Expressions: collecting, expanding and factorising
Algebra is a language for structure. Once you can see an expression as a small number of building blocks, manipulating it becomes reading rather than guessing.
Part of your national curriculum
- Algebra: Simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket and factorising into a single bracket
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
- • Like terms
- • Expanding brackets
- • Factorising into a single bracket
- • Checking equivalence
This lesson at a glance
- 30 minutes
- 25 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 25
Visual introductionPicture this
Expressions: collecting, expanding and factorising
Algebra is a language for structure. Once you can see an expression as a small number of building blocks, manipulating it becomes reading rather than guessing.
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 25
Before we start
What you already know
You should already be confident with negative numbers, times tables and substituting a value into a simple expression.
Key words
Part 3 of 25
Key words
Words you'll need today
- Term
- A single part of an expression, such as 3x or -5.
- Like terms
- Terms with exactly the same letters and powers.
- Expand
- Multiply out a bracket.
- Factorise
- Put an expression back inside a bracket.
- Equivalent
- Two expressions that give the same value for every input.
Reset break
Part 4 of 25
Pause
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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 5 of 25
Learning cycle 1 of 2
Collecting and expanding
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 6 of 25
Learn
Like terms
Terms combine only when their variable parts are identical: 4a + 7a = 11a, but 4a and 4a² are not like terms because a² means a × a. Careful students underline matching terms before combining, and keep each sign attached to the term that follows it.
Explore the idea
Part 7 of 25
Learn
Expanding brackets
Multiply the term outside by every term inside: 4(3x − 5) = 12x − 20, and −2(x − 6) = −2x + 12. The second is where marks are lost: multiplying two negatives gives a positive, so the constant becomes +12.
Reset break
Part 8 of 25
Pause
That's sitting 2 of 6 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 25
Quick check
Part 10 of 25
Learning cycle
Part 11 of 25
Learning cycle 2 of 2
Factorising and checking
A short piece of teaching, then a check to make sure it has landed.
Reset break
Part 12 of 25
Pause
That's sitting 3 of 6 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 13 of 25
Learn
Factorising into a single bracket
Factorising reverses expansion by taking out the highest common factor of every term, including letters. For 12x² − 18x the HCF is 6x, giving 6x(2x − 3). Taking out only 6 or only x is partial factorising and loses marks in GCSE questions that say 'factorise fully'.
Explore the idea
Part 14 of 25
Learn
Checking equivalence
Substitute a value such as x = 2 into the original and the simplified expression. If both give the same number, the manipulation is very likely correct; if not, an error is certain. This check takes ten seconds and catches most sign slips.
Quick check
Part 15 of 25
Reset break
Part 16 of 25
Pause
That's sitting 4 of 6 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 17 of 25
Explore the idea
Part 18 of 25
Worked example
Worked answer: expand and simplify 4(3x − 5) − 2(x − 6) (3 marks)
Expanding the first bracket gives 12x − 20 (1). Expanding the second gives −2x + 12, since −2 × −6 = +12 (1). Collecting like terms gives 10x − 8 (1). A common error is writing −2x − 12, which comes from ignoring the sign of the second term.
Common mix-ups
Part 19 of 25
Common mix-ups
Things people often get wrong
- People often think 3x and 3x² are like terms. The powers differ, so they cannot be collected.
- People often expand 2(x + 5) as 2x + 5. Everything inside the bracket is multiplied, giving 2x + 10.
Reset break
Part 20 of 25
Pause
That's sitting 5 of 6 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 21 of 25
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 22 of 25
Game · Recall cards
Simplify 5x+2x−3.
Card 1 of 4
Exit quiz
Part 23 of 25
Exit quiz
Show what you've learned
6 questions, marked together at the end. Nothing is timed.
1. Simplify 5a + 3b − 2a.
2. Expand 4(2x − 3).
3. Factorise 6x + 9.
4. Which pair are like terms?
5. Simplify 3(x + 2) + 2x.
6. The quickest way to test whether two expressions are equivalent is to...
Mastery quiz
Part 24 of 25
Marked quiz
End of lesson quiz: Expressions: collecting, expanding and factorising
4 questions, marked with the reasoning shown. No timer.
1. Simplify 5x+2x−3.
2. Expand 3(2x−5).
3. Factorise 8x+12.
4. Simplify 4x + 3 + 2x
Lesson round-up
Part 25 of 25
Lesson round-up
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Simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket and factorising into a single bracket