Relationships between variables in graphs

A graph is one of the most powerful ideas in mathematics, because it turns a relationship between two quantities into a picture you can read at a glance. How distance grows with time, how a savings balance changes month by month, how temperature rises through a day: each is a relationship between two variables, and a graph lets you see the whole story at once rather than as a list of numbers.

This year you learn to describe the relationships between variables shown in graphs, especially graphs built from real data. The skill is reading what the shape of a graph is telling you, and putting that meaning into words, which is the bridge between the algebra you have been writing and the world it describes.

Plotting a relationship

A graph uses two axes, one for each variable. The horizontal axis usually carries the variable you control, like time, and the vertical axis the variable that responds, like distance. Every point marks a pair of values: at this time, the object had reached that distance. When you plot a steady journey, the points line up in a straight line, because equal amounts of time always add equal amounts of distance.

The straightness itself carries meaning. A straight line means the relationship is constant, the same change in one variable always producing the same change in the other. If the line curved, it would tell you the rate was changing, perhaps a journey speeding up or slowing down. Reading the overall shape, straight or curved, rising or falling, is the first thing to do with any graph.

Reading meaning from a graph

Once a graph is drawn, you can read values straight off it. To find the distance at 2 hours, trace up from 2 on the time axis to the line, then across to the distance axis to read 100 kilometres. This lets a graph answer questions without any calculation. The direction of the line matters just as much: a line rising to the right means the variables increase together, while a line falling to the right means one decreases as the other increases, like water draining from a tank.

Comparing graphs adds another layer of meaning. On a distance-time graph, a steeper line means a faster speed, because more distance is covered in the same time. This is why graphs built from real data are so useful: a single picture can compare two journeys, show a trend over a year, or reveal a relationship that a column of figures would hide. Learning to describe these relationships in plain words, rising, falling, steady, steep, is exactly what this part of the curriculum asks of you, and it makes the link between numbers and the real world vivid.

When the line falls

Not every relationship grows. A tank draining over time gives a line that slopes downward, because the water left decreases as the time increases. Reading a falling line is the mirror image of reading a rising one: down to the right means one quantity shrinks while the other grows.

Comparing two relationships

Putting two lines on the same axes lets you compare them at a glance. On a distance-time graph the steeper line belongs to the faster car, because it covers more distance in the same time. Comparing slope is one of the most useful things a graph from real data lets you do.

Words and shapes go together

Describing a relationship means putting the shape of its line into words. A warming morning gives a rising line, an untouched savings balance gives a level line, and a draining battery gives a falling line. Learning to read rising, level and falling from a graph, and to say which is which, is the heart of this topic.

Teaching tip: graphs come alive when the data is the student own. Time how far they walk each minute, or track a savings total over a few weeks, then plot it together. Seeing a relationship they generated themselves turn into a line on a page makes the abstract idea of a graph concrete and personal.

Encourage describing a graph in words before reading exact values. Asking simply whether the line goes up or down, and whether it is steep or gentle, builds the habit of interpreting the overall story first, which is the heart of what the curriculum is asking for here.