Have A Hexagon

Task 53 ... Years 2 - 10

Summary

All the possible products which can be made by multiplying together the numbers on two dice are placed in the 18 sections of 3 hexagons. The game is to roll two dice and place a counter on the section represented by their product. Which hexagon is most likely to be covered first?

This cameo has a From The Classroom section which shows how one teacher turned the activity into a whole class investigation with her Year 3/4.

Materials

• Board with hexagons marked as shown
• Two numeral dice (rather than spot dice)
• At least 18 counters

Content

• times tables practice
• estimation and calculation of chance
• collection and organisation of data
• making and testing hypotheses

Iceberg

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

Label the hexagons Left, Middle and Right. Play the game several times and keep a record. Counter to the intuition of most students, the middle one is more likely to be covered first. The reasons is (in large part) due to the number of ways each product can be formed. For example the product 4 can be made from 1 x 4, 4 x 1 and 2 x 2. Checking out the possible ways to create each of the other products shows that there is a bias towards the middle hexagon. So the game is unfair - the middle one is likely to be covered first.

The challenge now becomes to reorganise the numbers on the hexagons so that the game is as fair as possible, ie: if you played 99 games, each hexagon would be covered first in about 33 of the games.

The investigation could be extended further by asking: What happens if we want to design a similar game to practise dice addition?. There are 11 possible totals, 2, 3, 4, ... 12. We could:

• use 2 hexagons and choose the total to be repeated.
• use 2 hexagons and choose to make one sector 'free' as if it were already covered.
• use 2 pentagons and choose which total to leave out.
Using any of these design approaches, is it possible to make a fair board?

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.

To turn this into a class investigation only requires multiple copies of the hexagon board and two dice per pair. The advantage of doing so is the swift collection of much more data, which means more firmly based hypotheses than are possible with only two students investigating. It doesn't take long for the class to see that the Middle hexagon is more likely to 'win'. It takes a little longer to find out why. It then becomes a considerable project to decide on the fairest way to redistribute the numbers to make three (almost) equally likely hexagons. Maths300 provides software that makes this project more manageable and efficient.

If you have Poly Plug, the red plugs make excellent counters for the activity. To see an example of the exploration in one class, see Learners At Work below.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 6, Have A Hexagon. This lesson also includes Investigation Guides and software to extend the investigation.

Visit Have A Hexagon 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 Have A Hexagon task is an integral part of:

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

The Have A Hexagon lesson is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 9 & 10

From The Classroom

Chandler Primary School

Jenny Tran
Year 3/4
Jenny Tran investigated Have A Hexagon with her Year 3/4 following a professional development session.