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SATs practice · Year 6Chapter 7 · Algebra and sequences
Year 6 · Chapter 7

Algebra and sequences

By the end of this chapter you will be able to

  • Find the next term and a missing term in a linear sequence
  • Find the nth term of a sequence, and write the formula for it
  • Substitute a value into a formula written in words or letters
  • Write and solve an equation for a missing number
  • List all the pairs that satisfy an equation with two unknowns

You should already know

  • Multiplication and division facts, and the four operations (Year 5 and Year 6 number)
  • Negative numbers and order of operations (Chapter 1 of this book)

Where this leads

  • Statistics and data (Chapter 8 of this book)

KS2 SATs practice for reasoning (Year 6). Original questions in the style of the national tests. Content: Generate and describe linear number sequences, including the term-to-term rule in words and the position-to-term rule (Year 5-6), Find missing terms in sequences, including sequences that go up and down (Year 5-6), Use simple formulae written in words and in letters, and substitute values into them (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, and list all possibilities in a systematic table (Year 5-6), Solve simple linear equations with one unknown, using inverse operations (Year 6), Explain a pattern and say whether a term is in a sequence (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWhat is the next number in the sequence 2, 4, 6, 8, __?
2
FWork out 3 x n when n = 4.
3
FSolve 5 + x = 9.
4
FTwo numbers add to make 10. One of them is 3. Write the other number.
5
FWork out 2 x 6 + 1.
7.1

Linear sequences and missing terms

Year 6
Generate and describe linear number sequences, including the term-to-term rule in words (Year 5-6), Find missing terms in sequences, including sequences that go up and down (Year 5-6)
Video lesson: Linear sequences and missing termsComing soon
Practice material. Do this without a calculator. This is practice in the style of the Year 6 maths reasoning test. It is not the official test, and no official test item is used. Marks and timings are a practice guide only.
Story. Priya is saving money for a school trip. She saves 5 pounds in the first week, 9 pounds in the second, 13 pounds in the third, and she wants to know how much she saves each week, so she can work out the next amounts.

Rule

A linear sequence goes up or down by the same amount each time. That amount is the difference.

The term-to-term rule says how to get from one term to the next, for example "add 4" or "subtract 3".

Key ideas

  • The difference is the same between every pair of terms in a linear sequence.
  • A sequence can go down: 20, 16, 12, 8 has the rule "subtract 4".
  • Check the rule on every pair of terms before you use it.
Worked example 7.1.1
The sequence starts 5, 9, 13, 17, ... Write the next term.

Method. Find the difference between terms, then apply it again.

  1. Find the rule. Each term is the one before with add 4.
    9 - 5 = 4
  2. Apply the rule to the last term. Take the last term, 17, and add 4.
    17 + 4 = 21
  3. Write the answer.
    21

Check. 21 - 17 = 4. ✓

Watch out. Check the difference between every pair of terms, not just the first two.

Worked example 7.1.2
Find the missing term: 10, 7, __, 1, -2.

Method. Find the difference between the known terms, then count to the gap.

  1. Find the difference. The terms go subtract 3 each time.
    difference = -3
  2. Work out the missing term. Count from a known term to the gap.
    10 - 6 = 4
  3. Write the answer.
    4

Check. 4 is in the same pattern: the difference is -3. ✓

Watch out. The term is missing from the list, so count the steps carefully.

Worked example 7.1.3
The sequence starts 30, 24, 18, 12, ... Write the next term.

Method. Find the difference between terms, then apply it again.

  1. Find the rule. Each term is the one before with subtract 6.
    24 - 30 = -6
  2. Apply the rule to the last term. Take the last term, 12, and subtract 6.
    12 - 6 = 6
  3. Write the answer.
    6

Check. 6 - 12 = -6. ✓

Watch out. Check the difference between every pair of terms, not just the first two.

Watch out

Watch out. Check the difference between every pair of terms. A sequence with a missing term may look wrong until you check.

Exam tip

Practice guide. Write the difference between two neighbouring terms above the sequence, then count it along.

Exercise 7A  Practice: linear sequences and missing terms

Non-calculator
1
FThe sequence starts 3, 8, 13, 18, ... Write the next term.
2
FThe sequence starts 20, 16, 12, 8, ... Write the next term.
3
FThe sequence starts 1, 8, 15, 22, ... Write the next term.
4
PSThe sequence starts 100, 92, 84, 76, ... Write the next term.
5
FFind the missing term: 4, 10, __, 22, 28.
6
FFind the missing term: 40, __, 30, 25, 20.
7
FFind the missing term: 7, 16, 25, 34, __.
8
FThe term-to-term rule for a sequence is "add 7". The first term is 12. Write the second and third terms.
9
PSThe term-to-term rule is "subtract 6". The fourth term is 9. What is the first term?
10
RExplain how you know that the sequence 3, 8, 13, 18 goes up by 5 each time.
11
FWhich is the correct term-to-term rule for 2, 6, 10, 14? Choose one: add 4, add 2 or multiply by 2.
12
RTick the sequences that go up in equal steps. Options: 2, 4, 6, 8 | 1, 3, 6, 10 | 5, 10, 15, 20 | 3, 5, 9, 11. Write the options you would tick, in the order given.
13
PSA sequence goes 50, 46, 42, __, 34. Find the missing term.
14
PSA bike shop changes the price of a bike by the same amount each week. It costs 240 pounds in week 1, 225 pounds in week 2 and 210 pounds in week 3. Write the price in week 5.
7.2

The nth term and the position-to-term rule

Year 6
Generate and describe linear number sequences, including the position-to-term rule (Year 5-6), Explain a pattern and say whether a term is in a sequence (Year 5-6)
Video lesson: The nth term and the position-to-term ruleComing soon
Story. A pupil builds a fence from square tiles. The first section needs 4 tiles, and each extra section needs 3 more. He wants to know how many tiles he needs for section 10, without drawing all of them.

Rule

The nth term is the term in position n. For a linear sequence, the nth term is first term + (n - 1) x difference.

A formula for the nth term looks like 3n + 1. The number in front of n is the difference.

Key ideas

  • nth term = first term + (n - 1) x difference.
  • The number in front of n in the formula is the difference.
  • To check a formula, put n = 1 and n = 2 in and compare.
Worked example 7.2.1
The sequence 4, 7, 10, ... continues in the same way. What is term number 10?

Method. Start at the first term and add the difference for each step after it.

  1. Find the first term and the difference. The first term is 4, and the difference is 3.
    4 and 3
  2. Count the steps to the nth term. The 10th term is 9 steps from the first term.
    10 - 1 = 9 steps
  3. Work out the term. Start at 4 and add 3 for each step.
    4 + 27 = 31
  4. Write the answer.
    31

Check. 4 + (10 - 1) x 3 = 31. ✓

Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.

Worked example 7.2.2
The sequence 4, 7, 10, ... has a position-to-term rule. Write the nth term in terms of n.

Method. The number in front of n is the difference. Then work out what to add or subtract.

  1. Find the number that goes with n. The term goes up by 3 each time, so the formula has 3n.
    3n
  2. Check the first term. When n = 1, 3 x 1 + (the rest) must equal 4.
    3 + 1 = 4
  3. Write the formula. Use the number that makes the first term correct.
    3n + 1
  4. Write the answer.
    3n + 1

Check. When n = 2, the term is 7. ✓

Watch out. Check your formula with n = 1 and n = 2.

Worked example 7.2.3
Is 41 a term of the sequence 5, 9, 13, 17, ...? Write yes or no.

Method. Check whether the difference from the first term is a multiple of the difference.

  1. Find the difference from the first term. 41 - 5 = 36.
    36
  2. Divide by the difference. 36 / 4 = 9, a whole number.
    9
  3. Decide. So 41 is term number 10.
    term 10
  4. Write the answer.
    yes

Check. The terms are 5 + 4 x (whole number). ✓

Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.

Watch out

Watch out. The 10th term is 9 steps after the 1st term, not 10 steps.

Exam tip

Practice guide. Test your formula with n = 1. If it gives the first term, it is probably right.

Exercise 7B  Practice: the nth term

Non-calculator
1
FThe sequence 6, 10, 14, ... continues in the same way. What is term number 8?
2
FThe sequence 2, 7, 12, ... continues in the same way. What is term number 20?
3
FThe sequence 50, 47, 44, ... continues in the same way. What is term number 12?
4
PSThe sequence 3, 7, 11, ... continues in the same way. What is term number 50?
5
FThe sequence 7, 11, 15, ... has a position-to-term rule. Write the nth term in terms of n.
6
FThe sequence 1, 7, 13, ... has a position-to-term rule. Write the nth term in terms of n.
7
PSThe sequence 20, 16, 12, ... has a position-to-term rule. Write the nth term in terms of n.
8
FIs 57 a term of the sequence 3, 9, 15, 21, ...? Write yes or no.
9
FIs 52 a term of the sequence 4, 9, 14, 19, ...? Write yes or no.
10
PSIs 50 a term of the sequence 1, 8, 15, 22, ...? Write yes or no.
11
FWhich formula fits the sequence 4, 7, 10, 13? Choose one: 3n + 1, n + 3 or 4n.
12
RSam says 90 is a term of the sequence 4, 9, 14, 19. Is he right? Explain.
13
PSThe 1st, 2nd and 3rd terms of a pattern of matchsticks are 5, 8, 11. Write the 20th term.
14
RExplain why the 100th term of the sequence 1, 3, 5, 7 is 199.
15
PSWrite the formula for the nth term of 12, 9, 6, 3.
7.3

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
Story. A trip to the science museum costs a fixed amount for the coach, plus a fixed price for each ticket. The teacher writes a formula so that she can work out the total for any number of pupils.

Rule

A formula shows how one amount depends on others. Put the value in place of each letter, then calculate.

Multiply before you add: 3n + 2 with n = 4 is 3 x 4 + 2 = 14.

Key ideas

  • Replace each letter with its value.
  • Follow the order of operations: multiply, then add.
  • The letter n usually stands for a number of things.
Worked example 7.3.1
A cost is worked out using the formula T = 4n + 10, where n is the number of tickets. Find T when n = 6.

Method. Replace n with the number, then multiply before you add.

  1. Write the formula. Total = 4 x n + 10.
    T = 4n + 10
  2. Substitute the value. Put n = 6 into the formula.
    4 x 6 + 10
  3. Work out. Multiply first, then add.
    24 + 10 = 34
  4. Write the answer.
    34

Check. 24 + 10 = 34. ✓

Watch out. Multiply before you add. 3n + 2 with n = 4 is 14, not 20.

Worked example 7.3.2
The perimeter P of a rectangle is found by P = 2l + 2w. Find P when l = 7 and w = 4.

Method. Substitute l = 7 and w = 4.

  1. Substitute. P = 2 x 7 + 2 x 4.
    2 x 7 + 2 x 4
  2. Work out. 14 + 8 = 22.
    22
  3. Write the answer.
    22

Check. 2 x 7 = 14 and 2 x 4 = 8. ✓

Worked example 7.3.3
The area of a triangle is A = bh / 2. Find A when b = 10 and h = 7.

Method. Substitute b = 10 and h = 7, then divide by 2.

  1. Substitute. A = 10 x 7 / 2.
    10 x 7 / 2
  2. Work out. 70 / 2 = 35.
    35
  3. Write the answer.
    35

Check. 35 x 2 = 70. ✓

Watch out

Watch out. 2n means 2 x n. If n = 5, then 2n = 10, not 25.

Exam tip

Practice guide. Write the formula with the numbers in place of the letters before you calculate.

Exercise 7C  Practice: formulae

Non-calculator
1
FA cost is worked out using the formula T = 2n + 5, where n is the number of tickets. Find T when n = 9.
2
FA cost is worked out using the formula T = 6n + 3, where n is the number of tickets. Find T when n = 7.
3
FA cost is worked out using the formula T = 7n + 0, where n is the number of tickets. Find T when n = 6.
4
FA cost is worked out using the formula T = 4n + 12, where n is the number of tickets. Find T when n = 10.
5
FWhich is the value of 3a + 2 when a = 5? Choose one: 17, 21 or 13.
6
PSThe formula for the total seats in a hall is S = 6r + 2, where r is the number of rows. Find S when r = 9.
7
PSThe cost of a day trip is C = 5n + 12 pounds, where n is the number of pupils. How much does a trip cost for 8 pupils?
8
FA rectangle has length 9 cm and width 5 cm. Use P = 2l + 2w to find P.
9
RExplain why 2n + 3 with n = 4 is 11, not 14.
10
PSThe cost of hiring a bike is C = 3h + 4 pounds, where h is the number of hours. Maya pays 22 pounds. How many hours did she hire the bike?
11
PSA cost is worked out using the formula T = 9n + 1, where n is the number of tickets. Find T when n = 3.
12
RWrite the formula in words for the total cost: a bus costs 4 pounds for each child plus 10 pounds for the driver.
7.4

Missing numbers, equations and pairs

Year 6
Express missing number problems algebraically, for example a + b = 10 (Year 5-6), Solve simple linear equations with one unknown, using inverse operations (Year 6), Find pairs of numbers that satisfy an equation with two unknowns, and list them in a systematic table (Year 5-6)
Video lesson: Missing numbers, equations and pairsComing soon
Story. At the bakery, Sam buys some buns and some rolls. The bill comes to 26 pounds, and he remembers the number of items, so he writes an equation to work out how many of each he bought.

Rule

An equation has an equals sign. To solve it, undo the operations in reverse order: subtract, then divide.

For two unknowns, such as a + b = 12, list the pairs in order in a table so that none are missed.

Key ideas

  • Undo addition with subtraction, and multiplication with division.
  • Write the equation first, then solve it.
  • List pairs in order, starting with the smallest value, so none are missed.
Worked example 7.4.1
Two numbers add to make 25. One of them is 9. Write the other number.

Method. Subtract the known number from the total.

  1. Write the number sentence. 9 + b = 25.
    9 + b = 25
  2. Find the missing number. Take 9 from 25.
    b = 25 - 9
  3. Write the answer.
    16

Check. 9 + 16 = 25. ✓

Worked example 7.4.2
A number is multiplied by 3, then 5 is added. The answer is 26. What is the number?

Method. Write an equation with n, then solve it by undoing the operations.

  1. Write the equation. Let the number be n. Multiply by 3, add 5, and get 26.
    3n + 5 = 26
  2. Subtract the added number. Take 5 from 26.
    3n = 21
  3. Divide by the multiplier. Divide 21 by 3.
    n = 7
  4. Write the answer.
    7

Check. 3 x 7 + 5 = 26. ✓

Watch out. Undo the operations in reverse order.

Worked example 7.4.3
Solve the equation 4x + 5 = 29.

Method. Use inverse operations: subtract, then divide.

  1. Undo the addition. Take 5 from both sides.
    4x = 29 - 5 = 24
  2. Undo the multiplication. Divide both sides by 4.
    x = 24 / 4 = 6
  3. Write the answer.
    6

Check. 4 x 6 + 5 = 29. ✓

Watch out. Undo the addition or subtraction first, then the multiplication.

Watch out

Watch out. Do the inverse operation to both sides. Subtract 5 from both sides of 4x + 5 = 29 before you divide by 4.

Exam tip

Practice guide. Check your answer by putting it back into the original equation.

Exercise 7D  Practice: missing numbers, equations and pairs

Non-calculator
1
FTwo numbers add to make 100. One of them is 37. Write the other number.
2
FTwo numbers add to make 56. One of them is 19. Write the other number.
3
FA number is multiplied by 2, then 9 is added. The answer is 33. What is the number?
4
FA number is multiplied by 5, then 4 is added. The answer is 39. What is the number?
5
FSolve the equation 3x + 7 = 31.
6
FSolve the equation 6x - 6 = 18.
7
PSSolve the equation 2x + 1 = 23.
8
PSHow many pairs of whole numbers a and b, both at least 1, have a + b = 12 and a < b? Write the number.
9
PSHow many pairs of whole numbers a and b, both at least 1, have a + b = 15 and a < b? Write the number.
10
PSWhich pair of numbers adds to 15 and has a difference of 3? Choose one: 6 and 9, 5 and 10 or 7 and 9.
11
FTick the pairs (x, y) that satisfy x + y = 12. Options: (3, 9) | (5, 6) | (4, 8) | (10, 1). Write the options you would tick, in the order given.
12
RExplain how a systematic table helps you to find all the pairs that add to 10.
13
PSBen has some counters. If he adds 7, then doubles the total, he gets 34. Write an equation with n for the number of counters, and solve it.
14
FSolve the equation 5x - 4 = 26.
15
PSTwo numbers add to make 81. One of them is 45. Write the other number.
7.5

Explaining patterns and checking terms

Year 6
Explain a pattern and say whether a term is in a sequence (Year 5-6)
Video lesson: Explaining patterns and checking termsComing soon
Story. Mia and Dan are studying a pattern of dots. Mia says the pattern goes up by 3 each time, and Dan says it goes up by 4. They must decide who is right, and explain why.

Rule

To explain a pattern, show the difference between two terms and check it on at least three pairs.

To say whether a number is a term, check that the difference from the first term is a whole-number multiple of the difference.

Key ideas

  • Check the pattern on several pairs of terms before you explain it.
  • A number is a term only if it can be reached by adding the difference.
  • One counter-example is enough to show a rule is wrong.
Worked example 7.5.1
Dan says the sequence 2, 4, 8, 16 goes up by 2 each time. Is he right? Write yes or no.

Method. Check the difference between each pair of terms.

  1. Differences. 4 - 2 = 2, but 8 - 4 = 4 and 16 - 8 = 8.
    2, 4, 8
  2. Decide. The differences are not the same, so the sequence does not go up by 2 each time.
    no
  3. Write the answer.
    no

Check. The differences 2, 4, 8 are not equal. ✓

Worked example 7.5.2
Is 17 a term of the sequence 2, 5, 8, 11, ...? Write yes or no.

Method. Check whether the difference from the first term is a multiple of the difference.

  1. Find the difference from the first term. 17 - 2 = 15.
    15
  2. Divide by the difference. 15 / 3 = 5, a whole number.
    5
  3. Decide. So 17 is term number 6.
    term 6
  4. Write the answer.
    yes

Check. The terms are 2 + 3 x (whole number). ✓

Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.

Worked example 7.5.3
Write the term-to-term rule for 30, 27, 24, 21.

Method. Find the difference between two terms.

  1. Difference. 27 - 30 = -3.
    -3
  2. Rule. Each term is 3 less than the one before, so subtract 3.
    subtract 3
  3. Write the answer.
    subtract 3

Check. 27 - 3 = 24. ✓

Watch out

Watch out. A pattern that works for the first two terms is not always right. Check the next pair too.

Exam tip

Practice guide. Write a short explanation with the difference, the check, and the answer.

Exercise 7E  Practice: explaining patterns and terms

Non-calculator
1
FWrite the term-to-term rule for 15, 12, 9, 6.
2
PSIs 50 a term of the sequence 5, 8, 11, 14, ...? Write yes or no.
3
PSIs 100 a term of the sequence 7, 13, 19, 25, ...? Write yes or no.
4
RIs 10 a term of the sequence 3, 7, 11, 15? Write yes or no.
5
RTick the statements that are true about the sequence 2, 5, 8, 11. Options: It goes up by 3 each time | The 4th term is 12 | The 5th term is 14 | The difference is 4. Write the options you would tick, in the order given.
6
RTom says: "The 10th term of 2, 4, 6, 8 is 10." Is he right? Write yes or no.
7
RExplain why the sequence 1, 4, 9, 16 cannot be a linear sequence.
8
PSThe first term of a sequence is 8 and the difference is -5. Is 58 a term of the sequence? Write yes or no.
9
RRavi says the number 1 is always a term in a sequence that starts at 1 and goes up by 2. Is he right? Explain.
10
PSThe sequence 1, 3, 5, ... continues in the same way. What is term number 100?

Chapter summary

SkillMethod
Linear sequenceThe same difference between every pair of neighbouring terms.
Term-to-term ruleSay how to get from one term to the next, in words: add 4, subtract 3.
nth termFirst term + (n - 1) x difference.
FormulaWrite n in place of the position: 3n + 1; the number in front of n is the difference.
SubstitutionReplace each letter with its value, then multiply before you add.
Solving an equationUndo the operations in reverse order, with the same change to both sides.
Pairs with two unknownsList all pairs in a systematic table, in order.
Is a term in a sequence?Is the difference from the first term a whole-number multiple of the difference?

Can you…?

  • I can find the next term and a missing term in a linear sequence
  • I can find the nth term and write a formula for it
  • I can substitute values into a formula written in words or letters
  • I can write and solve an equation for a missing number
  • I can list all the pairs that satisfy an equation with two unknowns
  • I can say whether a number is a term of a sequence, and explain why

Key terms

term
one number in a sequence
linear sequence
a sequence where the same difference is added or subtracted each time
difference
the amount added or subtracted each time in a linear sequence
formula
a rule that connects a result to a number, written with letters or words
substitute
put a number in place of a letter
inverse operation
the operation that undoes another, such as subtract for add

Exercise 7Z  Check-up

Non-calculator
1
The sequence starts 9, 16, 23, 30, ... Write the next term.
2
Find the missing term: 12, 8, 4, __, -4.
3
The sequence 7, 12, 17, ... continues in the same way. What is term number 15?
4
The sequence 3, 5, 7, ... has a position-to-term rule. Write the nth term in terms of n.
5
Is 31 a term of the sequence 4, 7, 10, 13, ...? Write yes or no.
6
A cost is worked out using the formula T = 5n + 8, where n is the number of tickets. Find T when n = 6.
7
Solve the equation 4x + 3 = 39.
8
A number is multiplied by 3, then 7 is added. The answer is 40. What is the number?
9
How many pairs of whole numbers a and b, both at least 1, have a + b = 14 and a < b? Write the number.
10
Two numbers add to make 60. One of them is 24. Write the other number.

Total 19 marks. Allow about 19 minutes.

Exercise 7X  Practice test questions

Non-calculator
1
FThe sequence starts 4, 10, 16, 22, ... Write the next term.[1 mark]
2
PSThe sequence 10, 13, 16, ... continues in the same way. What is term number 12?[2 marks]
3
FThe sequence 5, 9, 13, ... has a position-to-term rule. Write the nth term in terms of n.[2 marks]
4
FA cost is worked out using the formula T = 3n + 9, where n is the number of tickets. Find T when n = 8.[1 mark]
5
PSSolve the equation 5x + 2 = 37.[2 marks]
6
PSA number is multiplied by 4, then 6 is added. The answer is 46. What is the number?[2 marks]
7
PSHow many pairs of whole numbers a and b, both at least 1, have a + b = 16 and a < b? Write the number.[2 marks]
8
PSIs 30 a term of the sequence 2, 9, 16, 23, ...? Write yes or no.[2 marks]
9
RA sequence has the nth term 4n + 1. Explain why 30 is not a term of this sequence.[3 marks]
10
FWrite the missing numbers in this sequence, which goes down: 40, 35, __, 25, 20. Write the missing term.[2 marks]
← Chapter 6: Position and coordinatesChapter 8: Statistics and data →

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