Exemplars
Fair’s Fair?
Ann and Jane are playing a game. They each
have a penny to toss on their desk. Ann will
win 2 points when the pennies are tossed, and
the sides facing up match. Jane will win 3
points if the pennies do not match. Is this a
fair game?
Exemplars
TM
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair
- Page 1-
Exemplars
Grade Level 3–5
Fair’s Fair?
Ann and Jane are playing a game. They each have a penny to toss on their desk. Ann will
win 2 points when the pennies are tossed, and the sides facing up match. Jane will win 3
points if the pennies do not match. Is this a fair game?
Context
This task was given to fourth grade students who were studying probability and learning
how to make decision trees.
What This Task Accomplishes
This task allows the students to apply knowledge of probability to a problem–solving
situation. Students can use experimental and/or theoretical probabilities to reach a
conclusion.
Time Required for Task
One 45 minute period
Interdisciplinary Links
This task can be tied to a unit on games, or fairs and carnivals. Also, this unit would be a
good introduction for a class discussion on the concept of fairness.
Teaching Tips
Before giving this task to students, provide them with many experiences finding
experimental and theoretical probabilities. Some vocabulary you may want to teach around
probability includes: likely; unlikely; chance; equally likely; certain; uncertain; probable;
possible; impossible; possibility; sample; data; outcome; mutually exclusive; relative
frequency theory; independent event; dependent event; and probability of joint occurrences.
Students could create their own probability dictionaries as well as their own games of chance.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair (cont.)
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Exemplars
Students could also do the following mini-assignments for homework:
1. Mr. Smith wants to borrow a CD from Mrs. Smith. Mrs. Smith has 6 rock CDs, and 2 John
Denver CDs. Mr. Smith is in a hurry, so he picks one without looking. What is the
probability he will choose a John Denver CD?
2. The fourth grade students cannot decide which movie to pick for their next work
completion party. Mrs. Smith puts the titles of students’ choices on separate pieces of
paper and puts them in a bag. There are 2 adventure movies, 3 comedies, and 1 mystery.
Mrs. Green will pull one title out of the bag at random. What is the probability that she
will choose a comedy?
3. At a local restaurant, you go to order dessert. The menu has 3 choices for dessert. The
waitress tells you that one dessert on the menu is no longer available. What is the
probability that in making a random choice you choose that unavailable dessert?
4. Mr. Getty was in a batting cage. Out of 100 balls, he hit 30 of them. How many can he
expect to hit in his next 30 tries?
5. A bag has 10 new pencils: 3 red, 4 yellow, 1 blue, and 2 green. Find the possibility of
randomly choosing the following color pencils from the bag:
a) green
b) brown
c) red or green
d) red, yellow, or blue
e) blue or red
f) yellow
Suggested Materials
Pennies, graph paper
Possible Solutions
There is a 50% chance of tossing heads–heads or tails–tails. There is a 50% chance of tossing
heads–tails or tails–heads. Therefore both tosses are equally likely. To decide if the game is
fair, one looks at the points given, and since they are not equal, the game is not fair.
Benchmark Descriptors
Novice
The novice will not be able to find a mathematical strategy. Little or no understanding will be
evident. There will be no work to support the solution, and only basic math language will be
used.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair (cont.)
- Page 3-
Exemplars
Apprentice
The apprentice will more likely use an experimental model which may not get at the
underlying mathematics in the task. The apprentice may not have a full understanding of the
task, focuses only on the points involved, and does not consider the theoretical outcomes.
Practitioner
The practitioner will solve the problem theoretically, finding a correct answer to the problem.
S/he will use the math language of probability to communicate. The practitioner will create
an accurate and appropriate representation in which to record her/his approach and decision
making.
Expert
The expert will solve the problem theoretically, and may even verify the solution
experimentally. The expert will make mathematically relevant observations, and will use
precise and accurate math language. The representation will be complete and accurate to the
student’s solution.
Author
This task was written by Deb Armitage, K–8 Mathematics Assessment Consultant at the
Vermont Department of Education, and piloted by her in collaboration with Vermont
teachers.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair (cont.)
- Page 4-
Exemplars
Novice
There is no evidence
of an approach.
It is unclear how the
student arrived at
ten points.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student shows little or
no evidence of
understanding the problem.
Fair’s Fair (cont.)
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Exemplars
Apprentice
Representation
is labeled.
The student does not justify
the fairness considering the
game did not end in a tie.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair (cont.)
The student uses experimental
probability to find a solution.
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Exemplars
Practitioner
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student achieves
a correct solution.
The student uses
probability language
throughout.
The student explains and
documents his/her approach.
Representations are
accurate and labeled.
Fair’s Fair (cont.)
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Exemplars
Expert
The student shows
his/her work.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student achieves
a correct solution.
Fair’s Fair (cont.)
Work and representations
are labeled.
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Exemplars
Expert (cont.)
The student verifies his/her
solution through
experimental probability.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Fair’s Fair (cont.)
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