One Variable, Two Variable, Three Variable, More

Why do this problem?

This problem offers students an opportunity to revise what they know about representing data, drawing together what they have learnt in mathematics and other subjects.  It then asks them to creatively extend what they have learnt to a wider class of problems.

Possible approach

The lesson could simply follow the questions as offered in the problem.

Alternatively, the lesson could begin by asking students to recall all ways they can think of for representing data.  They could then be asked to classify the representations according to whether they would be used for quantitative or qualitative variables, and the number of each.  (It is probable that the only bivariate diagram that students will offer at this point is the scattergraph.)

There are likely to be enough diagrams for single variable data that there could be a brief discussion about discrete and continuous quantitative variables.  The lesson could then continue by asking about the case of two variables where only one is quantitative, and then where they are both qualitative.

It would also be useful for students to suggest contexts in which these might arise.  The "big data sets" provided by the English examination boards (which can be obtained from AQA, Pearson/Edexcel and OCR - the links are correct at the time of writing) might be helpful for this, or downloading data from other sources such as the UK Government's Open Data site or the JSE site given in the problem.  These may also provide motivation for the question of why we would be interested in displaying two or more variables simultaneously: we are interested in understanding how they are related.

Key questions

• How can we display data when we are dealing with more than one variable?
• What could these displays allow us to see in the data which we might not otherwise have noticed?

Possible extension

There are many websites and software packages available which allow easy visualisation of data.  As noted in the problem, CODAP is free and designed for student use.  An excellent free software package is RStudio; though this requires learning to write some simple computer code, it is very powerful once this has been done.  Encouraging students to explore real data using modern visualisation software will help them relate to data as something meaningful that has immediate real-world application; it is not a purely-calculational set of mechanical rules.

A possible homework task for students is to look through newspapers or websites for examples of data representation, and to bring them back to class to discuss.  When looking at them, students could consider questions such as:

• How many variables does this data contain, and what type of variables are they?
• How else could the data have been presented?

Possible support

• For the one qualitative and one quantitative variable case, consider the different ways we have of representing one quantitative variable.  How could these be adapted to show the same variable but for two or more distinct groups?
• For the two qualitative variables, students could think first about how they could represent this data clearly using numbers.  Can any of their representations easily be converted into graphical displays?