My colleague found an activity to do with his 5th grade class, similar to **this one**. Basically, he gave the students 10 coins each, and asked them to put the 10 coins on a number line (with numbers from 1 to 12) with a partner. Each round they roll 2 six-sided dice, find the total, and remove a coin from their number line if it matches the roll. They keep going until one of the two students has no coins on the number line.

At first, most student’s starting positions looked like this:

or this:

At the end of the lesson, we played a 5 coin version of the game, and one student’s paper (after 1 round) looked like this:

Unfortunately, although most of the students did notice that some numbers came up more frequently than others, as evident by their distributions looking a bit less flat, and bit more centred on 7 on the number line, many of the students still had obvious misconceptions of probabilty. Students made comments like:

"If I spin around twice before rolling, I get a more lucky roll."

"I got a few 11s last game, so I’m going to put a few more coins on 11."

"8 is my lucky number! I’m going to put 3 coins on 8."

"I need to spread out my numbers so I have more chance of getting a coin taken on each roll."

Our plan for next class is to have students switch up groups, discuss insights they’ve had on the game, play a couple of rounds, switch them up again, while we walk around and see what strategies they use. Hopefully we’ll see less students spinning around in order to improve their dice luck…

I also asked for resources on Twitter for using a probability game as part of a lesson on probability, and had the following 5 games recommended (all of these look good because they are relatively easily produced and used in an elementary school classroom).

**Pig**via @delta_dc**Can’t stop**via @Fad23**Decimal race**via @mathhombre**Farkle**via @bennettscience and @mathhombre**Press your luck**via @mathhombre

**Update:**

I wrote a simulation to test to see what distribution of coins is the best. I am somewhat surprised by the results. **Check it out here**.

Why am I surprised by the results? My intuition about how this game works failed me. I thought that the likelihood of each number coming up was the most important factor in deciding a good strategy for the game. It turns out that one has to balance out the knowledge of the likelihood of each number being rolled with having a selection of numbers available. It may be that the probability of a 7 being rolled is 6/36 but the probability of a 6 or a 7 being rolled is 11/36.

In fact, it turns out that having a selection of different rolls available is fairly important. I updated the simulation so I could choose the number of coins to place, and with 3 coins placed, choosing 4, 5, 6 is better than having all three coins on the number 7. With 2 coins, it is better to choose 6 and 8 than to put both of the coins on 7 (although 6, 7 is better than 6, 8 – but only slightly).

There are three messages I get from running this simulation.

- One should, in general, not make too many intuitive assumptions about how probability works, particularly with somewhat complicated examples.
- One should be careful how one uses even simple games to teach probability. All of the students saw that 7 was the most common number, but their intuition of "choosing a wide spread" is a valuable one for this game, but it doesn’t help us get at the idea of some numbers being more likely than others. I think we’d need a different game for that.
- It is probably a good idea to build the simulation before you play the game with students, if at all possible.

## Dan Pearcy says:

This looks like a great lesson. My favorite part is the fact that you have simply noted their comments and are going to give them more time to construct this concept in their own minds/with peers. I think that sometimes I get in there too quickly and start trying to guide them towards the answers before they may be ready for them.( I guess that can be due to having an overloading curriculum)

I did quite a similar thing but with 3 dice with an older group. They each drew out a three by three grid and inserted numbers between 3-18. Then I let them investigate the best set of numbers to go into the grid(I wrote a post on this if you’re interested). However, I have to say that I like the use of the coins. Whether you show them the probability distribution of 2 dice now or whether they see it later in shcool, they will be reminded of this game in which they put the most coins around the number 7..

Excellent stuff!

May 13, 2012 — 10:25 am

## Hank says:

A colleague and I are team teaching a unit to our 9th graders on probability using games. We used the same sum-of-2-die game with “payouts” as an intro to expected value. We have made use of a spreadsheet (Google Docs) and a projector as a way to “crowd-source” data and create a jumping off point for discussions. Getting all the data in one place seems to negate the “unlikely” results that some groups can get. Maybe that wouldn’t work as well with the little ones, but who knows, sure worked wonders for us.

Our version of the game: http://goo.gl/z9KBF

Our classes data: http://goo.gl/nvn3N

Both links are G Docs.

May 14, 2012 — 6:25 am

## David Wees says:

We plan on pooling data later – I agree, more data = more reliable results. However, in my experience, students are much less likely to share their misconceptions in a large group, so we end up not being able to tackle individual student misconceptions as easily. So we start with the small group activities, and gather evidence of misconceptions. We try and address some of these in the second class, and in the third class, we pull everyone together to tabulate some data.

May 14, 2012 — 8:54 am

## Hank says:

I like the idea of actively searching for misconceptions, something I do better in physics than in math. I have a very social group of 9th graders so getting them to share ideas (right or wrong) has not been much of an issue.

May 15, 2012 — 4:35 am