Against All Odds

To meet a need for resources for the new Math 10 curriculum, the Saskatchewan Teachers' Federation in cooperation with Saskatchewan Education Training and Employment, initiated the development of teacher-prepared unit plans.

A group of teachers who had piloted the course in 1992-93 were invited to a two and a half day workshop in August, 1993 at the STF. The teachers worked alone or in pairs to develop a plan for a section of the course.

Jim Beamer, University of Saskatchewan and Lyle Markowski, Saskatchewan Education Training and Employment acted as resource persons for the workshop.

Topic: Probability

Subject: Math 10

Overview

*management of a gambling establishment

Rationale

A. Foundational Objectives

• Appreciate the role of probability in understanding everyday situations
• Communicate a summary of financial projections in appropriate reports, tables, and graphs
• To support students in treating themselves, others, and their environment with respect
• Promote ideas, processes, experiences, and objectives in meaningful math contexts

B. Outcomes

• List sample space and events in a random experiment
  
• Calculate experimental probability for simple events
  
• Calculate the theoretical probability of an event and the probability of its complement.

• participation 10%
• group quiz 20%
• report evaluation 70% (35%group evaluated/35% teacher evaluated)

While this will produce a "numerical mark", evaluation through checklists can evaluate a student's social skills, work habits, contribution to group, analytical ability, etc. We would recommend that the group as a whole do self-evaluation based on the statement, "The highest mark should go to the group with the highest profit on the most ethical game." The idea is for groups to consider both issues and then come up with an evaluation system that, democratically, the class can live with!

Note: This is only a suggestion. Feel free to use a method you feel comfortable with. This evaluation is used to introduce some of the techniques mentioned in the Evaluation Handbook.

Common Essential Learnings

Adaptations

While we have chosen to learn only the basics, students will tend to come up with projects that may involve permutations and combinations. You will have to find a level you are comfortable enough with. If you want a challenge, allow your groups to tinker with the concepts of permutations/combinations. However, we would encourage you to allow students to do some discovery learning or perhaps some individual learning rather than having you teach the units.

Resources

Assessment Techniques

Teacher Assessment

2. The end report
Report is designed to show students' thought processes in trying to sell you their game. Our evaluation would concentrate on the areas of:
(a) Did the game meet the initial criteria?

(b) Did the report clearly show and explain how the odds were calculated?

(c) Did the group explain their balancing of the concepts of ethical vs profit?

(d) The overall organization and presentation of the report should be well thought out and organized.

Report might be worth 70% or so.

3. Performance Assessment
While we wanted to come up with a numerical grade, we wanted to try some checklists as well. The 3 checklists we chose to use are found on page 84-86 of the Student Evaluation Handbook. We would encourage you to revise these to fit your own areas of importance. They are good for creating anecdotal records to keep in a student's file.

Group Evaluation

1. Other Groups' Games
   It is important that the criteria be laid out clearly before the evaluation begins. We specifically wanted students to go through the process of developing a group evaluation but the teacher will need to carefully monitor the results so that the instrument works. We would encourage a system that is quite simple. For example:
   Amount of Profit Ethics Enjoyment of Game Overall
   __/10 ___/10 ___/10 ___/10
   

   >From this the groups should be able to defend each ranking including reasoning for giving their "number"
   
   

2. Self-evaluation
   Have students evaluate each other's performance within the group by using a scale such as the one on page 85 of the Student Evaluation Handbook, but we recommend they also give a number out of 10 to show how they feel about an individual's overall contribution to the group. You can make it clear that you will use this mark as part of your final numerical evaluation.

1. Introduce the concept of probability by having students brainstorm for possible areas where probability might be used or observed. The use of dice, spinners, thumbtacks works well as introduction. Eg. Surveys, lotteries, games, genetics. 10 minutes
2. Introduce the 10 minute film "Of Dice and Men" (National Film Board) and show it. 12-15 minutes
3. Debrief the film to extract the terms
   • probability )
     
   • simple event ) Record these
     
   • sample space )
     
   • experimental probability )
     
   • theroretical probability )
     
   10 minutes
   
4. Discuss gambling, lotteries, games of chance, etc.
   
   
   

   ---

   TEACHER TIP #1
   
   Note: Introduce the notion of (a) "fairness of the game," (b) "addition to the game,"
   (c) who are the players? 10 minutes
   

   ---

   
   
   
   
5. Focused Imaging (Have students close their eyes and imagine as you speak) 5 minutes.

• It is intended to serve novices to gaming; it must be simple!
• It should be fun to play and make the player want to come back.
• It should have a fair chance of being won at any time. No ties permitted.
• It should (on the long term) make money for the Bingo Palladium.
• Evaluation will be based on the statement: Most Profit, Most Fun, Best Ethics.

(Example) Rolling a die to get a 6.

• Simple events are 1, 2, 3, 4, 5, 6
• Sample space is (1, 2, 3, 4, 5, 6)
• Desired event is (6)
• Probability of rolling 6 is: Desired Event = 1/6
• Sample Space

• simple event
• sample space
• P (E) (Probability of an event)
• Experimental Probability
• (ie) actual = P (Experimental Event)
  attempted

• Have groups elect a leader and a recorder.
• Arrange papers and pencils, etc...functional arrangements.
• These groups will become the "Bingo Palladium" owners and their staffs.
• Distribute a paper copy of the focused imaging scenario from day 1 as well as the rules of engagement (see Hour #4 page 9).
• Review the situation briefly and allow the students the balance of the period to discuss, invent, and "write out" the description of their game.

1. A description of the game - be detailed - use examples.
   
2. The sample space of all possible simple events . List these if possible.
   
3. The probability of at least 5 selected outcomes that would happen playing your game.
   
4. A financial plan
   
   • What does it cost individuals to play it once.
     
   • How much can a player win on any one play...examples
     
   • Predict what the "Bingo Palladium" can expect to make if 1000 people play the game once each. (or otherwise project profits and validate the amount)
     
   • If your game or the playing of it depends upon factors or events out of the Palladium's control.
     
   
   
5. Estimate by writing down what you think will be the effect on a customer...will he/she want to play again?
   
6. This project write-up is due in Hour #4 (game day) and will count in your evaluation.

C. Allow the balance of the period for group discussions and planning.

Instruction:

1. Projects and write ups are due tomorrow (next day)
   
2. Materials for game day are student responsibilities
   
3. Bingo/Poker chips or monopoly money for the players will be provided by the teacher.

Rules of Engagement:

1. Set up your group's game on a table somewhere in the room.
   
2. Designate 2 players to work first half of period, 2 players for the second half.
   
3. Rest of players in group become players.
   
4. Group receives $10,000 in play money or chips in denominations of $50, $100, $500.
   
5. Top wager is $500.
   
6. Each player carries a piece of paper for each of the other group's games. On it he/she records the amount wagered on each bet, the win or loss, and how much lost (if any).
   
7. The playing lasts 20 minutes.
   
8. You may set a maximum limit as to how many players can participate each time.
   Option: If time allows and you can line up some prizes (pop, chocolate bars, etc.), have an auction and let groups bid for them. Adds a little "incentive" to the day.
   
9. At end total your group's wins/losses for each game. Send each group your stats on their game.
   
10. If time allows, discuss your feelings about each of the games, including your own. Were they ethical? Were they money makers?

B. Teacher writes on the board: (judgement criteria)
Which are: "Most Profit, Most Fun, Best Ethics"
Now we initiate a group process which seeks to evaluate the results. This may be a bit chaotic and may not produce definitive results.
Suggestions:

1. Obtain an exact tally of each student's winnings or losses (Post this).
   
2. Obtain an exact tally of each group's winnings or losses (Post this).
   
3. Make a checklist of value judgements on the "play worthiness" of games in general. Circulate the checklist (one for each game) (Post the results).
   
4. Initiate a class discussion which addresses ethics of the gambling event:
   
   
   • Degree of profit making
     
   • Moral tone of gaming/gambling
     
   • Gambling as a tax on the poor
     
   
   
5. Review the mathematical foundation of each game (time permitting)
   
   That is: Probability of winning ) could tie in to
   Probability of not winning (the house wins) ) complement
   Percentages - wins/losses
   
   
   
6. Circulate a page where each student writes his/her name followed by a ranking of groups
   
   

   	Ex. Joe Smith: 2, 5, 1, 3, 4 (first to last)
   
   ...the class can then do a totalling exercise such as:
   
   		first - 5 points )
   
   		second - 4 points ) The totals create an overall
   
   		third - 3 points ) ranking
   
   		fourth - 2 points )
   
   		fifth - 1 point )