Chapter 6 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.6 Chapter 6

Steepness: the gradient

How steep is a line? Can I put a number on it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the gradient of a line from two points on it
  • Say how the number in front of x changes the line

A hill can be gentle or steep. We can put a number on how steep a straight line is. The number is called its gradient.

Pick two points on a line. Go across from the first to the second. That distance across is the run. Then go up. That distance up is the rise.

The gradient is the rise divided by the run. A steep line rises a lot for each step across, so its gradient is big. A gentle line has a small gradient.

Draw it first

Find the gradient of a line

  1. A grid with the x axis across and the y axis up. Step 1: two dots A (1, 1) and B (3, 5). Step 2: the straight line through them, y = 2x − 1. Step 3: a step triangle from A, with the run 2 along the bottom and the rise 4 up the side.AB1234561234560xy

    1 Choose two points on the line. Here they are A (1, 1) and B (3, 5).

  2. A grid with the x axis across and the y axis up. Step 1: two dots A (1, 1) and B (3, 5). Step 2: the straight line through them, y = 2x − 1. Step 3: a step triangle from A, with the run 2 along the bottom and the rise 4 up the side.ABy = 2x − 11234561234560xy

    2 Draw the line through them with a ruler.

  3. A grid with the x axis across and the y axis up. Step 1: two dots A (1, 1) and B (3, 5). Step 2: the straight line through them, y = 2x − 1. Step 3: a step triangle from A, with the run 2 along the bottom and the rise 4 up the side.ABrun 2rise 4y = 2x − 11234561234560xy

    3 From A go 2 across, then 4 up to B. The run is 2 and the rise is 4. The gradient is 4 ÷ 2 = 2.

Worked example

Find the gradient of the line through (1, 1) and (3, 7).

  1. Find the run. Going from 1 to 3 across, the run is 3 − 1 = 2.
  2. Find the rise. Going from 1 to 7 up, the rise is 7 − 1 = 6.
  3. The gradient is the rise divided by the run: 6 ÷ 2 = 3.

Answer: 3

A grid with the x axis across and the y axis up. The line through A (1, 1) and B (3, 7). A step triangle from A shows the run 2 and the rise 6.ABrun 2rise 6123456123456780xy
The run is 2 and the rise is 6. The gradient is 6 ÷ 2 = 3.

Now a line that goes down as you move to the right. Going from the first point to the second, the rise is negative, because you go down. So its gradient is negative.

Worked example

Find the gradient of the line through (1, 5) and (4, 2).

  1. Find the run. Going from 1 to 4 across, the run is 4 − 1 = 3.
  2. Find the rise. The up number goes from 5 to 2, so the rise is 2 − 5 = −3. The line goes down.
  3. The gradient is −3 ÷ 3 = −1.

Answer: −1

A flat line has a rise of 0, so its gradient is 0 ÷ run = 0. An upright line has no run, so it has no gradient.

A grid with the x axis across and the y axis up. The line through A (1, 5) and B (4, 2), going down to the right. A step triangle from A shows the run 3 and the rise −3.ABrun 3rise −31234561234560xy
The line goes down to the right, so the gradient is negative.

Now look at the rule. In y = 2x, the number in front of x is 2. It is the gradient. In y = 3x it is 3. For lines that go up to the right, a bigger number is steeper. For any line, the bigger the number without its sign, the steeper the line. A negative number gives a line that goes down to the right.

A grid with the x axis across and the y axis up. Three lines through the origin: y = 2x is the steepest, y = x is less steep, and y = 0.5x is the gentlest.y = 2xy = xy = 0.5x1234561234560xy
For lines that go up, the bigger the number in front of x, the steeper the line.

See it three ways

With things

A steep bank rises 4 m for every 2 m along the ground. Walk up it and feel how steep it is.

As a picture

A grid with the x axis across and the y axis up. A bank drawn as a straight line on a grid with along (m) across and up (m) up. A step triangle from the dot S (1, 2) to the dot T (3, 6) shows the run 2 m and the rise 4 m.STrun 2 mrise 4 m12345612345670along (m)up (m)

In symbols

The gradient is 4 ÷ 2 = 2. The bank rises 2 m for every 1 m along.

Try it

Measure the steepness of a plank

You need: a plank or a flat board, some books, a ruler.

  1. Rest one end of the plank on a stack of books. The other end stays on the floor.
  2. Measure how high the top end is above the floor. This is the rise.
  3. Measure along the floor from the bottom end to the point just under the top end. This is the run.
  4. Divide the rise by the run. Then make the stack taller and measure again.

What you will see: A taller stack gives a bigger rise, so a bigger gradient. The plank is steeper. A rise of 20 cm over a run of 40 cm gives a gradient of 0.5.

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. A plank drawn as a straight line from the origin, on a graph with along the floor (cm) across and height (cm) up. Dashed guides show that the top end is 40 cm along the floor and 20 cm high.010203001020304050height (cm)along the floor (cm)

A wrong idea many people have

The gradient is the run divided by the rise.

Why it feels right: In a coordinate pair the across number comes first, so it seems we should divide across by up.

The test that breaks it: A steep line rises 6 while it runs 1. Run ÷ rise would give 1 ÷ 6, a small number. But a steep line should have a big gradient.

What is true: The gradient is the rise divided by the run. Up divided by across. A steeper line has a bigger gradient.

Key words

gradient
how steep a line is. It is the rise divided by the run.
rise
how far up you go between two points. It is negative if you go down.
run
how far across you go between two points.

Back to your guess

The answer is A: A line rises 4 while it runs 2 across. Its steepness is 2.

The steepness of a line is called its gradient. It is the rise divided by the run. Here 4 ÷ 2 = 2. The number 0.5 is the run divided by the rise, which is the wrong way round.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the gradient of the line through (1, 2) and (4, 8)?

    A grid with the x axis across and the y axis up. A line through the dots A (1, 2) and B (4, 8). A step triangle shows the run 3 and the rise 6.ABrun 3rise 612345671234567890xy
    Tap your answer.

    1. Not this one. You divided the run by the rise: 3 ÷ 6. The gradient is rise ÷ run. Go to M3.9.6. Try another.

    2. Not this one. You added 3 and 6. The gradient is the rise divided by the run. Go to M3.9.6. Try another.

    3. Not this one. That is the rise only. Divide it by the run, 3. Go to M3.9.6. Try another.

    4. Yes, that is right. The run is 4 − 1 = 3 and the rise is 8 − 2 = 6. The gradient is 6 ÷ 3 = 2.

    See the answer

    2

    The run is 4 − 1 = 3 and the rise is 8 − 2 = 6. The gradient is 6 ÷ 3 = 2.

    • If you chose A: You divided the run by the rise: 3 ÷ 6. The gradient is rise ÷ run. Go to M3.9.6.
    • If you chose B: You added 3 and 6. The gradient is the rise divided by the run. Go to M3.9.6.
    • If you chose C: That is the rise only. Divide it by the run, 3. Go to M3.9.6.

  2. What is the gradient of the line through (0, 6) and (3, 0)?

    A grid with the x axis across and the y axis up. A line through the dots A (0, 6) and B (3, 0), going down to the right.AB12345612345670xy
    Tap your answer.

    1. Yes, that is right. The run is 3 − 0 = 3. The rise is 0 − 6 = −6, because the line goes down. The gradient is −6 ÷ 3 = −2.

    2. Not this one. The size is right, but the line goes down to the right. A line going down has a negative gradient. Go to M3.9.6. Try another.

    3. Not this one. You divided the run, 3, by the rise, 6. The gradient is rise ÷ run: −6 ÷ 3. Go to M3.9.6. Try another.

    4. Not this one. That is the run. The rise is −6, and the gradient is rise ÷ run. Go to M3.9.6. Try another.

    See the answer

    −2

    The run is 3 − 0 = 3. The rise is 0 − 6 = −6, because the line goes down. The gradient is −6 ÷ 3 = −2.

    • If you chose B: The size is right, but the line goes down to the right. A line going down has a negative gradient. Go to M3.9.6.
    • If you chose C: You divided the run, 3, by the rise, 6. The gradient is rise ÷ run: −6 ÷ 3. Go to M3.9.6.
    • If you chose D: That is the run. The rise is −6, and the gradient is rise ÷ run. Go to M3.9.6.

  3. Which of these lines is the steepest?

    Tap your answer.

    1. Not this one. The number in front of x is 1. A bigger number makes a steeper line. Go to M3.9.6. Try another.

    2. Not this one. The number in front of x is 3. A bigger number makes a steeper line. Go to M3.9.6. Try another.

    3. Yes, that is right. The number in front of x is the gradient. The biggest gradient is 5, so y = 5x − 1 is the steepest.

    4. Not this one. The number in front of x is 2. A bigger number makes a steeper line. Go to M3.9.6. Try another.

    See the answer

    y = 5x − 1

    The number in front of x is the gradient. The biggest gradient is 5, so y = 5x − 1 is the steepest.

    • If you chose A: The number in front of x is 1. A bigger number makes a steeper line. Go to M3.9.6.
    • If you chose B: The number in front of x is 3. A bigger number makes a steeper line. Go to M3.9.6.
    • If you chose D: The number in front of x is 2. A bigger number makes a steeper line. Go to M3.9.6.

  4. A line goes down as you move to the right. What can you say about its gradient?

    Tap your answer.

    1. Yes, that is right. Going to the right, the rise is negative because the line goes down. So the gradient is negative.

    2. Not this one. A positive gradient goes up to the right. This line goes down. Go to M3.9.6. Try another.

    3. Not this one. A gradient of zero is a flat line. This line goes down. Go to M3.9.6. Try another.

    See the answer

    It is negative.

    Going to the right, the rise is negative because the line goes down. So the gradient is negative.

    • If you chose B: A positive gradient goes up to the right. This line goes down. Go to M3.9.6.
    • If you chose C: A gradient of zero is a flat line. This line goes down. Go to M3.9.6.

  5. What is the gradient of the line y = −2x + 3?

    Tap your answer.

    1. Yes, that is right. The number in front of x is the gradient. In y = −2x + 3 it is −2.

    2. Not this one. That is where the line crosses the y axis. The gradient is the number in front of x. Go to M3.9.6. Try another.

    3. Not this one. You dropped the minus sign. The number in front of x is −2. Go to M3.9.6. Try another.

    4. Not this one. The number in front of x is −2. The 3 is added on its own. Go to M3.9.6. Try another.

    See the answer

    −2

    The number in front of x is the gradient. In y = −2x + 3 it is −2.

    • If you chose B: That is where the line crosses the y axis. The gradient is the number in front of x. Go to M3.9.6.
    • If you chose C: You dropped the minus sign. The number in front of x is −2. Go to M3.9.6.
    • If you chose D: The number in front of x is −2. The 3 is added on its own. Go to M3.9.6.

  6. What is the gradient of the flat line y = 4?

    Tap your answer.

    1. Not this one. The 4 tells how high the line is. The gradient is the rise divided by the run, and a flat line has no rise. Go to M3.9.6. Try another.

    2. Yes, that is right. On a flat line the rise is 0 for any run. The gradient is 0 ÷ run = 0.

    3. Not this one. Only an upright line has no gradient. A flat line has a run, and its rise is 0. Go to M3.9.6. Try another.

    See the answer

    0

    On a flat line the rise is 0 for any run. The gradient is 0 ÷ run = 0.

    • If you chose A: The 4 tells how high the line is. The gradient is the rise divided by the run, and a flat line has no rise. Go to M3.9.6.
    • If you chose C: Only an upright line has no gradient. A flat line has a run, and its rise is 0. Go to M3.9.6.

  7. Find the gradient of the line through (1, 2) and (4, 11).

    See the answer

    3

    The run is 4 − 1 = 3. The rise is 11 − 2 = 9. The gradient is 9 ÷ 3 = 3.

  8. A steep slope rises 12 m while it runs 4 m along the ground. What is its gradient?

    See the answer

    3

    The gradient is the rise divided by the run. 12 ÷ 4 = 3.

  9. From an earlier chapter

    Where do the lines x = 2 and y = 6 cross?

    See the answer

    (2, 6)

    The line x = 2 gives the across number, 2. The line y = 6 gives the up number, 6.

Keep it

Copy this into your exercise book.

Gradient = rise ÷ run. A line going up to the right has a positive gradient. A line going down has a negative one. In y = mx + c the number in front of x is the gradient.

Draw: Draw a line through (1, 1) and (3, 5). Draw the step triangle. Write run 2, rise 4 and gradient 2.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (761 KB). It opens on any phone.