Lesson 18 Prediction and correlation

Episode 1

Reasoning Resources: Worksheet 1, Worksheet 1 on OHT (optional)
Whole class introduction
The concepts of probability and correlation need to be understood both in terms of mathematics and in their everyday application as an aspect of causality.

Probability here is integrated with the correlation methods by which outcomes can be explored.
Ask the pupils what they wish to be when grown up, leading quickly to classifying careers into salaried jobs e.g, teachers and office workers, and freelance or ‘own business'. Ask for advantages and disadvantages of each type, gradually focusing on security or steady income versus uncertainties or risks. Move to an example of a family newspaper shop and its profits in the first few weeks of starting the business.

Give out Worksheet 1 and work in whole class mode. Look at the table of shop profits and ask pupils why they think the shop started with weekly losses, and what the pattern of the first six weeks appears to be.On the board or OHP. plot the points for the first four weeks, inviting pupils to show where points should be plotted. Ask them to plot all the values on their own copy of the graph.
Pair work
In Worksheet 1 pupils handle one stochastic (probability) variable (weekly profit), in relation to weeks as a time series.

Here the ideas of likely profit and likely range of profits introduce pupils to the idea that any stochastic variable would always have two parameters, corresponding to the mean and the range of deviation from it.
Ask pupils to consider the following questions, using their graph:

• Is it likely that the shop made more than £2000 profit in week 9? Explain.
• Estimate the maximum and minimum profit expected in week 9.
• What do you think is the most likely profit at the end of week 9?
• By which week do you feel the shop is not going to lose money any more?
• Describe the pattern of the profit. What can you add to the graph to show that pattern?
• Describe the likely pattern from week 13 onwards.

As pupils tackle the questions, many will need help to develop their own ideas about the limits of probable variations in profit. Questions such as Could it have been as high as... or as low as...?, for a given week, are probably sufficiently leading. They should take the variation in the first six weeks as a basis for estimating this.
• Whole class sharing and discussion
For most pupils, correlation is not a concept that can be handled by conventional instructional methods alone. But the idea of a probabilistic (stochastic) variable and relationships as distinguished from deterministic ones is important, and we ought to touch on it in ways accessible to all pupils at different depths of understanding.

Progress in the understanding of correlation for most pupils is enhanced through the collaborative learning style of CAME methods. However, despite the value of the peer group as a whole to each pupil, he or she will only go so far, at the time, as her ‘zone of proximal development’ allows, so there cannot be a single instructional aim of the lesson for every pupil.
Discuss each question in turn, and draw a small table on the board to compare each pair's estimates. For example, the table on the right shows possible maximum and minimum estimates for week 9. For question 4, many pupils will ‘play safe’ and answer ‘Week 12' even though the variation (£200) suggests week 9 or earlier. For question 5, pupils may notice a ‘see-saw’ pattern, which some would associate with ‘stocking every fortnight’ or some other knowledge. They may examine that closely and find it not accurate, so help them to concentrate on upper and lower limits of the variations, indicating a general rising with time as well as the unpredictable up and down... So the middle line is a sort of smoothing or averaging. Prompt pupils, e.g. We need to know how good or how bad it may be any week, until pupils have produced some idea of likely profit and range of probable profits as two aspects of what is changing with time. Encourage pupils to illustrate this ‘smoothed’ middle line on the graph, sketching a regression ‘ribbon, or a band with upper- and lower-limit straight line through the scatter of the points.

Use question 6 as an opportunity to hear pupils’ ideas about why they might expect some levelling-off (for example, because of ideas about the size of the neighbourhood), or of continuous expansion (for example, because of new branches). The discussion should lead to an understanding that all businesses have to make such predictions, and that prediction is always based on some understanding of ranges of data and its variation.