By Alie Doolittle and Kimberly Gonzales

 

Grade Level: 9th grade Algebra

 

Materials:

• Worksheet provided
• 6 sided cube dice (one per team)
• 4 sided tetrahedron dice (one per team)
• Mini M&Ms (enough for 20 m&ms per student or student group)

 

Learning Goals:

• Build intuition about probabilities
• Explore the difference between expected and theoretical probability
• Develop strategies using probabilities

 

 

NCTM Standards:

• Understand and apply basic concepts of probability
• Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases;

 

Preparations:

This game involves little preparation.  The students are handed the worksheet that provides specific instructions to play.  Students can work in teams of two or one.  Each group receives one 6 sided dice and one 4 sided dice.   The teacher should walk around the room and make sure they are actively engaged in the activity and thinking deeply about the questions. If done properly, students will explain their point of view and agree with other students.  If discussion gets too heated, the teacher should give the students a hint about what to think about.

 

This lesson should come in after three to five day of probability lessons.  It should add a different dimension and be great entrance into talking about experimental and theoretical profanity and sample spaces.

 

Wrap Up Debrief Questions/ Teaching Points:

 

What would happen if the experiment was repeated many more times?  What trend would you see?

 

What are some factors that affect the probability that you came up with in your groups?

 

What if we used two other die with different values?  (Outline general steps   Do not give an example.)

 

Player 1: _______________________

 

Player 2: _______________________

 

Date: _______________

Probability Game and Worksheet

 

Game Instructions:

 

Place ten M&Ms under any of the 10 numbers.  You can place more than one under a number.

Player 1 roles the two different dice and sums the numbers.  The sum is tallied on the chart given below. When a number comes up with an M&M below it, Player 2 removes one M&M from the number.  Stop when all the M&Ms have been removed.

 

---------------

 

M&M Chart

 

2	3	4	5	6	7	8	9	10

 

------------------

 

Tally of Dice Sums

 

2	3	4	5	6	7	8	9	10

 

Total number of trials: ________

 

--------------------

Questions:

 

Calculate the observed probability from the dice tally (tally number / total number of trials):

 

2	3	4	5	6	7	8	9	10

 

Do some numbers come up more than once?

 

 

 

 

 

How many different ways are there to make 2 or 10 with two dice?

 

 

 

 

 

How many different ways are there to make 6?

 

 

 

 

Calculate the expected probability :(leave as a fraction)

Hint: Divide number of ways to get each individual number over by the total number ways the dice can be rolled

 

2	3	4	5	6	7	8	9	10

 

Would it be optimal (best strategic move) to place all your M&Ms on one number?

 

 

 

 

 

What are some reasons that the expected probability differs from the probability you calculated from your game?

 

 

 

 

 

Ms. Gonzales and Ms. Doolittle have decided to have the whole the class play the game and they will give a prize to who ever gets rid of their M&Ms first.  Using the expected probability and the probability you calculated from the game, develop a strategy to win the game.

 

 

 

 

 

 

Replay the game using the strategy you have observed and see if you can get the total number of trials to decrease.

 

 

 

 

 

Player 1: _______________________

 

Player 2: _______________________

 

Date: _______________

ANSWER KEY Probability Game and Worksheet

 

Game Instructions:

 

Place ten M&Ms under any of the 10 numbers.  You can place more than one under a number.

Player 1 roles the two different dice and sums the numbers.  The sum is tallied on the chart given below. When a number comes up with an M&M below it, Player 2 removes one M&M from the number.  Stop when all the M&Ms have been removed.

 

---------------

 

M&M Chart

 

2	3	4	5	6	7	8	9	10

 

------------------

 

Tally of Dice Sums

 

2	3	4	5	6	7	8	9	10
1	1	2	3	4	3	2	1	1

 

Total number of trials: __18______

 

--------------------

Questions:

 

Calculate the observed probability from the dice tally (tally number / total number of trials):

 

2	3	4	5	6	7	8	9	10
1/18	1/18	1/9	1/6	2/9	1/6	1/9	1/18	1/18

 

Do some numbers come up more than once?

 

Yes the middle numbers (4,5,6,7 and 8) appear the most

 

 

How many different ways are there to make 2 or 10 with two dice?

 

 

There is only one way to make each of the numbers.

 

 

How many different ways are there to make 6?

 

There are 4 different ways to make 6.  (2,4), (3,3)  (4,2) and (5,1).

 

 

Calculate the expected probability :(leave as a fraction)

Hint: Divide number of ways to get each individual number over by the total number ways the dice can be rolled

 

2	3	4	5	6	7	8	9	10
1/23	2/23	3/23	4/23	4/23	3/23	3/23	2/23	1/23

 

Would it be optimal to place all your M&Ms on one number?

 

No it would not be optimal to place all of your M&Ms on one square due to observed probability ,which does not match the expected probability at such a low number of trials.

 

 

What are some reasons that the expected probability differs from the probability you calculated from your game?

 

The expected probability would differ from the calculated probability, as there are a small number of trials.  At this low number of trials one outlier can skew the data away from the expected.

Ms. Gonzales and Ms. Doolittle have decided to have the whole the class play the game and they will give a prize to who ever gets rid of their M&Ms first.  Using the expected probability and the probability you calculated from the game, develop a strategy to win the game.

 

 

The best strategy would be to place the M&Ms in the center numbers with the highest probability of appearing per dice roll.

 

 

Replay the game using the strategy you have observed and see if you can get the total number of trials to decrease.

 

 

This lessons is written by students at Massachusetts Institute of Technology (M.I.T.), as part of their coursework for 11.124, Introduction to Teaching and Learning Science and Math.