Against All Odds
by
Murray Sanders
Indian Head School Division
and
Eric Hamm
Northern Lights School Division
CELs
1993
S105.7
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.
Time Allotment:
5 class periods
Topic:
Probability
Subject:
Math 10
Overview
Students will be very briefly introduced to the concept of probability
and then placed into a group learning situation. The group will be asked
to present a report that develops a new game of chance to be used with
new players. The group will design a game, present charts showing the odds
of winning and the probability of the 'house'* making money. After the
games are tested, the other groups will evaluate the game and discuss its
merits. Discussion will then proceed to consider whether a player can actually
make money and the impact gambling has on the individual and society in
general. The students will learn the math (almost implicitly) while doing
some serious thinking about profit/loss and the effects of gambling on
society.
*management of a gambling establishment
Rationale
There are few good opportunities in the math classroom where we
can spend time on the Common Essential Learnings skill of Personal and
Social Values and Skills. This unit is ideally suited for it by studying
the impact of gambling on the individual and on society. While we realize
that this unit could be done during one or two class periods we strongly
recommend that the extra time be allotted so that some of these outside
considerations will be examined. This unit does not advocate gambling but
has the students consider its implications. The students learn the math
skills on an implicit level, which we believe is very effective in this
situation.
Because this unit is often a major change of pace from the regular math
classroom, it can be used as a breather between two heavy units. The students
appreciate the change of pace and will react quite positively (well - most
of the time!) to this unit. The discussion over the establishment of the
evaluation criteria, the new game, and the discussion of the merits and
effects of each game will create lively classrooms and you will need to
be ready to "rein them in".
Good Luck and have fun!
Objectives
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.
Evaluation
Evaluation will be done on an on-going and end-product basis and
will be based on 3 categories.
- 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
This unit actually uses all of the CELs with the possible exception
of Technological Literacy. The stress would be on the three CELs of Critical
and Creative Thinking, Communications, and specifically Personal and Social
Values and Skills. The stress is deliberately on the Personal and Social
Values and Skills CEL and we would encourage you to stress this as well.
Adaptations
We have deliberately chosen to leave this unit slightly vague in
terms of instructional approaches. It is conducive to group work if you
are wishing to try some co-operative learning. We would encourage you to
use this concept but to fit it into a style you are comfortable with.
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
This unit is designed to allow you a wide range of resources. The
film Of Dice and Men is a good introduction and we do encourage
you to borrow it. Most of our probability material comes right out of current
texts in our Book Bureau such are:
Algebra and Trigonometry: Structureand Method Book 2, Mary P.
Dolciani and others, 1982 and Addison-Wesley
Mathematics 10, Brendan Kelly and others, 1987.
Other useful resources are:
MECC-Probability Kit-; F.Y.I. For Your Imagination, Instructional
Strategies Series, Saskatchewan Professional Development Unit and Saskatchewan
Instructional Development and Research Unit;
Teaching Statistics and Probability,
National Council of Teachers of Mathematics, 1981;
Exploring Probability
by Newman, Obremski, and Scharfer (Seymour Publications ISBN: 0-86651-333-7
c1987).
You may also want to line up some dice (you can use dice with any
number of faces you wish), coins, cards, thumb tacks, spinners, etc. for
students to explore with.
Assessment Techniques
There is a lot of room for flexibility here but this is a possible
option for you to use.
Teacher Assessment
1. The Quiz on day 4
It is designed to ensure that the essential math skills be checked.
We are encouraging group dynamics by having the group be responsible for
ensuring that all students know and can show how to do each of the questions.
The individual student in the group will be responsible for only 1 or 2
questions but the group score will be a compilation of all students' scores.
Quiz might be worth 20% (don't be afraid to change these values.)
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
- 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"
- 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.
Hour #1 Suggested Activities
- 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
- Introduce the 10 minute film "Of Dice and Men" (National
Film Board) and show it. 12-15 minutes
- Debrief the film to extract the terms
- probability )
- simple event ) Record these
- sample space )
- experimental probability )
- theroretical probability )
10 minutes
- 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
- Focused Imaging (Have students close their eyes and imagine as you
speak) 5 minutes.
For teacher background see: F.Y.I. For Your Imagination.
Hypothetical Situation:
Imagine yourself living as part of a community in the north. The area
is forested and the ground is rocky (Precambrian shield area). People here
grow up, have babies, play hockey, shop for groceries, and work for a living
like anywhere else except that here, the copper mine is the main employer.
Many people here enjoy outdoor recreation but they also like to gamble
. . . many folks play "Bingo" for fun. You are the owner of the
"Bingo Palladium" and today you've decided to introduce a brand
new game to some of your patrons.
Unfortunately the game hasn't been invented yet and you must invent it
with the help of your staff. The criteria/rules of the game are to be:
- 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.
Tomorrow you will meet with your staff and come to a decision.
Hour #2 - Suggested Activities
A. Perhaps try a little handout/chalkboard example, concept attainment
activity, or direct teaching to familiarize the students with the simple
math inherent in the probability of an event.
TEACHER TIP #2
Note: This is a good place to try an alternate teaching style such
as concept formation by showing students examples that work and examples
that did not and having them guess how the desired event was calculated.
15-25 minutes
(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
Make sure students know:
- simple event
- sample space
- P (E) (Probability of an event)
- Experimental Probability
- (ie) actual = P (Experimental Event)
attempted
One could try a few experiments here with dice, cards, or spinners.
B. Group the students into groups of 5 or so.
- 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.
TEACHER TIP #3
Note: Students will need a fairly clear set of instructions on
what the game write-up might entail. Because these are suggested activities
you may want to "tailor make" required outcomes. Here are a few
you could try:
Write a game report that includes:
- A description of the game - be detailed - use examples.
- The sample space of all possible simple events . List these if
possible.
- The probability of at least 5 selected outcomes that would happen playing
your game.
- 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.
- Estimate by writing down what you think will be the effect on a
customer...will
he/she want to play again?
- 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.
TEACHER TIP #4
Note: You may need to provide practice materials so it is a good
idea to have an assortment available (eg) cards, dice, spinners, multifaceted
die (>6), etc...
Note: Also that on game day the "Bingo Palladium" will provide
all necessary materials; that is, the student groups themselves arrange
and prepare for this.
Hour #3 - Getting Ready
A. Today's objectives include reinforcing the underlying mathematics
inherent in basic probability problems. In addition we want to allow additional
planning time to prepare "the game."
Arrange the seating into the groups as before and provide each with blank
paper, pencils, etc. Once settled, briefly review (a) simple events, (b)
sample space (c) P (Event) (d) Experimental probability and introduce the
probability of the complement of an event; P(E). One can quickly relate
this to winning a coin toss or losing it and at the same time explore the
total range of probabilities from 0 to 1 (i.e. impossibility and certainty).
25-45 minutes
The teacher needs to have previously prepared 10 questions on
probability
that reflect the concepts to be learned. These should be typed and copied,
ready for distribution.
Note: See Dolciani "Modern Algebra Book 2", page 600-603 as an
example.
Instruction:
Each person in the "Bingo Palladium" (group of 5) must
now learn to do the 10 questions just distributed. The groups themselves
may decide how this gets done. Evaluation of their learning will amount
to the teacher randomly selecting students and posing any two of the questions
to the selected student(s). Note the mark awarded to the group will be
the mark awarded to the individual(s) on this skill.
B. The balance of time remaining in this period is devoted to refining
the "game" in the groups. The individual teacher may decide how
to conduct the question evaluation above. This can be done prior to or
concurrently with the group work.
C. Near the end of this period, remind your students
- Projects and write ups are due tomorrow (next day)
- Materials for game day are student responsibilities
- Bingo/Poker chips or monopoly money for the players will be provided
by the teacher.
Hour #4
Objective: Evaluate each of the games presented
Rules of Engagement:
- Set up your group's game on a table somewhere in the room.
- Designate 2 players to work first half of period, 2 players for the
second half.
- Rest of players in group become players.
- Group receives $10,000 in play money or chips in denominations of $50,
$100, $500.
- Top wager is $500.
- 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).
- The playing lasts 20 minutes.
- 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.
- At end total your group's wins/losses for each game. Send each group
your stats on their game.
- If time allows, discuss your feelings about each of the games, including
your own. Were they ethical? Were they money makers?
Hour #5 - Judgement Day
Objective:Today we will explore a variety of traditional and other
evaluation methods. We will end up with a mark/number or letter.
A. Rearrange your students into the groups and, as before, assure that
they have pen, paper, etc.
***** The record of winnings, losses, and profits must be available to
today's groups. You may even want to duplicate these results for each group.
Calculators will be handy as will a white or chalk board.
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:
- Obtain an exact tally of each student's winnings or losses (Post
this).
- Obtain an exact tally of each group's winnings or losses (Post this).
- Make a checklist of value judgements on the "play worthiness"
of games in general. Circulate the checklist (one for each game) (Post
the results).
- 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
- 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
- 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 )
Note: This is a suggested evaluation only. Individuals may, and likely
will, choose to modify the process.
This unit comes from the The Stewart Resources Centre which provides
library resources and teacher-prepared materials for teachers in Saskatchewan.
To borrow materials or obtain a free catalogue listing unit and lesson
plans contact :
Stewart Resources Centre,
Sask. Teachers' Federation,
2317 Arlington Avenue,
Saskatoon, SK S7J 2H8;
phone 306-373-1660; fax 306-374-1122,
e-mail src@stf.sk.ca.
http://www.stf.sk.ca/
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