View examples of student solutions by clicking on a level below:

Fair Game??? Dilemma

A few students want me to play a game with them. They will give me a dime for each odd sum I roll with two die. I have to give them a dime for each even sum they roll with two die. I think I'm going to get cheated! I noticed that I can't roll one of my odd numbers - 1! I only get a choice of 5 odd numbers (3, 5, 7, 9, 11) but they will get a choice of 6 even numbers (2, 4, 6, 8, 10, 12). Should I play this game with the students? Using as much mathematical language and representation as you can, show me that this is or is not a fair game.

Suggested Grade Span:

Grades 6-8

Grade(s) in Which Task Was Piloted:

Grade 6

Task

Alternative Versions of Task

More Accessible Version:

A few students want me to play a game with them. They will give me a dime for each odd sum I roll with two die. I have to give them a dime for each even sum they roll with two die. I think I'm going to get cheated! I noticed that I can't roll one of my odd numbers - 1! I only get a choice of 5 odd numbers (3, 5, 7, 9, 11) but they will get a choice of 6 even numbers (2, 4, 6, 8, 10, 12). List all of the possible ways of getting each sum using the digits 1 - 6. Then determine the probability of getting an even and odd sum. Use the information to draw a conclusion; is this a fair game to play with the students?

More Challenging Version:

A few students want me to play a game with them. They will give me a dime for each odd sum I roll with two die. I have to give them a dime for each even sum they roll with two die. I think I'm going to get cheated! Should I play this game with the students? Using as much mathematical language and representation as you can, show me that this is or is not a fair game.

NCTM Content Standards and Evidence

Data Analysis and Probability Standard for Grades 6-8

Instructional programs from Pre-Kindergarten through grade 12 should enable students to...

Understand and apply basic concepts of probability.

NCTM Evidence A: Understand and use appropriate terminology to describe complementary and mutually exclusive events.
Exemplars Task Specific Evidence A: This task encourages students to communicate thinking using appropriate terminology and notation of probability such as "P(even)", "probability", "chance", "experimental", and/or "theoretical probability."
NCTM Evidence B: Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.
Exemplars Task Specific Evidence B: This task requires students to determine and then compare the probabilities of rolling even and odd sums with two die.
NCTM Evidence C: Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
Exemplars Task Specific Evidence C: To solve this task, students will need to list all possible outcomes to achieve the sums 2-12 using two die.

Time/Context/Qualifiers/Tip(s) From Piloting Teacher

This is a medium to long length task. Links This task could link to a unit on games around the world.

Common Strategies Used to Solve This Task

Most students will solve the task either experimentally or theoretically. Some will use a combination of the two approaches to verify solutions.

Possible Solutions

Original Version:

There is a 50% chance of rolling an odd sum, and a 50% chance of rolling an even sum, so the game is fair.

More Accessible Version:

The solution is the same as in the original task. This version is just more scaffolded.

More Challenging Version:

The solution is the same as in the original task. This version is just less scaffolded.

Task Specific Assessment Notes

General Notes: Most students will list combinations. There is the opportunity for students to use notation of probability to communicate solutions.

Task Specific Rubric/Benchmark Descriptors Click on a level for student example.
Novice	The novice will have no understanding of the mathematics of probability, and no approach will be evident.
Apprentice	The apprentice will show some understanding of probability but will achieve an incorrect solution.
Practitioner	The practitioner will show understanding of probability concepts and will achieve a correct solution.
Expert	The expert will show understanding of theoretical and experimental probability and will use both to verify accuracy of the solution.

Author

Clare Forseth has taught sixth grade math at the Marion Cross School in Norwich, Vermont for 26 years. She is a member of her school's Mathematics Curriculum Committee, Technology Committee, Assessment Committee, and Local Standards Board for teacher relicensure.