PSLE Learner’s Guide · Volume 0814
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Three consecutive whole numbers cover remainders 0, 1 and 2 in some order when divided by 3, so exactly one is divisible by 3.
Series routes: Use the complete PSLE Learner’s Guide volume directory, the original PSLE Learning Guide and the Mathematics learning hub.
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. Define consecutive precisely
- 2. Use the remainder cycle
- 3. Prove exactly one term is divisible
- 4. Locate the divisible member efficiently
- 5. Use the rule for products
- 6. Use the rule for counting blocks
- 7. Do not overextend the rule
- 8. Control examination reasoning
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Define consecutive precisely
Three consecutive whole numbers have the form n, n + 1 and n + 2. Their difference is one at each step; three merely nearby or ordered values do not qualify.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check consecutive step of one.
- Check complete remainder cycle modulo 3.
- Check exactly one divisible term.
- Check scope of the three-number block.
CHAPTER 2 OF 12 · Back to contents
2. Use the remainder cycle
A whole number leaves remainder 0, 1 or 2 when divided by 3. Adding one moves to the next remainder, so three consecutive numbers visit all three remainders in some order.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Prove exactly one term is divisible
Among the three remainders, only 0 marks divisibility by 3. Because the block contains each remainder once, exactly one of its members is divisible by 3.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Locate the divisible member efficiently
Divide the first number by 3. If its remainder is 0, choose the first; if 1, choose the third; if 2, choose the second. No full division of all three terms is required.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Use the rule for products
The product of three consecutive whole numbers contains a factor divisible by 3, so the product is divisible by 3. This conclusion does not say each factor is divisible by 3.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Use the rule for counting blocks
When a consecutive range is split into complete blocks of three, each complete block contributes one multiple of 3. Handle any leftover one or two numbers separately.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Do not overextend the rule
The claim needs three consecutive whole numbers. It does not apply automatically to three consecutive odd numbers, three arbitrary numbers or a block whose step is not one.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control examination reasoning
Write the consecutive form, identify the remainder cycle, state exactly one rather than at least one, and keep product or counting conclusions separate from the original divisibility claim.
Build the explanation by naming consecutive step of one and complete remainder cycle modulo 3. 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 exactly one divisible term and returning to scope of the three-number block. 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: Which of 25, 26 and 27 is divisible by 3?
Controlled answer. 27. The remainders are 1, 2 and 0.
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. Check the next block. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “27” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Which of 38, 39 and 40 is divisible by 3?
Controlled answer. 39. The middle term has remainder 0.
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 from the first term’s remainder. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “39” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Which of 51, 52 and 53 is divisible by 3?
Controlled answer. 51. The first term is already a multiple of 3.
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 the quotient. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “51” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Are 9, 11 and 13 consecutive whole numbers?
Controlled answer. No. The step is two, not one.
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. Classify them as consecutive odd numbers instead. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “No” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Repair: all three consecutive numbers are divisible by 3.
Controlled answer. Exactly one is divisible by 3. The other two leave remainders 1 and 2.
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 an example to refute the claim. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Exactly one is divisible by 3” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Repair: at least one of three consecutive numbers is divisible by 3.
Controlled answer. The stronger exact statement is that exactly one is divisible. A complete block contains one of each remainder.
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 two cannot occur. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The stronger exact statement is that exactly one is divisible” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: If n leaves remainder 1, which of n, n+1, n+2 is divisible by 3?
Controlled answer. n + 2. The remainder sequence is 1, 2, 0.
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. Test n = 10. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “n + 2” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: If n leaves remainder 2, which term is divisible?
Controlled answer. n + 1. The next remainder is 0.
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. Test n = 14. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “n + 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: If n is divisible by 3, which term is divisible?
Controlled answer. n. The following remainders are 1 and 2.
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. Test n = 21. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “n” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: Is the product 14 × 15 × 16 divisible by 3?
Controlled answer. Yes. Fifteen supplies a factor of 3.
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. Do not claim the other factors are multiples of 3. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Why is any product of three consecutive whole numbers divisible by 3?
Controlled answer. Exactly one factor in the block is divisible by 3. A product containing a multiple of 3 is divisible by 3.
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. Apply the rule without multiplying. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Exactly one factor in the block is divisible by 3” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Do three consecutive multiples of 2 contain exactly one multiple of 3?
Controlled answer. Not always under this rule. Their step is two, so they are not consecutive whole numbers as defined.
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. Test 2, 4, 6 and 4, 6, 8. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Not always under this rule” 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 consecutive step of one, complete remainder cycle modulo 3, exactly one divisible term, scope of the three-number block. 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 multiples of 3 are in 1 through 12?
Explained answer. Four. There are four complete blocks of three.
Delayed transfer. List them as a check. 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 multiples of 3 are in 1 through 14?
Explained answer. Four. The four complete blocks contribute four; 13 and 14 add none.
Delayed transfer. Handle the leftover terms explicitly. 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: Can two adjacent whole numbers both be divisible by 3?
Explained answer. No. Distinct multiples of 3 differ by at least 3.
Delayed transfer. Connect this to exactly one in the block. 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: What are the possible remainders on division by 3?
Explained answer. 0, 1 and 2. A remainder must be non-negative and smaller than the divisor.
Delayed transfer. Order them in a cycle. 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: Why does adding 1 cycle the remainder?
Explained answer. It moves to the next residue, with 2 followed by 0. Crossing a multiple of 3 resets the remainder.
Delayed transfer. Illustrate with 8, 9, 10. 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: Does exactly one divisible term identify its position without knowing n?
Explained answer. No. It may be first, second or third depending on the starting remainder.
Delayed transfer. State the three cases. 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 condition should be checked first?
Explained answer. That the values are whole numbers increasing by one. Without that structure the remainder-cycle proof may not apply.
Delayed transfer. Repair a nonexample. 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. Write n, n+1 and n+2, use the remainder cycle 0, 1, 2, and conclude exactly one term is divisible by 3. Keep the conclusion tied to a block of three consecutive whole numbers.
Delayed transfer. Use it once for a product and once for counting. 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.