Discrete+Mathematics


 * __Discrete Mathematics__**

Write the questions for students on the board and have them discuss them with each other. Ask students to share their responses to each of these scenarios. Ask students why their responses may be different from their classmates. Ideally the class discussion will mirror some of the concepts which follow. Every day each of us must make choices like those described above. The choices we make are based on the chance that certain events might occur. We informally estimate the probabilities for events by using a variety of methods: looking at statistical information, using past experiences, asking other people's opinions, performing experiments, and using mathematical theories. Once the probability for an event has been estimated, we can examine the consequences of the event and make an informed decision about what to do. Making the connection between choice and chance is basic to understanding the significance and usefulness of mathematical probability. We can help middle school students make this connection by giving them experiences wherein choice and change come into play followed by tasks that cause them to think about, and learn from, those experiences. The game of SKUNK presents middle-grade students with an experience that clearly involves both choice and chance. SKUNK is a variation on a dice game also known as "pig" or "hold'em." The object of SKUNK is to accumulate points by rolling dice. Points are accumulated by making several "good"(anything but the SKUNK number) rolls in a row but choosing to stop before a "bad"(the SKUNK number) roll comes and wipes out all the points. SKUNK can be played by groups, by the whole class at once, or by individuals. The whole-class version is described following an explanation of the rules.
 * **Learning Objectives**
 * practice decision making skills
 * investigate winning strategies
 * gain a basic understanding of experimental probability
 * **Materials**
 * *** Number cubes (dice)**
 * **Activity Sheet**
 * **Instructional Plan**

At the top of a piece of paper the word SKUNK will be written down. Draw a vertical line in between each letter in the word SKUNK making columns for each letter; this is the student's game board. Before the game starts, the teacher will roll the dice once to come up with the SKUNK number; this is the number that students do NOT want rolled for the rest of the game. The game starts out in the //S// column. The teacher will roll the dice and whatever number is rolled, the students will write down under their column //S//. The teacher will keep rolling and the students will write down the number being rolled. If the teacher happens to roll the SKUNK number, any player who is still playing the column being rolled for will lose all of their points. The students decide when they want to stop chancing all of their points under a column. For example, let us construct a game. I am the teacher and I roll a 5 for the SKUNK number before the game starts. I tell the students the SKUNK number. We are under column //S// and I roll a 6. A student will write their 6 down and has the choice to continue playing or withdraw from the column. When a student withdraws, they keep their 6 points they earned and wait until column //U// to play. Other students might choose to stay in the game under column //S// to accumulate more points but when doing so, they are risking losing all of their points for that column if a 5 is rolled. Students can choose to withdraw between each roll. When the SKUNK number, a 5, is rolled, the game moves to the 2nd column, //K// column where a new SKUNK number will be rolled for that column. The game will continue at this pace until we spell the entire word skunk. At the end of the game, the students add up their column points to determine who has the most points.
 * ** Introduction to Skunk **

This rating chart was devised by assuming that, on average, a "one" happens on about the third dice roll and the average score per good roll is "8." Therefore, with a strategy of "roll twice then stop" on each round, a person might get about 16 points on perhaps four out of five rounds for a total score of about 64. The 20 point intervals used for each category are arbitrary. Whichever rating scale students create, they should justify their reasoning for the intervals. The teacher may have each group of students devise their own game involving choice and chance. Writing up consistent, clear rules for their game will involve mathematical logic. Have each group present its game to the rest of the class. The teacher can eitherlet the whole class choose which of the games they would like to play or take a chance by selecting one of the games at random. Choice or chance? It's decision time! In groups, students can create a "rating chart". For example:
 * **Questions for Students**
 * I might make more money if I was in business for myself; should I quit my job?
 * An earthquake might destroy my house; should I buy insurance?
 * My mathematics teacher might collect homework today; should I do it?
 * **Assessment Options**
 * **Extensions**

0 to 20 - needs improvement 21 to 40 - you might do better 41 to 60 - average 61 to 80 - good over 80 - outstanding This rating chart was devised by assuming that, on average, a "one" happens on about the third dice roll and the average score per good roll is "8." Therefore, with a strategy of "roll twice then stop" on each round, a person might get about 16 points on perhaps four out of five rounds for a total score of about 64. The 20 point intervals used for each category are arbitrary. Whichever rating scale students create, they should justify their reasoning for the intervals. The teacher may have each group of students devise their own game involving choice and chance. Writing up consistent, clear rules for their game will involve mathematical logic. Have each group present its game to the rest of the class. The teacher can eitherlet the whole class choose which of the games they would like to play or take a chance by selecting one of the games at random. Choice or chance? It's decision time! **Game “//Constructing Blocks//”** Our original game was a game where we had students get into teams. Then one team rolls the dice. Teams take the number rolled and construct a “tower” with that many blocks trying to achieve the desired task. Steps 1-2 will be repeated changing who rolls the dice and the desired task. We then had students try to figure out what their surface area of the object was. When we played this game with students in Selah, Washington the game was too difficult. Many of the students were younger than anticipated though. I was under the impression that they would be middle school students. We felt like”//Construction Block//” game was not very fun, so we ended up making a new game. Our new game is called “Skunk” and we feel as this game should be more interesting to students while still aligning with the discrete mathematics standards.
 * Teacher Reflection**


 * Process Standards**
 * [|Representations]