Mathematics Developmental Continuum
First Experiences with Chance: 1.25
Supporting Materials
Indicator of Progress
Success depends on students beginning to develop a qualitative understanding of the attribute of chance, which enables them to compare the likelihood of some events.
Prior to this, students recognise both predictability and uncertainty in events.
Chance and probability are sophisticated concepts that do not fully develop until Level 4 and beyond. In terms of the Measurement Phases, this indicator of progress is an early stage of identifying the attribute.
Developing the mathematical concept of proportion is essential for quantifying chance, and it in turn depends on understanding of multiplication.
At this level, students build an intuitive basis for both chance and proportion.
Three phases of teaching chance
Illustration 1: Language of chance and predictability
Students can appreciate both predictability and uncertainty in events, and indicate this verbally. For example, “I am sure that my mummy will come to get me, but I am not sure exactly when she will come. She might be the next one to come or she might come later”.
Illustration 2: Informal appreciation of proportions
The concept of chance depends on the concept of proportion, so an important pre-cursor is for students to look not just at absolute numbers, but also at proportion. They need to be able to distinguish large numbers and large proportions. For example, they may say “A lot of students at our school support Carlton and a lot of students support Collingwood, but most students here support Collingwood”, commenting on both the absolute numbers and the proportion.
These are difficult ideas to express before multiplication is fully established, so the language of proportion, as with chance, is only qualitative at this level.
Illustration 3: The first understandings of likelihood
Some students may believe that some events are more likely to occur than others, when in fact the events are equally likely. For example, some students may think that it is much harder to roll a 6 on a die than any other number. This may be because they have taken part in games that require a 6 to start or do something else desirable. We regularly have our attention focussed on the difficulty of getting a 6, but not the equal difficulty of getting any other specific number (e.g. a 3). We tend to compare the lower probability of getting a 6 with the higher probability of getting any other number (i.e. just not a 6).
Students need a wide variety of chance experiences to help them understand when some things are likely to occur and others are not. These experiences develop intuition which will support more formal learning of chance at higher levels.
Teaching Strategies
Early teaching strategies to establish students’ awareness of the attribute ‘chance’ include:
- experiencing play and practical activities involving chance
- learning the language of chance
- comparing events on the basis of chance and
- distinguishing chance from other attributes.
These teaching strategies parallel those used for establishing awareness of other measurement attributes.
The concept of chance depends on understanding of proportion, which is a multiplicative concept. Until multiplication is strongly established, students can develop qualitative ideas of both concepts: chance and proportion.
Because of the preliminary stage of development of this concept, the basic teaching strategy is to address chance as it arises in a variety of classroom activities, rather than making it a focus of instruction at this stage. Only Activity 6 is an explicit activity – the others point out how the concepts of chance and proportion and some associated language can be developed incidentally through other classroom activities and events.
Activity 1: Sharing or choosing with a random element suggests that students observe relative proportions of items (e.g. as part of building counting skills) and link this to chance.
Activity 2: Playing games with a chance element highlights how teachers can use everyday classroom games to highlight aspects of chance.
Activity 3: Observing chance events from the world around us shows how teachers can use discussion of everyday situations to talk about events that are more or less likely than others.
Activity 4: Chance and sport discusses situations where teachers can focus on how to make a game fair by ensuring that all players have an equal chance of winning.
Activity 5: Linking chance with proportion points out everyday classroom opportunities for developing these two concepts in tandem.
Activity 6: Tallying dice rolls is intended to help students move away from the misconception that some numbers are more likely to come up than others when a die is rolled.
Activity 1: Sharing or choosing with a random element
There are many classroom and home occasions when young students are given something at random e.g. giving out coloured pencils, giving out small gifts or sweets. Sometimes there will be equal numbers of each colour/ type etc, so the chances of getting particular types are equal, and sometimes there will not be. Drawing attention to the proportions and chances involved supports and extends the Number learning that is the main focus at this stage.
Example: Giving out coloured pencils
Students can count beforehand how many of each sort there are, and then observe which item is the most common. The teacher can comment that this means that an individual student is most likely to get a red pencil, or have the same chance of getting a red pencil as a blue pencil etc. At this level, the most important learning outcomes are the Number outcomes such as accurate counting into the tens and knowing which number is larger (e.g. knowing that 25 is less than 47). Having students look at the whole set of pencils will help develop ideas of proportion, as well as absolute numbers. They can see, without counting, that most of the pencils are red.
Students might say “there are a lot more red pencils than blue pencils; there are less blue pencils than red pencils; most of the pencils are red; more of the pencils are red; probably I will get a red pencil”.
Example: Random processes for choosing
Students can be chosen for classroom activities using random processes. If they each put their name in a hat then they will have an equal chance of being selected. Students can use words such as ‘fair’ to describe this. “Everyone has got the same chance of being chosen, because we all wrote our names on only one card. It would not be fair if someone put their name in a hundred times but mine was only in once.”
Teachers might highlight this by carrying out the experiment!
Activity 2: Playing games with a chance element
Many simple games that are useful for developing mathematical skills (e.g. number facts) involve throwing dice, using spinners or choosing a card at random. These games provide opportunities to use chance language and to informally compare the likelihood of events.
For example, a student playing Snakes and Ladders (with one die) is 2 away from a ladder and 5 away from a snake. The teacher can ask:
- whether both of these are possible outcomes of the next throw (yes)
- whether one is more likely than the other (no) and
- how students have decided.
At this stage, students are likely to give ‘magical’ explanations, for example “I think I will land on the snake because I am not lucky today”, and the teacher can make counter predictions such as “I think you are just as likely to land on the ladder”.
If two students want to decide who goes first in a game, they can use a spinner to decide. You might like to ask them which spinner they would prefer to use to decide: one that is divided into two halves or one that has two very unequal regions. Get them to try both, noting that with the unfair spinner sometimes the smaller region will win … although most of the time it won’t.
When conducting dice games that require rolling a certain number to start or to have something ‘good’ happen, vary the requirement so that it isn’t always a 6 or a 1. This will help students avoid the misconception that 6s or 1s are somehow ‘unlucky’ or ‘lucky’ (see Illustration 3).
Activity 3: Observing chance events from the world around us
Many everyday events involve chance, so students will be exposed to chance language at home and school. For example, after the class has created a weather chart students can qualitatively compare the chance that tomorrow it will snow, rain, be hot, warm or cold. If it is often rainy and hardly ever snows where we live, it is more likely that it will rain tomorrow.
Teachers will model the use of the language of chance and of proportion when discussing the chart. Students may say “It is likely to be wet tomorrow”; “There is a good chance it will be fine for the swimming carnival”; “I can see just by looking at the chart, that it was sunny on most days this month”; “There are more sunny days than wet days where we live.”
Activity 4: Chance and sport
Sport provides many opportunities to think about chance. For example, students know that it is not as likely that a student will score a goal (whether football, basketball or throwing a bean bag into a hoop) if they have to stand further away than another student does, or if they have to get the ball into a smaller goal.
Students can discuss how to make a game fair by giving everyone the conditions for an equal chance. Teachers might discuss how it might be fair to give someone a head-start in a race, for example if a teacher or a Year 6 student was running in the race, because otherwise that person would have little or no chance of winning.
Activity 5: Linking chance with proportion
The concept of chance depends on the concept of proportion, so an important pre-cursor is for students to look not just at absolute numbers, but also at proportion. Students need to be able to distinguish large numbers and large proportions. These are difficult ideas to express before multiplication is fully established, so the language of proportion and work on it is only qualitative at this level.
Teachers can take opportunities arising in other classroom situations to make links between proportions (a qualitative concept only at this stage) and chance. Instances where there are implied proportions arise in situations such as the following.
A large number might still be a small proportion
“From the photo of school assembly, I can see that most kids wear green sunhats, and not as many wear gold sunhats. But there are still a lot of kids like me wearing gold sunhats.”
“Most of the blocks in the tub are yellow, and there are not as many blue ones, but there are still plenty of blue blocks for us all to build blue houses”.
The link between proportion and chance
“I hardly ever miss school, so it was unlucky that I was sick when the snake show was on.”
“They had mostly pink balloons at the party, so I had a good chance to get one.”
Informally comparing underlying proportions
“The boys cried just as much as the girls when they got their injections.”
“I nearly always eat my lunch, but she often throws hers in the bin.”
“The big kids have got a much better chance of getting the front seat on the bus.”
Activity 6: Tallying dice rolls
Have students work in pairs. Get one student to roll the die and the other person to mark a cross each time the number is rolled in the corresponding numbered column in a tallying table like the one below. Roll and record the results 20 times.
Have each pair share their tabled findings to the group, highlighting which numbers came up more and less. Discuss how one pair got more 4s, another pair got more 3s and so on.
Repeat the activity later in the week. Highlight that this time different numbers were ahead and behind.
There is no need to talk about formal probability, as such, or talk about equal likelihood. These experiences will build students’ understanding that there is no number that is especially likely to turn up or not turn up.
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Further Resources
The following resource contains sections that may be useful when designing learning experiences:
Digilearn object *
Spinners: predict and test – students race two cars along a track. Students use a coloured spinner (dial with pointer) to determine which car moves along the track. Students use a spinner with two or three equal-sized sectors, each coloured red or blue. Students test the spinner over a number of spins.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4401)
* Note that Digilearn is a secure site; DEECD login required.