Mathematics • Year 8 • Unit 2 • Lesson 9
Finding Gradient from Two Points
Build fluency with the gradient formula m = (y₂ − y₁) / (x₂ − x₁). One worked example, one guided example with blanks, then eight independent problems including negatives and special cases.
1. I do — fully worked example
Given two coordinates, you can find the gradient without ever drawing the line.
Problem. Find the gradient of the line through (2, 3) and (6, 7).
Step 1 — Label your points.
(x₁, y₁) = (2, 3) (x₂, y₂) = (6, 7)
Reason: pick either point as "1" — just be consistent for both subtractions.
Step 2 — Calculate the rise (y₂ − y₁).
Rise = y₂ − y₁ = 7 − 3 = 4
Reason: "y on top" of the gradient fraction.
Step 3 — Calculate the run (x₂ − x₁).
Run = x₂ − x₁ = 6 − 2 = 4
Reason: "x on bottom".
Step 4 — Compute m.
m = rise / run = 4 / 4 = 1
Answer: m = 1 (positive — line slopes uphill at 45°).
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Find the gradient through (1, 2) and (5, 14).
Step 1 — Label points: (x₁, y₁) = (______, ______) and (x₂, y₂) = (______, ______).
Step 2 — Rise = y₂ − y₁ = ______ − ______ = ______.
Step 3 — Run = x₂ − x₁ = ______ − ______ = ______.
Step 4 — m = rise / run = ______ / ______ = ______.
Sign check: y went up as x went up, so m 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 — apply the formula
3.1 Find m through (1, 3) and (4, 9). 1 mark
3.2 Find m through (0, 0) and (5, 10). 1 mark
3.3 Find m through (2, 7) and (5, 1). 1 mark
3.4 Find m through (3, 5) and (7, 5). State the gradient type. 1 mark
Standard — including negatives
3.5 Find m through (0, 0) and (−2, 6). 2 marks
3.6 Find m through (−3, −2) and (1, 4). 2 marks
Extension — simplify and decide
3.7 Find m through (−1, 2) and (3, −4). Give your answer as both a simplified fraction and a decimal. 2 marks
3.8 Show that the gradient through (1, 3) and (4, 9) is the same whether you label (1,3) as (x₁,y₁) or as (x₂,y₂). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do
(x₁, y₁) = (1, 2), (x₂, y₂) = (5, 14). Rise = 14 − 2 = 12. Run = 5 − 1 = 4. m = 12/4 = 3. Sign check: positive.
3.1 — (1, 3) and (4, 9)
m = (9 − 3)/(4 − 1) = 6/3 = 2.
3.2 — (0, 0) and (5, 10)
m = (10 − 0)/(5 − 0) = 10/5 = 2.
3.3 — (2, 7) and (5, 1)
m = (1 − 7)/(5 − 2) = −6/3 = −2.
3.4 — (3, 5) and (7, 5)
m = (5 − 5)/(7 − 3) = 0/4 = 0. Gradient type: zero (horizontal line).
3.5 — (0, 0) and (−2, 6)
m = (6 − 0)/(−2 − 0) = 6/(−2) = −3.
3.6 — (−3, −2) and (1, 4)
m = (4 − (−2))/(1 − (−3)) = 6/4 = 3/2 (= 1.5).
3.7 — (−1, 2) and (3, −4)
m = (−4 − 2)/(3 − (−1)) = −6/4 = −3/2 = −1.5.
3.8 — Order independence
If (x₁,y₁) = (1,3) and (x₂,y₂) = (4,9): m = (9−3)/(4−1) = 6/3 = 2.
If (x₁,y₁) = (4,9) and (x₂,y₂) = (1,3): m = (3−9)/(1−4) = −6/−3 = 2.
Both give m = 2. The order doesn't matter as long as you subtract consistently top and bottom.