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.
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
1.2, Road sign. A road sign says "10% grade", meaning the road rises 10 metres for every 100 metres travelled horizontally. On a graph, let A(0, 0) be the bottom of the hill and B(100, 10) be the point 100 metres along it; x measures horizontal distance to the right and y measures height.
(a) Find the gradient from A to B as a fraction in simplest form AND as a decimal.
(b) As x increases from A to B, does the hill have POSITIVE or NEGATIVE gradient?
(c) A truck travels downhill from B to A. Calculate the gradient using B then A, and explain whether reversing the travel direction changes the line's gradient. 3 marks
1.3, Ski run. A ski run joins point A(0, 0) at the bottom of a mountain to point B(800, 200) near the top. The x-axis measures horizontal metres to the right and the y-axis measures height in metres.
(a) Find the gradient between A and B.
(b) A skier travels downhill from B to A. Does that make the gradient negative? Explain what the sign of the gradient means as x increases.
(c) An expert run drops 200 m vertically over 400 m horizontally. Find the magnitude of its gradient and decide which run is steeper. 3 marks
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
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
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.
How did this worksheet feel?
What I'll revisit before next class:
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) From A(0, 0) to B(100, 10), m = (10 − 0)/(100 − 0) = 10/100 = 1/10 = 0.1.
(b) As x increases from A to B, height increases, so the gradient is positive (+1/10).
(c) From B(100, 10) to A(0, 0), m = (0 − 10)/(0 − 100) = (−10)/(−100) = +1/10. Both height and horizontal distance changes reverse, so the gradient of the same line does not change with the direction of travel.
1.3, Ski run
(a) From A(0, 0) to B(800, 200), rise = 200 and run = 800, so m = 200/800 = +1/4.
(b) No. From B back to A, rise = −200 and run = −800, so m = (−200)/(−800) = +1/4. Reading the graph left to right, x increases from A to B and height increases, so the gradient is positive. The skier's travel direction is right to left; downhill travel in that direction does not make the line's left-to-right gradient negative.
(c) The expert run's gradient magnitude is 200/400 = 1/2. Since 1/2 > 1/4, the expert run is steeper (twice the gradient magnitude).
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.