Mathematics • Year 10 • Unit 2 • Lesson 16
Gradient in the Real World
Translate real-world rate-of-change problems — wheelchair ramps, road grade, growth rates and cooling — into gradients. Then explain why a "1 in 14" ramp is equivalent to a gradient of about 0.071.
1. Word problems
For each, identify the rise and run carefully and state the gradient with units (m/s, $/hour, °C/min, etc.).
1.1 — Wheelchair ramp. A wheelchair ramp rises 30 cm over a horizontal run of 4.2 m. (Australian standard is a maximum gradient of 1 in 14, i.e. m ≤ 1/14 ≈ 0.071.)
(a) Find the gradient of the ramp (use the same units — convert 30 cm to 0.30 m).
(b) Compare to the standard 1/14 — is the ramp compliant? Show your numbers. 3 marks
1.2 — Road grade. A mountain road climbs from sea level (0 m) to 200 m altitude over a horizontal distance of 2.5 km. Road grade is gradient expressed as a percentage.
(a) Find the gradient (rise in metres / run in metres).
(b) Express it as a percentage. (Hint: m × 100%.)
(c) Is this a "steep" road by Australian standards (steep = ≥ 8%)? 4 marks
1.3 — Bank balance. Mai's savings rose linearly from $200 at week 0 to $680 at week 12.
(a) Find the gradient (savings per week).
(b) Interpret what the gradient means in everyday words.
(c) At this rate, what will her balance be at week 20? 3 marks
1.4 — Cooling coffee. A cup of coffee starts at 85 °C and cools to 60 °C after 10 minutes (assume linear over this short range).
(a) Find the gradient (°C per minute). Include the sign and explain why it's negative.
(b) Predict the temperature after 15 minutes. (Note: real cooling isn't linear forever, but it's a fair approximation for short times.) 3 marks
1.5 — Two points from a graph. A printer's "pages printed vs minutes" graph passes through (3, 90) and (8, 240). The relationship is linear.
(a) Find the gradient (pages per minute).
(b) State the printer's print speed in real-world units (pages/min). 3 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 A classmate says: "A 1-in-14 wheelchair ramp means it rises 1 m every 14 m, so the gradient is 14." Using the words rise, run and fraction, explain (i) what the "1 in 14" notation really means, (ii) why the gradient is actually 1/14 ≈ 0.071 (not 14), and (iii) what the gradient 14 would physically mean if it existed (sketch the physical impossibility).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Wheelchair ramp
(a) Rise = 0.30 m, Run = 4.2 m. m = 0.30/4.2 = 1/14 ≈ 0.0714. (b) 1/14 = 1/14. This ramp exactly meets the standard (compliant). Just compliant.
1.2 — Road grade
(a) Rise = 200 m, Run = 2500 m. m = 200/2500 = 0.08 (or 2/25). (b) 0.08 × 100% = 8%. (c) Exactly at the steep threshold — yes, this is "steep" by Australian convention (≥ 8%).
1.3 — Bank balance
(a) m = (680 − 200)/(12 − 0) = 480/12 = $40/week. (b) Mai saves $40 every week. (c) At week 20: B = 200 + 40(20) = $1000.
1.4 — Cooling coffee
(a) m = (60 − 85)/(10 − 0) = −25/10 = −2.5 °C/min. Negative because temperature is decreasing over time. (b) At t = 15: T = 85 + (−2.5)(15) = 85 − 37.5 = 47.5 °C.
1.5 — Printer
(a) m = (240 − 90)/(8 − 3) = 150/5 = 30 pages/min. (b) The printer prints at 30 pages per minute.
2.1 — Explain (sample response)
(i) "1 in 14" means for every 14 units of horizontal run, the ramp gains 1 unit of vertical rise. It is a ratio rise:run = 1:14, not the gradient itself. (ii) Gradient = rise/run as a fraction = 1/14 ≈ 0.071. The classmate has put run/rise instead. (iii) A gradient of 14 would mean 14 units up for every 1 unit across — almost vertical, more like a cliff face than a ramp. No wheelchair user could climb it. The lesson rule is rise over run, not the other way around.
Marking: 1 for explaining the 1:14 ratio meaning; 1 for the correct gradient 1/14; 1 for pointing out the inversion mistake; 1 for the physical-impossibility description.