Lesson 20 Heads and tails

# Episode 3

Reasoning Resources: Coin data from Worksheet 1, Worksheet 4, Worksheet 5 on
A2/A3 or OHT
Pair work on getting more out of the data
The idea that individual events do not depend on the previous history is sometimes counter-intuitive. This is investigated by counting the runs of heads and thinking about the fact that the observed probability of a run of halves each time we increase the number of heads by one. Ask pupils: If you get five heads in a row, is the chance that the next toss will be a head greater? Give out Worksheet 4.

Explain that questions 1 and 2 are to help in the process of gathering the whole class's data on the question, so that they do that quickly to give the results to the teacher. You may need to explain what is meant by a 'run,' and that the runs can continue across the dividing lines between the 10s on Worksheet 1.
Whole class collection of data
Collate the whole class data into a table on the board as quickly as possible.
Some of these ideas have been touched on, in a qualitative way, in TM 16: Three dice, but here the relation between theoretical and experimental probability is made explicit and quantitative. Pupils will get the chance to encounter the ideas involved here again in TM 24: Data relations Show the whole class how the pattern develops by drawing a bar chart of the results on Worksheet 5. You can either fill the chart at the same time as you write in the table, or wait to find the totals for all the class.

The pupils should now discuss the class results, as shown in the chart, with their partner(s), by considering questions 1 to 4 on Worksheet 4.
Whole class sharing and discussion
The agenda here is to move from general recognition that there is a decrease in the heights of the columns as the run increases to the pattern of the decrease, then to quantifying the decrease as approximately halving each step. Use the words ‘roughly’ ‘nearly’ almost, ‘the difference between', ‘About how many times’ and ‘How much bigger’.

• Why does the pattern look like a halving one?
• What is the shape of the halving pattern? (an exponential decrease)
• Do past actions affect future ones? In what sense ‘yes’? In what sense ‘no’? Leave it hanging!
• End of Lesson Reflection
Pupils may recognise that they have met the issue of sample size before. They can also explore other seemingly puzzling properties of probability.

## License

Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.