Number in Probability 1

Purpose

These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation (see copymaster of investigation write up) and may become longer and more involved tasks with subsequent recording/reporting. Typically an exercise is a 10 to 15 minute activity.

Specific Learning Outcomes
  • conduct experiments with equally likely results and others where the results are not equally likely
  • using tally charts and frequency tables to summarise the results of an experiment
  • recording the outcome of an experiment as a fraction
  • the difference between theoretical probabilities and experimental estimates of probability
Description of Mathematics

Probability Level 5

Required Resource Materials
Excel spreadsheet

calculator, random number tables

dice, coins, drawing pins, spinners

Copymaster (PDF or Word)

Practice exercises with answers (PDF or Word)

Activity

Prior knowledge.

  • explain the part-whole fraction construct
  • convert between fractions, decimals and percentages.
  • recognise the symmetry in objects
  • average a set of numbers

Background

Probability concepts build on an understanding of number, in particular fractions. Early understandings to develop are the language of probability, and the idea that the probability of an event is always between 0 and 1, including 0 (or never) and 1 (always). Later on we use fractional numbers, decimals and percentages in equivalent ways when we speak of the probability of an event. For example, the likelihood of tossing a head can be variously described as being 0.5 or 50% or 1/2 or 1 out of 2.

Other big ideas include the concept of variance, and simulation to model real-life situations.

Probability is a very practical mathematical undertaking and teachers must develop these concepts through practical activities and language.

Comments on the Exercises

Exercise 1: Trials
Asks students to record the results of coin tosses on a triangular grid and discuss their findings.  This exercise is a practical experiment that introduces students to some of the fundamental ideas of probability. On one level they are learning to undertake an experiment, get an outcome and express it in the language (and numbers) of probability. On another level they are learning the concepts of fair trials, the value of repeating the experiment to establish a reliable result, and the unpredictability of small sample sizes.

The language of probability used in this exercise:
Experiment = the probability task being undertaken (for example, the toss of a coin ten times)
Trial = one repetition (for example one toss of a coin)
Outcome = one of the results (like getting a head).

The use of fractions in probability can be problematic if the idea of variance is not connected to the fraction early on. For example, when we toss a coin, we expect to get heads half of the time. However, this is not a result we can get if we toss a coin once, we can either get a head, or a tail, and we cannot predict in advance what the result of any toss will be. The figure of ½ only is useful if we toss a coin a lot of times.

In terms of this experiment the figure of one half means there should be the same number of left movements as right movements ending up at the bottom with 0.5. In practice this is not a very common experience; results vary and occur in a random fashion over a range that is generally close to the expected outcome. This experiment of tossing a coin ten times is also an example of an experiment with a small sample size. In such experiments, the small number of trials leads to very varied results. For example, the chance of getting exactly 5 heads and 5 tails is only about 25%, so on average one experiment in four (in the class as a whole) is likely to have this result. Here are the likelihood of some other results: 6 heads and 4 tails (or vice versa) = 0.2051
7 heads and 3 tails (or vice versa) = 0.1172 (or roughly one in eight)
8 heads and 2 tails (or vice versa) = 0.0439 (or roughly one in twenty)

Students generally struggle with matching 1/2 (or what they expect to happen) to what actually happens when coins are tossed, so it is important to discuss the different (varied) results that have occurred. Starting the debrief with a show of hands of who got what in one of their experiments. This should establish the concept of variance in the results, and that everyone got varied results. Collating the results from the class as a whole in a tally chart (or frequency table) is then worthwhile. Firstly it introduces these tools for collating and displaying results, secondly they should show a distribution of results that is roughly symmetric around the (expected) ‘five head and five tails’ result. A next step could be to ask the question like ‘how could we make the experiment more reliable?’ This may be worthwhile as it might elicit the idea that if we threw the coin more times (like 30 or 50 times in each experiment) the results might be closer to one half. Alternatively, it may lead to the idea that by comparing everyone’s results we can average out ‘the answers’ to see what has happened overall. These ideas can be returned to in later exercises if they do not come up at this time.

Exercise 2: Tribulations
Asks students to record the result of a dice throw on a triangular grid and discuss their findings.

Teaching lessons preceding this exercise could include:

  • Discussing the outcomes from tossing a die, and introducing the concept of an event (one outcome or a collection of outcomes – for example, getting a prime number when a die is tossed).

 This exercise is designed to parallel exercise 1, but with students having a higher understanding of the key concepts. (In other words, they should know what they are doing this time). They should be able to explain that they would expect different pathways (and even different end results) if the die is only thrown ten times. They should be able to suggest using a tally chart to summarise the number of sixes thrown by different class members, to get a better picture of what is happening. This experience can be then be built upon to introduce the concepts of expectation and random variation– that we expect to get a six, if we toss the die often enough, one out of six times. However the results are unpredictable (random) if we only throw it a few times. Going back to the pathways from exercise 1 can illustrate this point – there are 210 or 1024 different pathways that can be followed in conducting the experiment, so with only 100 pathways from the class, very few are likely to be the same.

A useful follow-up can be to introduce the concept of a simulation – using a mathematical object that mirrors the results and the chance of those results happening. For example, using a die to simulate tossing a coin by ignoring the 3, 4, 5 and 6 (or by taking odd numbers to mean the result is a head, and even numbers for a tail) and repeating exercise 1 with a die.

The equivalence of fractions, decimals and percentages in describing probability is important, and should be highlighted with students who are at stage 7 and 8, though simpler equivalences can be used with students at lower levels. For example, 1/2 means 50%.

Making predictions about outcomes is very important. Correct predictions shows internalisation of this type of thinking, while incorrect predictions allow misconceptions to be challenged, and the student’s understanding of probability to develop higher levels of sophistication.

Exercise 3: Drawing Pins
Asks students to record the results of tossing a thumb tack and discuss their findings. Teaching lessons preceding this exercise could include:

  • discussion about the outcomes of tossing a drawing pin
  • deciding what is up and what is down
  • deciding if tossing one 30 times is the same as tossing 30 once.

This exercise repeats exercises 1 and 2, but challenges the understanding that has developed in those 2 exercises by introducing unequally likely results. This will develop students’ understanding as they should discover that predicting results is not always straightforward, especially when the results have a different chance of happening. In such a case, a small experiment of ten trials will not produce a very reliable result, though averaging the results of a number of experiments can, as can working out the probability based on a large number of trials. With a drawing pin (or die), 100 trials will produce a relatively stable probability.

Stress to students that the only way to work out the probability when the outcomes are not equally likely is to actually undertake the experiment – or to simulate it.

Exercise 4: The Language of Probability
Asks students to work with a classmate to discuss the language of probability. This exercise is in two parts. The first is a good reintroduction of students to probability, as it can build on students’ prior knowledge as well as their experience (and misconceptions about probability) from everyday life. This activity can lead onto other discussions around probability. One such conception worth discussing is that of luck. Students may well believe that if they are lucky, they will win a raffle, whereby the probabilistic model would say they have the same chance as everyone else.

Getting teachers to do the first activity is also interesting, as it is rare to find that they have similar results (and disagreements can develop if two or more are trying to negotiate meanings around some of the words). Some see certain words as being interchangeable, while others say one is more likely to happen. This highlights that the way we use and understand language is very individual, and that developing a shared understanding that can be used in the mathematics classroom is something that needs continuous work. Here it is the discussion and developing a common language for probability that is important – rather than an exact interpretation of each word. Similar meanings are likely to develop, but it is unlikely that everyone will agree to order the words in a single line of increasing probability.

A good follow-up (for homework) is to ask students to investigate other words that are used when talking about the probability of something happening. Asking adults can be a good start. Note that there are many other words and phrases, including sayings like ‘once in a blue moon’ (which happens to be the second full moon in a month, which is why it does not happen very often).

This sort of exercise can be an ongoing event and could be a class discussion followed by independent group work.

Exercise 5: Mathematical Probability 1
Asks students to explore theoretical probability of tossing a coin and recording the results. Teaching lessons preceding this exercise could include:

  • a discussion of objects with symmetry and the impact of symmetry on likelihood
  • discussion of what is not equally likely

 Develop the idea of mathematical probability can be calculated from the symmetries inherent within the tool being used, but also highlight that visible symmetry does not always tell the true picture. Note that a piece of lead drilled into one side of a wooden die can change the likelihoods of different outcomes. (This can be a fun thing to try – make a die, then drill out the numbers on the six and embed lead pellets, and see how much the probability changes. This can also be done with different amounts of lead...)

An additional activity is to make and use spinners with different areas for different numbers and dice with different shapes to get different probabilities. Warning…complexity is very easy to develop and may well cloud the basics.

Exercise 6: Mathematical Probability 2
Asks students to explore theoretical probability with a die. 
This is a consolidation of the previous exercise and should indicate learning. This exercise can be repeated for many different objects depending on student interest.

Exercise 7: Experimental Estimates of Probability 1
Asks students to conduct an experimental trial with a coin and record results. Teaching lessons preceding this exercise could include a discussion of:

  • the notion of repetition leading to reliability
  • “the Law of Large Numbers”

This graph was traditionally drawn by hand, which caused much grief for year ten students as many could not get the idea of keeping a running tally of the proportion of heads, or could not convert the fractions to decimals or could not plot the decimals they worked out, let alone work out what it all meant. Many of these problems can be avoided by getting the students to create a table in Excel, then graphing the data with a line graph.

This activity is also the door to much internet investigation. There are many simulations that are fun to find and use.

Exercise 8: Experimental Estimates of Probability 2

Asks students to conduct an experimental trial with a die.This is the same exercise as in exercise 7 but uses a die. Students should be stating ideas of their own that involve probability. Different students experimenting with a different number of trials is also valuable here as it allows them to compare results and make a decision about how many trials are needed before the graph becomes reasonable stable. Wisdom is around 30.

Exercise 9: Simulation Design
Asks students to design an experiment to simulate a situation.  An example is given to follow. Teaching lessons preceding this exercise could include:

  • discussing and exemplifying the simulation design process
  • linking repetition to “The Law of Large Numbers”
  • discussing how many is enough when doing an experiment

Have students investigate their own simulation. An idea is to simulate getting All Black (or Silver Ferns) cards from packets of wheat biscuits. If there are 10 different cards in a full set, how many packs do I need to buy if I want to make sure I get a full set? Note that having more than ten in a set may mean that the simulation tool needs to change. Ten sided dice are easy to get hold of, but not so 15 sided dice. An alternative would be to put 15 cards, with numbers one to fifteen, in a bowl, and have a student stir them before making blind draws. (Draws without looking or peeking.) Each card needs to be returned to the bowl after each draw.

Exercise 10: Random Numbers
Asks students to explore the concept of random numbers by completing a table of random numbers. Teaching lessons preceding this exercise could include:

  • the notion of randomness

This is a counting exercise but the interpretation if the count is a more complex notion. Random events are not predictable. Unfortunately for some people, this means they think that this means there is not pattern to the results, whereas in the toss of a coin 5 times, HTHTH is one possible outcome, while HHHTT is another. Both of these results are as equally likely as any other individual combination of H and T. However, getting a result that does not show such orderliness is more likely, as there are far more combinations that do not show order than do show order. One way around this issue is to deal with larger sample sizes, as there are far more combinations, and proportionately fewer combinations that show order. This is also found in the law of large numbers, unusual ‘runs’ of results, generally being cancelled out in the long run. (For example, the chance of getting ten heads in a row is small, one in 1024, or one in a thousand. This idea is used when checking the randomness in a set of numbers by taking a largish sample of them and seeing if the frequency of each number is roughly the same.


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