PurposeLesson activity

DescriptionThis VMCAnalytic explores the progression of reasoning in Math used by a group of 6th-grade students as they work on dice game problems. The students are asked to decide whether or not a game in which points are given based on the sum of the numbers shown when two dice are rolled is fair. If is not fair, meaning the players have an unequal chance of winning, the students are to discern which player has the advantage, as well as how the game could be changed so that the players do have an equal chance of winning. The students explore this question without using a background in probability and know to defend their decisions and thought processes along the way with various types of reasoning. Throughout the clip, the students are shown playing the dice games and recording their results. They are prompted by the researcher to conjecture which player has a greater chance of winning based on each game’s rules, which is often worded as “Which player would you rather be?” The students explore ideas of probability without naming them, as seen in the clips. The dice games invite students to be engaged with the mathematics and work together in natural ways. A study conducted by Miami University found that activities and games in math lead to improved understandings of course concepts, and dice games especially were shown to provide increased student engagement and interest (Maurer, Hudiburgh & Werwinski, 2019).

As shown in the events, the 6th-grade students show their progression of reasoning throughout their work on the Dice Game problem in the analytic. In the beginning, the students spend time playing the Game for Two Players and work on finding the possibilities for rolling each sum. Stephanie finds the number of ways each number can be rolled by creating a chart with 1-6 for the first dice on the left and 1-6 for the second dice on the top. She wrote the sums inside her table. Her chart shows that it is impossible to roll a sum of 1, because the smallest sum is 2 (if each dice rolls a 1). There is only one way to roll a sum of 2, which is by rolling a 1 on both dice. A sum of three can be rolled in two ways, which are 2 and 1 and 1 and 2. The students hold a long discussion about whether these rolls are considered the same or whether both should be included. The discussion is very fruitful as students justify their reasoning about whether or not both should be considered. Interesting questions come up, such as if 1 and 1 would also have two cases, and these questions are tackled through student discussions and examples using the physical die.

Students also have discussions about how to make the game “fair”. They work on distributing sums to each player and trying to be fair in that each person has equal chances of winning the game. Stephanie’s chart is referenced frequently to accomplish this. Considerable growth in understanding possible sample space outcomes is made through the session: (1) Stephanie’s organized charts; (2) the discussion of the outcome of rolling a 2 on one dice and a 1 on the other as compared with rolling a 1 on the first dice and a 2 on the other; and (3) coming up with ways to make the game fair. Stephanie’s foundational conjecture helped the students work through understanding the outcomes. Stephanie’s chart showed that it is impossible to roll a sum of 1; a sum of 2 can be rolled only in one way; a sum of 3 can be rolled in two ways; a sum of 4 can be rolled in three ways; a sum of 5 can be rolled in four ways; a sum of 6 can be rolled in five ways; a sum of 7 can be rolled in six ways; a sum of 8 can be rolled in five ways; a sum of 9 can be rolled in four ways; a sum of 10 can be rolled in three ways; a sum of 11 can be rolled in two ways; a sum of 12 can be rolled in one way. This helped the students determine which player had a higher chance of winning, that is, the player whose sum can be found with more combinations of rolls. The students also show an understanding of ideas of theoretical probability versus experimental probability in their discussions. Stephanie makes many references to this idea, saying that, "There’s a theory that player two, player B, has more of an advantage even though it is a game of luck. No way you could change it so it’s not a game of luck. No matter what you do, it’s gonna be a game of luck." Theoretically, certain sums have higher chances because there are more ways to make them than others. However, experimentally, anything can happen once the die are rolled. Any player can end up winning the game, because a higher chance does not guarantee a win and a lower chance does not guarantee a loss. The students agree when Stephanie explains this near the end when presenting her charts on the screen. They conclude that although there is a higher chance for certain sums to be rolled than others, is still a game of luck in the end.

References:

1. Maher, C. A. (2010). The longitudinal study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 3-8). Springer: New York, NY.

2. Maher, C. A. & Alston, A. S. & (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty & J. T. Zilliox (Eds.), Proceedings of the 27th Annual Conference of the International Group for the Psychology of Mathematics Education, (2), 25-32. Honolulu, HI: CRDG, College of Education, University of Hawaii.

3. Maurer K, Hudiburgh L, Werwinski L. What do students gain from games? Dice games vs word problems. Teaching Statistics. 2020;42:41–46. https://doi.org/10.1111/test.12214

Video References:

1. B49, Dice games for two players (student view), Grade 6, March 21, 1994, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3NG4NR2

Problem Statement:

Game for Two Players:

Roll two dice. If the sum of the two is 2, 3, 4, 10, 11, or 12, Player A gets 1 point (and Player B gets 0). If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point (and Player A gets 0). Continue rolling the dice. The first player to get 10 points is the winner.

Final Game:

Winner Takes All -Roll two, six sided, dice. If the sum is 7, Player A wins the game. If the sum is 8, Player B wins. Continue rolling the dice until there is a winner. Suppose you have a choice to be Player A or Player B. Which would you choose?

As shown in the events, the 6th-grade students show their progression of reasoning throughout their work on the Dice Game problem in the analytic. In the beginning, the students spend time playing the Game for Two Players and work on finding the possibilities for rolling each sum. Stephanie finds the number of ways each number can be rolled by creating a chart with 1-6 for the first dice on the left and 1-6 for the second dice on the top. She wrote the sums inside her table. Her chart shows that it is impossible to roll a sum of 1, because the smallest sum is 2 (if each dice rolls a 1). There is only one way to roll a sum of 2, which is by rolling a 1 on both dice. A sum of three can be rolled in two ways, which are 2 and 1 and 1 and 2. The students hold a long discussion about whether these rolls are considered the same or whether both should be included. The discussion is very fruitful as students justify their reasoning about whether or not both should be considered. Interesting questions come up, such as if 1 and 1 would also have two cases, and these questions are tackled through student discussions and examples using the physical die.

Students also have discussions about how to make the game “fair”. They work on distributing sums to each player and trying to be fair in that each person has equal chances of winning the game. Stephanie’s chart is referenced frequently to accomplish this. Considerable growth in understanding possible sample space outcomes is made through the session: (1) Stephanie’s organized charts; (2) the discussion of the outcome of rolling a 2 on one dice and a 1 on the other as compared with rolling a 1 on the first dice and a 2 on the other; and (3) coming up with ways to make the game fair. Stephanie’s foundational conjecture helped the students work through understanding the outcomes. Stephanie’s chart showed that it is impossible to roll a sum of 1; a sum of 2 can be rolled only in one way; a sum of 3 can be rolled in two ways; a sum of 4 can be rolled in three ways; a sum of 5 can be rolled in four ways; a sum of 6 can be rolled in five ways; a sum of 7 can be rolled in six ways; a sum of 8 can be rolled in five ways; a sum of 9 can be rolled in four ways; a sum of 10 can be rolled in three ways; a sum of 11 can be rolled in two ways; a sum of 12 can be rolled in one way. This helped the students determine which player had a higher chance of winning, that is, the player whose sum can be found with more combinations of rolls. The students also show an understanding of ideas of theoretical probability versus experimental probability in their discussions. Stephanie makes many references to this idea, saying that, "There’s a theory that player two, player B, has more of an advantage even though it is a game of luck. No way you could change it so it’s not a game of luck. No matter what you do, it’s gonna be a game of luck." Theoretically, certain sums have higher chances because there are more ways to make them than others. However, experimentally, anything can happen once the die are rolled. Any player can end up winning the game, because a higher chance does not guarantee a win and a lower chance does not guarantee a loss. The students agree when Stephanie explains this near the end when presenting her charts on the screen. They conclude that although there is a higher chance for certain sums to be rolled than others, is still a game of luck in the end.

References:

1. Maher, C. A. (2010). The longitudinal study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (pp. 3-8). Springer: New York, NY.

2. Maher, C. A. & Alston, A. S. & (2003). Modeling outcomes from probability tasks: Sixth graders reasoning together. In N. A. Pateman, B. J. Dougherty & J. T. Zilliox (Eds.), Proceedings of the 27th Annual Conference of the International Group for the Psychology of Mathematics Education, (2), 25-32. Honolulu, HI: CRDG, College of Education, University of Hawaii.

3. Maurer K, Hudiburgh L, Werwinski L. What do students gain from games? Dice games vs word problems. Teaching Statistics. 2020;42:41–46. https://doi.org/10.1111/test.12214

Video References:

1. B49, Dice games for two players (student view), Grade 6, March 21, 1994, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3NG4NR2

Problem Statement:

Game for Two Players:

Roll two dice. If the sum of the two is 2, 3, 4, 10, 11, or 12, Player A gets 1 point (and Player B gets 0). If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point (and Player A gets 0). Continue rolling the dice. The first player to get 10 points is the winner.

Final Game:

Winner Takes All -Roll two, six sided, dice. If the sum is 7, Player A wins the game. If the sum is 8, Player B wins. Continue rolling the dice until there is a winner. Suppose you have a choice to be Player A or Player B. Which would you choose?

Created on2022-04-30T18:27:25-0400

Published on2022-07-11T09:25:28-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-c3d5-5692