In this unit we play several games based on coloured cubes and coins. The purpose is to get some idea of the relative rate at which things happen and think about the concept of a fair game.
- Recognise that not all things occur with the same likelihood.
- Observe that some things are fairer than others.
- Explore adjusting the rules of games to make them fairer.
A fair game is a game in which there is an equal chance of winning or losing. We say that a game is fair when the probability of winning is equal to the probability of losing. Changing the rules of a game can affect the likelihood of winning or losing, and therefore whether the game is fair.
This unit can be differentiated by altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners For example
- Simplify the challenges by using smaller numbers of cubes or segments than suggested. Cube challenges III and IV could be parallel tasks to cube challenges I and II, using different coloured cubes.
- Expect students to share their thinking about the fairness of the challenges, accepting that some students may be describing their experience of playing the challenge rather than comparing relative likelihoods.
The challenges in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:
- spinners could be created with school colours or the colours of a favourite team. Spinners can be made and tested online using the Spinner learning object.
- students could select items to use instead of cubes in the cube challenges. This could be anything that appeals to their interests and experiences, such as All Blacks cards, although they need to be things that are equally likely to be selected.
- Multi-link cubes
- Access to online Spinner learning object
The first four sessions of this unit are structured around a number of challenges. The cube challenges involve randomly taking one cube from a bag of coloured cubes. To win the challenge you need to take a cube of a particular colour from the bag. Similarly, the spinner challenges involve one spin on a spinner and are won by landing on a particular colour.
For each challenge:
- Introduce the challenge and discuss students’ ideas about whether the challenge is fair, and why.
- Have the students play the challenge in pairs, recording how many games they play, and how many of these they win.
- Discuss how students’ ideas about whether the challenge is fair have changed now that they have tried it.
- If the challenge is unfair, ask students to suggest how the rules could be changed to make it fair, and then try the challenge with some of the rules suggested.
- Discuss students’ experiences of playing with the changed rules, and whether they think the challenge is now fair.
When discussing students’ ideas about whether each challenge is fair, support them to consider the probability of all possible events in a challenge, ordering them from most likely to least likely and identifying events that have the same likelihood of occurring. Students need not know the theoretical probabilities involved but should be able to explain their reasoning.
Session one challenges
Cube Challenge I:
Bag contents: one red and one blue multi-link cube
Choose one cube
To win the challenge: take a red cube
This challenge is fair, because there is an equal likelihood of winning (by selecting a red cube) or losing (by selecting a blue cube).
Cube Challenge II:
Bag contents: one red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube
This is not a fair challenge because it is more likely that a blue cube will be taken than a red cube. In fact, players are twice as likely to lose the challenge as to win it.
The challenge will be fair if there are an equal number of red cubes and blue cubes. The easiest way to change the challenge so that you win more often, is to add more red cubes. The more red cubes you add, the more likely you are to win the challenge.
Session two
Cube Challenge III:
Bag contents: one red, one blue and one green multi-link cube
Choose one cube
To win the challenge: take a red cube
This is not a fair challenge. There are three equally likely events: take a red, take a blue, or take a green. In terms of the challenge, players are more likely to lose by taking a blue or a green cube, than they are to win by taking a red cube.
The challenge will be fair if there are an equal number of red cubes and cubes that are not red. The easiest way to change the challenge so that you win more often, is to add more red cubes. The more red cubes you add, the more likely you are to win the challenge.
Cube Challenge IV:
Bag contents: three red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube.
This is not a fair challenge. There are two events: take a red, or take a blue, and taking a red is more likely than taking a blue. As far as the challenge is concerned players are more likely to win by taking a red (three out of five times) than they are to lose by taking a blue (two out of five times).
The challenge will be fair if there are an equal number of red cubes and cubes that are not red, so the easiest way to change this into a fair challenge is to add one blue cube.
Session three
Spinner Challenge I:
Spinner:
Spin the spinner once
To win the challenge: spinner lands on green
This a fair game as there is an equal likelihood of winning by landing on a green segment, and losing by landing on a red segment.
Spinner Challenge II:
Spinner:
Spin the spinner once
To win the challenge: spinner lands on green
This is not a fair game. There are three equally likely events: land on green, land on red, or land on blue. In terms of the challenge, players are more likely to lose by landing on red or blue, than they are to win by landing on green.
The challenge will be fair if there are an equal number of green segments and segments that are not green. One way to make the challenge fair is to divide the blue segment in half, and colour half of it red, and half of it green.
Session four
Work with Spinner Challenge III and Spinner Challenge IV. For each challenge have students play the game, suggest adaptations to the rules to make the game more fair, and try the new rules out. Discuss their ideas about whether the game is fair and why, throughout.
Spinner Challenge III:
Spinner:
Spin the spinner once.
To win the challenge: spinner lands on green.
This is not a fair challenge. There are two events: land on red, or land on green, and landing on green is less likely than landing on red. As far as the challenge is concerned players are more likely to lose by landing on green (two out of five times) than they are to lose by landing on red (three out of five times).
The challenge will be fair if there are an equal number of green segments and segments that are not green. The easiest way to change this into a fair challenge is to divide one of the red segments in half, and colour half of it green.
Spinner Challenge IV:
Spinner:
Spin the spinner once.
To win the challenge: spinner lands on green.
This is a fair challenge because there is an equal likelihood of winning (by landing on green) or losing (by landing on a colour other than green).
Session five
- Review a few of the challenges from the week. Ask students to think about the probability of all possible events in a challenge, ordering them from most likely to least likely and identifying events that have the same likelihood of occurring. Students need not know the theoretical probabilities involved but should be able to explain their reasoning.
- Ask students to work in pairs to make a new challenge using cubes, spinners, or something else that they select. In each case they should have some idea of whether their challenge is fair or not.
- Have students swap challenges and play them.
- Talk about students’ challenges, and have students explain whether they think particular challenges are fair, and why.
Dear parents and whānau,
This week, in pairs, we have been using coloured cubes and spinners to play probability games that we have called 'challenges'. We have been deciding whether or not the challenges are fair. Here is an example.
Cube Challenge: One red and two blue cubes are in a bag. One person chooses one cube. To win the challenge the person must take out a red cube.
This is not a fair 'challenge'. The most likely event is that a blue cube will be taken. Choosing red is less likely. In fact players here are twice as likely to lose the 'challenge' as to win it.
Ask your child to tell you more about the games. It would be good if you and your child could invent and play a 'challenge' of your own. Is it a fair 'challenge'? In other words is it equally likely that you would win or lose this 'challenge'?
Enjoy the challenge!