Digging Deeper into … Chance (3rd – 6th)
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Chance is one of the most fascinating areas of primary mathematics, since it is concerned with the outcomes of random processes. Thus, the conceptual foundations for areas of mathematics such as probability and combinatorics, can be found in this strand unit.
The big ideas about Chance:
- When considering random events and/or processes, we can use what we know (eg past experience and/or knowledge of the variables involved) to estimate/predict the likely outcome(s).
- If we identify all the possible outcomes in advance, we can refine and/or express our prediction using mathematical language.
- However, no matter how accurate the mathematical prediction, the actual outcome(s) is not certain (except in the unlikely case where there is only one possible outcome); that is the element of chance!
- If we collate the results from repeated identical investigations of a specific random process, the actual outcomes (experimental probability) are more likely to reflect the original mathematical predictions (theoretical probability).
Predicting Outcomes: Terminology
When beginning to discuss and predict the likelihood of various outcomes, the initial focus should be on the language of chance, and the terminology that accompanies it.
It can be very useful for the children to identify the various terminology, to discuss their interpretation of it and to explore the contexts in which the terminology is used in everyday parlance.
And while some of the phrases are more objective (e.g. impossible, never, certain, sure, definite), much of the language can be more ambiguous and is open to personal interpretation (possible, might, there’s a chance, (highly) likely, (highly) unlikely, not sure, uncertain).
It can be helpful to try to organise this language across a continuum for the children to interpret and establish their meanings in relation to the other phrases. Ask the children to identify terminology that is used when describing the likelihood of something occurring. Use questions/statements to elicit from the children the vocabulary for chance that they already have; this can be the language that they would use to answer the questions from the text above or could be from their responses to questions such as the following:
- What is the chance that it will rain today?
- What is the chance that it will be hot today?
- What is the chance that it will be dark tonight?
- What chance does my team have of winning the league?
- What chance does my county have of winning the All-Ireland Championship?
Ask the children to write this terminology on pieces/slips of paper. Sort the pieces of paper into groups and/or order them along a line (continuum), as shown in the images below, with words that have similar or identical meaning together.
This task is a perfect example of a low threshold, high ceiling task, in that all children can participate and there is no limit to the complexity of terminology that can be incorporated. If mathematical values such as percentages and/or fractions (eg 1 in 2 chance) are suggested, the children should be encouraged to incorporate these, as they see fit.
Indeed, in fifth and sixth class the children should be encouraged to use a continuum which is graded from 0-100% and/or 0-1, and to associate and align the vocabulary with mathematical values (eg impossible/never =0%, might or might not/even chance = 50%, definite/certain = 100% etc).
Predicting Outcomes Mathematically
Irrespective of whether it is tossing a coin, rolling a dice, spinning a spinner, picking from a bag, choosing a card, etc., the children should always be encouraged to identify all the possible outcomes, to predict outcomes that are more or less likely, and to justify their predictions.
The children can also be encouraged to make more mathematical predictions based on their understanding of the variables involved e.g. if we repeated this investigation 30 times, how many times would you expect each colour would be picked? What about 60 times? 120 times? Express the fraction of the total number number of “picks”, that you would expect for each colour. Can you express any of these as a percentage?
When predicting the outcomes of random processes that involve a combination of variables, it can be very useful to use a type of pictorial structure, such as branching (NB these can also be referred to as tree diagrams), to illustrate the possible outcomes. For example, when predicting the outcomes of a double coin toss, children will often think that each of the three outcomes have an equal chance, when in fact there is double the chance (ie 2 in 4 or 1 in 2 chance) of getting a heads and tails combination, than either both heads or both tails (see diagram below).
However, it is worth noting that, unless the children come up with a similar structure to predict outcomes of combinations, it is preferable to hold back on showing such a structure until they have conducted an investigation, similar to above, where their predicted outcomes did not align to the actual outcomes.
Conducting the investigations
Once all appropriate predictions have been recorded, we can move on to the most exciting part, the investigating! When conducting chance investigations, it is important that the children recognise that that they need to be conducted fairly and recorded clearly, similar to scientific investigations.
Encourage the children to consider what factors need to be kept the same each time, and how practices could affect the reliability of the results eg:
- When picking items (eg cubes from a bag, cards from a deck) does the chosen item need to be returned each time? Why/why not?
- How many times does an investigation needs to be repeated in order to get a reliable result?
To generate sufficient data, while not spending too much time on each investigation, ten can be a suitable number of turns per child. It can also be a good idea to organise the children into groups of three with rotating roles eg the first child has their turn, the second child records the outcome of each turn and the third child keeps count of how many turns the first child has had, and roles are rotated after ten turns.
Recording and reflecting on results
As mentioned previously, the children should be encouraged to consider how best to record results. Tally charts and frequency tables can be very useful and link in well with the strand unit of Representing and Interpreting Data. Results of investigations can be displayed in various types of graphs and charts. Children in fifth and sixth classes could also be asked to calculate the average value for each outcome, when all the results of a class group are considered; for example, in the double coin toss, what was the average number of heads, tails and heads-tails combination per group.
Once the results have been collated, it is very important that the children be given time to reflect on the results and to compare them to their predictions. While we would expect an equal number of heads and tails in a single coin toss (ie theoretical probability), the actual results may not resemble these predictions (experimental probability). Such is the element of chance! And this can be a difficult concept for the children to accept, particularly the notion that, even though the mathematics behind their predictions was accurate, the actual outcomes are different.
To explore this further, using a spreadsheet, such as Google Sheets or Microsoft Excel, to collate the results of the entire class can be a great way demonstrate, that when we combine all the investigations, experimental probability (ie the results) is more likely to mirror theoretical probability (the predictions). This can often help reassure the children that the “maths” behind this does indeed work!
Further Reading & Resources
- The PDST has a lot of resources for Data and Chance, including a booklet, slides and task cards for activities.
- Playing dice, card, spinner games, or indeed any type of chance-based games, can be a great way to get students thinking about probability, while also providing practice with mental computation, estimation, subitising and experience of problem-solving via strategic thinking.
- Don’t forget to check out the games bank in your Operation Maths TRB and/or the last page of the Number Facts books for examples and ideas.
- Check out this Mathswire page for more games that focus on probability.
- iTools has a great set of interactive tools for probability that cover coin and dice throws, pulls from a bag, among other random processes. As well as being very customisable, they compare the theoretical and experimental probability, using various visual structures including tables and branching (tree diagrams); the latter is used particularly well to illustrate possible outcomes in compound events (e.g. double coin toss or double dice throw) as well as combinations and arrangements.
- For a fifth and sixth class who are exploring combinations, Mashup Math has two excellent videos (view both below) which demonstrate how tree diagrams and area models can be used to identify all possible combinations; both video use contexts to which the children could readily relate.
- Johnnie’s Math Page has lots of resources for probability including interactive spinners and dice.