In this unit we play several games based on coloured cubes and spinners. The purpose is to investigate chance and think about the concept of a fair game.
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:
The challenges in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:
Te reo Māori vocabulary terms such as tōkeke (fair) and tūponotanga (chance) could be introduced in this unit and used throughout other mathematical learning.
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:
When discussing whether each challenge is fair, support ākonga 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. Ākonga do not need to know the theoretical probabilities involved. However, they should be able to explain their reasoning.
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.
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.
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.
Work with Spinner Challenge III and Spinner Challenge IV. For each challenge have ākonga 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).
Dear parents and whānau,
This week we have explored the fairness of different games. The games, or "challenges", we have played involved using coloured cubes and spinners to test out the probability of winning each game. 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!
Printed from https://nzmaths.co.nz/resource/cube-and-spinner-challenges at 12:42am on the 5th May 2024