In this unit we play probability games and learn about sample space and a sense of fairness.
Three important ideas underpin this unit:
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.
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:
Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. For example:
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).
We introduce the unit by rolling dice and investigating the numbers that come up.
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 | lll |
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 |
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 totals 240 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.
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:
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.
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?
Dear parents and whānau,
This week in maths we have been playing probability games, discussing if they are fair and what likely outcomes might be. We played the Bunny Hop game in class and we would like to share this with you.
Bunny Hop Game
The winner is the player who is on the highest number after 10 tosses each. Before you play, talk together about where you think the counters are most likely to be after 10 tosses each.
5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 |
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/s-not-fair at 8:47pm on the 26th February 2024