Have you ever walked or cycled up a steep hill? How did it feel compared to walking on a flat road?
Step-by-step solution
Idea: Steepness compares how much you go up with how far you go along. A flat road has no rise at all; a steep hill has a big rise for a short distance along.
- On a flat road you move forward without climbing at all, so little effort is needed.½ mark
- On a hill, every step forward also lifts you upward. You have to work against gravity, so it feels tiring.½ mark
- The steeper the hill, the more height you gain for each metre forward, and the harder it feels.½ mark
- This “height gained per distance along” is what mathematicians call the slope of a line: slope = riserun. A flat road has slope 0; going uphill (left to right) the slope is positive, downhill it is negative.½ mark
Check: Compare two ramps that both rise 1 m: one over 2 m along and one over 10 m along. 12 is bigger than 110, so the first is steeper, which matches how much harder it feels.
Answer to write in the exam
Flat road: no rise ⇒ least effort (slope 0)
Hill: rise with every step forward ⇒ more effort
Steeper hill ⇒ more rise per unit run ⇒ more effort
∴ Steepness = riserun = slope (gradient)
Common mistakes that cost marks
- Thinking a longer hill is always steeper. Steepness depends on rise compared with run, not on length alone.
- Calling a downhill road “no slope”. Going down from left to right is a negative slope.
How this can come in the exam
Which road is the steepest?
- Rises 2 m over 20 m
- Rises 3 m over 15 m
- Rises 5 m over 50 m
- Rises 1 m over 8 m
Show answer
(B) Rises 3 m over 15 m
Slopes: 0.1, 0.2, 0.1, 0.125. The largest is 0.2.
Try one yourself
A path rises 6 m over a horizontal distance of 30 m. What is its slope?
Show answer
630 = 15 (0.2).
More questions like this
- 1. Consider the point on the line AB whose x-coordinate is 8. What is the y-coordinate of this point? Express this as an ordered pair.
2. What is the x-coordinate of the point on the line AB whose y-coordinate is 6? Express this as an ordered pair. - Consider the straight line that passes through A (−1, 0), B (1, 2), and C (4, 5). Plot the points and find the slope of the line.
- Consider the line in the figure which passes through A (−4, 5) and B (−1, 2). Find the slope of the line.
- But why does ‘m’ represent the slope?
- Consider the line 5x + y = 3. Rewrite it as y = −5x + 3. Can you now find its slope? What does it mean?
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