Mathematics • Year 8 • Unit 2 • Lesson 8
Gradient from a Graph (Rise and Run)
Build fluency with the rise/run method: choose two clean lattice points, count rise and run, write the gradient as a fraction or decimal. One worked example, one guided fill-in, then eight independent problems.
1. I do — fully worked example
Gradient measures steepness: rise ÷ run. Always read left-to-right.
Problem. A line passes through the lattice points (1, 1) and (5, 3). Find its gradient.
Step 1 — Choose two clean points.
(1, 1) and (5, 3) — both sit exactly on grid corners.
Reason: lattice points give whole-number rise and run, so no estimation error.
Step 2 — Count rise (vertical change) and run (horizontal change).
Rise = 3 − 1 = 2 Run = 5 − 1 = 4
Reason: rise = (later y) − (earlier y); run = (later x) − (earlier x).
Step 3 — Write the gradient and simplify.
m = rise ÷ run = 2/4 = 1/2 = 0.5
Reason: simplify the fraction (HCF of 2 and 4 is 2), then convert to decimal if useful.
Answer: m = 1/2 = 0.5 (positive — line rises gently).
2. We do — fill in the missing steps
Same shape as Section 1, but the working is faded. Fill in each blank. 4 marks
Problem. A line passes through (1, 4) and (5, 1). The line slopes downhill. Find its gradient.
Step 1 — Identify the two points: (1, 4) and (5, 1).
Step 2 — Calculate rise and run (left-to-right):
Rise = 1 − 4 = ______ Run = 5 − 1 = ______
Step 3 — Gradient = rise ÷ run:
m = ______ / ______ = ______ (decimal: ______)
Sign check: Because the line slopes downhill, the gradient should be ______________ (positive / negative).
3. You do — independent practice
Show your working under each problem. First four are foundation, next two are standard, last two are extension.
Foundation — rise and run
3.1 A line rises 4 units and runs 2 units. What is its gradient? 1 mark
3.2 A line drops 3 units and runs 6 units. What is its gradient (with sign)? 1 mark
3.3 A line passes through (0, 0) and (3, 6). Find its gradient. 1 mark
3.4 A line passes through (0, 0) and (4, 6). Find its gradient as a simplified fraction. 1 mark
Standard — two-step gradient
3.5 A line passes through (2, 1) and (6, 9). Find rise, run and gradient. 2 marks
3.6 A line passes through (1, 5) and (4, 2). Find rise, run and gradient. State whether the line slopes uphill or downhill. 2 marks
Extension — choose your points wisely
3.7 A graph shows a straight line through (0, 2), (2, 5), and (4, 8). Confirm the gradient is the same no matter which pair of points you use. 2 marks
3.8 A line passes through (−2, −1) and (2, 7). Find its gradient. Write it as both a fraction and a decimal. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do
Rise = −3, Run = 4. m = −3 / 4 = −3/4 (decimal −0.75). Sign check: negative, matches downhill slope.
3.1 — Rise 4, Run 2
m = 4/2 = 2.
3.2 — Drops 3, Runs 6
Rise = −3, Run = 6 → m = −3/6 = −1/2 (= −0.5).
3.3 — (0, 0) to (3, 6)
Rise = 6 − 0 = 6, Run = 3 − 0 = 3 → m = 6/3 = 2.
3.4 — (0, 0) to (4, 6)
Rise = 6, Run = 4 → m = 6/4 = 3/2 (= 1.5).
3.5 — (2, 1) to (6, 9)
Rise = 9 − 1 = 8, Run = 6 − 2 = 4 → m = 8/4 = 2.
3.6 — (1, 5) to (4, 2)
Rise = 2 − 5 = −3, Run = 4 − 1 = 3 → m = −3/3 = −1. Line slopes downhill.
3.7 — Three collinear points
(0,2)→(2,5): m = (5−2)/(2−0) = 3/2. (2,5)→(4,8): m = (8−5)/(4−2) = 3/2. (0,2)→(4,8): m = (8−2)/(4−0) = 6/4 = 3/2. All three pairs give m = 3/2, confirming the line is straight.
3.8 — (−2, −1) to (2, 7)
Rise = 7 − (−1) = 8, Run = 2 − (−2) = 4 → m = 8/4 = 2 (= 2.0 as a decimal).