Mathematics • Year 8 • Unit 2 • Lesson 8
Steepness in the Real World
Apply rise/run to real situations where two quantities change together — ramps, distance-time graphs, ski runs, water tank levels, school enrolments. Read meaningful gradients off practical graphs.
1. Word problems
Each scenario gives you two clean data points on a real-world graph. Use rise/run to find the gradient and explain what it represents in the situation. Show your working — final-answer-only earns half marks.
1.1 — Wheelchair ramp. A school's ramp rises 30 cm over a horizontal run of 360 cm.
(a) Find the gradient of the ramp as a simplified fraction.
(b) The Australian standard for an accessible ramp is at most 1/14. Is this ramp compliant? 3 marks
1.2 — Distance-time graph. A jogger's distance-time graph passes through (0 min, 0 m) and (10 min, 1500 m).
(a) Use rise/run to find the gradient.
(b) State the units of the gradient.
(c) What does the gradient represent in the story? 3 marks
1.3 — Ski run. A short ski run drops from a height of 80 m at the top to 20 m at the bottom over a horizontal distance of 240 m.
(a) Find the rise (with sign) and the run.
(b) Calculate the gradient as a simplified fraction.
(c) In one sentence, say which is steeper: this run or a beginner run with gradient −1/8. 3 marks
1.4 — Water tank draining. A water tank holds 500 L. A graph of volume vs time passes through (0 min, 500 L) and (20 min, 100 L).
(a) Calculate the gradient.
(b) State its sign and what that sign tells you about the tank.
(c) State the units. 3 marks
1.5 — School enrolment. A school's enrolment graph passes through (2020, 800) and (2025, 1100).
(a) Find rise and run.
(b) Calculate the gradient and state its units.
(c) Predict the enrolment in 2027 if growth stays linear. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate finds the gradient of a line by reading two points off the graph that "look about right" but aren't quite on grid corners — they get m ≈ 1.8 instead of the true m = 2. In your own words, explain (i) why this can give a wrong answer, (ii) what they should do differently, and (iii) why the gradient should come out the same no matter which two points on the line you pick (as long as they ARE on the line). Use the phrase "lattice point" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Ramp
(a) m = 30/360 = 1/12.
(b) 1/12 ≈ 0.0833 and 1/14 ≈ 0.0714. Since 1/12 > 1/14, the ramp is steeper than the standard allows — not compliant.
1.2 — Jogger
(a) Rise = 1500, Run = 10. m = 1500/10 = 150.
(b) Units: metres per minute (m/min).
(c) The gradient represents the jogger's average speed = 150 m/min (= 9 km/h).
1.3 — Ski run
(a) Rise = 20 − 80 = −60 m. Run = 240 m.
(b) m = −60/240 = −1/4.
(c) |−1/4| = 1/4 = 0.25 and |−1/8| = 0.125. This run is steeper (twice as steep as the beginner run).
1.4 — Water tank
(a) Rise = 100 − 500 = −400 L. Run = 20 min. m = −400/20 = −20.
(b) Sign is negative — the tank is losing water (draining).
(c) Units: litres per minute (L/min). So 20 L drains every minute.
1.5 — School enrolment
(a) Rise = 1100 − 800 = 300. Run = 2025 − 2020 = 5.
(b) m = 300/5 = 60 students per year.
(c) From 2025 to 2027 is 2 more years × 60 = +120 students. Predicted 2027 enrolment ≈ 1100 + 120 = 1220 students.
2.1 — Explain your thinking (sample response)
Reading points that aren't quite on a grid intersection means the classmate is guessing the y-coordinate, which introduces error into both the rise and the run. They should always pick two lattice points — points where the line crosses a grid corner exactly — so the rise and run are whole numbers they can count, not estimate. The gradient should be the same no matter which pair of lattice points they pick because a straight line has a constant rate of change: the rise/run ratio is the same anywhere along the line, so picking different points just gives different-sized "gradient triangles" with the same shape (the same ratio).
Marking: 1 mark for explaining how off-grid reading creates error; 1 mark for naming/lattice points as the fix; 1 mark for the constant-gradient property; 1 mark for using "lattice point" correctly in a full sentence.