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Lesson 8 ~30 min Unit 2 · Linear Relationships +90 XP

Gradient from a Graph

Use rise and run to find the steepness of a straight line on a coordinate grid. Draw gradient triangles and read gradient as a fraction or decimal.

Today's hook: Every ski slope, ramp and road has a gradient. Knowing how to read gradient from a graph means you can instantly tell whether a line is steeper than a black ski run (gradient > 1) or gentler than a bike lane (gradient < 0.05).
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Think First
warm-up

Look at this line on a grid. How many units does it rise for each unit it runs?

0 1 2 3 4 5 1 2 3 4 run = 2 rise = 2
Record your answer in your workbook.
Show answer

The line rises 2 units for every 2 units it runs, so rise per unit run $= \frac{2}{2} = 1$. The gradient = 1.

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What You'll Master
objectives

Know

  • Gradient can be found from a graph by drawing a gradient triangle
  • Rise is the vertical change ($\Delta y$)
  • Run is the horizontal change ($\Delta x$)

Understand

  • Any two points on a straight line give the same gradient
  • How to choose convenient grid intersections for accurate counting
  • Why far-apart points give more accurate results

Can Do

  • Draw gradient triangles on graphs
  • Count rise and run from grid lines
  • Calculate gradient as a fraction and a decimal
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Words You Need
vocabulary
Gradient TriangleA right-angled triangle drawn on a line to measure rise and run.
RiseThe vertical change between two points on a line ($\Delta y$). Positive uphill, negative downhill.
RunThe horizontal change between two points on a line ($\Delta x$). Always left to right (positive).
Grid SquareA unit square on the coordinate grid used for counting rise and run.
Gradient from GraphFinding $\frac{\text{rise}}{\text{run}}$ by counting grid squares on a plotted line.
Counting MethodUsing the grid to directly count vertical and horizontal changes between two grid-intersection points.
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The Gradient Triangle Method
+5 XP

To find the gradient of a line on a graph, draw a gradient triangle — a right-angled triangle that uses the line as its hypotenuse.

0 1 2 3 4 5 6 1 2 3 4 run = 2 rise = 1 gradient = 1/2

The gradient formula is:

$$\text{Gradient}=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$$

4-Step Method 1. Pick grid-intersection points 2. Draw right-angled triangle 3. Count rise & run 4. Write rise/run, simplify Any 2 points → same answer!
rise = vertical ↑↓  run = horizontal ↔
Grid intersections
Choose points where the line crosses exact grid corners for easy counting.
Any two points work
On a straight line, any two points give the exact same gradient.
Far apart = accurate
Points far apart reduce the effect of counting errors.
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Choosing Good Points & Expressing Gradient
+5 XP

Not all points are equally easy to work with. For the most accurate gradient, follow these guidelines:

0 1 2 3 4 5 6 1 2 3 4 good: run=4 rise=2 1/2

DO: Pick points where the line crosses exact grid intersections.

DO: Choose points that are far apart to reduce counting errors.

DON'T: Guess points that are not on exact grid crossings.

DON'T: Use points very close together. A half-square error over run=1 is a 50% error; over run=6 it's only 8%.

Expressing gradient — always simplify the fraction:

Form Example Note
Fraction (simplest)$\frac{3}{4}$Divide by HCF
Decimal$0.75$Useful to compare steepness
Mixed number$1\tfrac{1}{2}$When numerator > denominator
Whole number$2$$\frac{4}{2} = 2$
Negative$-\frac{2}{3}$Downhill left to right

$$\frac{6}{8}=\frac{3}{4}=0.75 \qquad \frac{-4}{6}=-\frac{2}{3}\approx -0.67$$

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Spot the Trap
heads-up
Swapping rise and run
Counting rise as horizontal and run as vertical gives the reciprocal of the gradient.
Fix: rise = vertical side (up/down), run = horizontal side (left/right).
Forgetting the negative sign on downhill lines
Lines that go downhill from left to right have negative gradients because the rise is negative.
Fix: check the direction of the line before writing your answer.
Not simplifying fractions
Leaving $\frac{4}{2}$ instead of $2$, or $\frac{6}{8}$ instead of $\frac{3}{4}$.
Fix: always divide top and bottom by their highest common factor (HCF).
Choosing points not on the line
Your two points must lie exactly on the line — a point that looks close but isn't on the line will give a wrong gradient.
Fix: trace carefully along the line to check both points lie exactly on it.

Watch Me Solve It · 2 examples

Watch Me Solve It · Positive gradient
+15 XP per step
Q1
PROBLEM
Find the gradient of the line through $(1, 1)$ and $(5, 3)$ shown on the grid.
0 1 2 3 4 5 6 1 2 3 (1,1) (5,3)
  1. 1
    Choose two clear grid-intersection points
    $(1, 1)$ and $(5, 3)$ — both on exact grid corners
    Pick points that are far apart for accuracy.
  2. 2
    Find rise and run
    Rise $= 3 - 1 = 2$    Run $= 5 - 1 = 4$
    Rise = change in $y$ (vertical). Run = change in $x$ (horizontal).
  3. 3
    Calculate gradient and simplify
    $$\text{Gradient}=\frac{\text{rise}}{\text{run}}=\frac{2}{4}=\frac{1}{2}=0.5$$
    Simplify $\frac{2}{4}$ by dividing top and bottom by HCF = 2.
AnswerGradient $= \dfrac{1}{2} = 0.5$
Watch Me Solve It · Negative gradient
+15 XP per step
Q2
PROBLEM
Find the gradient of the line through $(1, 4)$ and $(5, 1)$.
0 1 2 3 4 5 1 2 3 4 (1,4) (5,1)
  1. 1
    Choose two clear points
    $(1, 4)$ and $(5, 1)$ — the line goes downhill
  2. 2
    Find rise and run — careful, line goes down!
    Rise $= 1 - 4 = {-3}$    (negative, goes down)    Run $= 5 - 1 = 4$
    Rise is negative because $y$ decreases as $x$ increases.
  3. 3
    Calculate gradient
    $$\text{Gradient}=\frac{-3}{4}=-0.75$$
    The negative sign confirms the line slopes downhill from left to right.
AnswerGradient $= -\dfrac{3}{4} = -0.75$
Copy Into Your Books

Formula

$$\text{Gradient}=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$$

4-Step Method

  1. Choose two grid-intersection points
  2. Draw a right-angled triangle
  3. Count rise (vertical) & run (horizontal)
  4. Write rise/run and simplify

Remember

  • Rise is positive uphill, negative downhill
  • Run is always left to right (positive)
  • Any two points on a straight line give the same gradient
  • Pick far-apart points for best accuracy

Brain Trainer · 10 problems

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Brain Trainer · Rise and Run
10 problems

Calculate gradient = rise ÷ run. Simplify all fractions!

  1. 1 Rise = 3, Run = 6. Gradient = ?

    $\dfrac{1}{2}$

  2. 2 Rise = 4, Run = 2. Gradient = ?

    $2$

  3. 3 Rise = 5, Run = 5. Gradient = ?

    $1$

  4. 4 Rise = $-6$, Run = 3. Gradient = ?

    $-2$

  5. 5 Rise = $-2$, Run = 8. Gradient = ?

    $-\dfrac{1}{4}$

  6. 6 Rise = 9, Run = 6. Gradient = ?

    $\dfrac{3}{2}$

  7. 7 Rise = $-8$, Run = 4. Gradient = ?

    $-2$

  8. 8 Rise = 7, Run = 14. Gradient = ?

    $\dfrac{1}{2}$

  9. 9 Rise = 0, Run = 5. Gradient = ?

    $0$ (horizontal line)

  10. 10 Rise = 6, Run = 9. Gradient = ?

    $\dfrac{2}{3}$

Quick Check · 5 questions

1
What is the gradient of a line that rises 4 units and runs 2 units?
+10 XP
run=2 rise=4
2
A line goes down 3 units and across 6 units. What is its gradient?
+10 XP
run=6 down 3
3
Which gradient triangle gives the correct gradient for the line through $(0,0)$ and $(3,2)$?
+10 XP
run=2 rise=3

A

run=3 rise=2

B

4
What is the gradient of a line through $(0, 0)$ and $(4, 6)$?
+10 XP
0 1 2 3 4 1 2 3 4 5 (0,0) (4,6)
5
Estimate the gradient of this line.
+10 XP
run = 5 rise ~ 2.7

Show Your Working · 3 questions

Show Your Working
8 marks total
Apply Medium 3 MARKS

Q6. Find the gradient of the line through $(0, 1)$ and $(4, 4)$ shown on the grid.

0 1 2 3 4 5 6 1 2 3 4
Answer in your workbook.
Apply Medium 3 MARKS

Q7. A line passes through $(2, 3)$ and $(6, 9)$. Draw a gradient triangle and find the gradient.

0 1 2 3 4 5 6 7 1 2 3 4 5 6 (2,3) (6,9) run=4 rise=6
Answer in your workbook.
Reason Hard 2 MARKS

Q8. Explain why choosing two points far apart gives a more accurate gradient than two points close together.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — Gradient = rise/run = 4/2 = 2.

2. B — Rise = −3 (downhill), Run = 6. Gradient = −3/6 = −1/2.

3. B — For (0,0) to (3,2): run = 3 (horizontal), rise = 2 (vertical). Option A swaps them.

4. B — Rise = 6, Run = 4. Gradient = 6/4 = 3/2. Simplify by HCF = 2.

5. C — Rise ≈ 2.7, Run = 5. Gradient ≈ 0.54, closest to 3/5 = 0.6.

Show Your Working Model Answers

Q6 (3 marks): Choose (0,1) and (4,4) [1]. Rise = 4−1 = 3, Run = 4−0 = 4 [1]. Gradient = 3/4 = 0.75 [1].

Q7 (3 marks): Rise = 9−3 = 6, Run = 6−2 = 4 [1]. Draw triangle correctly [1]. Gradient = 6/4 = 3/2 = 1.5 [1].

Q8 (2 marks): If we miscount by 0.5 squares and the run is 1, the error is 0.5/1 = 50% [1]. If the run is 6, the same error is only 0.5/6 ≈ 8%. Far-apart points reduce the relative effect of any counting mistake [1].

Stretch Challenge · +25 XP, +10 coins

Gradient Puzzles

  1. A line has gradient $\dfrac{2}{3}$ and passes through $(3, 1)$. Use the gradient triangle to find another point on the line.
  2. Two lines on the same grid have gradients $\dfrac{3}{4}$ and $\dfrac{4}{5}$. Which line is steeper? Explain using decimal equivalents.
  3. A line passes through $(1, 2)$ and $(4, -1)$. Find its gradient and explain what the negative value means in terms of the graph's direction.
Reveal solutions

1. From $(3,1)$: rise=2, run=3 → another point is $(6, 3)$. Going backwards: $(0, -1)$.

2. $\frac{3}{4}=0.75$ and $\frac{4}{5}=0.8$. Since $0.8>0.75$, the line with gradient $\frac{4}{5}$ is steeper.

3. Rise $= -1-2=-3$, run $= 4-1=3$. Gradient $= \frac{-3}{3}=-1$. The negative means the line slopes downhill from left to right (for every unit right, it drops 1 unit).

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Quick Review

Formula

Gradient = rise ÷ run = Δy ÷ Δx

4-Step Method

Pick grid points → draw triangle → count rise & run → simplify

Positive gradient

Rise is positive — line goes uphill left to right

Negative gradient

Rise is negative — line goes downhill left to right

Simplify

Always divide by HCF: 6/4 = 3/2, not 6/4

Far-apart points

Larger run means counting errors matter less

DAILY CHALLENGE · TODAY
The Gradient Sprint
Draw 5 different gradient triangles in 2 minutes and correctly calculate all 5 gradients for double coins!
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Lesson 9 — Finding Gradient from Two Points
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Unit overview · Maths subject page