Teacher guide: Linear relations

The idea in plain words

A rule such as $y = 3x + 1$ turns each value of $x$ into a value of $y$. Plot the pairs as points, $(0, 1)$, $(1, 4)$, $(2, 7)$, and they lie on a straight line. Any rule of the form $y = mx + c$ gives a straight line.

The two numbers do different jobs:

• $m$, the number in front of $x$, is the gradient: how far the line goes up for each step of 1 to the right. A bigger $m$ is steeper; $m = 0$ is flat; a negative $m$ goes down.
• $c$, the number on its own, is where the line crosses the $y$-axis (the value of $y$ when $x = 0$).

Why it matters

Steepness is a rate: dollars per hour, metres per second, litres per minute. Reading $m$ as "how much more for each one more" is the idea students will use in science graphs, budgets and modelling.

What students get wrong

• Swapping m and c. Reading the number on its own as the steepness, or expecting a bigger $m$ to lift the line. The board shows the line turning about the same point on the $y$-axis as $m$ changes.
• Any minus sign means "down". $y = 3x - 2$ still goes up: only a negative $m$ makes the line go down.

The words

gradient ($m$): up for each step right. $y$-intercept ($c$): where the line crosses the $y$-axis. steeper: goes up (or down) more for each step right.