Year 9 · Unit 3 Linear relationships · Lesson 2 of 7

How steep is a line?

Drag two points and watch rise and run: the gradient is the constant change from Lesson 1, found from any two points on the line.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 1) In a linear table, x goes up by 1 and y goes 5, 9, 13. What is the constant change?
    Answer4.
  2. Work out (−6) − 4.
    Answer−10.
  3. Work out 12 ÷ (−3).
    Answer−4: a positive divided by a negative gives a negative.
Words used in this lesson
gradient
how steep a line is: the rise divided by the run. For example: a gradient of 2 rises 2 for every 1 across.
rise
the change in y between two points. For example: from y = 2 to y = 6 is a rise of 4.
run
the change in x between two points. For example: from x = 1 to x = 5 is a run of 4.
slope
another word for gradient. For example: the slope of a road.
horizontal
flat, going straight across. For example: a line with gradient 0.
vertical
straight up and down. For example: a line with no gradient.
undefined
has no value, because the working cannot be done. For example: a rise over a run of 0.

Which road is steeper?

A road engineer compares two roads. Road A climbs 12 m over a horizontal distance of 100 m. Road B climbs 7 m over a horizontal distance of 40 m. Steepness depends on both numbers at once: how much you go up, and how far across it takes.

Predict first: which road is steeper? Then use the first figure to measure steepness on a grid: set the two points A and B and read the rise and the run.

−3
−2
3
2
4rise, from A to B
6run, from A to B
2/3gradient = rise ÷ run

From A(−3, −2) to B(3, 2): rise 4, run 6. Gradient = 4/6 = 2/3. The line rises from left to right.

Show me the steps
  1. A(−3, −2) and B(3, 2): rise 2 − (−2) = 4, run 3 − (−3) = 6. Gradient 4/6 = 2/3.
  2. A(−2, −3) and B(2, 5): rise 8, run 4. Gradient 8/4 = 2: steeper than the first.
  3. A(−3, 4) and B(3, −2): rise −2 − 4 = −6, run 6. Gradient −6/6 = −1. The line falls from left to right.
  4. A(−4, 2) and B(3, 2): rise 0, run 7. Gradient 0/7 = 0. A horizontal line.
  5. A(2, −3) and B(2, 4): rise 7, run 0. Dividing by 0 is not possible, so the gradient is undefined. A vertical line.
  6. A(3, −2) and B(−3, 4): the points of the third state in the other order. Rise 4 − (−2) = 6, run −3 − 3 = −6. Gradient 6/(−6) = −1: the same.

The gradient of a line is the rise divided by the run: the change in y divided by the change in x, taken in the same direction between two points. With A(x1, y1) and B(x2, y2), gradient = (y2 − y1) ÷ (x2 − x1).

A positive gradient rises from left to right, a negative one falls, and a horizontal line has gradient 0. A vertical line has a run of 0, and you cannot divide by 0, so it has no gradient.

Road A: 12 ÷ 100 = 0.12. Road B: 7 ÷ 40 = 0.175. Road B is steeper, even though it climbs less in total.

A common mistake: mixing the directions: taking y2 − y1 on the top but x1 − x2 on the bottom. That flips the sign. Go from the same point to the same point in both: the first figure's last state shows that swapping A and B changes both signs and leaves the gradient alone.

Any two points will do

A straight line has one constant rate of change, so every pair of points on it must give the same gradient. Slide A and B along the fixed line and see whether the gradient changes, even when the triangle gets bigger.

−2
2
6rise, from A to B
4run, from A to B
1.5gradient = rise ÷ run

A(−2, −4) to B(2, 2): rise 6, run 4. Gradient 6 ÷ 4 = 1.5.

Show me the steps
  1. A(−2, −4) and B(2, 2): rise 6, run 4. Gradient 6 ÷ 4 = 1.5.
  2. A(−4, −7) and B(0, −1): rise 6, run 4. The same gradient, 1.5, from a different pair.
  3. A(0, −1) and B(2, 2): rise 3, run 2. Gradient 3 ÷ 2 = 1.5: a smaller triangle, the same ratio.
  4. A(−4, −7) and B(4, 5): rise 12, run 8. Gradient 12 ÷ 8 = 1.5.
  5. A and B on the same point: no rise and no run, so there is nothing to divide.

Worked example: a mountain trail

On a trail in Nepal the height is 640 m at 2 km along and 1240 m at 8 km along. What is the gradient, in metres of height per km?

  1. Rise = 1240 − 640 = 600 m.
  2. Run = 8 − 2 = 6 km.
  3. Gradient = 600 ÷ 6 = 100 m per km. Check: for 6 km the height rises 6 × 100 = 600 m.

Your go. The first step is given. A solar lamp's battery shows 85% after 1 hour and 25% after 4 hours. What is the gradient, in per cent per hour?

  1. Rise = 25 − 85 = −60 (the charge falls, so the rise is negative).
  2. Now find the run and divide, then check.
Check the answer

Run = 4 − 1 = 3 hours, so the gradient is −60 ÷ 3 = −20: the battery loses 20% each hour.

Check before you go on

1. What is the gradient of the line through (−2, 5) and (2, −3)?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. What is the gradient of the line through (1, 2) and (5, 4)? Give it as a decimal.

3. A glacier is 180 m thick at the start and 150 m thick 6 years later. Find its rate of change, in metres per year. Use a minus sign for a fall.

Level 2 · Building

4. What is the gradient of the line through (−4, 3) and (2, −6)? Give it as a decimal.

5. What is the gradient of the line through (−3, 4) and (5, 4)?

6. A bicycle's trip meter reads 12 km after 30 minutes and 33 km after 90 minutes. What is the gradient of distance against time, in km per minute? Give it as a decimal.

Level 3 · Stretch

7. The line through (2, 1) and (6, k) has gradient 2. Find k.

8. A(1, 2) and B(3, 8) are on a straight line. The point C(5, t) is on the same line. Find t.

Something to think about

A line has gradient 2/3. Two different points on it are 12 apart across (the run). How far apart are they up the page (the rise)? Why does it not matter which two points you picked?

One way to think about it

A run of 12 is four lots of 3, and each lot rises 2, so the rise is 8. Any two points on a straight line give the same gradient, because the line has one constant rate: a bigger triangle on the line is the same triangle scaled up.

Exit ticket

9. A line passes through the origin (0, 0) and the point (2, 3). Which of these points is also on the line?