# How many ways are there to roll a 10?

I taught a class at MSU, Mankato titled “Math 483: Advanced Viewpoints on 5-8 Mathematics”. The class had a variety of goals, including pedagogical and mathematical ones. On the pedagogy side of things, we planned lessons, we read The Teaching Gap, we viewed the TIMSS videos and others, etc. On the mathematics side, we worked problems that came directly from middle school curricula, and we investigated questions that go deeper than we would expect middle school students to go, but that form an important foundation for making instructional decisions with middle school students.

In this last category, I tried something new last semester that I wanted to share with a larger audience. I would love critical feedback and questions about the activity and readers’ ideas about the mathematics involved.

My Math 483 students (I will refer to them as my “483 students” from here on out although there are not 483 of them) as a group had quite limited experience with middle school students-this was their first class examining the teaching and learning of mathematics, and most of them have tended to envision becoming high school, not middle school, teachers. One of the roles I played in class was the voice of a middle school student.

We were playing two dice games in class this spring: the Sum Game and the Product Game. For those without experience with these two games, here is how they work. Two players are playing against each other, one is player A, the other is player B. They alternate turns rolling two dice. In the Sum Game, no matter who rolls, if the sum is odd, player A gets 1 point. If the sum is even, player B gets 1 point. The Product Game is the same except we use the product instead of the sum of the dice. In either case, the players roll some set of number of times (say 20) and the person with the most points at the end wins.

In analyzing whether each game is fair (in the sense of each person having the same probability of winning), my students made the claim that the probability of rolling an even sum is 18/36 because there are 36 equally likely outcomes and 18 of them are even.

My inner middle schooler questioned this calculation. My experience with seventh graders and probability is that they commonly consider (4,6) and (6,4) to be the same outcome: a 4 and a 6. The idea that the order of the dice matters is not intuitive to many middle school students. So I posed the question to my 483 students, “How would you convince a seventh grader that (4,6) and (6,4) are different rolls?”

As we worked through a variety of strategies, I came to realize that this wasn’t quite the right question. One of these might be closer to what I intended:

(1) How do you know your model (there are 36 different equally likely rolls of two six-sided dice) is the correct one? How do you really know that?

…or maybe…

(2) What evidence would it take to convince you that your model is incorrect?

…or maybe…

(3) Imagine we were not sure which model was correct, what experiment could we perform that would help us to decide?

In the next post, I’ll share my students’ answers to my original question, and the statistical tests they concocted to answer question number 3.