First Down The Mountain

Task 87 ... Years 4 - 10

Summary

Climbers on a mountain begin at different heights above the ground. Students roll a pair of dice and the sum shows which climber may move one space down the mountain. It's a race between climbers! Which climber is more likely to reach WIN first?

Materials

• A3 size playing board
• 11 'climbers' and 2 dice

Content

• basic number skills
• probability - long run frequency
• probability distributions
• conditional probability
• statistical inference
• expectation
• combination theory

Iceberg

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

The card asks only that 10 games are played and begins a discussion of the fairness of the game. If it was a fair game, what would the students expect to happen in these ten games? Is ten games enough to make decision about fairness with any certainty? How much data would be enough?

If you keep a class record of the ten games played by each pair, the current pair can add their results and then discuss again whether they think the game is fair.

• One level of reasoning may be that 2 and 12 are closest to the WIN position so they are more likely to reach it first. Therefore the game is not fair.
• At another level of reasoning students might see that the heights of the columns have been arranged to match the number of ways of making the total at the top of the column.
  Total 2 3 4 5 6 7 8 9 10 11 12
  No. of Ways 1 2 3 4 5 6 5 4 3 2 1

  Consequently the argument might be that the board has been arranged to give each climber an equal chance of reaching the square just before WIN, so it will be the 'next' roll that matters and that is most likely to fall as a sum to seven, because there are more ways to make 7. Therefore the game is not fair because 7 is most likely to win.

However the second level of reasoning doesn't match the growing empirical results. It is in fact the 2 and the 12 that are more likely to win, but the explanation is more sophisticated than 'because they are closer to start with'.

An indication of the reason comes by considering a race between the 2 and the 7, ignoring all other rolls, where the 2 has to go one step and the 7 has to go six steps.

From the table above, there are only 7 roll results which would be of interest. Six of these are wins for Climber 7 and only one is a win for Climber 2. So, Climber 7 would win ⁶/₇ of the time. Climber 2 can win on any throw, but the only way Climber 7 can win is to roll 6 successive 7s before a single 2 is rolled. The theoretical chance of this is:

⁶/₇ x ⁶/₇ x ⁶/₇ x ⁶/₇ x ⁶/₇ x ⁶/₇ = 39.6%.

So in 1,000 trials we would expect Climber 7 to win 396 times which would mean Climber 2 would win 604 times.

What happens if we change the mountain. For example, if to win a climber only had to reach the line one above WIN?

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.

Introduce students to the game. This First Down The Mountain board should be useful. Collect guesses from everyone as which climber they think will win. This will give you some idea of the students' current thinking.

Students then play the game and add their results to a growing class record. Use the Iceberg Information to develop the lesson further. If your school is a Maths300 member, there is software to assist in collecting more data quickly.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 4, First Down The Mountain, which also includes three Investigation Guides.

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 First Down The Mountain task is an integral part of:

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 9 & 10

The First Down The Mountain lesson is an integral part of:

• MWA Chance & Measurement Years 9 & 10

This task is also included in the Task Centre Kit for Aboriginal Students and the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.