Lesson 24 Data relations

# Episode 2

Reasoning Resources: Pupils' household data
Pair and small group work
The agenda is to move from just trying descriptive models — like a three bar chart of more, equal and fewer - to realising that you have to think about the mathematical models used, and to choose one that will reveal the information you want to know.

Conduct the lesson so that at the most the teacher asks the questions, and the pupils have to construct possible ways of addressing them, and are also encouraged to think about the possible real-life causality involved.
Let the groups try their methods of choice from the suggestions in Episode 1. Use then following questions if necessary.

How can you use the data your class has collected to investigate the idea that the number of girls in a household stays the same or similar from one generation to another? Start with the class data for mother’s household and discuss with your partner(s} one or more different ways of using mathematics on this question.

With lower and middle attaining classes you may need to give more direction. You could advise them directly to write ‘more; ‘equal’ and ‘less’ alongside the numbers in their table before they count them, or compare the columns.

Emphasise that there are two variables: Current household and mother’s household. How can we use mathematics to see if there is a relation between them?

In all cases steer groups towards forms of correlation they have encountered in TM22: Comparing correlations. Rehearse promising lines with some groups, especially the two-by-two confirming and disconfirming table using more/more, fewer/fewer, fewer/more and more/fewer.In order to do this, pupils need to go through all the data, assigning each data pair to one cell in the grid.
Here, if the majority of the entries lay on the central diagonal they could be taken as confirming a correlation, but if they were scattered at random among the nine cells the evidence would suggest no correlation.
Whole class sharing/discussion
Some pupils may wish to fractionate the data into a 25-cell table, built up of ‘all girls’ ‘more girls’ equal’: ‘fewer girls’; ‘no girls! In this case a little discussion might reveal that ‘no girls’ for the mother’s family is impossible, so it would have to be a 20-cell table. The extra data on the fathers’ families is collected also (but not initially addressed) in case some of the pupils realise there is a problem of bias. Allow all groups to explain their methods of interpreting the data, what the problems are, and what conclusions they have reached, if any. The suggestion should emerge that using the mother's data will tend to produce a bias, because it will always include one girl. Groups may wish to compare the father’s data with ‘own family’ data, or combine mother’s and father’s data into one amalgamated group. Discuss the advantages of different methods: the simpler the data the easier it is to find a conclusion, but the fuller data gives a more accurate picture. Partial conclusions and other insights into the data unconnected with the hypothesis can also be seen as useful. Whatever conclusion the class arrives at will be a genuine and real conclusion for them. A discussion of whether this will be the same picture elsewhere could lead to consideration of sample size (the agenda of TM20: Heads and tails),
Extension
With a more able class, it may be possible to continue the investigation by converting all ratios to decimals, and plotting them as points on a scatter graph. Here the discussion will be about the degree of likelihood indicated by the estimated line of best fit.
End of Lesson Reflection
Depending on the issues that have interested the class during the activity, pupils should discuss the value or usefulness of this work, including the ambiguities and sensitivities of the data collection.