PSLE Learner’s Guide · Volume 0798
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If N = paqb…, the number of positive factors is (a+1)(b+1)… because each exponent is chosen independently.
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This guide uses original Primary 6 practice and does not predict a live question. Consult the SEAB PSLE formats examined in 2026 page for official information.
Open the full chapter index
- 1. Start with a complete prime factorisation
- 2. Count the choices for each prime
- 3. Multiply independent choices
- 4. Include 1 and the number itself
- 5. Treat prime numbers as a simple case
- 6. Use square numbers as a parity check
- 7. Handle the number 1 separately
- 8. Control examination work
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Start with a complete prime factorisation
Write the number as a product of prime powers before counting. For 72 = 2³ × 3², every positive factor is built only from powers of 2 and 3 within those exponent limits.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check complete prime factorisation.
- Check allowed exponent choices.
- Check multiplication principle.
- Check positive-factor boundary.
CHAPTER 2 OF 12 · Back to contents
2. Count the choices for each prime
For 2³, a factor may use 2⁰, 2¹, 2² or 2³: four choices. For 3², it may use 3⁰, 3¹ or 3²: three choices. Zero exponents matter because a factor may omit a prime.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Multiply independent choices
Each allowed power of 2 can be paired with each allowed power of 3. Therefore 72 has 4 × 3 = 12 positive factors. Adding the choices would miss combinations.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Include 1 and the number itself
Choosing exponent zero for every prime gives 1. Choosing every maximum exponent gives the original number. Unless a question explicitly excludes them, both are positive factors and belong in the count.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Treat prime numbers as a simple case
A prime p has factorisation p¹, so it has 1 + 1 = 2 positive factors: 1 and p. This matches the definition of a prime and provides a quick check on the rule.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Use square numbers as a parity check
Most factors pair as d and N ÷ d. For a square number, the square-root factor pairs with itself, so the total number of positive factors is odd. A non-square has an even factor count.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Handle the number 1 separately
The number 1 has exactly one positive factor, itself. It has no prime factors, so state this boundary case directly instead of forcing an ordinary non-empty prime-factor expression.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control examination work
Show the prime factorisation, write one bracket for each exponent, multiply the brackets and label the result positive factors. If time allows, check whether an odd count agrees with square status.
Build the explanation by naming complete prime factorisation and allowed exponent choices. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking multiplication principle and returning to positive-factor boundary. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 9 OF 12 · Back to contents
9. Worked examples
Worked case 1: How many positive factors does 72 have?
Controlled answer. 12. Since 72 = 2³ × 3², the count is (3+1)(2+1)=12.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. List factor pairs to verify. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “12” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: How many positive factors does 36 have?
Controlled answer. 9. Since 36 = 2² × 3², the count is 3 × 3 = 9.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Use its square status as a check. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “9” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: How many positive factors does 24 have?
Controlled answer. 8. Since 24 = 2³ × 3, the count is 4 × 2 = 8.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Identify the four factor pairs. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “8” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: How many positive factors does 100 have?
Controlled answer. 9. Since 100 = 2² × 5², the count is 3 × 3 = 9.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Explain why the count is odd. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “9” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: How many positive factors does 45 have?
Controlled answer. 6. Since 45 = 3² × 5, the count is 3 × 2 = 6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Write all factors as a check. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: How many positive factors does 48 have?
Controlled answer. 10. Since 48 = 2⁴ × 3, the count is 5 × 2 = 10.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Compare with 2⁵ alone. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “10” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: How many positive factors does 84 have?
Controlled answer. 12. Since 84 = 2² × 3 × 7, the count is 3 × 2 × 2 = 12.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Explain every exponent choice. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “12” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Repair: 72 has 3 + 2 = 5 factors.
Controlled answer. 72 has (3+1)(2+1)=12 positive factors. The exponents give counts of choices including zero, and independent choices are multiplied.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Show two combinations the addition method misses. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “72 has (3+1)(2+1)=12 positive factors” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Repair: 60 = 2² × 3 × 5, so it has 2 × 1 × 1 factors.
Controlled answer. It has 3 × 2 × 2 = 12 positive factors. Each exponent count needs one extra choice for exponent zero.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Name the factor created by all-zero choices. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “It has 3 × 2 × 2 = 12 positive factors” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: A number is p⁴ for prime p. How many positive factors?
Controlled answer. 5. The exponent may be 0, 1, 2, 3 or 4.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. List them symbolically. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “5” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: A number is p²q³ for distinct primes p and q. How many positive factors?
Controlled answer. 12. There are 3 choices for p’s exponent and 4 for q’s exponent.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Build a small choice table. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “12” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Why is 1 included in every positive-factor count?
Controlled answer. Every whole number is divisible by 1. It corresponds to choosing exponent zero for every prime.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Locate the original number in the exponent choices. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Every whole number is divisible by 1” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck complete prime factorisation, allowed exponent choices, multiplication principle, positive-factor boundary. Decide whether the evidence proves the conclusion or only supports a limited statement.
CHAPTER 10 OF 12 · Back to contents
10. Independent practice
Practice 1: How many positive factors does 120 have?
Explained answer. 16. Since 120 = 2³ × 3 × 5, the count is 4 × 2 × 2 = 16.
Delayed transfer. Check that 120 is not a square. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 2: How many positive factors does 96 have?
Explained answer. 12. Since 96 = 2⁵ × 3, the count is 6 × 2 = 12.
Delayed transfer. Compare its count with 72. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 3: A prime number has how many positive factors?
Explained answer. 2. Its factorisation p¹ gives exponent choices 0 and 1.
Delayed transfer. Contrast with 1 and a composite number. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 4: The count of positive factors is odd. What can be concluded?
Explained answer. The number is a perfect square. One factor, the square root, is unpaired while other factors form distinct pairs.
Delayed transfer. Test the claim on 36. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 5: Does every number with 12 factors have the same prime factorisation?
Explained answer. No. Patterns such as p¹¹, p⁵q¹ and p³q² can each produce 12 choices where valid.
Delayed transfer. Give small numerical examples cautiously. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 6: How many positive factors does 1 have?
Explained answer. 1. Its only positive factor is 1; treat it as a separate boundary case.
Delayed transfer. Explain why it is neither prime nor composite. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 7: What is the quickest audit after counting factors?
Explained answer. Check the prime factorisation and whether odd or even count matches square status. The audit catches missed primes, missing plus-one choices and impossible parity.
Delayed transfer. Apply it to 100 and 120. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 8: State the full method.
Explained answer. Prime-factorise completely, add one to each exponent, multiply the choice counts, include 1 and the number, then check square parity. The method follows the independent ways a factor can choose powers of each prime.
Delayed transfer. Create and solve a three-prime example. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
CHAPTER 11 OF 12 · Back to contents
11. Delayed transfer
On day one, solve clear examples. On day three, mix examples, near-nonexamples and incomplete evidence. On day seven, solve without headings or a word bank. Delay reveals whether the method can be reconstructed independently.
Keep an error log for the first failed stage: reading, representation, rule, execution or checking. The earliest failure identifies the repair.
Interleave this skill with neighbouring ideas that share vocabulary. Classify each item only after naming the distinguishing condition.
In timed work, mark the relationship, apply the rule, write the conclusion and audit one condition. If checking changes the answer, record the overlooked feature.
Create a teach-back card with a near-nonexample on the front and the repaired mechanism on the back. Explain why the tempting answer fails.
Finish with one basic example, one boundary case and one transfer case. Supply explanations that a classmate can verify.
For a second delayed test, reverse the direction: start from a conclusion and invent evidence that would justify it, then evidence that would be insufficient. This exposes the boundary between relevance and proof.
CHAPTER 12 OF 12 · Back to contents
12. Family guidance and FAQs
Guidance for parents and tutors
Ask for the decisive condition before confirming the answer. Use prompts such as “What exactly happened?”, “Which stage or meaning is this?” and “What would prove it?”
Fade support from labelled examples to mixed questions and delayed open response. Independent retrieval after time and topic change is the target.
Praise precise claim control. Narrowing an answer when evidence is incomplete is scientific and academic strength.
Frequently asked questions
Should I memorise the rule?
Memorise the compact rule, then practise examples and near-nonexamples so the conditions remain visible.
What if two answers seem possible?
Return to the exact relationship and choose the option satisfying every stated condition without adding facts.
Why delay practice?
Immediate repetition may depend on the recent explanation; delay tests reconstruction.
How should an error be corrected?
Name the first failed stage and practise it with a changed example.
When is the skill ready?
When the learner can explain, apply and transfer it after a delay.
What is the final audit?
Check relationship, mechanism, claim strength and whole-question fit.
Next routes
Continue at the Mathematics learning hub or return to the series directory.