# 12 Counters

### Task 117 ... Years 2 - 10

#### Summary

A game situation encouraging students to consider more likely and less likely events. They make a decision about the boxes into which they will place their counters, given that counters will be removed one at a time depending on the sum of two dice. The person who removes all their counters first is the winner.

This cameo includes an Investigation Guide.

#### Materials

• Playing board
• 24 counters and two dice

#### Content

• language of chance
• likely and less likely outcomes
• identifying all possible outcomes of an event
• expected number of trials to reach a success
• collecting, representing & interpreting information
• noticing patterns
• making & testing hypotheses
• testing all possible combinations

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Since this task is a game, there is no solution as such. The main learning is to spread the 12 counters across the more likely totals, such as 6, 7, or 8 and to have fewer on the less likely possibilities. In discussion with students, teachers will hear comments such as:

• I just put them anywhere.
• I wished for some numbers to roll.
• Six came up a lot last time so I put 3 counters on that one this time.
• Twelve doesn't come up much because you can only have 6 and 6 - it's not very likely.
• Ten doesn't happen very often - but seven happens lots of different ways.
all of which offer assessment information about the student's current chance concepts. It is always interesting if a student puts a counter in Box 1.

Analysis of the theoretical probability depends on deciding all the possible outcomes. To do this it is useful to have dice of different colours.

• Suppose the red dice was showing 6, what numbers could partner it from the green dice?
This reasoning leads to 36 possibilities and the theoretical chances of:

 1 2 3 4 5 6 0/36 1/36 2/36 3/36 4/36 5/36

 7 8 9 10 11 12 6/36 5/36 4/36 3/36 2/36 1/36

As useful as this analysis is, it still doesn't resolve the question of the best strategy for placing counters. This is actually very hard to work out theoretically and is best tackled empirically - that is, by experiment. Even so, there are many placement strategies to test, and how much data is enough to be reasonably certain that one strategy is better than another? Software, if available, would at least speed up the process of running the trials.

One way to encourage students to continue to explore some of these questions is to follow the use of the card with the challenges of this Investigation Guide. The Answer Guide for teachers is also available.

These guides have been adapted from the Chance & Measurement Replacement Unit. A Replacement Unit is one form of unit plan that integrates the use of both the invitation and the whole class lesson lives of a task. The Chance & Measurement unit has been designed for Years 5 to 8 and is available to purchase as a resource. It doesn't require Maths300 membership, although it would be enhanced by it.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

There are several ways to introduce Twelve Counters as a whole class lesson. For example, it is easy enough to mark out the 12 boxes in the playground and use twelve students as the counters. In fact, use two teams of twelve (girls/boys, or sleeves up/down) asking each team to place its twelve in the way they think will get their team out first. Then check the predictions by carrying out a trial. The friendly competition involved will be enough to encourage deeper investigation in pairs to try to work out the 'best' placement strategy.

If outside isn't a possibility, then in the gym place 12 numbered pieces of paper along the edge of the basketball court. Or in the classroom use the 12 pieces of paper on 12 tables which become 'bases' where students stand. Following some physically involving exploration like this, it is easy for each pair to set up a table top version back in the classroom, especially if you have Poly Plug to use as counters. The Investigation Guide above can then lead the development of the lesson.

At this stage, 12 Counters does not have a matching lesson on Maths300.

Visit 12 Counters in Menu Maths Pack C.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The 12 Counters task is an integral part of:

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 7 & 8
This task is also included in the Task Centre Kit for Aboriginal Students.