Mathematics • Year 8 • Unit 2 • Lesson 5
Gradients (Slope) of a Line
Build fluency with gradient = rise ÷ run. Learn the four cases (positive, negative, zero, undefined) and the formula m = (y₂ − y₁) / (x₂ − x₁). One worked example, one guided example with blanks, then eight independent problems.
1. I do — fully worked example
Read every line. Gradient is just "how much y goes up for every 1 across".
Problem. Find the gradient of the line through A(2, 1) and B(6, 4).
Step 1 — Identify the two points and label them.
(x₁, y₁) = (2, 1), (x₂, y₂) = (6, 4).
Reason: subscripts just keep the two points organised — it doesn't matter which one you call "1".
Step 2 — Find the rise (vertical change).
rise = y₂ − y₁ = 4 − 1 = 3.
Reason: y₂ − y₁ tells you how far y went up (positive) or down (negative).
Step 3 — Find the run (horizontal change).
run = x₂ − x₁ = 6 − 2 = 4.
Reason: x₂ − x₁ tells you how far the line travelled across.
Step 4 — Apply the gradient formula.
m = rise / run = 3 / 4 = 0.75.
Reason: gradient = rise ÷ run. A gradient of 3/4 means y rises 3 units for every 4 units across.
Step 5 — Interpret the sign.
m = +3/4 > 0 → the line slopes UPHILL from left to right.
Answer: gradient = 3/4 (positive — line goes uphill).
2. We do — fill in the missing steps
Same shape as Section 1. Fill every blank. 4 marks
Problem. Find the gradient of the line through P(1, 5) and Q(4, −1).
Step 1 — Label:
(x₁, y₁) = ( __ , __ ), (x₂, y₂) = ( __ , __ ).
Step 2 — Rise:
rise = y₂ − y₁ = ____ − ____ = ____.
Step 3 — Run:
run = x₂ − x₁ = ____ − ____ = ____.
Step 4 — Gradient:
m = rise / run = ____ / ____ = ____.
Step 5 — Interpret: sign of m is ____, so the line slopes __________ (uphill / downhill / flat).
3. You do — independent practice
Show your reasoning. 3.1–3.4 are foundation (rise/run from a description). 3.5–3.6 are standard (use the formula with two points). 3.7–3.8 are extension (zero / undefined gradient).
Foundation — rise and run
3.1 A line rises 4 units and runs 2 units. Find its gradient and state whether it slopes uphill or downhill. 1 mark
3.2 A line has rise = 3 and run = 6. Find its gradient as a simplified fraction. 1 mark
3.3 A line drops 5 units while moving 2 units to the right. Find its gradient (include the sign). 1 mark
3.4 A line has rise = 6 and run = 4. Simplify the gradient as a fraction. 1 mark
Standard — use the formula
3.5 Find the gradient of the line through (1, 2) and (5, 10). Show the rise/run calculation. 2 marks
3.6 Find the gradient of the line through (−2, 4) and (3, −1). Show the rise/run calculation. 2 marks
Extension — zero and undefined
3.7 Find the gradient of the line through (1, 4) and (7, 4). What does this value tell you about the line's direction? 2 marks
3.8 Find the gradient of the line through (3, 2) and (3, 9). Explain why the answer is "undefined". 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do P(1, 5), Q(4, −1)
Step 1: (x₁, y₁) = (1, 5), (x₂, y₂) = (4, −1).
Step 2: rise = −1 − 5 = −6.
Step 3: run = 4 − 1 = 3.
Step 4: m = −6 / 3 = −2.
Step 5: sign is negative, so the line slopes downhill.
3.1 — Rise 4, run 2
m = 4/2 = 2. Positive → uphill.
3.2 — Rise 3, run 6
m = 3/6 = 1/2 (or 0.5). Positive → uphill.
3.3 — Drops 5, runs 2 right
Rise = −5, run = 2. m = −5/2 = −2.5 (or −5/2). Negative → downhill.
3.4 — Rise 6, run 4
m = 6/4 = 3/2 (or 1.5). Positive → uphill.
3.5 — Through (1, 2) and (5, 10)
rise = 10 − 2 = 8; run = 5 − 1 = 4. m = 8/4 = 2. Positive → uphill.
3.6 — Through (−2, 4) and (3, −1)
rise = −1 − 4 = −5; run = 3 − (−2) = 5. m = −5/5 = −1. Negative → downhill at 45°.
3.7 — Through (1, 4) and (7, 4)
rise = 4 − 4 = 0; run = 7 − 1 = 6. m = 0/6 = 0. A zero gradient means the line is flat (horizontal) — y stays constant at 4 no matter what x is.
3.8 — Through (3, 2) and (3, 9)
rise = 9 − 2 = 7; run = 3 − 3 = 0. m = 7/0 = undefined. You can't divide by zero. The line is vertical (x = 3 for every point on it), and vertical lines don't have a defined gradient.