This week, in second grade enrichment math, the students received a new assignment from Gregory Goodwin — they were charged with the responsibility of testing the game Pop Go the Sticks.
The students were each given a game board, a rule card, a paper clip game piece, and three popsicle sticks. They read the rules of the game to prepare to play. According to the rules, Player A moves two spaces if all the sticks land with their design side up, and one space if all the sticks land with their plain side up, whereas Player B moves two spaces if one stick lands with its design side up and the two other sticks are plain, and one space if the sticks land with one plain side and two design sides up.
I asked the students to explain if they thought the game was fair. All but one of the students answered that they thought the game was fair and that each player had an equal chance of winning. Then, they began to play the game.
Normally, my goal is for the students to play each game at least twice, so each student has an opportunity to be Player A and Player B. On this day, though, we stopped halfway through the first game to check in. In each class, we had four simultaneous games running. In all eight games, Player B was winning by a lot. We talked about how likely it would be for this to happen if it were a fair game.
Next, we listed the possible outcomes after tossing the three sticks. We made a tree diagram and came up with the following possible outcomes: ppp, ppd, pdp, pdd, ddd, ddp, dpd, dpp (where “d” stands for “design” and “p” stands for “plain”). Player A can only move if the sticks land ddd or ppp; Player B can move if the sticks land in any of the other configurations. Once the kids saw the information displayed this way, they were easily able to articulate that Player A had a 2 in 8 chance of being able to move, and Player B had a 6 in 8 chance of being able to move. I could see their faces change as they came to the realization that the game was far from fair.
The students’ final task was to write a letter to Gregory Goodwin, explaining their conclusion that the game is unfair, and then describing how the rules of the game could be changed to make it fair. The kids came up with a variety of ways to change the rules.
Each student brought home the game board and popsicle sticks. Some of them planned to challenge you to play the game with them at home (without letting you know that the game is unfair). They were quite tickled that they were pretty much guaranteed to win. Hey, you take a W wherever you can, right?