Lesson 18 Prediction and correlation

# Episode 2

Reasoning Resources: Worksheet 2 on OHT
Whole class introduction
Piaget described qualitative understanding of correlation as early formal, and a quantitative treatment (because it involves proportion) as mature formal, since it is one of the ways in which people proceed from possible models to seeing whether reality agrees with the model. Only about 20% of pupils by the end of Year 8 (without intervention) have access to early formal thinking (or above), so the teaching of correlation as a purely instructional aim is inappropriate for at least 70% of pupils of this age. Ask pupils about predicting school results. Give out Worksheet 2, explaining that each cross marks one person's grades in two subjects. They can be labelled: This could be Mary, and this Fatima, All the points can labelled with (lower case) letters if necessary.

Ask: How many got E grade in science? How many got grade E in maths? Draw a faint, narrow lasso to show the range in each case.
Pair work
However, the majority of 12 year olds recognise intuitively that scatter diagrams show relationships. Many can also estimate the strength of relationship by how close the scatter is to being a simple straight line. This episode can be carried out as whole class with pupils given short periods to work in pairs on one question at a time, with immediate feedback. Questions for paired work:

• Sally got grade E in maths, but she missed the science exam. What is the range of science grades she might have got? What is the most likely prediction for her grade in science?
• James has missed both exams. Both the maths and the science teachers said he was ‘about average’ in the class. Suggest a way to award James a grade in maths and a grade in science.
• What can you do to the graph to show the patterns you think there are? Where would a line of the most likely connection between maths and science grade be?
• Here we are dealing with descriptive models (concrete operations) where there is a direct relation between the form of a graph and the relation it is modelling. Question 1 asks pupils to suggest a middle grade in the range F to C,as an estimate for Sally. They should remember what a median is, including for an even number of cases. Pupils would opt for the pair of middle grades, keeping the uncertainty, but reducing it. For question 2, pupils would grope for a middle grade in a maths and a middle grade in science. They should order the grades for each to find the median grade in each case. For question 3, some may suggest enveloping the data in top and bottom lines, which allows the ‘best fit’ line (better described as ‘best prediction’ line) to be placed in between. Prompt them for other descriptions of the line, e.g. the values that are ‘least different from’ or ‘least wrong’ compared to the real ones.
End of Lesson Reflection
What do you think are the advantages and disadvantages of this kind of mathematics - the mathematics of correlation? Elicit the ideas that correlation allows us to make predictions, or rather shows us likely limits for a prediction, and that the predictions are not precise. Ask for the differences between this and the work in Episode 1.