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Improve Tuition · Foundation Maths · Year 6–10
Algebra · Multiplying

Algebra: Multiplying

Multiplying terms, using powers, and expanding brackets — built up one small step at a time.

6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure

Multiplying in algebra means combining terms by multiplying their numbers and their letters together, and writing a power whenever a letter is multiplied by itself (so x × x = x2). Pupils most often slip in two places: treating a power like a multiple — writing h3 as 3h — and, when expanding a bracket, multiplying only the first term instead of every term inside.

At Improve Tuition, a qualified teacher finds exactly where a pupil’s method breaks down and rebuilds it with small, worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.

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Lesson 1

Multiplying letters

Aim: Write multiplied letters and numbers as one tidy term.

In algebra the multiplication sign is usually left out. So 4 × m is written 4m, and a × b × c is written abc. The number goes at the front and the letters follow; their order does not change the value (ab is the same as ba).

a×b×c→abc
Drop the times signs and write the letters together: a × b × c = abc.
Worked example
Write 5 × p × q as a single term.
  1. The multiplication signs are left out when we write the term.
  2. Put the number at the front: 5.
  3. Then write the letters next to it: p and q become pq.
  4. So 5 × p × q = 5pq.
  5. Check: 5pq still means 5 × p × q. ✓
Watch out
The number goes first, then the letters: 5pq, not p5q.

Try one

Write a × 3 × b as a single term.

Show answer
Collect the number to the front: 3.
Write the letters after it: ab.
So a × 3 × b = 3ab.
Check: 3ab means 3 × a × b. ✓
Practise this now — Section A →
Lesson 2

Powers

Aim: Use a power to show a letter multiplied by itself.

A power counts how many times a letter is multiplied by itself. So h × h × h is written h3 (‘h to the power 3’). Be careful: h3 means three h’s multiplied — that is not the same as 3h, which means three h’s added.

h×h×h→h3three h’s multiplied — not 3h
A power counts how many are multiplied: h × h × h = h3, not 3h.
Worked example
Write d × d × d × d as a power.
  1. Count how many d’s are multiplied together: four.
  2. Write the letter once with the count as a power.
  3. So d × d × d × d = d4.
  4. Check: the power 4 tells you four d’s are multiplied. ✓
Watch out
A power means multiply, not add: d4 is not 4d.

Try one

Write y × y as a power.

Show answer
Two y’s are multiplied together.
Write y with a power of 2.
So y × y = y2.
Check: y2 means y × y, said ‘y squared’. ✓
Practise this now — Section B →
Lesson 3

Multiplying terms

Aim: Multiply terms by multiplying the numbers and the letters separately.

To multiply two terms, multiply the numbers together and the letters together. For 3a × 2b the numbers give 3 × 2 = 6 and the letters give a × b = ab, so the answer is 6ab. When the same letter is multiplied by itself you get a power: x × x = x2.

6x23x2xnumbers: 2×3 = 6; letters: x×x = x2
Multiply the numbers and the letters: 2x × 3x = 6x2 (the x’s make a power).
Worked example
Multiply 4r × 3s.
  1. Multiply the numbers: 4 × 3 = 12.
  2. Multiply the letters: r × s = rs.
  3. Put them together.
  4. So 4r × 3s = 12rs.
  5. Check: 12rs means 12 × r × s. ✓
Watch out
The same letter multiplied becomes a power: 2x × 3x = 6x2, not 6x.

Try one

Multiply 2x × 5x.

Show answer
Numbers: 2 × 5 = 10.
Letters: x × x = x2.
So 2x × 5x = 10x2.
Check: the same letter multiplied gives a power. ✓
Practise this now — Section C →
Lesson 4

Expanding a bracket

Aim: Expand a bracket by multiplying everything inside by the number outside.

A number outside a bracket multiplies every term inside it — this is called expanding. For 2(x + 3), multiply the 2 by the x and by the 3: 2 × x = 2x and 2 × 3 = 6, giving 2x + 6.

2x6x322(x + 3) = 2x + 6
The number outside multiplies every term inside: 2(x + 3) = 2x + 6.
Worked example
Expand 4(a + 5).
  1. Multiply the 4 by the first term: 4 × a = 4a.
  2. Multiply the 4 by the second term: 4 × 5 = 20.
  3. Write the results with the + between them.
  4. So 4(a + 5) = 4a + 20.
  5. Check: every term inside was multiplied by 4. ✓
Watch out
Multiply both terms, not just the first: 4(a + 5) is 4a + 20, not 4a + 5.

Try one

Expand 3(2y + 1).

Show answer
Multiply 3 by 2y: 3 × 2y = 6y.
Multiply 3 by 1: 3 × 1 = 3.
So 3(2y + 1) = 6y + 3.
Check: both terms inside were multiplied. ✓
Practise this now — Section D →
Lesson 5

A letter outside the bracket

Aim: Expand a bracket that has a letter outside, using powers where a letter meets itself.

When a letter sits outside the bracket it still multiplies every term inside, and when it multiplies itself you get a power. For x(x + 4): x × x = x2 and x × 4 = 4x, giving x2 + 4x.

x24xx4xx(x + 4) = x2 + 4x
A letter outside multiplies each term too, and x × x makes x2: x(x + 4) = x2 + 4x.
Worked example
Expand p(p + 6).
  1. Multiply p by the first term: p × p = p2.
  2. Multiply p by the second term: p × 6 = 6p.
  3. Write the results together.
  4. So p(p + 6) = p2 + 6p.
  5. Check: the letter multiplied itself to give a power, and the number gave 6p. ✓
Watch out
A letter times itself is a power: x × x = x2, not 2x.

Try one

Expand t(t − 3).

Show answer
Multiply t by t: t × t = t2.
Multiply t by −3: t × (−3) = −3t.
So t(t − 3) = t2 − 3t.
Check: the minus stays with the 3t. ✓
Practise this now — Section E →
Lesson 6

Expand and simplify

Aim: Expand more than one bracket, then collect like terms.

Some questions have two brackets. Expand each one separately, then collect the like terms to tidy the answer. Keep each term’s sign as you go — take special care when a bracket is being subtracted.

2x + 6+4x + 4=6x + 10expand each, then collect like terms
Expand both brackets, then collect like terms: 2(x + 3) + 4(x + 1) = 6x + 10.
Worked example
Expand and simplify 2(x + 3) + 4(x + 1).
  1. Expand the first bracket: 2(x + 3) = 2x + 6.
  2. Expand the second bracket: 4(x + 1) = 4x + 4.
  3. Collect like terms: 2x + 4x = 6x, and 6 + 4 = 10.
  4. So the answer is 6x + 10.
  5. Check: both brackets were expanded, then the like terms combined. ✓
Watch out
Expand first, then collect. Mind the signs when a bracket is subtracted.

Try one

Expand and simplify x(x + 2) + x(x + 4).

Show answer
Expand each: x(x + 2) = x2 + 2x, and x(x + 4) = x2 + 4x.
Collect: x2 + x2 = 2x2, and 2x + 4x = 6x.
So the answer is 2x2 + 6x.
Check: the x2’s and the x’s were each combined. ✓
Practise this now — Section F →
↑ Back to the lessons
Section A · Multiplying letters
LEARNING OBJECTIVE · GRADE 2 · CALCULATION

Write multiplied letters and numbers as one term.

Success criteria — I can:
  • leave out the times signs
  • number first, then letters
  • order of letters does not matter
1.Write 6 × k as a single term.
Answer:
2.Write a × b × c as a single term.
Answer:
3.Write 4 × p × q as a single term.
Answer:
4.Write m × 7 as a single term.
Answer:
5.Write 2 × x × 5 × y as a single term.
Answer:
↑ Back to the lessons
Section B · Powers
LEARNING OBJECTIVE · GRADE 2 · CALCULATION

Use powers for a letter multiplied by itself.

Success criteria — I can:
  • count how many are multiplied
  • write the letter with a power
  • a power means multiply, not add
6.Write t × t × t as a power.
Answer:
7.Write y × y as a power.
Answer:
8.Simplify k × k × k × k × k.
Answer:
9.Write 5 × a × a as a single term.
Answer:
10.Write 2 × m × m × m as a single term.
Answer:
↑ Back to the lessons
Section C · Multiplying terms
LEARNING OBJECTIVE · GRADE 3 · CALCULATION

Multiply terms together.

Success criteria — I can:
  • multiply the numbers
  • multiply the letters
  • same letter gives a power
11.Multiply 3a × 2b.
Answer:
12.Multiply 4p × 5.
Answer:
13.Multiply 2x × 3x.
Answer:
14.Multiply 7r × 2s.
Answer:
15.Multiply 3m × 3m.
Answer:
↑ Back to the lessons
Section D · Expanding brackets
LEARNING OBJECTIVE · GRADE 3 · CALCULATION

Expand a bracket with a number outside.

Success criteria — I can:
  • multiply every term inside
  • keep the signs
  • one term in, one term out
16.Expand 2(x + 3).
Answer:
17.Expand 5(a + 2).
Answer:
18.Expand 3(2y + 4).
Answer:
19.Expand 4(p − 1).
Answer:
20.Expand 6(2c + 5).
Answer:
↑ Back to the lessons
Section E · Areas and powers
LEARNING OBJECTIVE · GRADE 2 · WORD PROBLEM

Form expressions for areas using multiplication.

Success criteria — I can:
  • area = length × width
  • a side times itself is a power
  • simplify the result
21.A square has sides of length s. Write an expression for its area.
Answer:
22.A rectangle is 4 cm wide and x cm long. Write an expression for its area.
Answer:
23.One face of a cube has edges of length a. Write an expression for the area of that face.
Answer:
24.A rug is w metres long and 5 metres wide. Write an expression for its area.
Answer:
25.A square tile has sides of length 2k. Write an expression for its area, simplified.
Answer:
↑ Back to the lessons
Section F · Expanding in context
LEARNING OBJECTIVE · GRADE 3 · WORD PROBLEM

Form and expand a bracket from a situation.

Success criteria — I can:
  • write the bracket first
  • multiply by the number outside
  • expand every term
26.A bag holds (x + 2) sweets. Write an expression for the number of sweets in 3 bags, expanded.
Answer:
27.A ticket costs £(y + 4). Write the cost of 5 tickets, expanded.
Answer:
28.A rectangle has width 4 and length (x + 6). Write its area, expanded.
Answer:
29.Each box holds (2n + 1) pencils. Write the total number of pencils in 3 boxes, expanded.
Answer:
30.A shape’s perimeter is 2(p + 3). Write it expanded.
Answer:
↑ Back to the lessons
Section G · Expand and simplify
LEARNING OBJECTIVE · GRADE 3 · WORD PROBLEM

Expand and then collect like terms.

Success criteria — I can:
  • expand each bracket
  • keep the signs
  • collect like terms
31.A rectangle has width x and length (x + 5). Write its area, expanded.
Answer:
32.A rectangle has width t and length (t − 2). Write its area, expanded.
Answer:
33.Expand and simplify x(x + 3) + x(x + 1).
Answer:
34.Expand and simplify 2(x + 4) + 3(x + 1).
Answer:
35.Expand and simplify x(x + 4) − 2(x + 1).
Answer:
0 of 35 answered
Guide

What does multiplying algebra mean?

Multiplying in algebra means combining terms by multiplying their numbers and their letters together, and writing a power whenever a letter is multiplied by itself. So 3a × 2b = 6ab, and x × x = x2.

Expanding a bracket uses the same idea: whatever sits outside multiplies every term inside. For 2(x + 3) that gives 2x + 6, and for x(x + 4) it gives x2 + 4x. When two brackets are expanded, collect any like terms to finish.

Common mistakes. Treating a power like a multiple (writing h3 as 3h), multiplying only the first term inside a bracket, or losing a sign when a bracket is subtracted.

Common questions
How do you multiply algebraic terms?
Multiply the number parts together and the letter parts together. For example, 4p × 3q = 12pq, and 2x × 5x = 10x2.
What does expanding brackets mean?
It means multiplying every term inside the bracket by whatever is outside it. For example, 3(x + 2) expands to 3x + 6.
Why is h3 not the same as 3h?
h3 means h × h × h, three h’s multiplied, while 3h means h + h + h, three h’s added. They are usually different values.
How do you multiply a letter by itself?
Use a power to show how many are multiplied: x × x = x2, and x × x × x = x3.
How do you expand and simplify?
Expand each bracket first, then collect like terms. For example, 2(x + 3) + 4(x + 1) = 2x + 6 + 4x + 4 = 6x + 10.
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