Mathematics • Year 8 • Unit 2 • Lesson 5
Gradient — Mixed Challenge
Pull together everything from Lesson 5: rise ÷ run, the formula m = (y₂ − y₁)/(x₂ − x₁), the four cases (positive, negative, zero, undefined), and comparing steepness. Six mixed problems, one "find the mistake", and one open-ended puzzle.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 5. Decide BEFORE you write what's being asked. 3 marks each
1.1 Find the gradient of the line through (1, 3) and (5, 11). State whether the line slopes uphill or downhill.
1.2 Find the gradient of the line through (−1, 6) and (4, −4).
1.3 Find the gradient of the line through (−3, 5) and (4, 5). What kind of line is this?
1.4 Find (or describe) the gradient of the line through (2, −1) and (2, 8). Justify your answer.
1.5 Line L₁ goes through (0, 0) and (4, 6). Line L₂ goes through (0, 0) and (3, 6). Which line is steeper? Find both gradients and compare them.
1.6 A line passes through (−2, 1) and has gradient 3. (a) Find a third point on this line by going 1 unit right and "rise" units up from (−2, 1). (b) Verify your point lies on the line by re-calculating m through the two points.
2. Find the mistake
Another student tried to find the gradient of the line through A(1, 5) and B(4, 2). Their working is below. Exactly one line contains a mistake. Spot it, explain why, then re-do the working correctly. 3 marks
Student's working — gradient through A(1, 5), B(4, 2):
Line 1: rise = x₂ − x₁ = 4 − 1 = 3.
Line 2: run = y₂ − y₁ = 2 − 5 = −3.
Line 3: m = rise / run = 3 / −3 = −1.
Line 4: So the line has gradient −1, sloping downhill.
(a) Which line(s) contain the mistake? (Note: the final answer happens to be CORRECT — but the working has the rise and run swapped.)
(b) Explain in one or two sentences what the student mixed up.
(c) Write out the corrected working in full, with the correct labels for rise and run, and the correct final gradient.
Stuck? rise = vertical change = y₂ − y₁. run = horizontal change = x₂ − x₁. The student swapped them in Lines 1 and 2.3. Open-ended challenge — same gradient, different lines
This question has more than one valid answer. 4 marks
3.1 Find three different pairs of points that ALL produce a gradient of exactly +2. None of your pairs may share both points with another pair.
For each pair:
(i) Write the two points.
(ii) Show the rise and run.
(iii) Verify the gradient is +2.
Bonus: at least ONE of your three pairs must include a point with a NEGATIVE coordinate.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — (1, 3) and (5, 11)
rise = 11 − 3 = 8; run = 5 − 1 = 4. m = 8/4 = 2. Positive → uphill.
1.2 — (−1, 6) and (4, −4)
rise = −4 − 6 = −10; run = 4 − (−1) = 5. m = −10/5 = −2. Negative → downhill.
1.3 — (−3, 5) and (4, 5)
rise = 5 − 5 = 0; run = 4 − (−3) = 7. m = 0/7 = 0. This is a horizontal line (every point has y = 5).
1.4 — (2, −1) and (2, 8)
rise = 8 − (−1) = 9; run = 2 − 2 = 0. m = 9/0 = undefined. You cannot divide by zero. The line is vertical (x = 2 for every point on it), so it has no defined gradient.
1.5 — Two lines from the origin
L₁ through (0, 0), (4, 6): m₁ = 6/4 = 1.5. L₂ through (0, 0), (3, 6): m₂ = 6/3 = 2. Since 2 > 1.5, L₂ is steeper.
1.6 — Through (−2, 1), gradient 3
(a) Gradient 3 = 3/1. From (−2, 1), go 1 right and 3 up: new point (−1, 4).
(b) Verify: rise = 4 − 1 = 3; run = −1 − (−2) = 1. m = 3/1 = 3 ✓.
2 — Find the mistake
(a) The mistake is on Lines 1 AND 2: the student has swapped the labels for rise and run.
(b) rise is the VERTICAL change (y₂ − y₁), not x₂ − x₁. run is the HORIZONTAL change (x₂ − x₁). The student swapped them — and accidentally still got −1 because by coincidence the numbers happened to flip into the same answer for these particular points.
(c) Corrected working:
rise = y₂ − y₁ = 2 − 5 = −3.
run = x₂ − x₁ = 4 − 1 = 3.
m = rise/run = −3 / 3 = −1. ✓ (Negative → downhill.)
3 — Same gradient, different lines (sample solution)
We need pairs whose rise ÷ run = +2.
Pair 1: (0, 0) and (1, 2). rise = 2, run = 1, m = 2/1 = 2 ✓.
Pair 2: (3, 1) and (5, 5). rise = 4, run = 2, m = 4/2 = 2 ✓.
Pair 3 (with a negative coordinate): (−1, −3) and (2, 3). rise = 3 − (−3) = 6, run = 2 − (−1) = 3, m = 6/3 = 2 ✓.
Other valid pairs: (0, 5) and (4, 13); (−2, 0) and (1, 6); any two points where the rise is exactly twice the run.
Marking: 1 mark per valid pair with correct working (3 marks). 1 bonus mark for satisfying the negative-coordinate requirement on at least one pair.