Mathematics • Year 8 • Unit 2 • Lesson 8
Rise and Run — Mixed Challenge
Pull together everything from Lesson 8: rise/run, gradient triangles, choosing lattice points, simplifying fractions, sign and direction. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 8. Show all working. 3 marks each
1.1 Find the gradient of the line through (1, 2) and (4, 11) using rise/run.
1.2 Find the gradient of the line through (3, 7) and (6, 1). Write it as a simplified fraction.
1.3 A line passes through (−3, 2) and (1, 14). Find its gradient.
1.4 A line passes through (2, 5) and (8, 5). Find its rise, its run, and its gradient. State the gradient type.
1.5 Two ramps for a delivery dock are being compared. Ramp A rises 50 cm over 200 cm; Ramp B rises 60 cm over 300 cm. Find the gradient of each and state which is steeper.
1.6 A line passes through (0, −2) and (4, 6). Find its gradient using rise/run, and confirm by checking the gradient between (0, −2) and (2, 2) is the same.
2. Find the mistake
Another student has tried to find the gradient of a line through (1, 5) and (4, 11). Exactly one step is wrong. Spot it, explain why, then re-do correctly. 3 marks
Student's working:
Line 1: Two points: (1, 5) and (4, 11).
Line 2: Rise = (later y) − (earlier y) = 11 − 5 = 6.
Line 3: Run = (later x) − (earlier x) = 1 − 4 = −3.
Line 4: m = rise/run = 6/(−3) = −2. The line goes downhill.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working and the correct gradient.
Stuck? When you read left-to-right, the "later" x is the BIGGER one. Apply the same labelling rule to both rise and run.3. Open-ended challenge — three triangles, one line
This question has many valid answers. 4 marks
3.1 Consider the line that passes through the origin (0, 0) and has gradient m = 3/2.
(a) List three different pairs of lattice points on this line. Each pair must give the SAME gradient of 3/2.
(b) For each pair, show the rise and run.
(c) In one sentence, explain why all three pairs must give the same gradient, even though the rise and run values differ.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — (1, 2) to (4, 11)
Rise = 11 − 2 = 9, Run = 4 − 1 = 3 → m = 9/3 = 3.
1.2 — (3, 7) to (6, 1)
Rise = 1 − 7 = −6, Run = 6 − 3 = 3 → m = −6/3 = −2.
1.3 — (−3, 2) to (1, 14)
Rise = 14 − 2 = 12, Run = 1 − (−3) = 4 → m = 12/4 = 3.
1.4 — (2, 5) to (8, 5)
Rise = 5 − 5 = 0, Run = 8 − 2 = 6 → m = 0/6 = 0. Gradient type: zero (horizontal line at y = 5).
1.5 — Ramps A vs B
Ramp A: m = 50/200 = 1/4 = 0.25. Ramp B: m = 60/300 = 1/5 = 0.20. Ramp A is steeper (0.25 > 0.20).
1.6 — (0, −2) to (4, 6); check via (2, 2)
(0,−2)→(4,6): Rise = 6 − (−2) = 8, Run = 4 − 0 = 4 → m = 8/4 = 2.
(0,−2)→(2,2): Rise = 2 − (−2) = 4, Run = 2 − 0 = 2 → m = 4/2 = 2. ✓ Same gradient confirms collinearity.
2 — Find the mistake
(a) The mistake is on Line 3 (and the wrong conclusion follows in Line 4).
(b) The student used (1 − 4) for the run — they should subtract in the SAME ORDER as the rise. They used "later − earlier" for rise (11 − 5) but flipped the order for the run. Both subtractions must read left-to-right (or both must read right-to-left).
(c) Corrected: Run = 4 − 1 = 3. So m = 6/3 = 2. The line slopes uphill, not downhill.
3 — Three triangles, one line (sample solution)
Line: y = (3/2)x. Three pairs of lattice points on the line:
Pair 1: (0, 0) and (2, 3). Rise = 3 − 0 = 3. Run = 2 − 0 = 2. m = 3/2.
Pair 2: (2, 3) and (6, 9). Rise = 9 − 3 = 6. Run = 6 − 2 = 4. m = 6/4 = 3/2.
Pair 3: (−2, −3) and (4, 6). Rise = 6 − (−3) = 9. Run = 4 − (−2) = 6. m = 9/6 = 3/2.
(c) All three pairs give the same gradient because a straight line has a constant rate of change. Different pairs give different-sized gradient triangles, but the ratio rise:run is always the same — they are similar triangles sitting along the same line.
Marking: 1 mark for each valid distinct pair on the line with correctly calculated m = 3/2 (up to 3 marks). 1 mark for the constant-gradient / similar-triangles explanation.