Monday, July 24, 2017

Same Game, Different Grade

I love working with people. Given the choice of math together or apart, or teaching together or apart, I would pretty much always take together.

So I was thrilled when Joe Schwartz was willing to work with me for Twitter Math Camp this year. Among many shared math interests, we both love math games. I've learned a lot from Joe's math game posts, and his Twitter Math Camp 16 presentation on them blew my mind. Such great learning potential the way he approaches games with students.

E.H. Shephard illustrating A.A. Milne
One game with which we had both already done things was Fill the Stairs, which has been around for a while in many variations. I had done a version with Esther Billings and David Coffey in an inservice, that involved having cut up bags of numbers. Joe had done a version with digit cards 0 to 9 where students fill in a staircase with 10 on the bottom stair
and 100 on the top stair. Joe and I had connected on a version I came up with earlier called Decimal Pickle. As we talked about it, the spirit of Tracy Zager began again. She's been haunting me all year from her TMC16 keynote, where she challenged us to do cross-grade collaboration. "What if we did variations of the game across grades?" Joe wondered. So we were in.

Necessary references: Joe's first post about it, and the redux.

The idea for Decimal Pickle came from a need for comparing decimal numbers of different length. Why would we flip different numbers of cards? The colors are pretty intuitive there. Black? Flip again! It added a lot of excitement to the game, almost a black jack feel. For the mathematics, it was perfect for the 5th graders to compare tenths, hundredths and thousandths.

Talking with Joe got me thinking about the big topic. How does order show up across the grades? When I think about number sense, I see a few components. First, number as quantity. Or the numbers in context. But second only to that is comparison. Well, second is representation. But third only to those two is comparison. And comparison before computation, which is right out. But this idea of order is really an up and down the curriculum issue. As numbers grow more complex, how to order them is very relevant. It's the experiential aspect of number that we often ignore as we get farther up the curriculum. One of the strengths of this game is that it requires comparing more than two numbers. I think ordering a set is more complex and challenging task. There may be a component of number sense I haven't thought about at play, a kind of sense of distribution.

I like games to use easily accessible materials, so playing cards are great. I often use J as 0. (And if the kids are old enough, "You know what you have if you've got Jack?")  I'm not sure why I first tried having students make their own gameboard, but I love it, now. There are students for whom that's their in for the game. (Deep game design - there's probably a whole player psychographic aspect to this.)

The first step for me on this generalize the game journey was to fit it for 3rd graders. I wanted single digit and 2 digit (teachers asked for no three digit), but thought that half and half was a weird balance. I settled on turn over a card. On a diamond you stop, otherwise turn over another.

Kids are not as familiar with playing cards as they used to be, so we started with something halfway between notice and wonder and Which One Doesn't Belong.  Then, as I often introduce games, I played vs the entire class. Then they break up and play in 2 vs 2 teams. One of Joe's great ideas was to have students make a number line with their results. Great task, ripe for discussion, strong in representation, awesome assessment.

Thinking about how to go even younger, I was thinking about sorting single digit cards. But how to make a game out of it? First came the first grade variation. Fill five spaces. Flip cards like War to start in the the middle space. Higher goes first (advantage) but their card is probably too big for the middle space (disadvantage). Every flip you place in a spot. If it's the same as a card you have, cover that card. Cards have to stay in order. So if you have 3 __ 4  6  __   __ and you turn over a 5, you can cover the 4 or the 6, but not the blank between 3 & 4.

That requirement to not move cards was too much for Kindergarten, so they could move their cards around. That generated plenty of discussion, too.

Thinking about how to extend the game past high school was a challenge. I kept thinking about order of operations. One of my pet peeves is PEMDAS, as I want students to think about 4 levels, grouping - exponents - multiplication/division - addiition/subtraction. (GEMS if we need an acronym.) So I thought about doing the red/black for more cards. 2 cards minimum, you can add or subtract. Three cards, have to join with add/subtract and multiply/divide. Four cards, have to do an exponent or root. Five cards you have to use a grouping structure, parentheses, radical, fraction bar. I think this could work, but haven't got a chance to test it yet. The class in which I was going to get to test was a college algebra class working on exponents, so this became the variation.

Marcel Duchamp
It was great. The students were constantly surprised by their results, got a lot out of comparing these extreme numbers, and became more efficient at arranging the numbers to get the effect they wanted even in the course of an hour playing the game. Older students also need these play experiences, I think they just abstract from them more quickly than younger students.

We'll be talking about this at Twitter Math Camp 17, so I hope you can join us if you're there. Here's a page with downloads and resources:, regardless. If you have ideas for more variations or get to try one of these, let me know!

What games do you use that connect to a big idea in math?


  1. I love the choice in the exponential version of the game. Gameplay seems to involve both placing and choosing the order of your a, b, and c in the exponential expression. Thanks for sharing the game -- will keep a note of it for my Algebra 1 class.

  2. Thanks for the comment! Choice is where a lot of the engagement in games lives, and one of the reasons I like these Fill the Stairs games so much more than War variations. Choice is where strategy comes in, so bonus problem solving. And not unrelated to engagement!

  3. This is really cool! I like how you've taken one basic idea and come up with ways to make it both engaging and mathematically challenging across a wide range of grade levels. I'm sorry I won't be able to see you present on it at TMC!

    1. Thanks, Katherine. Maybe next year in Cleveland?