph, 149 mph, 148 mph, etc…) Maybe this shouldn’t be surprising but for some reason, IT WAS.
- Not one we changed, just one I LOVE: the prisoners in hats one. The 3 person one is easy, quick, satisfying, and the 20 person one is tough and GREAT. I still maintain that my solution (yelling or whispering depending on the color) would have worked and was technically within the parameters of the problem, though…
- After calculating the probability of being dealt each hand in poker (an activity we do with the class), deciding to calculate the probability of being dealt each hand in poker if you were playing with two decks of cards. (It creates some interesting new possibilities, like 5 of a kind.)
- Then, after being surprised at the small difference between the probabilities of One Pair and High Card, figuring out how many decks you would have to play with in order for it actually to be MORE likely to be dealt One Pair than nothing at all (High Card.)
- Playing that card game. (21 cards, 2 players. Each player can take 1, 2, or 3 cards on their turn. You want to take the last card.) I actually think I could use this in class if there’s ever some down-time (oh wait, what is that.) We did modify the game by changing the number of cards, as well as the number of cards allowed to be drawn, to be sure the students understood WHY the correct strategy worked and hadn’t just picked up on/mimicked the pattern.
- The scavenger hunt. Just so unbelievably great. I’ve always been a nerd for secret messages and codes and a little bit of mystery, and this hunt with it’s encrypted messages and other puzzles is one of the highlights of my summer. It was 100+ degrees on the day of the hunt this year, and running all over campus with a group of teenagers was exhausting, but wonderful.
- What I may like even more than the actual event, though, is the way we built the suspense by having the “trial run” puzzler the day before. Dawson and I, with no warning, didn’t show up to class that day. Instead, we left a note on the board that they should start on the puzzler at their seats. Each student had a bit of an encrypted message telling them to come find us (in another building) and we just hid there while they figured it out and rescued us. I would LOVE to figure out a way to kick off a class this year like this, provide some suspense/mystery/build-up to some mathematical activity or event I have planned.
- And finally, of course, the pirate problem. This one will always have a special place in my heart. And the extension? So. Unbelievably. Cool. (You can see the problem and an article about the extension on wikipedia here but be warned that they don’t set it up well, and reveal the solution all too quickly. Also, we use/prefer a slightly different version of the problem, where only a MAJORITY of votes will pass a proposal. This makes the numbers come out a little differently.)

So, those are the puzzles that stand out in my mind from the summer with Dawson. It was a great one.

In addition, once I made it back to D.C. I got to participate in a truly excellent Geometry workshop with Henri Picciotto (check out his site here, he has some stellar resources.) He provided us (and challenged us) with a variety of puzzles that were (to me, at least) inherently engaging and often designed to help us build understanding of a specific Geometry concept. I’m not really sure how to describe what the experience was like, but I often found myself sitting down antsy to tackle whatever problem or challenge he had posed, only to realize later that it was actually going to lead us somewhere significant, content-wise. It was a little bit like being tricked into learning/seeing Geometry (in a good way), and it was GREAT. I left the workshop every day wanting more time to finish the puzzles we had looked at each day.

Maybe I’ll do another bullet point list of the things that stood out to me? Why not!

- The puzzle on the first day with pattern blocks. What a way to kick off class. I love the idea of jumping into the math first, letting students play/engage in something both interesting and mathematical, and THEN introducing yourself, the class, etc. Sets the tone for what they can expect from you and your class.
- Oh, and also, that puzzle was great in and of itself. I actually did want to take more time to figure out why certain combinations were impossible, and maybe I should convince Bree to take a second look at it with me.
- Creating convex polygons with the pattern blocks. I used to DO this as a kid, without even thinking about what I was doing. We tried to create convex polygons using increasing numbers of blocks, and the perfectionist/OCD/need-to-complete-tasks person in me wants to return to this and see how far I could go. 100? 200? WE’RE GONNA NEED MORE PATTERN BLOCKS.
- Pentominoes! First, how many are there? Then, the memory trick for them. Finally, the games! I downloaded “Zentominoes” on my iPhone (a free app) and have been enjoying it for the past couple of weeks.
- Tangrams! First, how do you create them? Then, being challenged to create shapes out of the different number of pieces. Finally, the games! I downloaded “TanZen” (also a free app) and while this one hasn’t been getting as much play as the pentomino puzzles (the controls are weirder and the pieces don’t “click” into place in that satisfying way) it’s still great.
- Super tangrams! This one is Henri’s, and I really liked this. First, how many are there? (This was actually frustrating, to me. Why couldn’t I see all the possible pieces?) Then, solving the congruence mini-puzzles. Finally, “blowing up” the super tangrams. This worked both as a puzzle (filling in the shapes) and as a mathematical discussion. What were the scaling factors of area and of perimeter for each larger version of the same shape?
- The construction challenges on Cabri. I don’t know why I was so into these, but this was great. It’s too bad I don’t teach Geometry, because I think it would be appropriate to throw some of these at students. To my knowledge, we actually don’t do much with construction in the Geometry curriculum at my school.
- The soccer goal/angles activity. This wasn’t really a puzzler, but it certainly was PUZZLING to me where that player maximizes his angle on the goal. What does that shape look like? (Turns out it’s hyperbolic.)
- Introducing new vocabulary by demonstrating the concept with human bodies. i.e. “One person stand here. Another person stand here. Okay, now, everyone else in the class: your goal is to stand equally far away from those 2 people.” instead of just saying “A perpendicular bisector is blahblahblah…” I’m definitely going to try this with conic sections in Precalculus this year. Yes, I know I’m going to need string and LOTS OF SPACE for ellipse. Maybe also more human bodies. (cue me trolling the hallways for poor unsuspecting souls with a free period when I’m teaching…)
- And speaking of slices of 3-dimensional objects, the slicing a cube puzzle! I can’t tell you how frustrated I was by trying to precisely cut (out of construction paper) the slice that goes from the top corner on one end to the bottom corner on the other. Yes, it’s a rhombus. It shouldn’t have been so difficult for me to figure out how to construct it! This is another one I’d love to return to with Bree and try to actually construct all of the shapes he had listed (well, the ones that are possible.)
- Quadrilaterals from the inside out. This was a great activity, and his whole quadrilaterals unit sounds like something I’d love to steal in its entirety if/when I teach Geometry.
- Construction challenges on Cabri 3d. These were less math-y and more just kind of “can you find the right combination of clicks in the right order that will make this software do what you want it to” but still so cool!
- The visual “proof” of the formula for volume of a pyramid in Cabri 3d. This is actually something I had spent time before trying to visualize/rationalize using only my brain/imagination. Was never able to do it. I’ve got it, now, because I could look at it, rotate it, color-code it. Great.
- Finally, and only because I was SO DAMN PROUD of my beautiful Archimedean solid, thinking through and playing with creating Platonic and Archimedean solids.

Yeah. That workshop had me more than a little jazzed about mathematics. I really hope our school is able to host Center for Innovative Teaching workshops again next year, and that Henri comes back.

Last but not least. I was reminded of the dartboard problem a couple of weeks ago, which I had seen people talking about on twitter but had not actually tackled myself. Found myself facing a few hours spent on a bus and decided to give it a go. Holy moly. So great. If you haven’t attempted this problem yet, please do, because it is incredible.

I hung my work for the problem above my desk at work, 1) because it looks really cool and 2) to remind me that the combination of grinding frustration and thrilled elation when solving a really great puzzle is something I need to seek out. I want to be an intrepid explorer. I want to be mathematically fearless. And I want to embolden my students to do that, as well.

Rambling and totally ADHD in the best way possible.

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Jason Buellon 09.04.2011at 4:07 pm