Mathematics • Year 8 • Unit 2 • Lesson 5

Gradient in the Wild

Use gradient where it shows up in real life: roof pitch, road signs, ski runs, wheelchair ramps and water draining from a tank. Then explain what a "negative gradient" really means in plain English.

Apply · Real-World Maths

1. Word problems

For each scenario, calculate the gradient and interpret the sign. Show rise and run clearly.

1.1 — Roof pitch. A roof rises 1.5 m over a horizontal span of 6 m.

(a) Find the gradient of the roof.
(b) Express the gradient as a simplified fraction AND as a decimal.
(c) The next-door house has a roof with gradient 1/2. Which roof is steeper, and by how much (in fraction form)? 3 marks

Stuck? m = 1.5 / 6 = 1/4. Compare 1/4 to 1/2.

1.2 — Road sign. A road sign says "10% grade" — meaning the road rises 10 metres for every 100 metres travelled horizontally.

(a) Express the road's gradient as a fraction in simplest form AND as a decimal.
(b) Plot two example points (origin and the 100 m mark) and state whether the line going uphill has POSITIVE or NEGATIVE gradient.
(c) A truck driver coming DOWN the same hill would experience the opposite slope. What's the gradient from her perspective? 3 marks

Stuck? 10/100 = 1/10 = 0.1. "Uphill" = +1/10; the same hill seen from the top going down has rise = −10 over run = +100, so −1/10.

1.3 — Ski run. A ski lift takes you from point A(0, 0) at the bottom of a mountain to point B(800, 200), where the x-axis measures horizontal metres and the y-axis measures vertical metres.

(a) Find the gradient of the ski run (going DOWN from B to A).
(b) What does the negative sign tell you?
(c) A steeper, expert run drops 200 m vertical over only 400 m horizontal. Find its gradient and decide which run is steeper. 3 marks

Stuck? Going from B(800, 200) down to A(0, 0): rise = −200, run = −800, m = +1/4 (the formula handles signs automatically). But thinking of "skier going downhill left-to-right": going from a higher start to a lower end gives negative m. For comparison: 200/400 = 1/2 vs 200/800 = 1/4.

1.4 — Wheelchair ramp. Australian Standard AS1428 says a wheelchair ramp must have a gradient no steeper than 1/14 (one vertical unit up for every 14 horizontal units across).

(a) A ramp rises 0.5 m over a horizontal distance of 6 m. Find its gradient.
(b) Does this ramp meet the standard? Explain with a calculation.
(c) If the rise stays at 0.5 m, what is the SHORTEST horizontal distance the ramp could have and still meet the standard? 3 marks

Stuck? 0.5/6 = 1/12. Compare 1/12 vs 1/14: 1/12 is BIGGER (steeper) → fails. For "exactly meeting" the standard: 0.5/x = 1/14 → x = 7 m.

1.5 — Water draining. A tank's water level drops as it drains: at 0 minutes the depth is 60 cm; at 10 minutes it's 45 cm; at 20 minutes it's 30 cm; at 30 minutes it's 15 cm.

(a) Plot (or describe) the points (0, 60), (10, 45), (20, 30), (30, 15) and find the gradient of the line that joins them.
(b) In plain English, what does the negative gradient mean for the water level? 3 marks

Stuck? Each 10-minute step drops the depth by 15 cm → m = −15/10 = −1.5 cm per minute.

2. Explain your thinking

This question is about communication. Use full sentences. 4 marks

2.1 A classmate says: "A line with gradient 0 isn't really a line — it's just nothing." In your own words, explain (i) what a gradient of 0 actually means geometrically, (ii) the difference between a gradient of 0 and an UNDEFINED gradient, and (iii) give one real-world example of each. Use the phrase "horizontal vs vertical" somewhere in your answer.

Stuck? Revisit lesson § Key Terms — "gradient" zero vs undefined. Zero rise = horizontal; zero run = vertical.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Roof pitch

(a) m = rise/run = 1.5/6 = 0.25.
(b) As fraction: 1.5/6 = 1/4. As decimal: 0.25.
(c) Next-door roof = 1/2 = 0.5; ours = 1/4 = 0.25. The next-door roof is steeper, by 1/2 − 1/4 = 1/4.

1.2 — Road sign 10% grade

(a) 10/100 = 1/10 = 0.1.
(b) Going uphill, y increases as x increases → positive gradient (+1/10).
(c) Same physical hill but described as a descent: rise = −10 over run = +100, so m = −1/10.

1.3 — Ski run

(a) From B(800, 200) down to A(0, 0): rise = 0 − 200 = −200, run = 0 − 800 = −800. m = −200/−800 = 1/4. Equivalently, going from A up to B: m = 200/800 = +1/4 (positive uphill).
(b) A negative gradient from the skier's perspective means the line slopes downhill (y decreases as x increases).
(c) Expert run: m = 200/400 = 1/2. Since 1/2 > 1/4, the expert run is steeper — twice as steep, in fact.

1.4 — Wheelchair ramp

(a) m = 0.5/6 = 1/12 ≈ 0.083.
(b) Standard maximum is 1/14 ≈ 0.071. Our ramp is 1/12 ≈ 0.083, which is BIGGER → steeper than allowed. Does not meet AS1428.
(c) To exactly meet the standard, m = 1/14 = 0.5/x → x = 0.5 × 14 = 7 metres. The ramp needs at least 7 m horizontal to meet AS1428.

1.5 — Water draining

(a) Pick any two points, e.g. (0, 60) and (30, 15). rise = 15 − 60 = −45; run = 30 − 0 = 30. m = −45/30 = −1.5 (cm/min).
(b) A negative gradient means the depth is DECREASING: the water level drops by 1.5 cm for each minute that passes.

2.1 — Explain your thinking (sample response)

A gradient of 0 doesn't mean "no line" — it means a perfectly flat line: the rise is 0, but you can still run any distance you like. The line is horizontal, which is the boundary between uphill and downhill. An UNDEFINED gradient is the opposite extreme: the run is 0, so you can't divide rise by run at all. Geometrically, that's a vertical line. So horizontal vs vertical: gradient 0 = horizontal (flat, like a calm lake's surface), undefined = vertical (straight up, like a flagpole). Both are real lines — they just sit at the two extreme cases of the rise/run ratio.

Marking: 1 mark for explaining gradient 0 = horizontal; 1 mark for explaining undefined = vertical (rise/0); 1 mark for a real-world example of each; 1 mark for using "horizontal vs vertical" and writing in full sentences.