Simplifying, expanding and factorising
Algebra is not a new kind of arithmetic — it is the same arithmetic with letters standing in for numbers you do not know yet. Expanding and factorising are the same move in opposite directions.
Lesson overview
What you'll learn in this lesson
Simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket and factorising into a single bracket
Key learning points
- • Collecting like terms
- • Expanding a bracket
- • Factorising into one bracket
- • Checking your own work
This lesson at a glance
- 30 minutes
- 18 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 18
Watch first
Simplifying, expanding and factorising
An English teaching video on this topic from CAST. Watch it, then scroll down for the lesson.
In a nutshell
Simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket and factorising into a single bracket
What you already know
Part 2 of 18
Before we start
What you already know
You can substitute numbers into expressions and use the order of operations.
Key words
Part 3 of 18
Key words
Words you'll need today
- term
- A single part of an expression, such as 5x or −3.
- like terms
- Terms with exactly the same letters and powers.
- factorise
- Write an expression as a product, taking a common factor outside a bracket.
Learning cycle
Part 4 of 18
Learning cycle 1 of 2
Part 1 — Simplify and expand
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 5 of 18
Learn
Collecting like terms
5x + 3y − 2x + y = 3x + 4y. Only x terms combine with x terms. Watch signs: the minus belongs to the 2x. Note that 4x and 4x² are not like terms, because x and x² are different quantities — 4x + 4x² cannot be simplified at all.
Explore the idea
Part 6 of 18
Learn
Expanding a bracket
3(2x − 5) means every term inside is multiplied by 3: 6x − 15. With a negative outside, −2(x − 4) = −2x + 8; the two minuses give a plus. Expand first, then collect: 4(x + 2) − 3(x − 1) = 4x + 8 − 3x + 3 = x + 11.
Quick check
Part 7 of 18
Quick check
Simplify 7a − 3b − 2a + 5b.
Reset break
Part 8 of 18
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.
Quick check
Part 9 of 18
Quick check
Expand −4(2x − 3).
Learning cycle
Part 10 of 18
Learning cycle 2 of 2
Part 2 — Factorise
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 11 of 18
Learn
Factorising into one bracket
Factorising asks: what is the biggest thing every term shares? For 12x + 18, both share 6, giving 6(2x + 3). For 5x² − 15x, both share 5x, giving 5x(x − 3). Always take the highest common factor — 3(4x + 6) is not fully factorised.
Explore the idea
Part 12 of 18
Learn
Checking your own work
Every factorisation can be checked by expanding it again, and every simplification can be checked by substituting a number. Put x = 2 into both 4(x + 2) − 3(x − 1) and x + 11: both give 13, so the simplification is right. This trick catches sign errors instantly.
Quick check
Part 13 of 18
Quick check
Factorise 12y + 20.
Quick check
Part 14 of 18
Quick check
Factorise 6x² − 9x.
Common mix-ups
Part 15 of 18
Common mix-ups
Things people often get wrong
- Adding 4x and 4x² as if they were like terms.
- Multiplying only the first term inside a bracket.
- Stopping at a common factor that is not the highest one.
Challenge round
Part 16 of 18
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
Mastery quiz
Part 17 of 18
Marked quiz
End of lesson quiz: Simplifying, expanding and factorising
6 questions, marked with the reasoning shown. No timer.
1. Simplify 3x + 4x².
2. Expand 5(3n + 2).
3. Expand and simplify 2(x + 5) + 3(x − 1).
4. Factorise 15a − 25.
5. Best way to check a factorisation?
6. Simplify 4p × 3p.
Lesson round-up
Part 18 of 18
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 LunaDo 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.
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 outDo 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
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 18 · Watch & discover