Happy New Year everyone! I hope all of you are looking forward to your brand new set of 525,600 minutes.

For our very first blog post of 2014 I want to talk about *fair games*. In my consulting and advising work with teachers, I am often asked about how to teach basic probability concepts to elementary age children. Here in Virginia, we have probability and statistics included in each of the K-8 state standards of learning. Novice teachers are often surprised to see standards like: The student will...

- "use data from experiments to predict outcomes" in 2nd grade
- "investigate and describe the concept of probability as chance and list possible results of a given situation" in 3rd grade
- "predict the likelihood of an outcome of a simple event; and represent probability as a number between 0 and 1, inclusive" in 4th grade
- "make predictions and determine the probability of an outcome by constructing a sample space" in 5th grade

The Common Core State Standards have similar goals, though they are clustered around 7th grade and are much more robust:

1. Understand that the probability of a chance event is a number

between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.2. Approximate the probability of a chance event by collecting data on

the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.3. Develop a probability model and use it to find probabilities of events.

Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.4. Find probabilities of compound events using organized lists, tables,

tree diagrams, and simulation.

Teachers are thrilled when I provide them with a wonderful, easy, low-cost GAME that (1) illustrates wonderfully the difference between experimental and theoretical probability, (2) begins with a concrete concept that every child already is familiar with: "fair", (3) students LOVE to play, and (4) can be revisited year after year to scaffold learning and deepen understanding of the core concepts of chance. Plus, as an added bonus, this game also allows students to practice mental math and addition / multiplication facts!

**Pre-Game Pitch**

I love the open-ended inquiry approach to begin this lesson and usually advise teachers to start with a question like, "*How do you know if a game is fair or not?*" or "*What makes a game fair?*" Students usually have very good intuition about this concept and it provides a good starting point for the discussion on probability.

**Play**

The best way to experience the concept of fair games and basic probability is by *doing* so I always recommend a hands-on approach to this lesson. My favorite two are the Sum Game and the Product Game.

In my in-service training workshops for teachers, I love to introduce the games something like this: "*Stand up. Raise your right hand. Choose a number from 0 to 5 by holding up that many fingers. Now look around the room and find a partner that* (many choices here - they can include 'makes a sum less than 4', 'makes a product that's an even number', 'is an exact match to yours' etc.)." Once everyone has a partner, I give them a pair of dice and say, "*Look deeply into your partner's eyes and decide which of you is odd.*" It's always good for a chuckle (and an odd look - ha! Sorry, couldn't help it!), especially after I follow it with, "*Now once you've decided who's odd, the other person is obviously even. Now on a sheet of paper make a tally sheet with 'odd' and 'even' at the top so you can keep score.*"

Next, I give them directions for playing the game and make sure everyone knows that the instant someone wins I want to hear it. Usually, I make up something appropriate for my audience. For example, in working with a group of graduate students in UVA's master of education program, I might say, "*When you win, I want you to scream, 'Wahoo!' really loud.*" (since *wahoo* is the nickname for a UVA student). I let them know that once they indicate they've won a game, I will ask, "*Who won?*" and that the correct response is NOT, "*I did.*" :-) Instead, they will need to tell me whether they are even or odd so I can keep a class tally of who's winning.

I remind them that the reason we're keeping this tally is because we're interested in determining whether or not these games are fair. Then they play several games (the number of games each pair plays depends upon how many teachers or students I have in the room) and I keep track of who's winning on the board. Everyone loves these games! Students of all ages (including adults) really get into it and become very aware of whether they think the games are fair or not while they play them.

**Analyze**

The key to this lesson is what happens after the fun of the game. When doing this activity as part of a teacher training workshop, I will often ask the teachers to find another pair to work with and "prove" whether the games are fair or not. Students often need a bit more guidance so I will often do another activity with them (using both dice and coins) before we get to the part about determining whether or not the games are fair. It's beautiful because both teachers and students can clearly see the difference between experimental and theoretical probabilities in the context of the games once we start analyzing our class results.

**Next Steps**

As I mentioned earlier, what makes this game so great is its adaptability and flexibility to be used in many different grade levels and at many different points in the student's learning process. It's also wonderful because there are many variations in the game itself, including using different dice (e.g. 8-sided or 10-sided) or different rules (Example: What if you make a fraction by 1st die / 2nd die and Player A gets a point if the fraction is proper while Player B gets a point if the fraction is improper and both players get a point if the fraction = 1?). As you can begin to see, the possibilities are endless!

If you'd like the complete set of activity materials (teacher's instruction sheet, student information sheet, and notes sheet in PDF form), they are available for purchase here:

As always, I'd love to hear from you.

**References:**

Site where the blog graphic came from

Link for the cool animated dice gif

Virginia SOLs

Common Core State Standards in Mathematics

Cool site for online probability games

Another nice lesson on probability (with sample problems)

Chapter on fair games, betting, & probability

More activities on fair games & probability

Nice SlideShare on fair games & probability

**The Solver Blog**

**Author: Dr. Diana S. Perdue**