Mathematics • Year 10 • Unit 2 • Lesson 16
Gradient of a Line — Skill Drill
Build fluency with the gradient formula m = (y₂ − y₁)/(x₂ − x₁), recognise positive/negative/zero/undefined gradients, and extract the gradient from y = mx + c or from general form Ax + By + C = 0.
1. I do — fully worked example
Gradient through two points using m = rise/run.
Problem. Find the gradient of the line through A(2, 5) and B(6, 11).
Step 1 — Label coordinates.
(x₁, y₁) = (2, 5) (x₂, y₂) = (6, 11)
Step 2 — Apply the gradient formula.
m = (y₂ − y₁) / (x₂ − x₁) = (11 − 5) / (6 − 2) = 6/4
Step 3 — Simplify the fraction.
m = 3/2 = 1.5
Step 4 — Interpret the sign.
m > 0 → line slopes upward from left to right.
Answer: m = 3/2 = 1.5.
2. We do — fill in the missing steps
Extract the gradient from general form by rearranging to y = mx + c. Fill in each blank. 5 marks
Problem. Find the gradient of 4x − 2y + 8 = 0.
Step 1 — Isolate the y term:
−2y = ____ x − ____
Step 2 — Divide every term by −2:
y = ____ x + ____
Step 3 — Read off the gradient (coefficient of x):
m = ____
Step 4 — Read off the y-intercept: c = ____.
Step 5 — Sign of m: ____ (positive / negative) → slopes ____ from left to right.
3. You do — independent practice
Give exact fractions when needed; never just a decimal if a fraction is cleaner.
Foundation — single-skill recall
3.1 Find the gradient through (1, 2) and (5, 8). 1 mark
3.2 Find the gradient through (3, 4) and (7, 4). 1 mark
3.3 Find the gradient through (5, 1) and (5, 9). State if undefined. 1 mark
3.4 Find the gradient of y = −4x + 7. 1 mark
Standard — negative coordinates / rearrangement
3.5 Find the gradient through P(−3, 7) and Q(5, 3). Simplify the fraction. 2 marks
3.6 Find the gradient of 3x + 2y − 12 = 0 by rearranging to y = mx + c. 2 marks
Extension — sign meaning + missing coordinate
3.7 A line has gradient m = −2 and passes through A(1, 5) and B(4, k). Find k. (Use the formula m = (y₂ − y₁)/(x₂ − x₁).) 3 marks
3.8 Classify each of the following gradients as "positive", "negative", "zero" or "undefined", and describe the line's direction in one phrase: (i) m = 2/3, (ii) m = 0, (iii) m = −5, (iv) m undefined (vertical line). 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (4x − 2y + 8 = 0)
Step 1: −2y = −4x − 8. Step 2: y = 2x + 4. Step 3: m = 2. Step 4: c = 4. Step 5: positive → slopes upward.
3.1 — (1,2) to (5,8)
m = (8 − 2)/(5 − 1) = 6/4 = 3/2 = 1.5.
3.2 — (3,4) to (7,4)
m = (4 − 4)/(7 − 3) = 0/4 = 0 (horizontal line).
3.3 — (5,1) to (5,9)
m = (9 − 1)/(5 − 5) = 8/0 — undefined (vertical line).
3.4 — y = −4x + 7
m = −4 (coefficient of x).
3.5 — (−3,7) to (5,3)
m = (3 − 7)/(5 − (−3)) = −4/8 = −1/2.
3.6 — 3x + 2y − 12 = 0
2y = −3x + 12 → y = (−3/2)x + 6. m = −3/2, c = 6.
3.7 — Find k
−2 = (k − 5)/(4 − 1) → k − 5 = −6 → k = −1. So B = (4, −1).
3.8 — Classify
(i) m = 2/3 → positive, slopes upward gently from left to right.
(ii) m = 0 → zero, horizontal line.
(iii) m = −5 → negative, slopes downward steeply from left to right.
(iv) m undefined → undefined, vertical line (rise on zero run).