The purpose of this unit is to use games to gather and present data in a systematic way in order to determine the likely outcomes of some everyday events.
- investigate probability in common situations
- make and justify the probability of events in common situations
- theoretically and experimentally examine the probabilities of games of chance
The unit provides a focus on interpreting data displays, including analysing any distinctive features. It also focuses on moving students from intuitive and experimental ways of determining possible outcomes to the development of a more systematic approach. This movement from intuition to system and theory is a fundamental aspect of mathematics. It is also what distinguishes mathematics from other disciplines. Mathematics tries to quantify and organise the world on a sound and provable basis. Hence Pythagoras’ Theorem tells us without doubt that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. This can be proved without doubt. Similarly we can prove that the probability of getting a 3 when two dice are rolled is 1/18. This certainty does not occur for instance in other sciences. Think about the development of our knowledge of the Universe. Once we believed that the Earth was supported on the back of a turtle. Later we believed that the Earth was at the centre of the Universe. Our ideas have grown from there and are still growing as we meet obstacles to the latest theory. Naturally this does not make mathematics better in any way. It is rather just different and it is trying to do a slightly different job.
- Gameboard
- 2 dice
- Counters in two different colours, one for each player
- Ten cards from a suit from the ace = 1, to the 10
likelihood, outliers, mode, experimental, systematic
Getting Started
Beat-it
- Pose the following problem for discussion by the students (perhaps in groups of 4: What influences your choice as to whether or not you will cross the road?
What influences your decision as to whether you will climb on to the school roof and get a tennis ball that has landed up there?
The students should record their responses. - Introduce the game Beat-it described below, and focus on the necessity to make a choice – an important choice as it determines whether or not you can continue in parts of the game.
Game Resources
- 2 dice
Game Description – BEAT-IT: A grid, like the one below, is drawn with each column headed by letters from the name of the game.
B |
E |
A |
T |
I |
T |
- The game begins by focussing on the left hand column, and progressing through the other 5. To participate in the game all students need to be standing. Any time throughout the game they can choose to sit down.
- A pair of dice is rolled and the two numbers called out. Each player sums the number together and records the answer in the first column. If a 3 and a 5 were thrown the students record an 8 in the first column. Before the next throw, players decide whether to remain standing or sit down. If they sit down, they add up the total of the scores in the first column to give a total for that column. In our game that means they would end up with 8 as their total. If they choose to stay standing, they can record the next total to the first column.
B
E
A
T
I
T
8
6
4
5
5
7
6
8
9
10
5
3
6
5
7
2
11
8
9
12
2
5
5
3
9
5
12
7
Totals
14
0
38
0
49
0
Grand total = 101
- If at any time, other than the very first throw for each column, a total of 7 is thrown, all scores in that column are wiped and the column total goes to 0.
The game continues for each column until there are no more players, ie all the students are seated - either by choice or be being wiped out with a 7. The procedure is repeated for each of the columns. When the game is completed, all the totals of the 6 columns are added together to give the final score. - Play the game with the students and discuss the various strategies used when deciding when to sit down. These might be recorded so that they can be used for future games.
- This is an example of a set of class grand totals displayed as a stem and leaf graph.
5
4
6
0 9
7
1 1 3 5 7
8
5 5 8 9 2
9
4 7 8
10
1 5 7 9 9
11
7 9 2
12
6
13
2
- Pose questions to the students that focus them on the clustering of the results, the ranges of the results, and on any outliers. For example: What does this group of numbers tell? Were our results reasonably close? How might these (outliers) be explained? Discuss with the students how they could manipulate the data on the graph so that a mean average could be found. Similarly, ask them how the organisation of data might help them in determining the most common score(s) [mode], or finding the median average score. Ask the students for ways in which their initial strategies may be improved upon or to suggest new ones that might help improve the class average and overall performance. Discuss with the students how an improvement in overall class performance might be reflected on the stem and leaf graph.
- Repeat the game and compare the results by comparing stem and leaf graphs or using back to back stem and leaf plots. Performances may get better over a week period. (You might start each mathematics session that week with the game.)
- The students now become involved in exploring how mathematics might help them in developing or confirming a fairly reliable strategy. In pairs, give the students a pair of dice and ask them to devise an experiment that will give them an indication of how often a 1 will appear. In developing their experiment, ask them to consider the number of throws they might need to make in order to give a fairly reliable conclusion. The students can express their result as a percentage or fraction, but they must be able to interpret their result in terms of a strategy for playing the game.
Exploring
Game One: Who Gets Off First Wins In order to further develop the progression from experimental to a more systematic way of determining possible outcomes, introduce the game "Who gets off first wins".
Game Resources
- Gameboard
- Counters in two different colours, one for each player
- 2 dice between two students, the game is played in pairs
Game Description – Who gets off first wins: There are two players. Each player is given 10 counters to place on number on their "totals strip"- more than one counter may be placed on any of the numbers. Players take turns to throw a pair of dice and add the totals. If their total on the board has counters placed on it, they can remove ONE counter from that number.
First person to get all of their counters off the board wins.
Player One |
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2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
- Tell the students to play the game in pairs and write some questions that they might want to ask about this game.As they play the game ask questions that focus on the probabilities of the game:
- What do the popular totals seem to be?
- Where on the board are the totals that don’t seem to occur very often?
- Why might some totals occur more often than others?
- If you were to play the game again, where would you place the counters?
- Play the game again trying to place the counters in the "best" places. After the students have played the game ask them to comment on the effectiveness of their placements.
- Pose the question:
How can we determine the likelihood of different totals?
Discuss the need to have a "reasonable" number of trials and the ways that you could keep track of the totals. - Get each pair to roll the dice 50 times and record the results on a class chart. Determine the percentages for each total.
- Two students are having a "debate" about the number of different ways or combinations of getting a total of seven. Garry reckons there is only three ways of getting the total. Jackson thinks there are six.
What is the reasoning behind Garry and Jackson's thinking?
How might you help Garry and Jackson settle their debate? - Ask the pairs to make up a fair game that involves; 2 dice and the possible totals that can be made from them; a handful of counters, and 2 players.
Use mathematics to illustrate that your game is fair.
Explain your game to other groups.
It is hoped that what will emerge from this task are students’ ways of counting and recording possible combinations - This task gets the students to explore a particular way (if it has not already emerged) of recording information related to "Who Gets Off First Wins."
Provide the students with the following results from the game "Who Gets Off First Wins." ask them to interpret and make sense of the information and format. For example, you might ask the students to explain the ‘paired numbers’, and possibly to give a label to each axis based on what they represent.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
Provide the students with the following statement (or something similar) that is based on the information in the table:
Of 36 throws of the two dice 10 of them should give totals that are less than 6.
Ask the students to verify this and make some statements of their own from the table of results.
- This section provides a situation where the students can apply some of the ideas they have been developing in the unit so far.
- Begin with a discussion of the term "fair" and what it means to be fair. For example:
What is a fair game or result? How would you decide if a game was fair?
Should all games be fair?
What might be a good meaning for the term fair when analysing different games? - Introduce the game "Winning Differences". Allow the students to work in pairs and ask them to analyse the game using ideas developed over the unit. One requirement of their investigation is to check the fairness of the game. If it is not fair, they must provide some suggestions supported by evidence.
Game Resources
- 1 Dice
- Ten cards from a suit from the ace = 1, to the 10
Game Description
– Winning Differences:
This game uses 1 dice and ten cards from a pack of cards. The cards are 1 (an ace), 2, 3, …, 10. Children have 7 turns and these turns alternate.
Each turn consists of throwing a dice, choosing a card from the suit, and taking the difference of the two numbers.
Player 1 wins if they get a difference of 0, 1, 2, 3, 4.
Player 2 wins if they get a difference of 5, 6, 7, 8, 9.
Is the game fair? If not how can it be made fair?
Reflecting
- In the final session recap the games that have been played and try to work out theoretically the probabilities of success. Ask and determine the answers to such questions as:
How can we be sure what are the best places to put your counters in the game of Who gets off First?
What is the probability of winning in the game Winning Differences?
Was Winning Differences fair?
Can you justify your position? How can we be sure what are the best places to put your counters in the game of Who gets off First?
What is the probability of winning in the game Winning Differences?
Was Winning Differences fair? Can you justify your position? - Give the students a challenge:
Using any of the dice and or the cards, make up a game of your own.
Try it out with other students in the class, making any changes needed until it is a fair game.
Totals Board for Player One
Player Two |
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2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Dear parents and whānau,
This week in maths we have been playing games!
We have been studying the chance or probability linked to each game to see if we could find winning strategies. We have also looked at the games to see if they were fair. Play a game of Winning Differences with your child and help them think about whether the game is fair or not.
Home Activity
Winning Differences
Game Resources
- 1 Dice
- Ten cards from a suit from the ace = 1, to the 10
Game Description
This game uses 1 dice and ten cards from a pack of cards. The cards are 1 (an ace), 2, 3, …, 10. Players have 7 turns and these turns alternate.
Each turn consists of throwing a dice, choosing a card from the suit, and taking the difference of the two numbers.
Player 1 wins if they get a difference of 0, 1, 2, 3, 4.
Player 2 wins if they get a difference of 5, 6, 7, 8, 9.
Is the game fair? If not how can it be made fair?