Mathematics • Year 8 • Unit 2 • Lesson 9
Two Points, One Gradient — In Real Life
Apply the gradient formula m = (y₂ − y₁)/(x₂ − x₁) to two-point real-world data — population growth, fitness goals, fuel usage, plant growth, and savings. Always state the units in your final answer.
1. Word problems
Each scenario gives you exactly two data points. Use the gradient formula, then state what the gradient represents and its units. Show working — final answers alone earn half marks.
1.1 — Plant growth. A bean plant is 4 cm tall on day 2 and 16 cm tall on day 10.
(a) Write the two points as (day, height).
(b) Apply the gradient formula and state the units.
(c) What does the gradient represent in context? 3 marks
1.2 — Fitness goal. Maya can run 1 km in 6 minutes when she starts training, and 1 km in 4 minutes 8 weeks later.
(a) Write two points (weeks, minutes-per-km).
(b) Use the gradient formula to find m.
(c) Interpret the gradient sign and value in plain English. 3 marks
1.3 — Fuel usage. A car's fuel tank shows 60 L at the start of a trip (km 0) and 36 L after 300 km.
(a) Write the two points (km, litres).
(b) Find the gradient using the formula.
(c) State the units and interpret in context. 3 marks
1.4 — Town population. A regional town had 4500 people in 2015 and 6300 people in 2024.
(a) Write the two points (year, population).
(b) Calculate the average growth rate (gradient) using the formula.
(c) Predict the population in 2030 if growth continues at the same rate. 3 marks
1.5 — Savings. Karim's savings account holds $120 in week 3 and $390 in week 12.
(a) Write the two points (week, $).
(b) Calculate the gradient using the formula.
(c) Interpret the gradient in one sentence (include units). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes m = (x₂ − x₁)/(y₂ − y₁) — they've put x on top and y on bottom. For the points (1, 3) and (4, 9), they get m = 3/6 = 1/2. The correct answer is m = 2. In your own words, explain (i) the mistake, (ii) what the correct formula is, (iii) why their wrong answer is the reciprocal of the right answer, and (iv) a memory trick to avoid this mix-up. Use the phrase "y on top, x on bottom" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bean plant
(a) (2, 4) and (10, 16).
(b) m = (16 − 4)/(10 − 2) = 12/8 = 1.5 cm/day.
(c) The plant grows on average 1.5 cm taller each day.
1.2 — Fitness goal
(a) (0, 6) and (8, 4).
(b) m = (4 − 6)/(8 − 0) = −2/8 = −1/4 min/week (= −0.25).
(c) Maya's time per km is dropping by ¼ minute (15 seconds) per week — she's getting faster.
1.3 — Fuel
(a) (0, 60) and (300, 36).
(b) m = (36 − 60)/(300 − 0) = −24/300 = −0.08 L/km.
(c) The car uses 0.08 L per km (or 8 L per 100 km). Negative sign = fuel decreasing.
1.4 — Town population
(a) (2015, 4500) and (2024, 6300).
(b) m = (6300 − 4500)/(2024 − 2015) = 1800/9 = 200 people/year.
(c) 2030 is 6 more years after 2024. Predicted population = 6300 + (200 × 6) = 6300 + 1200 = 7500.
1.5 — Savings
(a) (3, 120) and (12, 390).
(b) m = (390 − 120)/(12 − 3) = 270/9 = $30/week.
(c) On average Karim is saving $30 each week.
2.1 — Explain your thinking (sample response)
The classmate has flipped the formula upside down: they put x on top and y on bottom, when the correct formula is m = (y₂ − y₁)/(x₂ − x₁) — "y on top, x on bottom". For the points (1, 3) and (4, 9), the correct gradient is m = (9 − 3)/(4 − 1) = 6/3 = 2, while their flipped formula gives (4 − 1)/(9 − 3) = 3/6 = 1/2. That's the reciprocal of 2 — flipping a fraction upside down gives 1/(original), and that's exactly what flipping the formula does. A memory trick: "rise OVER run, and rise is up so it goes UP top" — gradient is steepness, so the vertical change (y) must be on top of the fraction.
Marking: 1 mark for spotting the flipped formula; 1 mark for stating the correct formula; 1 mark for explaining the reciprocal relationship; 1 mark for a sensible memory trick using "y on top, x on bottom".