This week in Algebra 1, my students completed the first part of their inequalities unit with much success, but now storm clouds appear on the horizon – compound inequalities, where english class meets math class with talk of conjunctions – those pesky and’s and or’s. A dice game helped my students make sense of these compound ideas.
The 35 game: 3 students, 3 dice, 3 rounds.
In each round, a player rolls the 3 dice and records their sum. The goal: by the end of 3 rounds, to get as close to a total of 35, without going over. After round 2, each player has the choice to stop if they like, but highest score, closest to 35, wins the game. To help students understand the game, I gave the class time to play in their groups, record results, and think about strategy. The next day in class, we selected 3 students to play in front of the class. Players took turns rolling, and results were recorded on the board after each roll. After round 1, here is how a game between Mickey, Sam and Kim was shaping up:
Kim has taken a small lead. Round 2 rolls then go in order. We record them, then look at the round 2 sums.
Still pretty close, Sam now leading. It’s Mickey’s turn to roll. Mickey probably needs to roll in round 3, but what is he hoping for? Some rolls will cause him to go over. Will any rolls cause him not to take the lead? All students in class were equipped with number lines going from 3-18 which I made using Number Line Generator. Class discussion quickly yielded consensus that 3 was the lowest roll for Mickey, and 14 was the highest. How do we write these as inequalities, and how do these inequalities “play” together. The key word here is “and”, and all students recorded the possibilities:
After we agreed on the interval of possible “safe” values, Mickey made his round 3 roll – and was safe!
A total of 31 – not bad, but 2 other players yet to go. Moving on to Sam, students discussed her possible “safe” rolls, and I was surprised how quickly we were able to generate the inequality. Note, for ease of discussion, we made ties “safe”, as a tie would keep a player in the game (we’d do a new game after to break any ties).
How did Sam do in round 3?
Too much! And Mickey is still in the lead.
Moving to Kim’s turn, I changed the focus from the player to her opponent. Rather than find rolls which are advantageous to Kim, I asked students to think about Mickey: what is HE hoping for her to roll? Which rolls would cause him to win the game? This small twist took a bit more time in groups, and provided rich discussion of the difference between the conjunctions AND and OR. In this case, Mickey would be happy if Kim rolled less than 11 OR if she rolled more than 15. Shading these on the number line revealed a solutions set which looked different from the previous 2:
In the end, this simple game allowed for group discussion and a natural discussion of the conjunctions. In class the following day, we started once more with the game, and I stopped the game now and then to have students sketch solution sets of the game from differing perspectives.
One last note: there is a clear discussion of discrete vs continuous variables to be had here, and I brought it in when it seemed like the class could handle it. In our game, it’s not possible to roll a sum of 9.5, yet we shaded values between integers on the number line. A chance to bring in domain discussion here, where the domain of the game is limited to integers between 3 and 18 versus the real number line we often use, is welcome here – grab the opportunity to highlight the precise mathematical language.