Evens-Odds Game

Rationale: “Middle-grades students should have opportunities to experiment actively with situations that model probability including "making hypotheses, testing conjectures, and refining their theories on the basis of new information" (NCTM 1989, 111). These experiences should include a discussion of theoretical probabilities.” (Wiest and Quinn 1999, 358)

Intended Audience: 5th grade class. No prior knowledge of probability is assumed.

Objective: Given experience with the Evens-Odds Game, TLWD  knowledge that listing all possible outcomes of an experiment is helpful in predicting what might happen in future experiments.

Materials Needed:
Tetrahedron dice, 2 per pair (5 pairs- nice if each pair has two different colors)
Octahedron Dice for extension, 2 per pair
Game Board, 1 per person
Markers, as many as 20 per person
Ziploc bags (sandwich and large) to organize all materials and to make handouts easy
Summary sheet for outcome recording

Correlation with a Guiding Document: (EALR State of Washington)

1.4 Understand and apply concepts and procedures from probability:

“Know how to list all possible outcomes of simple experiments”
“Understand and use experiments to investigate uncertain events”

Offering the Lesson: (This is a modified ITIP (Instructional Theory Into Practice) Format.

Hook/set: "One of the things that I like to do is to play games on cold evenings on our kitchen table. I like to play games like _____ . What are a few board games, not computer games, that you enjoy playing?"

Wait for hands. Take a few responses.

_________ games you mentioned involved dice.

Content Objective/Teaching Point : Today we are going to explore a field of mathematics called probability through a board game that involves dice. Our goal is that by the end of today’s lesson you’ll be able to figure out a way to list of all the possible things that can happen and that you’ll find a way to use that information to help you win the game.

Instruction : Our game today is Called “Odds and Evens”. Let me show you what we’ll need to play and then I’ll go over the rules.

Here are the dice we’ll use: allow students to handle and comment

'How many sides?'        
Tetrahedron dice. Tetra is Greek for “Four”
'What numbers are on the sides?'
'How will we know what number has been rolled?' (Number that you can read upright on the bottom is the number that is used.)

We share a pair of dice between two people.

Show and explain gameboard. Everyone will have their own gameboard, although we’ll all have a partner to play against.

Show and explain that these are the markers, we start off with just 8. We use these to make our predictions.

Instruction/Modeling : How to play the game:

Each person gets their own board
Each person gets 8 markers
Place the markers any way you’d like between the even and the odd sides of the board (Try not to lead them in any way this first time.)
Players will take turns rolling the dice. WE them multiply the numbers on the dice and get the product. (CFU of term product)
If product is even, remove a marker from the even side. If product is odd, remove a marker from the odd side.
If you roll an odd product and you have no marker on the odd side, you simply lose your turn.
The WINNER is the first one to empty their playing board of chips.

Check For Understanding :  "Any questions about the rules?
With my board looking like this (Have something modeled) and I roll an even product, what should happen?"

Independent Practice : Students are paired and allowed to play.

I/Discussion:

Listen to the children tell you what happened:
Did one kind of product come up more often than the other? (Depending on your students, you may or may not want to go to WHY at this point. They may need another round for experience or they may start analysis of the game at this point. Follow as they lead.))
How would you place your chips for the next round?

Independent Practice: Play game again.

I/Discussion:

Compare what happened in this second round with the first. Were the results similar?
Now go for WHY-

Hopefully a student will suggest listing all of the possible outcomes, if not, ask whether or not a list might help.

Allow students to suggest a means of listing outcomes. Provide premade organizer sheet if deemed needed for these children.

Guide students through an evaluation of the list:

Even Products    Odd Products
3,4   4,4,   2, 3         1,1
1,4   2,2,    4, 3        3,1
2, 4  3, 2     4,2        1,3  
4,1    2, 1     1,2       3,3

Students should be able to state that there are 3 times more Even products than Odd.

They may be able to say that out of every four turns, three should be even products.

Discuss and help the students to reflect on their wording and their conclusions.

Check For Understanding: Now, If I gave you twelve markers, where would you place them? Listen to children’s explanations.

FORMATIVE ASSESSMENT MOMENT-

Are the children’s explanations based on their list of the sample space (Theoretical approach) or are their explanations based on their experience (Experiential approach) playing the game? Both reasons are acceptable at this early point in their experience.
Do the children suggest a 9 (even) and 3 (odd) distribution, which would show proportional thinking? Something close?

Do any of the children suggest recording their products so that they can test experimental probability against theoretical probability? This leads to the next question and the next lesson.

Check For Understanding: What should we do next in order to understand this game?

CLOSURE: Today we had a chance to play a new game. You found out that it can really help to make a list of all of the possible outcomes so that you can think about the game and make the best predictions. Next time I come we will do some more work with this game and we’ll get ready to teach the game to other people at Family Math Night.
References:

Bibliography:
Wiest and Quinn. (1999).  Exploring probability through an Evens-Odds Dice Game. Mathematics Teaching in the Middle School. 4, 358-362.