# Episode 1

Reasoning Resources: Coins, Worksheet 1
Whole class
There are two approaches to probability. The first is causal, where the causality is known in advance. Since we know that coins are symmetrical, we have a prior expectation that the probability of throwing a head should be 0.5. Introduce the activity as testing the expectation that there is an equal chance of getting heads and tails when throwing (tossing) a coin.

Carry out a class demonstration with a sample of 10 throws. Will 10 throws give 5 heads?

Discuss fairness in dice and coins, perhaps by asking pupils to demonstrate unfair and fair throws, so they understand how to avoid bias in the experiment.
Pair work
The second is observational (experimental probability). We actually count the occurrence of the possibilities (two alternatives in this case), and work out the probability from the sample of observations made. But in a small sample the proportions may vary wildly from what we expect. Give out Worksheet 1 to pairs of pupils. To reduce the noise pupils can spin the coin and hold it in one hand.

Suggest that pupils write H in the table if the coin shows heads andT if it shows tails. After 10 throws they can work out how many heads were in that batch. They should complete all 50 throws before answering the questions.

Help pupils to work out the percentages if they get confused. Most pupils will realise how to rewrite a number out of 10 as a percentage, and that they could simply double a number out of 50 to find the percentage. Show and leave examples on the board for scores such as ‘3 out of 10’ and ‘21 out of 50! As an alternative, they could calculate ‘parts of 1’and use these to compare proportions.

Some pairs may need to be shown how to plot their proportions on the graph.
Whole class sharing and discussion
Probability is one of the ten ‘formal operational schemata’ or reasoning patterns that Piaget described as characterising the formal stage. It involves comparing ratios so as to look for the underlying proportions determining the probability. It also involves making some kind of mental model of reality, and then seeing to what extent reality confirms or behaves in accord with the model. Collect the data from each group into a table on the board, recording the least number of heads in a sample of 10, the greatest number of heads in a sample of 10, and the percentage (proportion) of heads in 50 throws. For example: Allow pupils to express their ideas about why the variation in the total samples (50) is smaller than for samples of 10, All the pairs should check to see if this is true for their samples, appreciating that they need to use a percentage (proportion) to do that, while also referring to the raw scores for their samples of 10. Pupils would struggle to put in a chain of reasoning the ‘averaging effect’ of combining samples. For the 10 throw samples the variation is between 20% and 80%. In the 50 throw samples the variation is between 45% and 60%.

For some classes the numbers can be truncated or rounded to one decimal place only, with the difference qualitatively noted as a bit bigger’ or fair way bigger’

With a high attaining class, for whom the practicalities of the activity are straightforward, you could discuss ways of collecting data to investigate whether there is a 50-50 chance of a baby being a boy or girl.