# Difference of Squares Game

## A Game!

Finals week often has awkward down time for students. With that in mind, I made a game for my middle school students to play next week.

I’d love some feedback on the directions, the scoring system, and the game itself.

And if you play with your students, let me know how it goes!

## A Few Words About Points

The decision to award more points for even values was arbitrary (I could just as easily have chosen odds) but also intentional (I want to motivate students to observe patterns/behavior and use their observations to target certain values or types of numbers).

More points for higher numbers was un-arbitrarily intentional (I want to motivate students to tinker with larger numbers).

I want students to hunt in a clearly defined, finite space, hence the 1-100 boundaries. Good idea? Bad idea? I’m not sure yet. I’ll let you know how things go next week.

1. Hi Michael,

If my kids were playing this game, they’d all be drawing squares on grid paper and removing smaller squares from this original one. (They’re pretty good with visual patterns, wonder why. 🙂

The directions are fine. I’m wondering about the scoring system, it seems too much for middle school kids to keep track of the 4 different (really 5) ways to earn points. And instead of 10-50 points, could they just be 1-5 points?

Okay, since you called this a “game,” I’m looking for the FUN factor in it. I’m honestly having a hard time seeing that, so I HOPE you’ll try it out and tell me I don’t know what I’m talking about (which is 78% of the time). Maybe you have a big prize in mind for the winner to make it motivating?

I’m thinking of something like this: Have a sheet of paper filled with a ton of square numbers and non-square numbers, these numbers are written/typed so that they’re randomly spaced and oriented on the paper. In each round, BOTH players circle their own two square numbers and find the difference. But the player with the LARGER EVEN difference earns 1 point, other player earns 0. Game over when both players cannot each find a pair of square numbers.

This may make it more game-like as it’s faster paced (finding and circling best pairs before opponent) and places more emphasis on recognizing square numbers quickly. I don’t know. I’ll leave now. Thanks for sharing, Michael.

2. Yeah, what Fawn said. Not only do I think this is not fun, I’m not sure I see what the point of it is. Are students just practicing squaring and subtracting numbers? Or do you want them to see patterns and use strategies to obtain large/even answers?
I’m biased because I’m usually pretty anti-game. If you’re looking to give the kids a break because of finals week, do some kind of food-related math lab…Fawn did some kind of volume/doughnut thing…why not do that?
I’m coming off pretty negative here. Feel free to criticize stuff on my blog too!

3. Hi Michael,

I think you’re onto something here. However, I’m a little unclear about the learning goal or skill to be reinforced. I understand you want them to see patterns with squares. I’ll echo Fawn’s idea to have them draw these squares. If you have dice, maybe use them here. Some of the best dice I bought years ago were twelve-sided dice. The best part about the dice is a twelve-sided die inside another twelve-sided die. If twelve-sided dice aren’t an option, use two six-sided dice. Chances are you can get six-sided dice more readily so what about something like Get to Zero with the Difference of Squares?
Students have 10 rounds to get closest to zero. Each round, each student will roll the two dice twice. The student must determine if they’ll subtract the first roll from the second or the second roll from the first roll. Player1 goes and says they’ll subtract the second roll from the first. The first roll of dice add up to 7. Player1 squares 7 and has 49. Player1 rolls a second time with a sum of 9 so the square is 81. Subtracting 81 from 49 the student now has -32. It’s their opponent’s turn now. Say the next round Player1 has a difference of 48, their running score is at 16 (because of -32 + 48).
I really don’t know what I’m talking about, and I’m pretty sure that didn’t make any sense, but this idea just bounced in my head and I had to share it. I’m curious to see how your game goes. I echo Fawn, maybe reduce the scoring to single digits. Wait. Did I just agree with Fawn, not once, but twice?

4. Fawn, Nathan, and Andrew,

I really appreciate the time you all took to look at the game handout and offer feedback (on a Saturday, even!). I’ll probably miss something I wanted to respond to, but here goes…

My biggest takeaway from your comments (and it’s become borderline obvious even to me as I look at the game again) is that this won’t be fun for students, except for that strange, small percentage of kids who love numbers for the sake of numbers and/or games for the sake of games. In fact, I explained the rules to someone this evening after reading Fawn’s comments and almost cringed inside as I thought… “Who would want to play this?” (aside from me, of course).

Okay, so the game isn’t very fun. Ignoring that, what’s the deal with those points?!?! I don’t really know why, but I like scaling the score by 10 (10, 20, 30, etc., rather than 1, 2, 3, etc.). It could be a result of watching “Whose Line is it Anyway” on Comedy Central as a kid (the non-Drew Carey version) where large quantities of points were bandied about in a mostly arbitrary manner.

Too many scoring options to keep track of? Maybe. But I’m actually confident that any student who thinks the game is fun (this may be an empty set) would have no trouble mastering the scoring rules. I’ll let you know how things go next week.

What was the mathematical point to the game? Practice with squares, a bit. Practice with subtraction, not so much. Most of all, my hope (probably a foolish one, given the tedious nature of the game) is that students will look for patterns and turn them into strategies, including simple ones (“Hey, odd^2-odd^2=even, even^2-even^2=even, even^2-odd^2=odd, and odd^2-even^2=odd!”) as well as more complex ones, which will allow them to (1) more systematically find “repeats” and earn a 50 point bonus on some turns, and (2) select values that are impossible to duplicate. I’ll echo Andrew’s comment that I might not have made any sense just there. Hope that’s not the case.

Regarding Fawn and Andrew’s games specifically, these alternatives seem to have some qualities that my game lacks: Fast paced, more playful, actually feels like a game (rather than a semi-interesting math activity with the word “game” stamped at the top). If my game bombs next week, then I have two great alternatives to try, and a set of qualities to try to infuse into any future games I dream up.

Nathan, I hopped over to your blog and read a few of the most recent posts in the hopes of exacting some negative comment vengeance. I was sorely disappointed, since everything I read was awesome. However, I’ve got a new feed in Google Reader/Feedly, and I’ll be on the lookout for negative comment opportunities in the future. 🙂

Thanks again for your comments, everyone! I’m trying to get better at everything I do as a teacher (even random finals week “games”) and having input from others is invaluable.

5. Another option is to make a bingo-like card (could be larger than 5 x 5) with the results of various differences of squares. They would have to come up with combinations that would produce whichever one they need to make a line. More game-like, speeds it up and might encourage them to look at some of the patterns you mentioned. Just a thought!

6. It seems like there are lots of good patterns that could emerge from playing this game to generate some data. For example, offer 100 point bonus for a difference that is even but not a multiple of 4 (exercise: why should this be worth so much?).

I also like more points for even differences, since you get odd differences of consecutive squares and never even differences.

I also really like Fawn’s idea of mixing in squares and non-squares to build sensitivity. Less computing, more observing.

7. I don’t know about the “not fun” thing. I know a teacher who told her students on the last day of school that as soon as the whole class (six students) had done 300 math problems, they were done for the year. They totally got into it and she just handed out worksheets and said she should have done that more often… she said if they started going too fast she had harder worksheets. (Now that I think of it, ironically, it could have been novelty — she rarely used worksheets…) They finished ten minutes before the end of the day/year…

8. I would have worked in some visuals, too — and I am afraid I probably would not have been able to resist building in some way of making them see, many, many, many, many times, that 8^2 – 6^2 is totally different from (8-6)^2 … hmmm… like letting them choose between the two answers when they rolled the dice for something…