Dear Family, your Operation Maths guide to Chance

Dear Family, your Operation Maths guide to Chance

Dear Family, listed below are some practical suggestions as to how you might support your children’s understanding of the maths topic of Chance (for 3rd-6th classes only). Also below, are a series of links to digital resources that will help both the children, and you, learn more about Chance.

Junior Infants to Second Class
You can also find class specific tips in the Operation Maths Dear Family letters for third to sixth class.

Practical Suggestions for all Children

  • Chance is one of the most interesting, and fun, areas of primary maths, since it is mostly about probability, i.e. identifying the possible outcome(s) of random events. With your children, talk about ‘chance’ whenever you have the chance (excuse the pun!):
    • What is the chance that you’ll go to school today?
    • What is the chance that you’ll get homework today?
    • What is the chance that you’ll get to watch TV or get to play computer games?
    • What is the chance that it will be warm tomorrow, that it will rain, that it will snow?
    • What factors affect the likelihood of these events occurring? For example, the day that it is, the time of year, whether the child has done their chores etc.
  • In school, we use language such as impossible, (highly) unlikely, may or may not, possible, (highly) likely, certain, etc., to describe the likelihood of events occurring. Encourage your children to use these words as accurately as possible, especially the words impossible and certain. For example:
    • On a sunny day, what is the chance of rain? Unlikely or highly unlikely you could say, but it wouldn’t be correct to say impossible, because anything is possible!
    • On a day when you have organised to do something e.g. go shopping, what is the chance of it happening? Likely or highly likely, because it is already organised, but it is not certain, because again anything could happen to disrupt the well-made plans, like the car mightn’t start.
    • If I toss a 6-sided dice once, what is the chance of getting a 7? Now that’s impossible! What is the chance of getting a number from 1 to 6? That is certain!
  • Children in 5th and 6th classes are also encouraged to use more mathematical ways, including using fractions, decimals and percentages, to express probability e.g. 100% certain, a 1 in 4 chance, 50/50, etc. This type of language could also be included in your discussions at home.
  • So, 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! For example, when I toss a six-sided dice, each number has a equal chance of coming up. Therefore, if I do this repeatedly for a number of times, I could expect to see equal occurrences of each number. Yet that might not happen in reality! But, it is most often the case, that if you repeat this type of investigation enough times, the actual results WILL end up being very close to the predicted outcomes. In other words, the more you do something, the more likely it will happen as predicted. Some of the activities in the Operation Maths books are specifically designed to explore this. So try them and see!
  • Many games are designed around random outcomes so play board games, card games, dice games, any type of game where you can’t know from the outset who will definitely be the winner! Ask the children before you play, and as you play, who do they think will win and why; perhaps somebody in the family is a dab hand at rolling sixes, is a card shark or after a number of turns is already way ahead of everybody else. At the end of the game did that person win? Perhaps, on this occasion, a person was dealt “bad” cards, or the dice didn’t fall as hoped for, or another player caught up and overtook the early leader. Or maybe not! Experiences like this, help the children appreciate how lots of different factors can influence and affect an outcome, and that they can predict winners or outcomes based on the best information that they have at the time, but that the predicted outcome may or may not materialise.
  • Study the weather! Look at the sky and discuss the chances of rain, sun, snow, lightning etc. Look up Met Éireann’s website to find out the weather forecast for your area and then, afterwards, discuss whether the predicted weather arrived. Again, while meteorology, the study of weather, is a science in itself, it is still involves using the best scientific information available at the time to predict the weather, which, in the end, may or may not happen.
  • Sport provides us with an abundance of opportunities to discuss chance:
    • What are the chances of a particular team or individual winning a game, match, fight, competition or race?
    • Before the event could you predict an outcome?
    • What information about the competitors or teams might be useful to influence this predication?
  • Draw the children’s attention to any other situation where chance plays a role e.g. the chances of winning a raffle or the lottery.

Digital Resources for Third to Sixth Classes

Math is FunProbability: Background information on probability and chance from Maths is Fun

Random Dice!Interactive online chance tools: No dice at home? Don’t want to have to make up a spinner? There are lots of interactive tools and random chance games here.

Math Antics - Basic Probability - YouTubeBasic Probability: A video from Math Antics, that introduces the concept of chance, language of chance and the probability line.

Impossible? Unlikely? Get seriously foul, gross drinks in The Vile ...

The Vile Vendor Probability Game Use your understanding of chance to work out the likelihood of getting these vile drinks!

The slushy sludger: questionsThe Slushy Sludger Use your knowledge of probability to predict what kind of slushy you are likely to get from the choices on offer.

Measurement curiosities: Online games to practice probability

Probability Pond Selection of probability games based round a pond theme. 

Climber Probability Math Game for Kids | Toy TheaterClimber Probability Game: Help the climber reach the top by clicking on the colour that you think will win the spin.

Interactive Probability Math Game - Pull Objects From The BagProbability activities: There are a lot of activities here that range from simpler to more complex. Start with activity 1 and then go through them in order, until the content gets too difficult.

Adjustable SpinnerAdjustable online spinner: use this to make up your own spinner , predict the outcome and and then investigate the actual outcomes.

View details - ScootleSpinners Chance and Data Assessment Build spinners to show how well you understand chance and probability. Your answers will be saved to a report which you can review at the end.

PROBABILITY MODEL MATH ACTIVITY! - YouTubeUsing area models: For fifth and sixth class, Mashup Math has this excellent video which demonstrate how area models can be used to identify all possible outcomes.

Tree Diagrams Explained! - YouTubeUsing Tree Diagrams: Another excellent video from Mashup Math, this one demonstrates how tree diagrams can be used to identify all possible combinations.

Mathwire.com | Data Analysis & Probability GamesCheck out this Mathswire page for more games that focus on probability.

IXL | Maths and English Practice

Probability: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription.

Probability: Some more online practice games


Digging Deeper into … Chance (3rd – 6th)

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of chance, please check out the following post: Dear Family, your Operation Maths Guide to Chance

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).

FACT: To avoid ambiguity, some organisations have agreed on a consensus that equates this terminology with a fractional expression or percentage; you can view one such consensus here.

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.

From Operation Maths 5

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).

From Operation Maths 5
From Operation Maths 6

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!

TIP: To make life easier for you, we have created a sample spreadsheet for the Double Coin Toss, please click on the link to view (and save/copy). For further information on the values of using spreadsheets to record results please check out this informative article on Probability Experiments with Shared Spreadsheets from NCTM.

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.

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