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SavannahYear 8
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MathsAlgebraic reasoning30 min★★★ difficulty

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

Words to know

simplifymanipulatealgebraicexpressionscollecting

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Expressions: collecting, expanding and factorising

An English teaching video on this topic from mathantics. 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

2

What you already know

Part 2 of 19

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.

3

Key words

Part 3 of 19

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.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Collecting and expanding

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

5

Explore the idea

Part 5 of 19

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.

6

Explore the idea

Part 6 of 19

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.

7

Quick check

Part 7 of 19

Quick check

Simplify 4a + 7a − 3a².

8

Quick check

Part 8 of 19

Quick check

Expand −2(x − 6).

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

Factorising and checking

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

11

Explore the idea

Part 11 of 19

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'.

12

Explore the idea

Part 12 of 19

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.

13

Quick check

Part 13 of 19

Quick check

Factorise fully 12x² − 18x.

14

Quick check

Part 14 of 19

Quick check

Which is a valid check of a simplification?

15

Explore the idea

Part 15 of 19

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.

16

Common mix-ups

Part 16 of 19

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.
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: Expressions: collecting, expanding and factorising

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

  1. 1. Simplify 5a + 3b − 2a.

  2. 2. Expand 4(2x − 3).

  3. 3. Factorise 6x + 9.

  4. 4. Which pair are like terms?

  5. 5. Simplify 3(x + 2) + 2x.

  6. 6. The quickest way to test whether two expressions are equivalent is to...

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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    Simplify and manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket and factorising into a single bracket

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