# Greedy Pig

### Task 200 ... Years 4 - 8

#### Summary

This is a game that is won by getting the higher total after five rounds. It doesn't matter if you beat your partner, or lose, in one round. What matters is that you have the higher total after five rounds. In each round, players roll a dice (taking turns is fine) and both receive the number of points shown. They do this a second time and both receive the dice points. So now they have two scores. At this point, if a player wishes, they can quit the round and keep the total of these two scores. Or they can keep rolling and adding on points to create a progressive total until they either lose ... or decide to quit. A player loses if a 2 is rolled while they are still playing the round. If that happens their score for that round is immediately zero!

The challenge is to develop a strategy for deciding when to quit each round so that you get the highest possible score in the five round game.

#### Materials

• 1 playing board and 1 cube dice
• 2 marking pens and a cloth

#### Content

• average
• decimals, calculations
• mental arithmetic
• probability calculations
• probability, expected number
• probability experiences
• recording mathematics
• statistics, analysing data
• statistics, collecting & organising data
• statistics, frequency
• statistics, inference
• statistics, mode / median / mean

#### Iceberg

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

The task teaches the rules then expects the students to design the investigation. At first they will play - so does a mathematician when they become interested in a problem. Playing brings understanding of what is involved in this chance based investigation and leads to collecting data. The playing board records dice rolls as well as the total for each round. Over five rounds there is a significant amount of data about the distribution of the outcomes; even if one player quits early, the dice rolls are still recorded so the other player can calculate their total.

However, playing one five round game is not enough to make sound hypotheses about when to quit, so encourage students to summarise the data from the first game before they clean off the board and start again. After at least three games, encourage students to look at their data to begin to shape hypotheses about when to quit. There are several possible responses. The main testable ones are:

• Quit after two rolls.
• Quit after a certain number of rolls which includes the two free ones.
• Quit after a certain number of points.
Encourage students to select a strategy each and play a couple of games to compare them. Discuss whether this is enough data to be able to make a reasonable comparison. In fact it isn't and this task doesn't expect a definitive answer to the challenge of finding the best strategy for quitting. It does expect that students will see that:
• There are alternatives.
• An experiment could be designed to test the alternatives.
• The experiment would need to produce more data than can be delivered by two players in one session.
• The data would need to be organised and statistics like range, mode, median or mean would need to be used to compare strategies.
To encourage these insights, after the students have found at least two approaches, they could describe in their journal the experiment they would design to test their hypotheses.

#### Extensions

• Although at school level the investigation resolves best to long term data collection and analysis, the 'quit after 2 rolls' strategy can be explored theoretically using the concept of Expected Value. Expected value is the number of points you can expect to get on each roll. For the first two rolls there is no killer number, so, since you could get 1 or 2 or 3 or 4 or 5 or 6 points as a dice outcome, it is reasonable to expect the average of these, which is 21 ÷ 6 = 3·5. So the expected points for the 'quit after 2 rolls' strategy is 7 points per round, hence 35 points for a game. How does this theory compare with trial games using this strategy?
• What happens if the killer number is 6?
(First thoughts might be that it doesn't matter what the killer number is because each dice number has the same chance of being rolled. But if 6 is the killer number, it can only be used in a round total if it appears in the first two rolls. That will affect any points based strategy.)

#### 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.

As a whole class lesson, this is easy to state, easy to start and involves heaps of mathematics. Quickly, and almost carelessly, draw on the board a recording table like the one on the card. Ask each person to quickly sketch this on the top half of a blank piece of A4 paper. The only other equipment you will need is one, preferably large, dice. It also helps to have a monitor record your dice throws on the board as you play with the class.

As soon as possible - don't wait for the stragglers to finish their board; they will catch up - announce that you are going to roll the dice and give them all points which they have to record. Do it and ask the monitor to record.

Now explain the game.

Okay, stand up if you want to keep playing this round. When you decide to quit you must sit down. But, if you are standing when a 2 is rolled, your score is immediately zero. When everyone is either seated, or has been 'zero-ed' we start a new round.
After the first round declare the real interest as being in the highest total after 5 rounds, then play the next 4 rounds as quickly as possible. Ask students to find their grand total then invite them to the board to include their data on a stem-and-leaf graph. Examine the distribution together and ask the persons with the highest and lowest total to explain their strategy. Discuss the likelihood of those strategies being successful when the game is played again. Discuss other strategies that have been used and make notes as appropriate. Encourage thought about strategies that haven't been suggested, especially points or rolls strategies, and use the data recorded by the monitor to get an idea of how they would have performed.
Now I am going to give you two minutes to decide on a strategy for the next game. I want you to tell your partner the strategy you will use and I want you to stick to it in every round of the next game. I want to find out if discussing the mathematics will lead to us getting a better result as a class.
Ask several students to state their strategy, then play the game. Invite students to include their new total on the stem-and-leaf graph to create a back to back stem-and-leaf graph. It will be obvious if there is a change in the distribution. But how do we measure the amount of change? The students' own data now gives reason for introducing statistics such as range, mode, median and mean which are measures that can be used to compare the two distributions.

Ask students to record the investigation so far in their journals. Then, if there is continuing interest discuss and agree on two strategies for the class to test and compare - for example 'quit after 5 rolls' and 'quit after 21 points'. Carry out these experiments and ask the students to add to their journal entry. They will also need to comment on whether they believe the better of these two is indeed the best of all.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 5, Greedy Pig, which also includes an Investigation Guide, software which allows thousands of experiments in a short time and an extended discussion of learning features. If you have access to the software the class lesson can continue as each pair explores their own choice of 'best strategy' and develops the evidence to support their case.

#### 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 Greedy Pig task is an integral part of:

• MWA Chance & Measurement Years 3 & 4

The Greedy Pig lesson is an integral part of:

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