Good Learning Co.
SavannahYear 8
← Maths
MathsAlgebraic reasoning30 min★★★ difficulty

Straight-line graphs and what they mean

A straight-line graph turns a rule into a picture. Gradient and intercept are not just numbers on a page: in a real context they are a rate and a starting value.

Part of your national curriculum
  • Algebra: Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions

Lesson overview

What you'll learn in this lesson

Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions

Key learning points

  • • y = mx + c
  • • Finding gradient from two points
  • • Parallel and negative gradients
  • • Reading real meaning

This lesson at a glance

  • 30 minutes
  • 19 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

coordinatesquadrantssketchgraphslinear

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Straight-line graphs and what they mean

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions

2

What you already know

Part 2 of 19

Before we start

What you already know

You should already be able to plot coordinates in all four quadrants and substitute values into an expression.

Recap: Expressions: collecting, expanding and factorising
3

Key words

Part 3 of 19

Key words

Words you'll need today

Gradient
How steep a line is: change in y divided by change in x.
Intercept
Where the line crosses the y-axis.
Linear
Making a straight line when plotted.
Parallel
Lines with the same gradient that never meet.
Rate of change
What the gradient means in a real context.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Reading the equation

A short piece of teaching, then a check to make sure it has landed.

5

Explore the idea

Part 5 of 19

Learn

y = mx + c

In y = mx + c the gradient m tells you how much y changes for each increase of one in x, and c is where the line crosses the y-axis. For y = 3x − 4 the line rises 3 for every 1 across, and crosses the y-axis at (0, −4).

6

Explore the idea

Part 6 of 19

Learn

Finding gradient from two points

Gradient is the change in y divided by the change in x. Through (1, 5) and (4, 14) the gradient is (14 − 5) ÷ (4 − 1) = 3. Substituting one point back gives 5 = 3(1) + c, so c = 2 and the equation is y = 3x + 2.

7

Quick check

Part 7 of 19

Quick check

The gradient of y = −2x + 7 is

8

Quick check

Part 8 of 19

Quick check

Which point lies in the third quadrant?

9

Reset break

Part 9 of 19

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
10

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Comparing lines and meaning

A short piece of teaching, then a check to make sure it has landed.

11

Explore the idea

Part 11 of 19

Learn

Parallel and negative gradients

Parallel lines have equal gradients and different intercepts. A negative gradient means y falls as x rises, so y = −2x + 7 slopes downwards. A horizontal line has gradient zero and equation y = a constant; a vertical line has equation x = a constant and no defined gradient.

12

Explore the idea

Part 12 of 19

Learn

Reading real meaning

For a taxi fare F = 2.5d + 3, the gradient 2.5 is the cost per mile and the intercept 3 is the fixed charge before travelling. Answering 'what does the gradient represent' with a number alone gains nothing: it must be described in the units of the context.

13

Quick check

Part 13 of 19

Quick check

A line parallel to y = 4x − 1 could be

14

Quick check

Part 14 of 19

Quick check

For fare F = 2.5d + 3, the 3 represents

15

Explore the idea

Part 15 of 19

Worked example

Worked answer: find the equation of the line through (1, 5) and (4, 14) (3 marks)

The gradient is (14 − 5)/(4 − 1) = 9/3 = 3 (1). Substituting (1, 5) into y = 3x + c gives 5 = 3 + c, so c = 2 (1). The equation is y = 3x + 2, which checks against the second point: 3(4) + 2 = 14 (1).

16

Common mix-ups

Part 16 of 19

Common mix-ups

Things people often get wrong

  • People often read the gradient as just the change in y. It is the change in y for every one step in x.
  • People often think a negative gradient means the line is below the axis. It means the line falls from left to right.
17

Challenge round

Part 17 of 19

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Straight-line graphs and what they mean

6 questions, marked with the reasoning shown. No timer.

  1. 1. In y = 3x − 2, the gradient is...

  2. 2. In y = 3x − 2, the line crosses the y-axis at...

  3. 3. A line through (1, 2) and (3, 8) has gradient...

  4. 4. Which line is parallel to y = 2x + 1?

  5. 5. A negative gradient means the line...

  6. 6. On a distance-time graph, the gradient represents...

19

Lesson round-up

Part 19 of 19

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 19 · Watch & discover

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    Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions

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