lesson
A line has a rate and a starting value
1 · Learn
For a straight line y = mx + c, m is the gradient and c is the y-intercept. Gradient is change in y divided by change in x between two points on the line. A negative gradient means y decreases as x increases.
Use the graph's actual scales. Counting squares without reading units can give a wrong gradient when axes use different intervals. Context determines what the gradient's units mean.
A line may not be written as y=mx+c at first. For 2x+3y=12, subtract 2x from both sides: 3y=12-2x. Divide EVERY term by 3: y=4-(2/3)x, or y=-(2/3)x+4. The gradient is -2/3 and the vertical intercept is 4. Points (0,4) and (3,2) give the same gradient: change in y divided by change in x is (2-4)/(3-0)=-2/3. Rearranging, a table and the plotted line describe the same relationship.
2 · Worked example
For y = 3x - 2, the gradient is 3 and the y-intercept is -2. Points (0,-2) and (2,4) give gradient 6/2 = 3.
3 · Your turn
State gradient and y-intercept of y = -2x + 5, then find y when x = 4.
Check your answer
Gradient -2, intercept 5, and y = -3.
4 · Apply your learning
Plot a line using a table and verify its gradient with two well-separated points.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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