# Win At The Fair

### Task 133 ... Years 4 - 10

#### Summary

This fairground game situation has an interesting twist. It becomes clear reasonably quickly (usually before 100 trials) that the designer is paying out more than is being received. The challenge then is to 'fix' the game to either break even, or make a reasonable profit.

This cameo has a From The Classroom section which shows what Year 5/6 students can be interested in tackling investigations in an after school program.

#### Materials

• Board with hexagons marked as shown
• Two dice
• Two counters
• Recording Sheet

#### Content

• basic number skills
• combination theory
• probability - long run frequency and expectation
• statistics
• organising and displaying data
• statistical inference
• Working Mathematically process

#### Iceberg

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

It isn't hard to imagine this game spread out in larger size on a table at the school fête with the intention of the designers (a group of students of course) being to raise money for the school. The trouble is, they are much more likely to lose money. Students come to realise this as they play the game a few times and add their results to the on-going collection of class result sheets as explained on the card.

Answers to the first few questions on the card will depend on the on-going class results, but even a pair playing ten games are likely to see problems. Assuming it costs one dollar to play each game, here are the results from one group:

 Prize Wins 20¢ 1 50¢ 2 \$1 3 \$2 1 \$3 2 \$4 0 \$5 1 Total Paid \$17.20

It doesn't always happen like this in ten games, but playing 3 sets of 10 is very likely to produce one outstanding loss and the aggregate of the three is even more likely to be a noticeable loss.

Once the long term loss is established a key challenge is to redesign the board so that (for example) the game:

• breaks even, or
• produces an average of 25% profit.

#### 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.

Using this task with a whole class quickly develops the quantity of data necessary to realise the game is likely to produce a loss. You will need to design a board first, which is easy to do with standard shapes in most word processing software, and collect sufficient counters and dice. (Red Poly Plug make good counters for this investigation.)

Begin by introducing the investigation through the story shell of using the game at the school fête. Pairs play a few games and the growing data is recorded on the board. After a while, pause and examine the data in the light of paying \$1 to play each game.

Is the class doing well yet with its fund raising objective?
Continue gathering data and pausing every now and then to analyse it until the trend to loss becomes clear.
What could we change that would produce a profit for the school?
This question might bring several answers, such as:
• re-arrange the prize values
• change the prizes
• include a \$0 hexagon
• change the dice directions
• ...
One or more of these could then be investigated as a class, but they also offer opportunities for individual or partner projects. The psychology of game playing/design can come into this challenge. For example the board could have a 50¢ prize in every hexagon. That would ensure a profit if the cost of play was \$1, but would anyone play the game?

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 1, Win At The Fair. This lesson also includes an Investigation Guide and software. The software allows many trials quickly and has an option for changing prize values and/or dice directions.

#### 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 Win At The Fair task is an integral part of:

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

The Win At The Fair lesson is an integral part of:

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

## From The Classroom

#### Dubbo PCYC

Robert McGregor
Years 5 & 6
Robert volunteers his time once a week to run an after school maths program at the Police Citizens Youth Club (PCYC) in Dubbo, New South Wales. This is not a tutorial program to help students 'get through the text book' but rather a quality enrichment program assisting students to develop their self-esteem by realising that they can work like a mathematician (regardless of what they appear to be achieving in the school context). Robert draws much of his inspiration from Mathematics Centre resources and here he comments on the success of a group investigation based on the Maths300 companion lesson for Win At The Fair.

 On Monday afternoons after school I run a Maths enrichment program for students in Years 5 and 6 at the local PCYC. The aim of the program is to explore what 'working mathematically' means for the Maths classroom and to expose students to working in groups on open ended questions. Recently we completed two lessons using Win at the Fair. I would now like to offer the following feedback. Win at the Fair offered a problem that most school students are familiar with - a game to use at the school fete. I enjoyed this lesson because it could have been taken in so many directions, e.g. the tallying of results, putting data into a histogram, calculating the average payout and ratios (i.e. every dollar in vs. every dollar paid out) and the speed of running games manually vs. the speed of the simulation software. I gave one group the opportunity to explore the outcomes of rolling a pair of dice. Soon the class realized that when rolling two dice the numbers 5, 6, 7, and 8 have a far greater chance of being rolled than say a 2 or a 12. The class was really surprised about this finding. The lesson also allowed for a discussion about gambling, a common problem in rural New South Wales where Dubbo is located. The students were able to realize how easy it is to manipulate the outcomes of a game, in favour of the party running the game, but still have the game appear as being fair. It took a number of students quite some time to design a Win at the Fair game board that would have given a much fairer result. Making changes to the game board, playing their game ten times, realizing that changes had to be made again, then designing another board certainly took up a lot of time. Next time when teaching this lesson I would probably design 5 or 6 different boards myself (that I knew offered a fairer result) and offer these to the students to explore. We did have the software available but it was only on one small laptop so could not be used by all students at any one time. Also, if I wanted to take the exploration of 'Maths in Games' further I would probably move on to Lesson 183, Snakes and Ladders. Robert McGregor Dubbo After School Maths Program