Year 6 · Mathematics · Algebra
Patterns and unknown values
Describe and continue number patterns using a rule, and find unknown values in number sentences involving all four operations.
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- Describe a number pattern using a rule and use the rule to continue it.
- Use an input–output rule to find missing values in a table.
- Find unknown values in number sentences involving addition, subtraction, multiplication and division.
Success looks like
- I can write the rule for a pattern, such as “start at 4 and add 7”.
- I can use a rule to find the next terms or a missing term in a pattern.
- I can find the missing number in a number sentence and check that it makes the sentence true.
The big idea
A number pattern follows a rule. A full rule says where the pattern starts and what happens each time. For example, “start at 4 and add 7” gives 4, 11, 18, 25, 32, … and “start at 3 and multiply by 2” gives 3, 6, 12, 24, 48, …
To find a rule, look at what happens between terms. If the difference between terms is always the same, it is an adding (or subtracting) pattern. If the differences keep growing, check whether each term is being multiplied by the same number.
An input–output table uses a rule to change each input into an output. If the rule is “multiply by 3, then add 1”, an input of 2 gives 2 × 3 + 1 = 7, and an input of 10 gives 10 × 3 + 1 = 31. Rules like this let us jump straight to any term without listing all the ones before it.
A number sentence with an unknown value is like a balance. The equals sign means both sides are worth the same. To find the unknown, think about what number makes both sides balance.
Inverse operations undo each other and help us find unknowns:
- For ? + 39 = 100, undo adding by subtracting: 100 − 39 = 61.
- For 125 − ? = 78, think “125 take away what leaves 78?” That is 125 − 78 = 47.
- For 7 × ? = 63, undo multiplying by dividing: 63 ÷ 7 = 9.
- For ? ÷ 4 = 12, undo dividing by multiplying: 12 × 4 = 48.
Sometimes there are numbers on both sides, as in 4 × ? = 3 × 12. Work out the side you can: 3 × 12 = 36. Now solve 4 × ? = 36, so the unknown is 9. Always check by putting your answer back in.
Worked examples
Describing and continuing a pattern
Describe the rule and find the next two terms: 6, 15, 24, 33, …
- Find the differences: 15 − 6 = 9, 24 − 15 = 9, 33 − 24 = 9.
- The difference is always 9, so the rule is “start at 6 and add 9”.
- 33 + 9 = 42 and 42 + 9 = 51.
Answer: Start at 6 and add 9. The next two terms are 42 and 51.
Using an input–output rule
The rule is “multiply by 4, then subtract 1”. What is the output when the input is 6? What input gives an output of 39?
- Input 6: 6 × 4 = 24, then 24 − 1 = 23.
- For an output of 39, work backwards using inverse operations. Undo “subtract 1” by adding 1: 39 + 1 = 40.
- Undo “multiply by 4” by dividing by 4: 40 ÷ 4 = 10.
- Check: 10 × 4 − 1 = 40 − 1 = 39.
Answer: An input of 6 gives an output of 23. An input of 10 gives an output of 39.
Balancing a number sentence
Find the unknown value: 4 × ? = 3 × 12.
- Work out the side you know: 3 × 12 = 36.
- Now the sentence is 4 × ? = 36.
- 36 ÷ 4 = 9.
- Check: 4 × 9 = 36 and 3 × 12 = 36, so both sides balance.
Answer: The unknown value is 9.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
What is the rule for this pattern? 7, 13, 19, 25, …
Add 6 — Each term is 6 more than the one before: 7 + 6 = 13, 13 + 6 = 19, 19 + 6 = 25.
What is the next term in this pattern? 2, 6, 18, 54, …
162 — The rule is multiply by 3. 54 × 3 = 162.
Find the unknown value: ? × 8 = 72.
9 — 72 ÷ 8 = 9. Check: 9 × 8 = 72.
Which number makes this number sentence true? 150 − ? = 85
65 — 150 − 85 = 65. Check: 150 − 65 = 85.
Find the unknown value: ? ÷ 6 = 9.
54 — Undo dividing by multiplying: 9 × 6 = 54. Check: 54 ÷ 6 = 9.
A pattern starts at 3 and each term is multiplied by 2. What is the fifth term?
48 — The terms are 3, 6, 12, 24, 48. The fifth term is 48.
The rule is “multiply by 4, then subtract 1”. What is the output when the input is 7?
27 — 7 × 4 = 28, then 28 − 1 = 27.
Mia has some stickers. She gives away 15 and has 38 left. How many stickers did she have to start with?
53 — The number sentence is ? − 15 = 38. Undo the subtraction: 38 + 15 = 53.
Watch out for
- Treating the equals sign as “the answer comes next”. In 4 × ? = 3 × 12, some learners write 12. Explain that = means “is the same as”, and that both sides must balance.
- Using the wrong inverse, such as solving ? ÷ 4 = 12 by dividing 12 by 4. Ask the learner to check the answer by putting it back into the original sentence.
- Describing only the step and not the start, for example “add 7” for 4, 11, 18. A full rule says where the pattern begins, which matters when two patterns share the same step.
Tips for parents
- Use a kitchen scale or a seesaw to show that the equals sign means balance: both sides must weigh the same.
- Turn unknown-value problems into everyday stories, such as sharing biscuits or saving pocket money.
- Encourage the learner to always check by substituting the answer back into the number sentence.
Go further
- Build a growing pattern with toothpicks (1 square needs 4 toothpicks, 2 squares in a row need 7, 3 squares need 10). Find the rule and predict how many toothpicks 20 squares need.
- Play “Guess my rule”: one person secretly chooses a two-step rule such as “double and add 3”, and the other gives inputs until they can work it out.
- Write your own number sentences with an unknown on each side of the equals sign, such as 5 × ? = 40 + 5, and swap them with a family member to solve.
About 45 minutes
You'll need
- Paper and pencil
- Counters, blocks or toothpicks for building growing patterns
- A calculator for checking (optional)
Key words
- Rule
- A description of how a pattern changes from one term to the next, or how an output is made from an input.
- Term
- Each number in a pattern. The first term is the first number in the pattern.
- Input and output
- The number that goes into a rule and the number that comes out after the rule is applied.
- Unknown value
- A missing number in a number sentence, often shown by a box, a question mark or a letter.
- Equals sign
- The symbol = that shows both sides of a number sentence have the same value.
Still tricky?
For parents
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