Cat & Mouse

Task 223 ... Years 2 - 10

Summary

The idea of this game is quite simple. A cat is lying in wait for a mouse beside a network of paths. The mouse is hunting cheese stored in the same network. Either the cat will catch the mouse, or the mouse will get the cheese (with eating likely to be the outcome of either situation). These are the two events of interest. The investigation is based around which event is most likely, given a particular network design and a given rule for deciding what move the mouse makes when it has a choice.

The activity is an investigation for two students, not a game between them, but many students are soon barracking for one or other of the 'combatants'.

Materials

• One 'mouse' and one dice
• Two playing boards

Content

• probability calculations
• probability, concept of fairness
• probability, expected number
• probability experiences
• probability, sample space / sample size
• reasoning
• statistics, analysing data
• statistics, collecting & organising data
• statistics, confidence levels
• statistics, frequency

Iceberg

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

It is worth noting in the beginning that both boards supplied with the task were designed by primary children. Hence the invitation at the end of the card for students to design their own board.

One result for 10 games on Board A is Cat: 4, Mouse: 6. Not really enough data to be certain about any long term result, but perhaps enough experience to realise that:

• At each choice there are two equally likely outcomes for the mouse.
• At the first choice after the start, the mouse has a 1 in 2 chance of getting the cheese whichever way it travelled from the start. So, it has a 1 in 4 chance of getting the cheese in two moves.
To gather more data teachers might keep an on-going record on a class display, or in an electronic journal on the school web site. As they use the task, each pair adds the results of their 10 games to the class data. Over time a clearer picture will develop of the empirical evidence for the most likely outcome. At least as important as the total of the results, which supports an ever increasing intuitive confidence about the most likely outcome, is recording the Mouse/Cat pairs. This gives a clear picture of the meaning of natural variation and a feel for how it builds to make a long term pattern. In this case a pattern suggesting likelihood.

It is also often true that students decide for themselves that 10 games is not enough data and run more trials for themselves.

Recognition of the 1 in 2 chance at each choice is the basis of the challenge about imagining playing 64 games. The reasoning begins with a statement like:

• If we are playing 64 games we can expect that 32 times the mouse will go this way and 32 times the other way.
This is a very natural way of introducing, or reinforcing, the mathematician's concept of expectation.

A quick sketch of the board (or using a non-permanent whiteboard marker) and continuing this form of reasoning leads to this outcome for Board A:

This is the theoretical result. 68.75% of the time the mouse gets the cheese. How does the growing empirical evidence compare? Probably quite well based on the evidence from 6400 experiments with the equivalent board using the software designed for the companion Maths300 lesson.

Board B is an extension of the task, as is the invitation for students to design (and analyse?) their own board.

Further investigation might be built around creating boards with the same non-equal choice at each node of the network. That level of investigation leads the problem into Year 10 and beyond.

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.

Converting this task to a whole class investigation helps the teacher gather class data more quickly. You will need sufficient copies of the game board(s) and one dice for each pair. In this form the lesson can be used to highlight the natural variability and confidence levels as described above and extend into the meaning of expectation if appropriate. Given the boards in the task were designed by primary students (Milgate Primary School). Lots more investigation will develop from your students own boards.

Infant teachers will see possibilities for the lesson at their level in this snippet from Robyn Floyd's classroom recorded in Calculating Changes Stories.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 155, Cat and Mouse which also includes a reproducible copy of the board. The lesson offers a powerful extension of the investigation through companion software.

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 Cat & Mouse task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 7 & 8

The Cat and Mouse lesson is an integral part of:

• MWA Chance & Measurement Years 7 & 8