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KS3 Maths · Year 7Chapter 4 · Algebraic notation and substitution
Year 7 · Chapter 4

Algebraic notation and substitution

By the end of this chapter you will be able to

  • Write algebra using the shorter notation: leave out the times sign, put the number first, and use powers for repeated letters
  • Write fractions, brackets and division with letters in the correct form
  • Substitute positive numbers into expressions and formulae, using brackets where needed
  • Substitute negative numbers into expressions, and keep track of the signs of powers
  • Use the words expression, equation, inequality, formula, term, factor and coefficient correctly

You should already know

  • Multiplication and division facts, and the order of operations (Chapter 1)
  • Using a letter for an unknown number (Year 6, Chapter 4)
  • Fractions and decimals (Chapter 2)

Where this leads

  • Simplifying and expanding expressions (Chapter 5)
  • Solving equations and rearranging formulae (Chapter 6)

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: A1 notation: ab, 3y, a², a³, a²b, a/b, fractions as coefficients and brackets, A2 substitute numbers into formulae and expressions, including scientific formulae, A3 the vocabulary of expressions, equations, inequalities, terms and factors.

Exercise P  Do you remember?

Non-calculator
1
FWork out 3 × 7 + 4.
2
FWork out 20 − 3 × 4.
3
FWork out 5 × (2 + 3).
4
FWhat is 4 squared?
5
FWrite 12 ÷ 4 as a fraction in its simplest form.
4.1

Algebraic notation: multiplying and dividing

Year 7
A1
Video lesson: Algebraic notation: multiplying and dividingComing soon
A cinema charges £t for a ticket. If Priya buys 3 tickets, she pays 3 × t. Mathematicians do not write the times sign when a letter is involved, so we write this as 3t.

In algebra we use letters for numbers we do not yet know. We follow some rules to make the writing short and clear:

Key ideas

  • Leave out the × sign, and put the number first: 4 × y = 4y.
  • Repeated letters are powers: a × a = a2, a × a × a = a3.
  • Division with letters is a fraction: m ÷ n = m/n.
  • Do not write 3t as t3, because t3 looks like t cubed (t3).
Worked example 4.1.1
Write 5 × x × y in a shorter way.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Leave out the times signs. Algebra does not need the × sign between letters.
    5 x y
  2. Put the number first. A number is written in front of the letters.
    5 is already first
  3. Write the answer. The letters are written next to each other.
    5xy

Check. 5 × x × y gives 5xy, with no times signs. ✓

Watch out. Write the number first, then the letters in alphabetical order.

Worked example 4.1.2
Write a × a × b × b × b in a shorter way.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Group the repeated letters. a appears twice and b appears three times.
    a × a = a2, b × b × b = b3
  2. Write them together. Put the powers next to each other.
    a2 × b3
  3. Write the answer. The × sign is not written.
    a2b3

Check. a has power 2 and b has power 3, from the number of times each letter is used. ✓

Watch out. The power belongs to one letter only. a2b3 is not (ab)5.

Worked example 4.1.3
Write (n + 3) ÷ 2 as a fraction.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Write the division as a fraction. A division by 2 is written with 2 on the bottom.
    (n + 3)/2
  2. Check the brackets. The brackets keep n + 3 together, so the whole thing is divided by 2.
    (n + 3)/2 means (n + 3) ÷ 2
  3. Write the answer. The fraction line shows the division.
    (n + 3)/2

Check. Check: with n = 1, (1 + 3) ÷ 2 = 2. ✓

Watch out. Without the brackets, n + 3/2 would mean n + 1.5, which is different.

Watch out

Watch out: 3t is not t3. The number always goes first, so t3 would be read as t cubed. Also, 2 × a × b is written 2ab, not 2a + b.

Exam tip

Grown-up tip. Write a short sum in words first ("3 lots of t") before writing it in letters, so that the pupil sees why the times sign is dropped.

Exercise 4A  Writing algebra with multiplication and division

Non-calculator
1
FWrite 3 × a in a shorter way.
2
FWrite y × 7 in a shorter way.
3
FWrite b × c × 4 in a shorter way.
4
FWrite p × p in a shorter way.
5
FWrite x × x × x in a shorter way.
6
FWrite 2 × m × m × n in a shorter way.
7
FWrite a ÷ 3 as a fraction.
8
FWrite 5 × (x + 1) without the times sign, in the correct form.
9
FWrite each of these in a shorter way: (a) 4 × t (b) t × t × 4 (c) t ÷ 4
(a)(a) 4 × t
(b)(b) t × t × 4
(c)(c) t ÷ 4
10
PSPriya pays p pence for each pen. She buys n pens. Write an expression for the total cost in pence.
11
RWhich of these means the same as 2ab? A: 2 × a + b B: 2 × a × b C: a × b + 2
12
RSam writes 3 × x as x3. Explain why this is not good algebra, and write it correctly.
4.2

Fractions, brackets and powers in algebra

Year 7
A1
Video lesson: Fractions, brackets and powers in algebraComing soon
A rectangle has a length of 2a and a width of b. Its area is found by multiplying the length and width, and written in algebra it is 2ab. A pizza shared between n people is written as a fraction: each person gets 1/n of a pizza.

Algebraic fractions are written with a fraction line. A number on the top of a letter is called the coefficient: in 3y/4, the 3 is multiplied by y, and then the answer is divided by 4.

Brackets keep parts of an expression together. A number outside brackets multiplies the whole bracket: 4(x + 2) means 4 × (x + 2).

Key ideas

  • A fraction line is a division: p × 2 ÷ 5 = 2p/5.
  • A number outside brackets multiplies everything inside: 4(x + 2) means 4 × (x + 2).
  • Coefficient: the number multiplied by a letter, such as 3 in 3y.
  • Repeated letters are powers: 2 × a × a × 3 = 6a2.
Worked example 4.2.1
Write (2 × p) ÷ 5 as a fraction.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Write the multiplication. The 2 is multiplied by p.
    2 × p = 2p
  2. Write the division as a fraction. Dividing by 5 puts 5 on the bottom.
    2p ÷ 5 = 2p/5
  3. Write the answer. The fraction line shows the division.
    2p/5

Check. If p = 10, (2 × 10) ÷ 5 = 4, and 2p/5 = 4. ✓

Watch out. Keep the 2 on the top with the p.

Worked example 4.2.2
Write a rectangle of length 2a and width b as an area expression.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Multiply the length and width. Area = length × width.
    2a × b
  2. Write it in the shorter form. Leave out the times sign, and put the number first.
    2ab
  3. Write the answer. The area is 2ab.
    2ab

Check. If a = 3 and b = 2, the area is 2 × 3 × 2 = 12. ✓

Watch out. Do not write 2a + b for an area: that is a sum, not a product.

Worked example 4.2.3
Write 4 × (x + 2) in the correct form.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Keep the brackets. The bracket stays, and the times sign is left out.
    4 × (x + 2) = 4(x + 2)
  2. Check the meaning. The 4 multiplies the whole bracket.
    4(x + 2) = 4x + 8
  3. Write the answer. The brackets are kept.
    4(x + 2)

Check. With x = 1, 4 × (1 + 2) = 12 and 4x + 8 = 12. ✓

Watch out. Writing 4x + 2 would be wrong: the 4 must multiply the whole bracket.

Watch out

Watch out: 4(x + 2) is not 4x + 2. The 4 multiplies both the x and the 2 in the bracket.

Exam tip

Grown-up tip. Use a number to check each expression: put x = 1 into both sides. If the two answers differ, the notation is wrong.

Exercise 4B  Fractions, brackets and powers

Non-calculator
1
FWrite 3 × k × k in a shorter way.
2
FWrite a × 2 × a × 2 × a in a shorter way.
3
FWrite p × 1/2 as a fraction.
4
FWrite (3 × y) ÷ 4 as a fraction.
5
FWrite 4 × (x + 1) in a shorter form, keeping the brackets.
6
FWrite the perimeter of a square with side t as a shorter expression.
7
FWrite 2 × m × m × n in a shorter way.
8
FWrite 5 × p × q × q in a shorter way.
9
FWrite each of these in a shorter form: (a) a × a × a × a (b) b ÷ c (c) 2 × (m − n)
(a)(a) a × a × a × a
(b)(b) b ÷ c
(c)(c) 2 × (m − n)
10
PSCinema tickets cost £c each and drinks cost £d each. Write an expression for the cost of 3 tickets and 2 drinks.
11
RSam says 2(a + 1) is the same as 2a + 1. Is Sam right? Use a = 3 to explain.
12
RExplain why we write 4xy and not 4 × x × y with the times signs.
4.3

Substituting positive numbers

Year 7
A2
Video lesson: Substituting positive numbersComing soon
A plumber charges a call-out fee, then a fixed amount per hour. The total cost can be found by putting the number of hours into a formula. Replacing a letter with a number is called substitution.

To substitute, replace each letter with its value. Use brackets around each value, so that the working is clear: 3x when x = 4 becomes 3 × (4) = 12.

A formula is an equation that links quantities. In science, the formula v = u + at links the final speed v, the starting speed u, the acceleration a and the time t.

Key ideas

  • Substitute: replace each letter with its value, using brackets.
  • Follow BIDMAS: powers first, then × and ÷, then + and −.
  • A formula can be used for many different values of the letters.
  • Check your answer by working out the other way, or by estimating.
Worked example 4.3.1
Find 3x + 2 when x = 4.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    3x + 2
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    3 × 4 + 2
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    3 × 4 = 12; 12 + 2 = 14
  4. Write the answer. Check that it is the value of the expression for the values given.
    14

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Worked example 4.3.2
Find 5ab when a = 2 and b = 3.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    5ab
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    5 × 2 × 3
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    5 × 2 = 10; 10 × 3 = 30
  4. Write the answer. Check that it is the value of the expression for the values given.
    30

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Worked example 4.3.3
Use the formula v = u + at to find v when u = 3, a = 4 and t = 5.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    v = u + at
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    3 + 4 × 5
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    4 × 5 = 20; 3 + 20 = 23
  4. Write the answer. Check that it is the value of the expression for the values given.
    23

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Watch out

Watch out: 2 × 3 + 1 is 7, not 9. Multiplication comes before addition, so work out 2 × 3 first.

Exam tip

Grown-up tip. Ask the pupil to write the value in brackets above each letter before they start, so that they never forget a number.

Exercise 4C  Substituting positive numbers

Non-calculator
1
FFind 2x + 5 when x = 3.
2
FFind 4a − 7 when a = 6.
3
FFind 3y2 when y = 2.
4
FFind 5(p + 3) when p = 1.
5
FFind 2ab when a = 4 and b = 5.
6
FFind x/4 + 1 when x = 12.
7
FThe perimeter of a rectangle is P = 2(l + w). Find P when l = 7 and w = 3.
8
FThe speed formula is s = d/t. Find s when d = 150 and t = 2.5.
9
FA formula for temperature in degrees Fahrenheit is F = 9C/5 + 32. Find F when C = 20.
10
FUse v = u + at to find v when u = 12, a = 2 and t = 6.
11
FFind the value of: (a) x2 + 2x when x = 5 (b) 3(x + y) when x = 2 and y = 7
(a)(a) x2 + 2x when x = 5
(b)(b) 3(x + y) when x = 2 and y = 7
12
RSam says that if n = 3, then 2n2 = 36. Is he right? Explain.
4.4

Substituting negative numbers and powers

Year 7
A2
Video lesson: Substituting negative numbers and powersComing soon
A submarine is 12 m below the surface. Its depth is written as a negative number, −12. When a formula uses a negative value, the signs must be kept carefully.

Put every negative value in brackets when you substitute, so that you can see its sign. For example, 3x when x = −2 is 3 × (−2) = −6.

Be careful with powers. (−4)2 means −4 × −4 = 16, but −42 means −(4 × 4) = −16.

Key ideas

  • Put a negative value in brackets: x = −2 becomes (−2).
  • A negative times a negative is positive: (−2) × (−2) = 4.
  • Only the number directly before a power is squared or cubed: −x2 means −(x × x).
  • Subtracting a negative is the same as adding: a − (−6) = a + 6.
Worked example 4.4.1
Find 3x − 5 when x = −2.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    3x − 5
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    3 × (−2) − 5
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    3 × (−2) = −6; −6 − 5 = −11
  4. Write the answer. Check that it is the value of the expression for the values given.
    −11

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Worked example 4.4.2
Find x2 when x = −4.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    x2
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    (−4)2
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    (−4) × (−4) = 16
  4. Write the answer. Check that it is the value of the expression for the values given.
    16

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Worked example 4.4.3
Find 2a − b when a = −3 and b = −6.

Method. Replace each letter with its value, using brackets. Then work out the answer, following the order of operations.

  1. Write the expression or formula. This is what we need to work out.
    2a − b
  2. Replace each letter with its value. Brackets keep a negative number together, so its sign is not lost.
    2 × (−3) − (−6)
  3. Work out the calculation, in order. Powers first, then multiplication and division, then addition and subtraction.
    2 × (−3) = −6; −6 − (−6) = −6 + 6 = 0
  4. Write the answer. Check that it is the value of the expression for the values given.
    0

Check. Check by replacing the letters again and working out the sum a second time. ✓

Watch out. Put brackets around each value before you multiply or square. Without brackets, −3 squared looks like −(3 squared).

Watch out

Watch out: −x2 is −(x × x), so if x = 5 then −x2 = −25, not 25. Also (−3)2 = 9, but −32 = −9.

Exam tip

Grown-up tip. Ask the pupil to write the value in brackets before they do anything else, and to say "a negative times a negative is positive" out loud.

Exercise 4D  Substituting negative numbers

Non-calculator
1
FFind 4x + 1 when x = −3.
2
FFind 2a − 7 when a = −5.
3
FFind x2 when x = −6.
4
FFind −x2 when x = 5.
5
FFind 3y − y2 when y = −2.
6
FFind a − b when a = −1 and b = −6.
7
FFind 5(x − 4) when x = −2.
8
FFind mn when m = −4 and n = −7.
9
FFind t/x when t = 12 and x = −3.
10
FFind the value of: (a) p3 when p = −2 (b) (−p)3 when p = −2
(a)(a) p3 when p = −2
(b)(b) (−p)3 when p = −2
11
PSFind 2a + 3b when a = −4 and b = 5.
12
RSam says that if x = −3, then x2 = −9. Explain his mistake.
4.5

The words of algebra

Year 7
A3
Video lesson: The words of algebraComing soon
A newspaper says: "The cost is 2x + 5 pounds. If the total is more than 20 pounds, we charge extra." Each part of this sentence has a different name in algebra: an expression, a formula and an inequality.

An expression has no equals sign, such as 3x + 4. An equation has an equals sign, such as 3x + 4 = 10. An inequality uses a sign such as <, > ≤ or ≥, such as 3x + 4 > 10. A formula is an equation that links two or more quantities, such as A = lw.

A term is a part of an expression separated by + or −: in 3ab + 4c − 5 there are three terms. A factor multiplies to make a term: the factors of 3ab are 3, a and b. The coefficient is the number in front of a letter: in 7x − 2y, the coefficient of x is 7.

Key ideas

  • Expression: no equals sign (3x + 4). Equation: has = (3x + 4 = 10). Inequality: has <, >, ≤ or ≥.
  • Formula: an equation that links quantities, such as A = lw.
  • Term: a part separated by + or −. The terms of 3ab + 4c − 5 are 3ab, 4c and −5.
  • Factor: a number or letter multiplied to make a term. Coefficient: the number in front of a letter.
Worked example 4.5.1
Which of these is an equation: 2x + 3, 2x + 3 = 11, 2x + 3 > 11?

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Look for an equals sign. An equation has =.
    2x + 3 has no =; 2x + 3 = 11 has =; 2x + 3 > 11 has >
  2. Check the others. The > sign makes an inequality, and no sign makes an expression.
    2x + 3 is an expression; 2x + 3 > 11 is an inequality
  3. Write the answer. Only one has an equals sign.
    2x + 3 = 11

Check. 2x + 3 = 11 has an equals sign. ✓

Watch out. An equation has an equals sign. An inequality uses < > ≤ or ≥.

Worked example 4.5.2
List the terms of 3ab + 4c − 5.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Find the parts separated by + or −. Each term is a part of the expression between the + and − signs.
    3ab, 4c and −5
  2. Count them. There are three terms.
    3 terms
  3. Write the answer. Keep the sign with the term.
    3ab, 4c and −5

Check. Three terms: 3ab, 4c and −5. ✓

Watch out. The − sign belongs with the 5, so the third term is −5, not 5.

Worked example 4.5.3
Find the factors of 4mn.

Method. Use the shorter notation rules: leave out the times sign, put the number first, and write repeated letters as powers.

  1. Find the numbers and letters multiplied together. A factor multiplies to make the term.
    4 × m × n = 4mn
  2. List the factors. The factors are the numbers and letters that are multiplied.
    4, m and n
  3. Write the answer. Each factor is a separate multiplication.
    4, m and n

Check. 4 × m × n = 4mn. ✓

Watch out. A factor is one of the things multiplied, not a sum.

Watch out

Watch out: x + 2 = 7 is an equation, but x + 2 is only an expression. Do not call every line of algebra an equation.

Exam tip

Grown-up tip. Ask the pupil to point at the = or < sign in each line. The sign tells them which word to use.

Exercise 4E  Expressions, equations, terms and factors

Non-calculator
1
FSay whether each is an (a) expression, (b) equation, (c) inequality or (d) formula: (a) 4x − 1 (b) 4x − 1 = 7 (c) 4x − 1 < 7 (d) A = lw
(a)(a) 4x − 1
(b)(b) 4x − 1 = 7
(c)(c) 4x − 1 < 7
(d)(d) A = lw
2
FWrite the terms of 5a + 3b − 2.
3
FWhat is the coefficient of y in 6y − 4?
4
FWhat is the coefficient of x in −x + 3?
5
FList the factors of 4mn.
6
RWhich is the odd one out: p + q = 7, 3p = 9, 2x < 10, a + b? Explain.
7
PSWrite an expression: multiply the number n by 3, then add 5.
8
FHow many terms are there in 4a + 2b − 7c + 5?
9
RSam says that 3x and x3 mean the same. Explain why we write 3x and not x3.

Chapter summary

SkillMethod
Shorter notationLeave out the times sign and put the number first: 4 × y = 4y
Powers in algebraRepeated letters: a × a = a2, a × a × a = a3
DivisionWrite a division with letters as a fraction: m ÷ n = m/n
BracketsA number outside a bracket multiplies everything inside: 4(x + 2)
SubstitutionReplace each letter with its value in brackets, then use BIDMAS
Negative valuesPut negative values in brackets; a negative times a negative is positive
Expression, equation, inequalityNo sign; has =; has < > ≤ or ≥
Terms, factors, coefficientsTerms are separated by + or −; factors multiply to make a term; the coefficient is the number in front of a letter

Can you…?

  • I can write algebra with the times sign left out, and the number first
  • I can write repeated letters as powers, and divisions as fractions
  • I can substitute positive numbers into expressions and formulae
  • I can substitute negative numbers, using brackets, and handle powers of negative numbers
  • I can use the words expression, equation, inequality and formula correctly
  • I can name the terms, factors and coefficients of an expression

Key terms

expression
a collection of terms with no equals sign, such as 3x + 4
equation
a statement with an equals sign, such as 3x + 4 = 10
inequality
a statement using <, >, ≤ or ≥, such as 3x + 4 > 10
formula
an equation that links quantities, such as A = lw
substitute
replace a letter with a number
term
a part of an expression separated by + or − signs
factor
a number or letter that is multiplied to make a term
coefficient
the number in front of a letter, such as 7 in 7x

Exercise 4Z  Check-up

Non-calculator
1
Find 3x + 7 when x = 4.
2
Find 5 − 2y when y = −3.
3
Find a2 + b when a = −5 and b = 3.
4
Find 4(m − 2n) when m = 7 and n = −1.
5
Find xy/2 when x = 6 and y = 5.
6
Find ab − c when a = 1.5, b = 4 and c = 2.
7
Find t2 − t when t = −3.
8
Write 3 × p × p × q in a shorter way.
9
Write (x + 2) ÷ 3 as a fraction.
10
Find 2x3 when x = 2.

Total 19 marks. Allow about 19 minutes.

Exercise 4X  Practice test questions

Non-calculator
1
FWrite 6 × m × n in a shorter way.[1 mark]
2
FFind 4x − 3 when x = 5.[1 mark]
3
FFind 2a + 5b when a = 3 and b = 4.[1 mark]
4
FFind x2 − 3x when x = 6.[2 marks]
5
FUse v = u + at to find v when u = 4, a = 3 and t = 6.[2 marks]
6
FFind 3(p − 5) when p = −2.[2 marks]
7
FPriya buys n pens at 35p each and one pencil at 20p. Write an expression, in pence, for the total cost.[2 marks]
8
FFind x2 − 2x when x = −4.[2 marks]
9
FThe perimeter of a rectangle is P = 2(l + w). Find P when l = 8.5 and w = 3.5.[3 marks]
10
RExplain why 2(a + 1) is not the same as 2a + 1. Use a = 3 to show it.[3 marks]
← Chapter 3: Standard form, rounding and estimationChapter 5: Simplifying, expanding and factorising →

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