# That's not fair!

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Purpose

In this unit we play probability games and learn about sample space and a sense of fairness.

Achievement Objectives
S2-3: Investigate simple situations that involve elements of chance, recognising equal and different likelihoods and acknowledging uncertainty.
Specific Learning Outcomes
• Use dice and related equipment to assign roles and discuss the fairness of games.
• Play probability games and identify all possible outcomes.
• Compare and order the likelihood of simple events.
Description of Mathematics

Three important ideas underpin this unit:

• The set of all possible outcomes of a random phenomenon is called the sample space.
• An event is any outcome, or set of outcomes of a random phenomenon.
• A fair game is a game in which there is an equal chance of winning or losing.

Students should be given lots of experience with spinners, coins, dice and other equipment that generates outcomes at random (e.g. drawing a name from a hat). The equipment can be used to play games, which should lead to a discussion of fairness (or otherwise) of the equipment and to finding the possible outcomes of using it. As they play games, record results and use the results to make predictions, they develop an important understanding - that with probability they can never know exactly what will happen next, but they get an idea about what to expect.

Students at this Level will begin to explore the concept of equally likely events, such as getting a head or tail from the toss of a coin, or the spin of a spinner with two equal sized regions. Students can handle simple fractions at Level 2, and assigning simple probabilities provides them with an interesting and useful application of these numbers. Students can understand that the probability of getting a head when tossing a coin is 1/2. Given a spinner that is marked off equally in three colours, students can also understand that the probability of getting any one of the colours is 1/3 because there are three equally likely events and one of them has to happen.

Opportunities for Adaptation and Differentiation

This unit can be differentiated by varying the scaffolding of the tasks or altering expectations to make the learning opportunities accessible to a range of learners. For example:

• working directly with students as they work through the probability games. Guide them to think through all possible outcomes, predict outcomes, record outcomes and reflect on results
• encouraging students to work at their own pace taking as long as they need to work through each game. Students do not need to complete all of the games listed
• expecting students to share their thinking about the fairness of the games, accepting that some students may be describing their experiences of playing the game rather than considering probability more generally.

Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. For example:

• in the game Putakitaki/Duck racing, native birds that are prevalent in your local environment could be used
• when students create their own games in the final session, encourage them to consider their friends and classmates when planning, and to create a game that will appeal to them and be fun to play. This could be achieved by incorporating favourite elements from other games, or items of current interest.

Te reo Māori vocabulary terms such as tūponotanga (probability), matapae (prediction) and tōkeke (fair) could be introduced in this unit and used throughout other mathematical learning. Another term that may be useful in this unit is Putakitaki (Paradise duck).

Required Resource Materials
Activity

We introduce the unit by rolling dice and investigating the numbers that come up.

1. Begin the session by showing the students the large die and asking them which number they think will come up if you roll it.
What number do you think I will roll?
Why do you think that?
Roll the die and see whether students' predictions were correct. Repeat a couple of times.
2. What are the possible numbers that I can roll?
List these on the board and tell the students that this list of all the possible outcomes is called the sample space.
3. What if I rolled the die twenty times. What do you think will happen? Why?
List these predictions on the board or on chart paper.
4. With the class, roll the die twenty times keeping track with a tally chart. Summarise onto a class chart.

 1 2 3 4 5 6 lll llll l lll llll l lll
5. Give pairs of students a die and ask them to work together to roll it 20 times. As they finish, ask them to record their results on the class chart.

 Pairs 1 2 3 4 5 6 Mr Tihi 3 4 1 3 6 3 Ben & Tane 2 5 3 2 4 4
6. Discuss the results with the class. Look back at their earlier predictions.
Why are all our results different?
If you rolled the die another twenty times what do you think would happen? Why?
7. Now let's add our results together.
What do you think that we will find?
Use a calculator to sum down each of the columns

Number rolled

 Pairs 1 2 3 4 5 6 Mr Tihi 3 4 1 3 6 3 Ben & Tane 2 5 3 2 4 4 Jay & Sarah 5 3 3 2 5 2 Class totals240 rolls 45 36 42 31 39 47

At this level it is likely that the students may attribute uneven results to "special luck". It is through continued experience that students come to appreciate the mathematics of probability.

#### Exploring

Over the next 3 days the students play a number of probability games. Copymasters for each game are attached to this unit.  They are encouraged to think about the sample space and to make predictions as they play the games. They will also begin to think about whether the games are fair.

Tell the students that they are going to play a number of games in pairs over the next 3 days and there are some general things they need to do with each game:

• as they play each game they are to write down the possible outcomes (the sample space). They are also to write a prediction about what they think will happen in the game
• play the game, recording the results
• compare what happens with their prediction.

Note: At this level do not expect the students to make mathematically sound predictions or systematically identify all possible outcomes. It is likely that they will make incomplete lists of possible outcomes. In future work, as they have similar experiences, their thinking will become more systematic and mathematically sound. The main aim of this unit is to start the students thinking about possible outcomes and notions of fairness.
Probability games to select from:

Bunny hop (Copymaster 1)
Sample space = {Heads, Tails}
As there is an equal chance of getting a head or a tail you would expect the bunny to be close to 0 after a number of turns.

Doubles (Copymaster 2)
Sample space

 + 1 2 3 4 5 6 1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6,

There are 6 ways of getting a double or 6 out of 36.

It is unlikely that the students will be this systematic about identifying the sample space. However they should identify that you are more likely to get non-doubles.

Pūkeko racing (Copymaster 3)

Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
The table shows the ways of getting each sum. 7 (6 ways) is the most likely followed by 6 and 8 (5 ways).

 + 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

Odds or evens (Copymaster 4)

Sample space = {1 3, 5 (odd) and 2, 4, 6 (even)}
For this game the probability of each player winning is equal.

Sums (Copymaster 5)
From the table for Pūkeko racing you can see that there are 24 ways of rolling a 5, 6, 7, 8 or 9 which is a probability of 24 out of 36 or 2/3. The probability of rolling a 2, 3, 4, 10, 11, or 12 out of 35 or 1/3.

Up or down (Copymaster 6)
Sample space = {heads, tails}
This is very similar to the bunny hop game. You expect the climber to be close to 0 after a number of turns.

At the end of each session have a class sharing time to discuss a couple of the games.

• Tell us about one of the games you played today
• What were the possible outcomes?
• What did you think would happen?
• What happened when you played the game?
• Did anyone else play the same game?
• Did you get the same results?
• Do you think that the game was fair? Why? Why not?

#### Reflecting

On the final day of the unit ask the students to invent their own games using either coins or dice. Share the games with others in the class.

Which number came up the most? Why do you think it did?
If we rolled the die again do you think we would get the same results? Why?