Mathematics • Year 8 • Unit 2 • Lesson 7
Gradient Sign in the Real World
Use positive, negative, zero and undefined gradients to describe real situations — hiking, returning home in a car, a thermometer reading, a parked vehicle. Then explain why the sign of the gradient matches the story.
1. Word problems
Each scenario describes a real relationship between two quantities. Identify the gradient sign (positive / negative / zero / undefined) and justify your answer with a one-sentence reason. Show any numbers you use.
1.1 — Hiker on a hill. A hiker walks up a hill, then across a flat ridge, then down the other side. Sketch (or describe) the height-vs-distance graph in three pieces.
(a) State the gradient sign for each piece.
(b) Which piece has a zero gradient and why? 3 marks
1.2 — Car driving home. Lin's distance from home is 60 km. She drives straight home at a constant speed, reaching home after 1 hour.
(a) Build a small table for t = 0 h and t = 1 h.
(b) What is the sign of the gradient of distance-from-home vs time? Justify with a one-sentence reason. 3 marks
1.3 — Heating water. A kettle's temperature rises from 20°C to 80°C in 2 minutes. Then it sits at 80°C while you make tea.
(a) Describe the gradient sign during the heating phase.
(b) Describe the gradient sign during the tea-making phase.
(c) Sketch (or describe) the overall temperature-vs-time graph. 3 marks
1.4 — Phone battery. A phone battery starts at 100% and drains evenly to 20% over 8 hours of use.
(a) What is the sign of the gradient of battery-% vs hours?
(b) Write an equation in the form B = mh + c for the battery percentage after h hours.
(c) State the value and sign of m. 3 marks
1.5 — Elevator. An elevator goes from the ground floor straight up to floor 8 (no in-between stops).
(a) On a height-vs-time graph, what is the gradient sign while it rises?
(b) An "express" lift travels nearly straight up on a height-vs-horizontal-position graph. What does its gradient look like in that picture?
(c) In one sentence, explain the difference between a zero gradient and an undefined gradient. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "A line with a negative gradient must lie below the x-axis." In your own words, explain (i) what is wrong with this statement, (ii) give a concrete counterexample with an equation and a check point, and (iii) describe what the gradient actually tells you (versus what the y-intercept tells you). Use the phrase "direction not position" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Hiker on a hill
(a) Up the hill: positive (height rises as distance increases). Flat ridge: zero (height stays constant). Down the other side: negative (height falls as distance increases).
(b) The flat ridge — height doesn't change with distance, so rise = 0 → gradient = 0.
1.2 — Car driving home
(a) Table: t = 0 → d = 60; t = 1 → d = 0.
(b) Negative gradient. Distance from home decreases as time increases, so d falls as t rises. (Equation: d = −60t + 60.)
1.3 — Heating water
(a) Heating phase: positive (temperature rising).
(b) Sitting phase: zero (temperature constant).
(c) Graph: rises sharply from 20 to 80 over 2 min, then flat horizontal line at 80°C — a "rising then flat" shape.
1.4 — Phone battery
(a) Negative (battery % decreasing).
(b) Drop = 100 − 20 = 80% over 8 hours → 10% per hour. So B = −10h + 100.
(c) m = −10 (% per hour), negative.
1.5 — Elevator
(a) Positive — height increases as time increases.
(b) On a height-vs-horizontal-position graph, the lift goes straight up with almost no horizontal movement, so the line is nearly vertical → undefined gradient.
(c) Zero = perfectly flat horizontal line (y = c). Undefined = perfectly vertical line (x = a). Zero has rise = 0; undefined has run = 0 (division by zero).
2.1 — Explain your thinking (sample response)
The classmate has confused gradient with position. A negative gradient only means the line slopes downhill from left to right — it tells you the direction not position. A clear counterexample is y = −x + 10: the gradient is m = −1 (negative), but at x = 0 the line passes through y = 10, well above the x-axis. So the line can have a negative gradient and still sit entirely in the positive-y region. The gradient describes which way the line tilts, while the y-intercept (the c in y = mx + c) describes where the line cuts the y-axis, i.e. its starting position.
Marking: 1 mark for spotting the mix-up; 1 mark for a valid counterexample with an equation; 1 mark for verifying with a check point above the x-axis; 1 mark for clearly contrasting gradient (direction) with y-intercept (position) and using "direction not position".