1. Fostering Students’ Probabilistic Thinking: Big Ideas and Connections Kyle T. Schultz James Madison University schultkt@jmu.edu

2. Welcome  Grab a cup of M&Ms and a game sheet with the 12 circles  Place 10 M&Ms in the circles in whatever arrangement you choose  You may put more than one M&M in a circle  You will get a chance to revise your choices for the next game

3. 10 M&Ms Game  What “rules” or strategies for placing the ten counters might students follow?

4. 10 M&Ms Game: Student Strategies  Be bold: All or nothing, load it up!  Focus on numbers rolled in the previous game.  Focus on unrolled numbers. “It’s due!”  Spread it out.  Whimsy (“Seven” rhymes with my boyfriend’s name, “Kevin”)  Visual Pattern

5. 10 M&Ms Game  What “rules” or strategies for placing the ten counters might students follow?  Mathematically, what is the best strategy for placing the counters?

6. 10 M&Ms Game  What “rules” or strategies for placing the ten counters might students follow?  Mathematically, what is the best strategy for placing the counters?  How does this game address some of the “big ideas” of probability?

7. Big Ideas of Probability  Chance has no memory.

8. Big Ideas of Probability  The probability of a future event occurring can be characterized on a continuum between impossible (0) and certain (1).

9. Big Ideas of Probability  The relative frequency of outcomes in experiments can be used to estimate the probability of an event. The more trials, the better the estimate.

10. Big Ideas of Probability  For some events, the exact probability can be determined by analyzing the properties of the event itself. (Theoretical Probability)

11. Big Ideas of Probability  Simulation is a technique used for answering real-world questions or making decisions in complex situations in which an element of chance is involved. Van de Walle, Bay-Williams, Lovin, & Karp (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 6–8. Boston, MA: Pearson.

12. Fraction Wall  On her or his turn, each player:  Rolls the dice  Shades in the bricks of the wall equivalent to what was rolled (Each row is one whole.)  Tries to fill in the entire wall before her/his opponent(s) do  Must roll the exact amount needed to complete the wall

13. Fraction Wall  What do students have to know to be successful at this game?  What strategies might students develop to be more successful at this game?  How can the big ideas of probability be addressed through playing and discussing this game?

14. Assessing Progress

15. Fraction Wall  Game Variations  Players cannot color in the value they rolled  “Freeze” one die when only one brick remains

16. Another Investigation The World Series Problem The Cardinals and Tigers are playing a best-four-of- seven series, with the winning team to collect a $1,000,000 bonus. When the Cardinals led two games to one, the umpires went on strike due to a labor dispute, causing the cancellation of the remainder of the games. If the Cardinals and Tigers are evenly matched and if the money will be divided in proportion to the team’s probabilities of winning the series, how should the money be divided?

17. World Series Task  Discuss in groups of two or three:  Do the simulation results affect your thinking about the task?  In what other ways could this task be represented?

18. Thanks  If you would like electronic copies of the materials used in this presentation, send me an email: schultkt@jmu.eduschultkt@jmu.edu  I would appreciate your feedback on this session.

19. Relevant Virginia SOLs 6.16 The student will a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events. 7.9 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event. 7.10 The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle. 8.12 The student will determine the probability of independent and dependent events with and without replacement.