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).