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SATs practice · Year 6Chapter 6 · Algebra and formulae
Year 6 · Chapter 6

Algebra and formulae

By the end of this chapter you will be able to

  • Write a formula in letters from a description in words
  • Substitute values into a formula and work out the answer
  • Describe a linear sequence with a term-to-term rule and find a missing term
  • Find a missing number in an equation with a letter
  • List all the pairs or combinations that fit a rule, in a systematic table

You should already know

  • Order of operations (Chapter 1 of this book)
  • Multiplication facts and missing number sums (Year 5 maths)

Where this leads

  • Measures, conversions and formulae for perimeter and area (Chapter 8 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Use simple formulae written in words and in letters, and substitute values into them (Year 5-6), Generate and describe linear number sequences, including the term-to-term rule in words (Year 5-6), Express missing number problems algebraically, for example a + b = 10 (Year 5-6), Find pairs of numbers that satisfy an equation with two unknowns (Year 5-6), Enumerate the possibilities of combinations of two variables in a systematic table (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWork out 3 × 4 + 2.
2
FWhat number makes 7 + ☐ = 12?
3
FContinue the sequence: 2, 4, 6, ... What is the next number?
4
FWork out 10 - 3 × 2.
5
FWhat is 5 × 6?
6.1

Formulae in words and letters

Year 6
Use simple formulae written in words and in letters, and substitute values into them (Year 5-6)
Video lesson: Formulae in words and lettersComing soon
Practice material. Non-calculator practice. This book is practice in the style of the Year 6 arithmetic test. It is not the official test, and no official test item is used. Do every question without a calculator. Marks and timings are a practice guide only.
Story. A taxi charges £3 to start, and then £2 for every mile. Ruby wants a formula she can use for any journey, so she writes the cost using letters: C for the cost in pounds and m for the number of miles.

Rule

A formula shows how one quantity depends on others. Write it in words first, then replace the words with letters: cost = 3 + 2 × miles becomes C = 3 + 2m.

A letter next to a number means multiply: 2m is 2 × m. For the full work on formulae, read Year 6 maths, chapter 6.

Key ideas

  • Replace each word with a letter, and keep the number in front of it.
  • 2m means 2 × m; the letter is never written as a number.
  • A formula with a total often uses the number of items times the price, plus any extra fixed amount.
Worked example 6.1.1
Write a formula for the cost, C pounds, of a taxi ride of m miles, if the taxi charges £3 plus £2 per mile.

Method. Write the cost in words, then replace the words with letters.

  1. Write it in words. Cost = £3 plus £2 for each mile.
    cost = 3 + 2 × miles
  2. Replace the words with letters. Cost becomes C and miles becomes m.
    C = 3 + 2 × m
  3. Tidy the formula. We do not write the multiplication sign with a letter.
    C = 3 + 2m
  4. Write the answer.
    C = 3 + 2m

Check. When m = 1, C = 5, which is £3 + £2. ✓

Worked example 6.1.2
Write a formula for the perimeter, P, of a rectangle with length l and width w.

Method. The perimeter is two lengths and two widths.

  1. Add the four sides. Two sides are l and two sides are w.
    l + w + l + w
  2. Group the same letters. l + l is 2l and w + w is 2w.
    2l + 2w
  3. Write the formula. Perimeter P is 2l + 2w.
    P = 2l + 2w
  4. Write the answer.
    P = 2l + 2w

Check. A 3 by 2 rectangle has P = 2 × 3 + 2 × 2 = 10. ✓

Worked example 6.1.3
Ali starts with 40 stickers and gives away s stickers. Write a formula for the number of stickers, N, that he has left.

Method. Start with 40 and take away the number given away.

  1. Write it in words. Left = 40 minus the number given away.
    left = 40 - given
  2. Replace with letters. Left is N and given away is s.
    N = 40 - s
  3. Write the answer.
    N = 40 - s

Check. If s = 15, N = 25. ✓

Exercise 6A  Practice: formulae in words and letters

Non-calculator
1
FWrite a formula for the total cost, T pence, of n pens at 3p each.
2
FA bag holds b sweets. Write a formula for the number of sweets, S, in 5 bags.
3
FTickets cost £5 for an adult and £3 for a child. Write a formula for the total cost, C pounds, for a adults and c children.
4
FWrite a formula for the total number of sweets, S, in b boxes of 12 sweets, plus 5 loose sweets.
5
FA plumber charges £4 for each metre of pipe, m metres, and a fixed £2 delivery charge. Write a formula for the cost C.
6
FWrite a formula for the number of days, D, in w weeks and 3 extra days.
7
PSWrite a formula for the total number of items, I, needed for c children if each child needs 2 pencils and 1 ruler.
8
PSA plant is 5 cm tall and grows 3 cm each week. Write a formula for its height, H cm, after w weeks.
9
PSWrite a formula for the total number of minutes, M, in h hours and m minutes.
10
PSWrite a formula for the perimeter, P, of a triangle with sides a, b and c.
11
RSam says 3n means n + n + n. Is he right? Explain.
12
RWrite a formula for the number of seats, S, in r rows of 8 seats, if 3 seats are broken.
6.2

Substituting values into formulae

Year 6
Use simple formulae written in words and in letters, and substitute values into them (Year 5-6)
Video lesson: Substituting values into formulaeComing soon
Story. Ruby uses the taxi formula C = 3 + 2m to work out the cost of a 6 mile journey. She replaces m with 6, then works out the answer.

Rule

Substitute means replace each letter with its value. Then work out the calculation, using the order of operations: multiply before you add or subtract.

Key ideas

  • Replace every letter with its value before you calculate.
  • 2m with m = 6 is 2 × 6 = 12.
  • Multiply before you add or subtract.
Worked example 6.2.1
Use C = 3 + 2m to find C when m = 6.

Method. Replace m with 6, then work out the calculation.

  1. Substitute. Replace m with 6: C = 3 + 2 × 6.
    C = 3 + 2 × 6
  2. Multiply first. 2 × 6 = 12.
    C = 3 + 12
  3. Add. 3 + 12 = 15.
    C = 15
  4. Write the answer.
    15

Check. The cost for 6 miles is £15. ✓

Worked example 6.2.2
Use a = 3b - 4 to find a when b = 9.

Method. Replace b with 9, then multiply, then subtract.

  1. Substitute. a = 3 × 9 - 4.
    a = 3 × 9 - 4
  2. Multiply. 3 × 9 = 27.
    a = 27 - 4
  3. Subtract. 27 - 4 = 23.
    a = 23
  4. Write the answer.
    23

Check. 27 - 4 = 23, so a = 23 when b = 9. ✓

Worked example 6.2.3
Find the value of 2x + 3y when x = 4 and y = 5.

Method. Replace x with 4 and y with 5, then work out each part.

  1. Substitute. 2 × 4 + 3 × 5.
    2 × 4 + 3 × 5
  2. Work out each part. 2 × 4 = 8 and 3 × 5 = 15.
    8 + 15
  3. Add. 8 + 15 = 23.
    23
  4. Write the answer.
    23

Check. 8 + 15 = 23. ✓

Exercise 6B  Practice: substituting values into formulae

Non-calculator
1
FWork out 4a + 1 when a = 3.
2
FWork out 10 - 2n when n = 3.
3
FWork out 3p + q when p = 2 and q = 7.
4
FWork out pq when p = 3 and q = 8.
5
FWork out (a + b) × 2 when a = 5 and b = 4.
6
FWork out 5x - y when x = 4 and y = 6.
7
PSUse P = 2l + 2w to find P when l = 7 and w = 3.
8
PSUse M = 60h + m to find M when h = 2 and m = 15.
9
PSUse C = 5a + 3c to find C when a = 2 adults and c = 4 children.
10
PSUse S = 12b + 5 to find S when b = 3.
11
RThe formula is T = 4x + 3. Find the value of x that makes T equal 19.
12
RFind the value of x that makes 2x + 5 = 17.
13
RAli says that if a = 2 and b = 3, then 2a + b = 7. Is he right? Explain.
6.3

Linear number sequences

Year 6
Generate and describe linear number sequences, including the term-to-term rule in words (Year 5-6)
Video lesson: Linear number sequencesComing soon
Story. Tom makes a tower of bricks. The first row has 5 bricks, and each row has 6 more than the row before. He wants to know how many bricks are in the tenth row.

Rule

A linear sequence goes up or down by the same amount each time. That amount is the term-to-term rule, which you can write in words: "add 3" or "subtract 4".

To find the 10th term of 5, 11, 17, ..., add the step 9 times: 5 + 9 × 6 = 59.

Key ideas

  • Find the step between terms: add or subtract the same amount each time.
  • The term-to-term rule says how to get from one term to the next.
  • To find a far term, use the step, not a long list.
Worked example 6.3.1
Find the next two terms of 4, 7, 10, 13, ...

Method. Find the step, then add it on.

  1. Find the step. 7 - 4 = 3, and 10 - 7 = 3.
    step = 3
  2. Add the step. 13 + 3 = 16 and 16 + 3 = 19.
    16, 19
  3. Write the rule. The term-to-term rule is add 3.
    add 3
  4. Write the answer.
    16 and 19

Check. The rule is add 3. ✓

Worked example 6.3.2
The sequence 20, 17, 14, 11, ... goes down each time. What is the term-to-term rule, and what is the 6th term?

Method. Find the step, then add or subtract it.

  1. Find the step. 17 - 20 = -3, so each term goes down by 3.
    step = -3
  2. Continue the list. 11 - 3 = 8, then 8 - 3 = 5.
    8, 5
  3. Read the 6th term. The 6th term is 5.
    6th term = 5
  4. Write the answer.
    subtract 3; 5

Check. 20, 17, 14, 11, 8, 5: the 6th term is 5. ✓

Worked example 6.3.3
Find the 10th term of 5, 11, 17, 23, ...

Method. Find the step, then add it 9 times to the first term.

  1. Find the step. 11 - 5 = 6, so the step is 6.
    step = 6
  2. Count the steps. The 10th term is 9 steps after the 1st term.
    9 steps
  3. Work it out. 5 + 9 × 6 = 5 + 54 = 59.
    59
  4. Write the answer.
    59

Check. The 9th step gives 5 + 54 = 59. ✓

Exercise 6C  Practice: linear number sequences

Non-calculator
1
FWrite the next term: 2, 7, 12, 17, ...
2
FWrite the next term: 30, 26, 22, 18, ...
3
FWrite the next term: 7, 10, 13, 16, ...
4
FWrite the next term: 50, 44, 38, 32, ...
5
FWrite the term-to-term rule for 6, 9, 12, 15, ...
6
FFind the missing term: 8, ☐, 16, 20, ...
7
FFind the 10th term of 3, 7, 11, 15, ...
8
PSFind the 8th term of the sequence 1, 3, 5, 7, ...
9
PSA sequence starts at 15 and goes down by 4 each time: 15, 11, 7, ... Which term is 3?
10
RTwo sequences are 4, 9, 14, 19, ... and 3, 9, 15, 21, ... Write the first two numbers that appear in both sequences, in order.
11
RIs 50 a term of the sequence 3, 8, 13, 18, ...? Write Yes or No.
12
RFind the 12th term of 100, 90, 80, 70, ...
13
RThe sequence 2, 4, 8, 16, ... goes up by doubling. Explain why it is not a linear sequence.
6.4

Missing numbers and two unknowns

Year 6
Express missing number problems algebraically, for example a + b = 10 (Year 5-6), Find pairs of numbers that satisfy an equation with two unknowns (Year 5-6)
Video lesson: Missing numbers and two unknownsComing soon
Story. Priya has 10 counters in two bags, bag A and bag B. She knows that a + b = 10, where a is the number in bag A and b is the number in bag B. She wants to find all the ways this can happen.

Rule

A letter stands for a missing number. In a + b = 10, if a = 3, then b must be 7.

When there are two unknowns, there can be several pairs of numbers that work. Test each pair, or solve the two equations together by adding or subtracting them.

Key ideas

  • A letter stands for one missing number.
  • If the sum is fixed, one number decides the other: b = 10 - a.
  • Two equations can be solved together by adding or subtracting them.
Worked example 6.4.1
a + b = 10. If a = 3, what is b?

Method. Replace a with 3, then find the missing number.

  1. Substitute. 3 + b = 10.
    3 + b = 10
  2. Find the missing number. b = 10 - 3.
    b = 7
  3. Check. Adding 3 and 7 gives 10.
    3 + 7 = 10
  4. Write the answer.
    7

Check. 3 + 7 = 10. ✓

Worked example 6.4.2
x + y = 12 and x - y = 4. Find x and y.

Method. Add the two equations together to remove y.

  1. Add the equations. (x + y) + (x - y) = 12 + 4, so 2x = 16.
    2x = 16
  2. Find x. x = 16 ÷ 2 = 8.
    x = 8
  3. Find y. Use x + y = 12: 8 + y = 12, so y = 4.
    y = 4
  4. Write the answer.
    x = 8 and y = 4

Check. 8 + 4 = 12 and 8 - 4 = 4. ✓

Worked example 6.4.3
Find all the pairs of whole numbers a and b, with a + b = 9 and a < b, where a and b are counting numbers.

Method. Try each value of a, then check the other number.

  1. Try a = 1. b = 8, and 1 < 8.
    (1, 8)
  2. Try a = 2, 3 and 4. b = 7, 6 and 5, all bigger than a.
    (2, 7), (3, 6), (4, 5)
  3. List the pairs. Stop at a = 4, since a must be smaller than b.
    (1, 8), (2, 7), (3, 6), (4, 5)
  4. Write the answer.
    (1, 8), (2, 7), (3, 6), (4, 5)

Check. Each pair adds to 9 with the smaller number first. ✓

Exercise 6D  Practice: missing numbers and two unknowns

Non-calculator
1
Fa + b = 15. If a = 6, what is b?
2
Fa + b = 12. If a = 7, what is b?
3
Fa + b = 20. If a = 4, what is b?
4
Fa + b = 16. If a = 9, what is b?
5
FFind x if x + 7 = 19.
6
FFind the missing number: 3a = 21.
7
PSx + y = 20 and x - y = 6. Find x.
8
PSa + b = 14 and a - b = 2. Find a and b.
9
PSTwo whole numbers add up to 20 and multiply to 96. Find the two numbers, the smaller first.
10
PSHow many pairs of whole numbers a and b have a × b = 24 with a < b?
11
Rx + y = 10 and 2x + y = 14. Find x and y.
12
RFind all the pairs of counting numbers x and y, both from 1 to 5, that satisfy 3x + 2y = 12.
13
RExplain why a + b = 10 has more than one pair of whole-number answers.
6.5

Systematic tables and combinations

Year 6
Enumerate the possibilities of combinations of two variables in a systematic table (Year 5-6)
Video lesson: Systematic tables and combinationsComing soon
Story. A shop sells 2p and 5p stickers. Ali wants to spend exactly 10p. He makes a table to list every way he can do it, so he does not miss one.

Rule

To list all the combinations of two variables, go through one variable in order (0, 1, 2, ...) and find the other variable each time. A table keeps the list in order, so no possibility is missed or repeated.

Key ideas

  • Go through one variable in order, and find the other each time.
  • A systematic table shows every possibility once.
  • Check each pair in the original equation.
Worked example 6.5.1
How many ways are there to make exactly 10p using 2p and 5p coins?

Method. Go through the number of 5p coins in order, and find the 2p coins each time.

  1. Try 0 five-pence coins. Then 2p coins make 10p, so 5 coins.
    (5, 0)
  2. Try 1 five-pence coin. Then 5p is left, which is not a multiple of 2.
    no
  3. Try 2 five-pence coins. Then no money is left, so 0 two-pence coins.
    (0, 2)
  4. Write the answer.
    2 ways

Check. There are 2 ways. ✓

Worked example 6.5.2
List all the pairs of whole numbers (x, y) with x + 2y = 6 and x, y ≥ 0.

Method. Make a table: try y = 0, 1, 2, 3 and find x.

  1. y = 0. x = 6 - 0 = 6.
    (6, 0)
  2. y = 1. x = 6 - 2 = 4.
    (4, 1)
  3. y = 2. x = 6 - 4 = 2.
    (2, 2)
  4. y = 3. x = 6 - 6 = 0.
    (0, 3)
  5. Write the answer.
    (6, 0), (4, 1), (2, 2), (0, 3)

Check. Each pair gives 6 when put into x + 2y. ✓

Worked example 6.5.3
Ali has 3 shirts and 2 pairs of trousers. How many different outfits can he make?

Method. Each shirt can go with each pair of trousers, so multiply.

  1. List the choices. Shirts: 3. Trousers: 2.
    3 and 2
  2. Multiply. There are 3 × 2 = 6 outfits.
    3 × 2 = 6
  3. Check with a table. Each of the 3 shirts can be paired with 2 trousers.
    6 pairs
  4. Write the answer.
    6

Check. The table has 6 cells. ✓

Exercise 6E  Practice: systematic tables and combinations

Non-calculator
1
FHow many ways can you make exactly 12p using 2p and 5p coins?
2
FHow many outfits can Ali make with 4 shirts and 3 pairs of shorts?
3
FComplete the table of pairs for x + y = 8 with whole numbers x and y from 0 to 8, where x is 0, 1, 2, 3 and y is the other number. Write the pairs for x = 0, 1, 2 and 3.
4
PSA cake costs 4p. Ali has 12p and buys cakes and 2p sweets, with no change. How many ways can he spend exactly 12p using 4p cakes and 2p sweets?
5
PSAdult tickets cost £4 and child tickets cost £2. A group pays exactly £12. List the possible numbers of adults and children, as (adults, children).
6
PSHow many different 2-digit numbers can be made from the digits 1, 2 and 3, using each digit at most once?
7
PSHow many pairs of numbers, x and y, both from 1 to 4, add to 5?
8
RWhich pair is NOT a solution of x + 2y = 8? Choose one: (8, 0), (4, 2), (2, 4) or (6, 1).
9
RExplain how a systematic table helps you find all the combinations, and why you should not miss one.

Chapter summary

SkillMethod
FormulaWrite the words first, then replace them with letters: C = 3 + 2m.
SubstituteReplace each letter with its value, then multiply before adding: 2 × 6 + 3 = 15.
Term-to-term ruleThe step between terms, in words: add 3 or subtract 5.
Missing numberUse the letter as the missing number: a + b = 10, so b = 10 - a.
Two unknownsAdd or subtract the two equations to find one unknown; check both.
Systematic tableGo through one variable in order, and find the other each time.

Can you…?

  • I can write a formula in letters from a description in words
  • I can substitute values into a formula and work out the answer
  • I can describe a sequence with a term-to-term rule and find a far term
  • I can find a missing number in an equation with a letter
  • I can list all the pairs or combinations in a systematic table

Key terms

formula
a rule that shows how one quantity depends on others, written with letters
substitute
replace a letter with a value
term
one number in a sequence
term-to-term rule
the rule that says how to get from one term to the next
linear sequence
a sequence that goes up or down by the same amount each time
systematic
in a clear, ordered way, so that nothing is missed

Exercise 6Z  Check-up

Non-calculator
1
Write a formula for the total cost, C pounds, of n notebooks at £4 each.
2
Use T = 2a + 9 to find T when a = 4.
3
Find the value of 5m - n when m = 3 and n = 4.
4
Find the next two terms of 9, 13, 17, 21, ...
5
Write the term-to-term rule for 30, 25, 20, 15, ...
6
Find the 10th term of 4, 9, 14, 19, ...
7
a + b = 16. If b = 9, what is a?
8
x + y = 11 and x - y = 3. Find x.
9
How many ways can you make 10p using 1p and 5p coins? Count the ways.
10
List all the pairs of whole numbers x and y with x + y = 4 and x ≥ y.

Total 24 marks. Allow about 24 minutes.

Exercise 6X  Practice test questions

Non-calculator
1
FWrite a formula for the cost, C pounds, of m metres of ribbon at £3 per metre.[1 mark]
2
FUse D = 6p - 2 to find D when p = 5.[2 marks]
3
FFind the value of 3a + 2b when a = 6 and b = 5.[2 marks]
4
FFind the next term of 40, 37, 34, 31, ...[1 mark]
5
FWrite the term-to-term rule for 2, 7, 12, 17, ...[1 mark]
6
PSFind the 10th term of 6, 10, 14, 18, ...[2 marks]
7
PSx + y = 15 and x - y = 3. Find x and y. Write x first.[2 marks]
8
PSHow many ways can you make 12p using 2p and 3p coins?[3 marks]
9
FWrite a formula for the total number of pupils, P, in c classes of 30 pupils plus 4 extra pupils.[2 marks]
10
RAli says that a + b = 10 and a - b = 4 gives a = 7 and b = 3. Is he right? Explain how to check.[3 marks]
11
RFind the value of 4x - 3 when x = 5, then find the value of x when 4x - 3 = 25.[3 marks]
12
PSHow many pairs of whole numbers a and b, both from 1 to 9, have a + b = 10?[2 marks]
← Chapter 5: Ratio and proportionChapter 7: Mental arithmetic →

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