Teaching context - List and represent outcomes of chance experiments
Students at this level are successful when they can comment on the likelihood of winning simple games based on chance and can determine these outcomes using both:
- Theoretical probability: number of favourable outcomes/total number of possible outcomes; or
- Experimental probability: number of favourable outcomes/ total number of trials.
It is important for students to engage in games which have equally likely outcomes that encourage them to state the sample space (which are the possible outcomes in an experiment), explore and measure probabilities, use key vocabulary and reason if games are fair or unfair. Some examples are listed below:
- Rock Paper Scissors - In this game students have a 1/3 chance of winning, the sample space is {rock, paper, scissor}.
- Coin tossing - When tossing a coin, the likelihood of landing on heads is the same as landing on tails ie p(heads) = 1/2 and p(tails) = 1/2 = 0.5 = 50%. The sample space for coin tossing {head, tail}.
- Rolling a six-sided die - Students can investigate the probability of rolling each number i.e. p(1) = 1_ 6 , rolling an even or odd number p(even) = 3/6 = 1/2 p(odd) = 3/6 = 1/2 , or a number greater than 3 or less than and equal to 3 p(no.>3) = 3/6 = 1/2 p(no.≤3) = 3/6 = 1/2 .
The sample space for rolling a six-sided die {1,2,3,4,5,6}.
- Using Spinners with equal sections ie each colour has a 25% chance. The sample space is {blue, red, yellow, green}.
The most common misunderstandings for students are:
- They tend to believe in luck (for example, they will have a better chance at rolling their favourite numbers).
- They make prediction based on their likes and interests (for example, their favourite colours for spinners).
- They do not realise that chance has no memory (for example, if a student has rolled four sixes in a row they often believe that the fifth roll cannot possibly be another six).
A way to overcome this is to have students frequently play and create games linked with chance. This will encourage students to use the language, test their findings, record their results and make generalisations. Students also need to experience playing unfair games for example, you could give students nontransitive dice where each represents a different set of numbers ie:
- Die 1: has the numbers 1 1 6 6 8 8
- Die 2: has the numbers 2 2 4 4 9 9
- Die 3: has the numbers 3 3 5 5 7 7
The aim is to roll the highest number. Have one student choose first as this would give the second student an advantage. Once students play a few rounds they can construct tables to see which die would have a better chance of winning.
Teaching idea - Rock, paper, scissors
This variation of the game is from NZMaths. In this task students play in groups of three. The scoring system is as follows:
- Player (a) scores a point if they all show the same sign
- Player (b) scores a point if only two players show the same sign
- Player (c) scores a point if they all show a different sign.
Have students play few rounds and record their results. Once they have finished pose the questions:
- Is this a fair game? Why/why not?
- Who do you think has the greater chance of winning?
- Explain why you think this is so and list the probabilities for each player.
Have students represent all possible outcomes using a tree diagram or by making lists. For example, if player one chose Rock first the following are possibilities:
- {RRR, RRP, RRS, RPR, RPP, RPS, RSR, RSP, RSS}
Have students determine all possibilities i.e. they have to continue the tree diagram to show the first player choosing Scissors or Paper as well. This will help them discover all 27 outcomes and answer the questions above.
Extension ideas
Explore more game options
Have students come up with a fourth (or fifth, sixth, and so on) option in the game after rock, paper, and scissors.
They can complete the activity as outlined above with this fourth option and with an additional person being added into the game. They can evaluate the difference that having a fourth option and an additional person makes to the game described above and also justify their ideas.
This adds complexity to the task for high-ability students as an additional option must be considered in their responses to the questions in the task. It requires students to engage in higher order thinking.
Explore the notion of ‘chance'
Students can create their own fake sequences of what they think twenty rounds of the rock, paper, and scissors game described above would look like. They can then watch and record real rounds of the game. High-ability students can respond to the following questions:
- What differences are there between the fake and the real data sets?
- Is it possible to spot the fake sequences? Why or why not?
This can be done in pairs or groups of like-ability students. Each high-ability student can create fake and real sequences and share these with their peers. Each student can see if they can spot ‘fake’ sequences and justify their choices. This task adds complexity, as it requires high-ability students to question the notion of ‘chance’. High-ability students need to engage in higher order thinking to respond to the task questions.
Explore the notion of ‘luck'
Have students explore the notion of ‘luck’ through evaluating the following statements about the rock, paper, and scissors game:
- If your opponent plays scissors twice in a row you can assume they won't play it a third time.
- Rock is used most often so play paper.
- Play scissors or rock in the first round as most players will not play rock in their first go.
High-ability students respond to the following questions using their knowledge of luck and chance to justify their decisions:
- Which, if any, of the above statements are good advice?
- Which, if any, of the above statements are poor advice?
This task adds complexity as it requires high-ability students to question the notion of ‘luck’. High-ability students need to engage in higher order thinking to respond to the task questions.
Original lesson plan available on the Maths Curriculum Companion