Counter Escape

Task 6 ... Years 4 - 8

Summary

We have three 'boxes' (A, B, C) and 3 counters. Any number of counters (up to 3) can be placed in any of the boxes. They are removed from a box by the roll of a dice according to the rules:
  • If the roll is 1, take a counter from A.
  • If the roll is 2 or 3, take a counter from B.
  • If the roll is 4, 5, 6, take a counter from C.
Problem: What is the best way to place the counters to remove them all in the least number of rolls? 		 

Materials

  • 6 counters in two colours - 3 for each player
  • 1 dice

Content

  • probability - experiencing chance
  • expressing chance as a fraction
  • combination theory
  • problem solving - testing all possible combinations
  • expected number of trials to reach a success
  • statistical inference
Counter Escape

Iceberg

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

The challenge of finding the best placement strategy can be tackled at several levels. At the less sophisticated level students might choose a placement strategy (eg: AAA) and carry out say 20 experiments to get an idea of the number of trials it would take to remove all the counters. The next step might be to compare this strategy to a different one (eg: AAB) with another 20 trials.

At a more sophisticated level students could try to detail all the possible placement strategies (an application of combination theory) and carry out a set of trials for each one. The next step might be to use the probabilities of rolling:

  • 1
  • 2 or 3
  • 4, 5 or 6
to calculate the expected number of rolls to remove all the counters for each strategy.

Once this problem has been explored, What happens if...? questions generate an even deeper investigation:

  • What happens if we change the number of boxes?
  • What happens if we change the number of counters?
  • What happens if we change the dice rules?

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. 		   

The whole class lesson for this task is detailed in Maths300 Lesson 88, Counter Escape. This lesson also extends the investigation with software. If you have Poly Plug you can use the plugs as counters in the boxes A, B & C.

An alternative approach to lesson plan detailed in Maths300 is to keep a class display record of the work done by each pair as they tackle the task in the form above. Each new pair must:

  1. refer to the class record to see if there is a placement strategy still to be tested,
  2. carry out a given number of trials on this new strategy, or,
  3. add a given number of trials to the data of a strategy already tested.
Over time this approach provides whole class first-hand data which becomes the basis for class discussion and analysis and the starting point for the What happens if...? questions.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 88, Counter Escape.

Visit Counter Escape on Poly Plug & Tasks.

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 Counter Escape task is an integral part of:

  • MWA Chance & Measurement Years 3 & 4
  • MWA Chance & Measurement Years 7 & 8

The Counter Escape lesson is an integral part of:

  • MWA Chance & Measurement Years 7 & 8

Green Line
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