# Dice Footy

### Task 232 ... Years 2 - 8

#### Summary

Students are involved in a dice game that is a mathematical model of the scores in an Australian Rules football game. (Note: It is not trying to model how the real football is played.) Previous knowledge of the code is not necessary because the scoring rules are explained. Namely, it is possible to score both goals (worth 6 points each) and behinds (worth 1 point each). A match is played over four quarters and the winner is the team with the higher points total at the end of a match. The task practises calculation of averages (with opportunity for asking Can I check it another way?) and then introduces a challenge involving probability considerations.

#### Materials

• Four dice - two different dice for each player
• Playing board
• Marking pen and wiping cloth

#### Content

• average
• mathematical modelling
• mental arithmetic
• multiplication, calculations / times tables
• probability calculations
• probability, expected number
• probability experiences
• statistics, analysing data
• statistics, collecting & organising data #### 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 probably worth noting to begin with that just as in a 'real' game of footy, the scoreboard is designed to record results cumulatively not as separate scores for each quarter, so there is no fifth line for the total. When the match is finished the winner is instantly known. No further calculation is needed.

Answers to questions 1 to 4 will vary from group to group and will involve a reasonable amount of calculation, mental and otherwise. Question 5 sets the stage for the challenge, so it may be worth mentioning to students that the double line on the card is a reminder to ask the teacher to check the work to that point. Not that you need to check all their calculations. Rather inquire of them how they did the average calculation and whether they were able to check it another way.

The Challenge implies that it is possible to predict the average goals, behinds and points in a match. Not exactly for their matches but in the sense of probability. The process is to realise that if a dice is rolled 6 times (in a perfect world) we could expect each face to appear once. Therefore the total expected 'score' for that experiment would be:

• 1 + 2 + 3 + 4 + 5 + 6 = 3 x 7 = 21
So the expected value for one roll would be 21 ÷ 6 = 3.5 and this is the same whether it is a roll for goals or a roll for points. So the expected average result for four quarters would be:
4 x 3.5 goals + 4 x 3.5 behinds
= 14 goals + 14 behinds
= 14 x 6 + 14 points
= 98 points
How close were the student result to this theoretical average?

The largest and smallest scores in the Dice Footy game are more straight forward to calculate. The largest would be rolling 6 every time for the whole match, thereby scoring 24 goals and 24 behinds for 168 points. The smallest would be rolling 1 every time for the whole match, thereby scoring 4 goals and 4 behinds for 28 points. This calculation has been requested in the hope of begging the question: What is the chance of scoring the maximum? One chance in every ... games?.

Again, if students recognise that each dice roll in this game is independent, then there are 8 events involved in a match. We can expect a 6 in the first event once in 6 rolls. But we can expect a 6 in the first and second events (6, 6) to happen once in 6 x 6 = 36 rolls. For three events we expect the outcome (6, 6, 6) once in 6 x 6 x 6 = 216 rolls. For four events to produce (6, 6, 6, 6) - to give the half time score - would be once in 6 x 6 x 6 x 6 = 1,296 rolls. Continuing this reasoning shows that the maximum score can be expected once in:

• 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 1,679,616 games
The result is exactly the same for the minimum score.

#### Extension

There are many extensions and variations possible in this game and the students might suggest some. One that many students find interesting is to play one more match and stop it at the point where Player B still has to play their final quarter. What is the chance that they will win? (Note: This means win not draw).
• There are 36 ways the game can finish from here.
How many of them will be a score that wins?
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

#### 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 requires 2 different dice for each student (a goal dice and a behind dice) and a score sheet. The students could easily sketch their own score sheet from an example on the board.

One way to begin the lesson is to ask the students to gather around one table as two students play the game through as a demonstration. The advantage of the whole class investigation arena is that so much more data can be collected in the same time and it can be displayed as class data on the board (perhaps using a projected spreadsheet). In this case you might see advantage in recording scores for the four separate quarters and using a fifth line for the total. Checking the total points another way then encourages using the horizontal and vertical checks often used when data is tabulated.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 161, Dice Footy 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 Dice Footy task is an integral part of:

• MWA Chance & Measurement Years 9 & 10

The Dice Footy lesson is an integral part of:

• MWA Chance & Measurement Years 9 & 10 