Mental calculation is not one trick performed faster. It is the ability to notice a useful relationship and choose a shorter route through the numbers. Primary 1 learners can build this flexibility with make-ten, doubles, near doubles, counting on, complements and inverse fact families.
This laboratory extends Mental Mathematics, Make Ten, Doubles, Counting On and Flexible Strategies, Addition & Subtraction Fact Fluency, and the Addition and Subtraction Repair Lab.
Fluency becomes flexible when the learner can explain why one strategy is useful for this problem and another for the next.
Session 1 | Counting On From the Larger Number
For 8 + 3, start at 8 and count three more: 9, 10, 11. This is more efficient than counting all eleven objects from one when the starting amount is already known.
Use counting on when one addend is small. It is less efficient for 8 + 7, where a make-ten or near-double strategy may shorten the reasoning.
Worked Example 1 | Count On
12 + 3: start at 12 and count 13, 14, 15. Therefore 12 + 3 = 15.
Session 2 | Make Ten
For 8 + 7, split 7 into 2 and 5. Eight plus two makes ten; ten plus five makes fifteen. The calculation becomes 8 + 7 = 8 + 2 + 5 = 15.
Make-ten works because ten is a useful landmark in the decimal number system. The split should preserve the original addend.
Worked Example 2 | 9 + 6
Split 6 into 1 and 5. Nine plus one makes ten; ten plus five gives 15. Therefore 9 + 6 = 15.
Session 3 | Doubles
Doubles are useful anchor facts: 4 + 4 = 8, 5 + 5 = 10, 6 + 6 = 12, 7 + 7 = 14, and so on.
Build doubles with paired rows or symmetric groups so the relationship is visible rather than memorised without meaning.
Worked Example 3 | Double 7
Two groups of 7 make 14. Therefore 7 + 7 = 14.
Session 4 | Near Doubles
For 7 + 8, use the known double 7 + 7 = 14, then add one more: 15. Or use 8 + 8 = 16, then subtract one: 15.
Both routes are valid. Ask which known double the learner finds easier to retrieve.
Worked Example 4 | 6 + 7
Double 6 is 12; one more gives 13. Therefore 6 + 7 = 13.
Session 5 | Complements to Ten
Practise pairs that make ten: 0 and 10, 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. These complements support make-ten addition, subtraction and missing-number equations.
Ask in both directions: “What goes with 8 to make 10?” and “10 minus 8 is what?” The same relationship should support addition and subtraction.
Worked Example 5 | Missing Complement
8 + □ = 10. The missing number is 2. This same fact gives 10 − 8 = 2 and 10 − 2 = 8.
Session 6 | Fact Families
From 8 + 5 = 13, generate related facts: 5 + 8 = 13, 13 − 8 = 5 and 13 − 5 = 8. The facts are connected because they use the same whole and parts.
This reduces the amount of isolated memorisation. One secure relationship can support several equations.
Worked Example 6 | Family of 6, 7 and 13
- 6 + 7 = 13
- 7 + 6 = 13
- 13 − 6 = 7
- 13 − 7 = 6
Choose a Strategy, Do Not Force One
For 9 + 2, counting on is efficient. For 9 + 7, make ten may be efficient. For 8 + 8, a double is immediate. For 8 + 9, a near double may be natural.
The goal is not to make every learner use the same named strategy on every problem. The goal is to recognise structure and choose a valid efficient route.
Compare Two Methods
For 8 + 6, make ten gives 8 + 2 + 4 = 14. A near-double route uses 7 + 7 = 14 if the learner can see that 8 + 6 redistributes one from 8 to 6 while preserving the total.
At Primary 1, the redistribution explanation is optional enrichment. The important idea is that different correct routes can lead to the same answer.
Subtraction by Thinking Addition
To solve 15 − 8, ask 8 + what makes 15? If 8 + 7 = 15 is known, then 15 − 8 = 7.
This can be more efficient than counting backward seven or eight individual steps when the fact family is secure.
Use Landmarks Beyond Ten
For 27 + 5, move 3 to reach 30, then add the remaining 2 to reach 32. The make-ten idea generalises to the next multiple of ten.
This connects mental calculation with place value and the Number Line Laboratory.
Worked Example 7 | Bridge Through 40
38 + 5: add 2 to reach 40, then add 3 more to reach 43.
Estimate Before Calculating
Before solving 28 + 7, notice that the answer must be greater than 28 and greater than 30. An answer of 25 is impossible before any detailed recalculation.
Reasonableness checking protects mental work from unnoticed slips.
Common Mental-Strategy Errors
| Error | Likely weak link |
|---|---|
| counts starting number as first added unit | counting-on convention weak |
| make-ten split changes total addend | decomposition not conserved |
| near-double answer off by two | adjustment direction confused |
| retrieves double but cannot use nearby fact | fact network shallow |
| knows addition fact but not related subtraction | inverse connection weak |
| uses long counting route for every fact | strategy selection not yet flexible |
Twenty Practice Questions
- Solve 8 + 3 by counting on.
- Solve 12 + 2 by counting on.
- Solve 8 + 7 by making ten.
- Solve 9 + 6 by making ten.
- Find double 6.
- Find double 8.
- Use a near double to solve 6 + 7.
- Use a near double to solve 8 + 9.
- Complete 8 + __ = 10.
- Complete 3 + __ = 10.
- Write the fact family for 8, 5 and 13.
- Use addition to solve 15 − 8.
- Use addition to solve 14 − 6.
- Which strategy is especially efficient for 9 + 2?
- Which strategy is especially efficient for 8 + 8?
- Which strategy is especially useful for 9 + 7?
- Solve 27 + 5 by bridging through 30.
- Solve 38 + 5 by bridging through 40.
- Is 25 a reasonable answer for 28 + 7?
- Show two different valid methods for 8 + 6.
Explained Answers
1. 11. Count 9, 10, 11. 2. 14. 3. 15. Split 7 into 2 and 5. 4. 15. Split 6 into 1 and 5. 5. 12. 6. 16.
7. 13. Double 6 is 12, then add one. 8. 17. Double 8 is 16, then add one. 9. 2. 10. 7. 11. 8 + 5 = 13, 5 + 8 = 13, 13 − 8 = 5, 13 − 5 = 8.
12. 7. Because 8 + 7 = 15. 13. 8. Because 6 + 8 = 14. 14. Counting on. 15. Doubles. 16. Make ten or a near-double route. 17. 32. 18. 43. 19. No. The sum must be greater than 28. 20. Examples: make ten or near-double/redistribution, if explained correctly.
A Strong Practice Progression
- Count on from the larger addend.
- Learn complements to ten.
- Use make-ten with visible models.
- Build and retrieve doubles.
- Derive near doubles.
- Link addition and subtraction fact families.
- Choose strategies without prompts.
- Bridge through multiples of ten.
- Compare methods.
- Check reasonableness before and after calculation.
What Adults Can Ask
- “What number could you start from?”
- “Can you make ten?”
- “Do you know a nearby double?”
- “Which fact in this family do you already know?”
- “Is there a useful multiple of ten nearby?”
- “Which strategy is shortest here, and why?”
Return to the Primary 1 Mathematics Learning Hub.
