PSLE Learner’s Guide · Volume 1058
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If a non-zero whole number divides two whole numbers exactly, it also divides their sum and their difference. The condition must hold for both original numbers.
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. State the condition fully
- 2. Build the factor proof
- 3. Require both original divisions
- 4. Keep difference direction controlled
- 5. Do not assume the converse
- 6. Use the property for checking
- 7. Connect to the HCF
- 8. Audit with remainders
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. State the condition fully
Let d be a non-zero whole number. If d divides a exactly and d divides b exactly, then d divides a plus b and the non-negative difference between a and b.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the two whole numbers candidate divisor exact quotients sum and non-negative difference.
- Check the factor representation a equals d times m and b equals d times n.
- Check whether the candidate divides both originals and whether a claimed converse has been tested.
- Check divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification.
CHAPTER 2 OF 12 · Back to contents
2. Build the factor proof
Write a equals d times m and b equals d times n. Then a plus b equals d times m plus n, while a minus b equals d times m minus n when a is at least b. Both results still contain d as a factor.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Require both original divisions
Knowing that d divides only one original number is insufficient. The other number may leave a remainder, and that remainder can remain in the sum or difference.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Keep difference direction controlled
For whole-number work, subtract smaller from larger unless a signed difference is intended. The common-divisor conclusion applies to the magnitude of the difference.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Do not assume the converse
A number can divide a sum without dividing either addend. For example, 5 divides 12 plus 8, but 5 divides neither 12 nor 8 exactly.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Use the property for checking
When a candidate is said to be a common divisor, test both originals, then use the sum and difference as confirmation. Agreement with only the derived values does not replace the original tests.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Connect to the HCF
Every common divisor of two numbers divides their difference. Repeatedly replacing a larger number by a difference can reveal the same highest common factor without changing the set of common divisors.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Audit with remainders
Divide both originals by d. Zero remainders combine to zero in the sum and difference. If either remainder is non-zero, the common-divisor condition has failed and the shortcut cannot be quoted.
Build the explanation by naming the two whole numbers candidate divisor exact quotients sum and non-negative difference and the factor representation a equals d times m and b equals d times n. 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 whether the candidate divides both originals and whether a claimed converse has been tested and returning to divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. 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: 6 divides 24 and 18. Does 6 divide their sum?
Controlled answer. Yes; 42 divided by 6 is 7. Both originals are multiples of 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. Check the difference as well. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Does 6 divide the difference between 24 and 18?
Controlled answer. Yes; the difference is 6. Subtracting two multiples of 6 leaves another multiple of 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. Represent both numbers as 6 times a whole number. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Prove the sum rule using a = dm and b = dn.
Controlled answer. a + b = d(m + n). The distributive property leaves d as a factor.
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 corresponding difference proof. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: 4 divides 28 but not 18. May the common-divisor rule be applied?
Controlled answer. No. The condition requires exact division of both originals.
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. Calculate the remainder of the sum to see the failure. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Does 5 dividing 12 + 8 prove that 5 divides 12 and 8?
Controlled answer. No. The sum is 20, but both addends leave non-zero remainders.
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 this as a counterexample to the converse. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: A number divides 45 and 30. Must it divide 75?
Controlled answer. Yes. Seventy-five is their sum.
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 all common divisors and verify. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: A number divides 45 and 30. Must it divide 15?
Controlled answer. Yes. Fifteen is their difference.
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. Connect the result to the HCF. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Why is the divisor required to be non-zero?
Controlled answer. Division by zero is not defined. The phrase divides exactly requires a valid non-zero divisor.
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. Keep zero as a possible original number instead. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Does every divisor of a difference divide both originals?
Controlled answer. No. For example, 7 divides the difference between 20 and 13, but divides neither original.
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 false converse. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: 8 divides 56 and 40. Find two derived multiples of 8.
Controlled answer. 96 and 16. The sum and difference preserve the common factor.
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. Divide both derived values by 8. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: If d divides a and b, does d divide 2a + b?
Controlled answer. Yes. Two a values and one b value are still a sum of multiples of d.
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. Extend to 3a – 2b when the result is non-negative. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: A learner checks only that 9 divides 63 + 27. What is missing?
Controlled answer. Separate checks that 9 divides 63 and 27. A divisible sum alone does not establish common divisibility.
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. Complete all three divisions. Solve the altered case from the mechanism rather than copying the previous wording.
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 the two whole numbers candidate divisor exact quotients sum and non-negative difference, the factor representation a equals d times m and b equals d times n, whether the candidate divides both originals and whether a claimed converse has been tested, divisor multiple factor exact division remainder sum difference absolute difference common divisor HCF proof distributive property converse counterexample condition and verification. 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: Find the HCF of 84 and 60 using a difference step.
Explained answer. 12. Replace 84 by 24; the common divisors of 84 and 60 are the common divisors of 60 and 24, whose HCF is 12.
Delayed transfer. Verify 12 against both originals. 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: Why does replacing the larger number by the difference preserve common divisors?
Explained answer. Every common divisor of the originals divides the difference, and a common divisor of the smaller number and difference also rebuilds the larger. Both directions can be checked by addition and subtraction.
Delayed transfer. Write the two implications 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: 3 divides 0 and 21. Does it divide their sum and difference?
Explained answer. Yes. Zero is divisible by every non-zero whole-number divisor, and both derived values are 21.
Delayed transfer. State why the divisor itself cannot be zero. 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: 7 divides 35 and 14. Does it divide 35 – 2×14?
Explained answer. Yes. The expression is a difference of multiples of 7 and equals 7.
Delayed transfer. Show the factorised form. 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: What remainder pattern proves the rule quickly?
Explained answer. Both original remainders are zero, so their sum and difference have remainder zero. Remainders preserve the condition.
Delayed transfer. Contrast remainders 2 and 3 modulo 5. 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: If two numbers leave the same remainder on division by d, what can be said about their difference?
Explained answer. Their difference is divisible by d. The equal remainders cancel.
Delayed transfer. Connect this related result without confusing it with common divisibility. 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: Can the property find a common divisor without testing the originals?
Explained answer. No. Derived divisibility can suggest a candidate but the original conditions still require confirmation.
Delayed transfer. Use a counterexample with a divisible sum. 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. Test that the non-zero candidate divides both whole numbers exactly, write each original as the divisor times a whole number, factor the divisor from the sum or non-negative difference, verify with division, and reject the converse unless the original divisions are separately established. The method preserves the condition and explains the mechanism.
Delayed transfer. Apply it after a delay to common-factor checking HCF reduction and a false-converse item. 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.