How to Perform in PSLE | Learner’s Guide Vol 0954 | Mathematics: Numbers With the Same Remainder Have a Difference Divisible by the Divisor

CHAPTER 1 OF 12 · Back to contents

1. Write each division statement

If A and B leave remainder r when divided by d, write A=dq+r and B=dp+r. The quotients may differ, but the divisor and remainder are shared.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

• Check the two dividends and common divisor.
• Check the equal remainder removed by subtraction.
• Check the difference as a multiple of the divisor.
• Check remainder bounds HCF candidate divisors zero differences and reverse verification.

Contents · Next

CHAPTER 2 OF 12 · Back to contents

2. Subtract to remove the remainder

A−B=d(q−p). The equal +r terms cancel, so the difference is divisible by d.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 3 OF 12 · Back to contents

3. Use the difference to find candidates

Any possible common divisor must be a factor of the positive difference. This narrows a search but does not by itself prove that both original remainders match.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 4 OF 12 · Back to contents

4. Check the remainder bound

A remainder must satisfy 0≤r

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 5 OF 12 · Back to contents

5. Extend to several numbers

If several numbers leave the same remainder, their pairwise differences are all divisible by d. The greatest possible divisor often comes from the HCF of the differences, followed by a remainder check.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 6 OF 12 · Back to contents

6. Keep same remainder separate from same quotient

Equal remainders do not require equal quotients. The numbers can be far apart by one or more multiples of the divisor.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 7 OF 12 · Back to contents

7. Reverse the relationship carefully

If d divides A−B, then A and B have the same remainder when divided by d. This reverse statement works because numbers differing by a multiple of d occupy the same remainder position.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 8 OF 12 · Back to contents

8. Control examination work

Subtract, factor or find an HCF, reject divisors that violate the remainder bound, test every stated number and explain why subtraction removed the shared remainder.

Build the explanation by naming the two dividends and common divisor and the equal remainder removed by subtraction. 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 the difference as a multiple of the divisor and returning to remainder bounds HCF candidate divisors zero differences and reverse verification. State why the tempting alternative is too strong, incomplete or about a different stage.

Contents · Previous · Next

CHAPTER 9 OF 12 · Back to contents

9. Worked examples

Worked case 1: 35 and 59 leave the same remainder when divided by d. What must divide their difference?

Controlled answer. d. Their difference 24 equals a whole number of groups 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. List candidate divisors of 24. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “d” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 2: If the common remainder is 3, which candidate divisors of 24 remain possible?

Controlled answer. 4, 6, 8, 12 and 24. A divisor must be greater than remainder 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. Test both original numbers. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “4, 6, 8, 12 and 24” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 3: Do 35 and 59 leave the same remainder when divided by 8?

Controlled answer. Yes. Both leave remainder 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. Verify the difference method. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 4: Do they leave the same remainder when divided by 6?

Controlled answer. Yes. Both leave remainder 5, and 24 is divisible by 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. Note that the common remainder changed with the divisor. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 5: Repair: the divisor must equal the difference.

Controlled answer. The divisor must be a factor of the difference. Several divisors can divide the same 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. Give three valid factors. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “The divisor must be a factor of the difference” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 6: Two numbers differ by 42. Can they have the same remainder when divided by 7?

Controlled answer. Yes. Forty-two is divisible by seven.

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. Construct a pair and verify. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 7: Can they have the same remainder when divided by 8?

Controlled answer. No. Forty-two is not divisible by eight.

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 by quotient-remainder form. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 8: 18, 48 and 78 have what useful differences?

Controlled answer. 30, 30 and 60. A common divisor for equal remainders must divide all differences.

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. Find the HCF. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “30, 30 and 60” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 9: What is the greatest divisor for which 18, 48 and 78 have the same remainder?

Controlled answer. 30. The HCF of the differences is 30, and each number leaves remainder 18, which is below 30.

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. Verify each division. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “30” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 10: Why is divisor 6 also valid for those three numbers?

Controlled answer. Six divides every difference. All three numbers leave remainder 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. List all valid divisors greater than any stated remainder condition. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “Six divides every difference” 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 11: If the stated common remainder is 8, can divisor 5 work?

Controlled answer. No. A remainder cannot be equal to or greater than its 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. State the bound. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 12: Numbers 47 and 71 differ by 24. Do they have the same remainder modulo 12?

Controlled answer. Yes. Their difference is two groups of twelve; both leave remainder 11.

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 modulo 4. 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 the two dividends and common divisor, the equal remainder removed by subtraction, the difference as a multiple of the divisor, remainder bounds HCF candidate divisors zero differences and reverse verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Contents · Previous · Next

CHAPTER 10 OF 12 · Back to contents

10. Independent practice

Practice 1: If A=B, what is their difference?

Explained answer. 0. Every non-zero divisor divides zero, and identical numbers have identical remainders.

Delayed transfer. Explain why the divisor still 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 2: Can the divisor be zero?

Explained answer. No. Division by zero is undefined.

Delayed transfer. Keep zero difference separate from zero divisor. 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: Three numbers have pairwise differences 36, 60 and 96. What is the greatest possible shared-remainder divisor before checking bounds?

Explained answer. 12. The HCF of 36, 60 and 96 is 12.

Delayed transfer. Test a supplied remainder against 12. 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: Why is checking one difference sometimes insufficient for three numbers?

Explained answer. A divisor may divide one pair’s difference but not the others. Every number must occupy the same remainder class.

Delayed transfer. Use differences from one reference 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 5: Reverse: 11 divides A−B. What follows?

Explained answer. A and B leave the same remainder when divided by 11. Their difference consists of whole groups of eleven.

Delayed transfer. Demonstrate with concrete numbers. 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: What is the nearest wrong shortcut?

Explained answer. Choosing the HCF and stopping. The candidate divisor and any stated remainder must still be checked in the original divisions.

Delayed transfer. Add a bound failure 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.

Practice 7: How can a model show the rule?

Explained answer. Arrange both numbers into groups of d and compare the equal leftovers. Subtracting removes matching leftovers and complete common groups remain.

Delayed transfer. Draw a bar model. 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. Subtract numbers with equal remainders, find factors or the HCF of differences, enforce 0≤r. Explain the cancellation rather than quoting a trick.

Delayed transfer. Solve a delayed reverse problem. 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.

Contents · Previous · Next

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.

Contents · Previous · Next

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.

Contents · Previous