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
- 20 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 20
Visual introductionPicture this
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.
In a nutshell
Manipulate algebraic expressions, solve equations and interpret graphs of functions.
Learning cycle
Part 2 of 20
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 20
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.
Reset break
Part 4 of 20
Pause
That's sitting 1 of 5 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.
Explore the idea
Part 5 of 20
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 6 of 20
Quick check
Part 7 of 20
Reset break
Part 8 of 20
Pause
That's sitting 2 of 5 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.
Learning cycle
Part 9 of 20
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 10 of 20
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 11 of 20
Reset break
Part 12 of 20
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 20
Explore the idea
Part 14 of 20
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 15 of 20
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
Reset break
Part 16 of 20
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 20
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.
Challenge round
Part 18 of 20
Game · Recall cards
(x + 4)(x − 6) expands to
Card 1 of 4
Mastery quiz
Part 19 of 20
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 20 of 20
Lesson round-up
Ready when you are
Quiz score
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Points this lesson
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Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaPart 1 of 20 · Watch & discover