Linear relationships

In Year 8 you plotted straight lines from a table of values, read the gradient and intercepts, and solved equations and inequalities. In this topic you'll treat lines as objects you can measure and describe exactly: finding the midpoint, gradient and length of an interval, writing the equation of any line from a graph or from two points, spotting parallel and perpendicular lines, and using linear rules to model real situations like hire costs, savings and fuel use.

What you'll learn

• 4.01 Midpoint: finding the point halfway between two points, and working backwards to find a missing endpoint.
• 4.02 Gradient: measuring steepness as rise over run, and using the gradient formula for any two points.
• 4.03 Distance between two points: using Pythagoras' theorem to find the length of an interval on the Cartesian plane.
• Investigation: transformations of linear graphs. Use graphing software to see what changing $m$ and $c$ in $y = mx + c$ does to a line, and how lines move, turn and flip.
• 4.04 Gradient-intercept form: reading and writing equations in the form $y = mx + c$, and rearranging other equations into that form.
• 4.05 Linear graphs: sketching lines using intercepts or the gradient and $y$-intercept, horizontal and vertical lines, and reading graphs of real situations.
• 4.06 Parallel and perpendicular lines: using gradients to tell whether lines are parallel or at right angles.
• 4.07 Equation of a straight line: writing the equation of a line from a point and a gradient, or from two points.
• 4.08 Direct proportion: relationships of the form $y = kx$, the constant of proportionality, and their graphs.
• 4.09 Model linear equations: building linear models from words, tables and graphs, using them to answer questions and judging how good they are.
• 4.10 Linear inequalities: solving inequalities, including when to reverse the sign, and using them in real problems.

Which level should I pick?

Every worksheet comes in three levels. Each one covers the same skills.

• Level 1 · Warm-up: smaller numbers and more help along the way. A good place to start.
• Level 2 · Core: the main worksheet for the lesson.
• Level 3 · Challenge: fractions, unknowns and problems with more steps, for when the Core level feels easy.

Not sure? Start with Level 2. If the first few questions feel tricky, drop to Level 1. If you finish quickly and get them right, try Level 3. Download the worksheet, print it, and write your answers in the boxes.

Practice tests

When you've finished the lessons, try the three practice tests. Each one is like the real topic test: 45 minutes, 40 marks, a section without a calculator and a section with one. Write your working under each question, then copy your final answers onto the answer sheet on the last page, just like you will in the real test.