Year 5 · Mathematics · Algebra
Number patterns and missing numbers
Find and continue patterns made by adding or multiplying, and work out unknown values in number sentences such as 6 × ? = 42.
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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
- Find the rule for a number pattern made by adding or multiplying.
- Continue a number pattern using its rule.
- Find unknown values in number sentences such as 6 × ? = 42.
Success looks like
- I can describe a pattern rule, such as “add 4” or “multiply by 2”.
- I can work out the next numbers in a pattern.
- I can use a related fact to find a missing number in a number sentence.
The big idea
A number pattern follows a rule. To find the rule, look at how each number changes to become the next one.
In the pattern 3, 7, 11, 15, … each number is 4 more than the one before. The rule is add 4, so the next number is 15 + 4 = 19. Patterns made by adding grow by the same amount every time.
In the pattern 2, 4, 8, 16, … the jumps are 2, 4 and 8, so it is not adding the same amount. Instead, each number is double the one before. The rule is multiply by 2, and the next number is 16 × 2 = 32. Patterns made by multiplying grow faster and faster.
To check a rule, test it on every pair of numbers, not just the first two. In 3, 6, 12, 24, “add 3” works from 3 to 6, but not from 6 to 12. “Multiply by 2” works for every step.
A number sentence with a missing number is like a puzzle. In 6 × ? = 42, ask yourself, “6 times what makes 42?” Our multiplication facts tell us 6 × 7 = 42, so the missing number is 7.
We can also use the inverse (opposite) operation. To solve ? + 27 = 63, undo the adding by subtracting: 63 − 27 = 36. To solve 48 ÷ ? = 6, think: 6 × 8 = 48, so 48 ÷ 8 = 6 and the missing number is 8.
- Adding is undone by subtracting.
- Multiplying is undone by dividing.
- Always put your answer back into the number sentence to check it is true.
Worked examples
An adding pattern
Find the rule and the next two numbers: 5, 13, 21, 29, …
- Find the differences: 13 − 5 = 8, 21 − 13 = 8, 29 − 21 = 8.
- Each number is 8 more than the one before, so the rule is “add 8”.
- 29 + 8 = 37 and 37 + 8 = 45.
Answer: The rule is add 8. The next two numbers are 37 and 45.
A multiplying pattern
Find the rule and the next number: 1, 3, 9, 27, …
- The jumps are 2, 6 and 18, which are not the same, so it is not an adding pattern.
- Try multiplying: 1 × 3 = 3, 3 × 3 = 9, 9 × 3 = 27. The rule “multiply by 3” works every time.
- 27 × 3 = 81 (because 20 × 3 = 60 and 7 × 3 = 21).
Answer: The rule is multiply by 3. The next number is 81.
Finding a missing number
Find the missing number: 9 × ? = 54.
- Ask: “9 times what makes 54?”
- Use known facts: 9 × 5 = 45 and 9 × 6 = 54.
- Check: 9 × 6 = 54 is true.
Answer: The missing number is 6.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
What is the next number in this pattern? 5, 12, 19, 26, …
33 — The rule is add 7. 26 + 7 = 33.
What is the rule for this pattern? 3, 6, 12, 24, …
Multiply by 2 — 3 × 2 = 6, 6 × 2 = 12 and 12 × 2 = 24. “Add 3” only works for the first step.
Find the missing number: 8 × ? = 56.
7 — 8 × 7 = 56, so the missing number is 7.
What is the next number in this pattern? 1, 3, 9, 27, …
81 — The rule is multiply by 3. 27 × 3 = 81.
Find the missing number: 45 ÷ ? = 9.
5 — 9 × 5 = 45, so 45 ÷ 5 = 9. The missing number is 5.
Find the missing number: ? − 18 = 25.
43 — Undo the subtraction by adding: 25 + 18 = 43. Check: 43 − 18 = 25.
What is the next number in this pattern? 100, 91, 82, 73, …
64 — Each number is 9 less than the one before. 73 − 9 = 64.
Each table at a party seats 6 people. 1 table seats 6, 2 tables seat 12 and 3 tables seat 18. How many people can sit at 9 tables?
54 — The pattern goes up by 6 for each table, so 9 tables seat 9 × 6 = 54 people.
Watch out for
- Checking a rule on only the first two numbers. In 3, 6, 12, 24 the rule “add 3” seems to work at first. Ask the learner to test the rule on every step.
- Assuming every pattern is an adding pattern. If the jumps keep getting bigger, try multiplying.
- Reading the missing-number sentence as an instruction, for example answering 48 for 6 × ? = 42 because 42 + 6 = 48. Ask the learner to say the sentence aloud as a question: “6 times what makes 42?”
Tips for parents
- Build patterns with blocks or matchsticks (for example, squares in a row) so the learner can see how each step grows.
- Turn missing-number questions into stories: “I had some apples, I ate 18 and now I have 25. How many did I start with?”
- Encourage the learner to check every answer by putting it back into the number sentence.
Go further
- Make up your own pattern with a secret rule, write the first four numbers and challenge someone in your family to find the rule.
- Investigate a doubling pattern: if you saved 1 cent on day 1, 2 cents on day 2, 4 cents on day 3 and so on, how much would you save on day 10?
- Explore patterns on a 100 chart: colour every multiple of 4, then every multiple of 8. What do you notice?
About 40 minutes
You'll need
- Counters, blocks or matchsticks
- Paper and pencil
- A 100 chart (optional)
Key words
- Pattern
- A sequence of numbers or objects that follows a rule.
- Rule
- The instruction that tells you how to get from one number in a pattern to the next.
- Term
- One of the numbers in a pattern.
- Unknown
- A missing number in a number sentence, often shown by a box or a question mark.
- Inverse operation
- The operation that undoes another. Subtraction undoes addition, and division undoes multiplication.
Still tricky?
For parents
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