MathsGCSE maths: algebra and graphs30 min★★★ difficulty

Expanding, factorising and algebraic proof

Algebraic manipulation is the grammar of GCSE maths. Fluent expanding and factorising unlocks equations, graphs, proof and most multi-step problems.

Lesson overview

What you'll learn in this lesson

Manipulate algebraic expressions, solve equations and interpret graphs of functions.

Key learning points

• Expanding brackets
• Factorising
• Algebraic proof

This lesson at a glance

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

Words to know

manipulatealgebraicexpressionsequationsinterpret

Scroll down — the lesson carries on below

Part 1 of 16 · Discover6%

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Part 1 of 16

Watch first

Expanding, factorising and algebraic proof

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

In a nutshell

Manipulate algebraic expressions, solve equations and interpret graphs of functions.

Learning cycle

Part 2 of 16

Learning cycle 1 of 2

Part 1 · Expanding brackets

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

Explore the idea

Part 3 of 16

Learn

Expanding brackets

Multiply every term inside by every term outside. For double brackets, (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15. For triple brackets, expand two first, then multiply the result term by term, keeping the working in neat columns to avoid losing signs.

Explore the idea

Part 4 of 16

Learn

Factorising

Always look for a common factor first. For a quadratic x² + bx + c, find two numbers that multiply to c and add to b. Recognise the difference of two squares: x² − 49 = (x + 7)(x − 7). When the coefficient of x² is not 1, split the middle term and factorise in pairs.

Quick check

Part 5 of 16

Quick check

(x + 4)(x − 6) expands to

Quick check

Part 6 of 16

Quick check

x² − 81 factorises to

Reset break

Part 7 of 16

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

Learning cycle

Part 8 of 16

Learning cycle 2 of 2

Part 2 · Algebraic proof

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

Explore the idea

Part 9 of 16

Learn

Algebraic proof

Use algebra to represent the general case: 2n is any even number, 2n + 1 any odd number, and consecutive integers are n, n + 1, n + 2. Manipulate to a form that makes the conclusion unavoidable, then state it. Testing a few examples is verification, not proof, and scores nothing.

Quick check

Part 10 of 16

Quick check

Any odd number can be written as

Quick check

Part 11 of 16

Quick check

Factorising x² + 7x + 12 gives

Explore the idea

Part 12 of 16

Worked example

Worked answer: prove that the sum of three consecutive integers is always a multiple of 3 (3 marks)

Let the integers be n, n + 1 and n + 2, where n is any integer (1). Their sum is n + (n + 1) + (n + 2) = 3n + 3 (1). This factorises to 3(n + 1), which is 3 multiplied by an integer, so the sum is always a multiple of 3 (1). Showing that 4 + 5 + 6 = 15 would not be a proof.

Challenge round

Part 13 of 16

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

Challenge round

Part 14 of 16

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

____ to a form that makes the conclusion unavoidable, then state it.

Mastery quiz

Part 15 of 16

Marked quiz

End of lesson quiz: Expanding, factorising and algebraic proof

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

1. (x + 4)(x − 6) expands to
2. x² − 81 factorises to
3. Any odd number can be written as
4. Factorising x² + 7x + 12 gives

Lesson round-up

Part 16 of 16

Lesson round-up

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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

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
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

Write your answer, notes or working here

Saved as you type.

Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 16 · Watch & discover

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