Analyzing Two Dice

Back at the beginning of the semester, I said that I wanted kids to get a picture in their heads about adding two dice. Here are some (belated) results.

On the first quiz, way back in January, I asked this question:

Aloysius says, “if you roll two dice and add, the chance that you get an even number is P(even) = 0.5 because half of all whole numbers are even.”

Use an area model to help show that he’s right that P = 0.5, but explain why his reasoning is wrong.

I expected students to add up the various probabilities for sums of 2, 4, 6, 8, 10, and 12—(1/36 + 3/36 + 5/36 + 5/36 + 3/36 + 1/36)— get 18/36 and say it was 1/2. How silly I was. They were much more creative.

I also hoped they would say that the reasoning was bad because the numbers are not equally likely. How silly I was.

Right Answers

Anyhow, some diagrams. Click to enlarge. First, the most popular. I hadn’t anticipated the “checkerboard” aspect of the area diagram and how easy it would be to see that half of the (equally-likely) combinations were even. It also tempts one to make an analogous problem with five-sided dice:

Here is the diagram; the legend is next.

The legend and explanation for the "checkerboard" response.

But then, several students used this kind of diagram, which is kind of brilliant and totally unexpected (by me at least):

A different approach: this student reasons about the sums of even and odd numbers rather than from the canonical diagram.

Wrong Answers

Of course, there were still islands of trouble, for example:

Here, the student does not get how the area model works despite having watched the videos.

We also have this one; where it come from no one (least of all the student) is sure. I include it for all you teachers out there who will nod and say, “yup, you never know what you’re gonna get.” Remember: click to enlarge.

An unusual area model. No one knows why the student used the sizes in the diagram.

What do we make of this? The best thing is that the student at least knew that something was wrong with the diagram, and owned up to it on the paper. This is something I have been asking them to do, and I get it really seldom.

But then, the denominator 441 is in fact the number of little squares in the diagram (21 x 21), but of course 220.5 of them are not colored in; that’s just the number you’d have to have to make the probability 0.5.

So I have an assessment problem. I’m reasonably convinced that the right answers show some level of understanding, but I can’t really tell, from the wrong answers, what’s going wrong.

Parte Deux: Why Aloysius was Wrong

Although most of the diagrams were good, most of the responses to why Aloysius’s reasoning was bad were not. Here’s one that makes me doubt myself to the core:

His reasoning is wrong because you never know what you’re going to roll but you do know that 50% of the possible sums are even but not that you’ll roll an even number/sum 50% of the time.

Here’s one that shows a good observation (but still doesn’t complete the catch):

Aloysius’s reasoning is wrong because in the spread of #s (sums) that you can get, which is 2–12, there are more even #s. What he should have said was that

I don’t know

Then an attempt to use the vocabulary:

The probability of rolling two dice and having an even sum is mutually exclusive…

There were also a handful of non-responses, a handful of good ones, and the rest something like the above. Many were enough longer that I will spare you reading them.

Anyhow, I really like the question in principle, so what should I do about this? On the second quiz, I put in another Aloysius question—I gave them a Fathom simulation with lots of mistakes, and asked them to identify the mistakes and fix them. I think that went better. I will also be insisting on corrections in order to do re-takes to improve the scores on the corresponding learning goals. That will at least force them to confront what went on and think about it again.

But now it’s time for dinner.