Teaching mathematics
Teaching mathematics

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Teaching mathematics

3.1 Variables

You have seen that an indeterminate quantity can be thought of as a specific quantity that happens to be unknown (for now), or as one of several unknown quantities. Extending this idea leads to thinking of a quantity as a variable, that is, as a single object that can change, or vary in some way.

Variables can be discrete or continuous.

A discrete variable can take only certain values, and it would be nonsensical to choose a value between them. The position number of a sequence is an example of a discrete variable. A position number can only be a whole number. It would be nonsensical to talk about the ‘3 and a halfth’ term.

A continuous variable is one which can take any value. The nature of a variable may depend on how it is measured. One of the common variables that learners will meet in school is time. Time is usually a continuous variable, but we can make it discrete by taking daily, or hourly, aggregate measurements. You will return to this idea in Week 8 (Working with data).

Activity 7 Representing covariance

Timing: Allow 15 minutes

Look at the four pairs of variables below.

  1. Are the variables discrete or continuous?
  2. Informally, how do you visualise each pair of variables changing together?
    • Time in hours / Time in minutes

    • Age / Number of hours of sleep

    • Number of people sharing a box of 24 chocolates / How many chocolates each person gets

    • Your age / Your elder sibling’s age.

  3. Now watch the video below which shows different mathematical representations of the first example.
Download this video clip.Video player: Video 2 Different representations of covariance
Skip transcript: Video 2 Different representations of covariance

Transcript: Video 2 Different representations of covariance

Covariance is an important concept. It describes the relationship between variables that change together. Mathematics has conventional ways of representing covariation, or you might have your own images.
Think about the two variables, time in hours and time in minutes. You might visualise them as the hour hand and the minute hands on a clock. Both measuring time, but with the minute hand moving faster than the hour hand. Or you might think about the variables of specific number equivalences. One hour is the same as 60 minutes. Two hours are 120 minutes. And so on.
This pattern gives us a rule. The number of hours, multiplied by 60, is equal to the number of minutes. The function machine is the closest mathematical representation of this rule. For a specific number of hours, I can calculate a specific number of minutes.
Another way of thinking about the two variables is to think of them as two lengths that change together. This representation is called a double number line. It shows the comparative size of each variable. Both times start at zero and they both increase. But the time in minutes grows 60 times as fast.
We're more used to seeing this comparison of two variables in the two dimensions of a graph. One variable is measured on the x-axis. The other is measured on the y-axis. Starting at zero, zero, each data point is represented at the corner of the rectangle formed by these two lengths. By convention, the first coordinate is measured on the x-axis and the second on the y-axis.
When the two variables have such different sizes, the scale on one axis has to be adjusted to fit the paper. Let's re-scale the x-axis. The points plotted here correspond to the values in this table. The top line is the x values. The second line is the y values.
Time is a continuous variable. So we can draw a straight line through these points to show all the intermediate points. The equation of the line is y equals 60 times x. This equation is the algebraic representation of the graph.
We can read intermediate values from the line graph. How many minutes is 2.5 hours? Start at naught, go along the x-axis to find 2.5, then up till you hit the line. Use a ruler or your finger to find the same height on the y-axis, and read off 150. 2 and 1/2 hours is 150 minutes.
We know what x represents in this graph. It is time in hours, measured on the x-axis. And y is the time in minutes. The number 60 is the gradient. It's defined as the ratio, increase in y divided by increase in x.
The gradient is important for describing covariance. It tells you exactly how y changes when x increases by 1. Here, the gradient is a constant. It's always 60.
On a graph, the gradient also tells you the slope of the graph. A constant gradient means that the graph of the function is a straight line. This graph shows a proportional relationship, of the kind you met in an earlier week. The graph of a proportional relationship is always a straight line that passes through zero, zero.
To summarise, then. You've looked at different ways of representing covariance. You might find a real life object that measures both variables. Or you can visualise specific pairs of equivalent numbers. You can put these pairs of numbers in a table as coordinates. You can visualise lengths growing together or on the two axes of a graph. You can write a rule in words and symbols for a function machine, or as an algebraic equation for the graph.
Learners should meet all these representations. One way of distinguishing types of covariance is to think about the gradient of the graph. The number that tells you how much y changes when x increases by 1.
Learners in the middle years should learn to recognise examples and the graphs of linear and proportional relationships and examples of non-linear relationships. They'll work with equations for straight line graphs only.
End transcript: Video 2 Different representations of covariance
Video 2 Different representations of covariance
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Time in hours is usually considered as a discrete variable since we only write whole numbers of hours (except perhaps half hours)

Time in minutes could be either discrete or continuous.

An age is discrete as you are likely to measure it in years or in years and months, but not with an accuracy of days or hours.

Number of people and chocolates are discrete variables.

You might have thought of graphs, or lines growing together, or imagined groups of chocolates.


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