How to Perform in PSLE | Learner’s Guide Vol 1010 | Mathematics: For Two Positive Whole Numbers, GCD × LCM Equals Their Product

CHAPTER 1 OF 12 · Back to contents

1. State the two-number identity

For positive whole numbers a and b, GCD(a,b) × LCM(a,b) = a × b. Write the numbers and the two named quantities before substituting.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

• Check the two positive whole numbers and whether the question really concerns exactly two numbers.
• Check their prime-factor exponents and the minimum or maximum chosen for GCD and LCM.
• Check the identity GCD times LCM equals the product and the divisibility checks on any recovered value.
• Check factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope.

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CHAPTER 2 OF 12 · Back to contents

2. See why prime factors prove it

For each prime, the GCD takes the smaller exponent and the LCM takes the larger exponent. Adding those exponents gives the total exponent in a × b.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 3 OF 12 · Back to contents

3. Use the identity to check work

After finding a GCD and LCM by listing or prime factors, multiply them. The result must equal the product of the original two numbers.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 4 OF 12 · Back to contents

4. Recover a missing LCM or GCD

Rearrange only after identifying the four quantities. LCM = a×b÷GCD, while GCD = a×b÷LCM.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 5 OF 12 · Back to contents

5. Recover a missing number carefully

If a, GCD and LCM are known, b = GCD×LCM÷a. The result must be a positive whole number and must actually produce the stated GCD and LCM.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 6 OF 12 · Back to contents

6. Check divisibility relationships

The GCD must divide both numbers. Both numbers must divide the LCM. These conditions detect many copied or rearrangement errors.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 7 OF 12 · Back to contents

7. Do not extend the formula blindly

The simple product identity is for two positive whole numbers. For three or more numbers, multiplying one overall GCD by one overall LCM does not generally equal the product.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 8 OF 12 · Back to contents

8. Audit the result

Confirm the scope, compute with prime factors or listing, apply the identity, test integrality and divisibility, and recalculate the GCD and LCM for any recovered pair.

Build the explanation by naming the two positive whole numbers and whether the question really concerns exactly two numbers and their prime-factor exponents and the minimum or maximum chosen for GCD and LCM. 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 identity GCD times LCM equals the product and the divisibility checks on any recovered value and returning to factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 9 OF 12 · Back to contents

9. Worked examples

Worked case 1: Find GCD and LCM of 12 and 18, then check the product.

Controlled answer. GCD 6 and LCM 36. Six times 36 is 216, equal to 12 times 18.

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 with prime factors. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 2: The numbers are 8 and 30. Their GCD is 2. Find the LCM.

Controlled answer. 120. LCM = 8 × 30 ÷ 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. Check that both numbers divide 120. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 3: The numbers are 24 and 40. Their LCM is 120. Find the GCD.

Controlled answer. 8. GCD = 24 × 40 ÷ 120.

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 common factors to 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 4: One number is 15, the GCD is 5 and the LCM is 60. Find the other number.

Controlled answer. 20. The missing number is 5 × 60 ÷ 15.

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. Recalculate GCD(15,20) and LCM(15,20). 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 5: One number is 14, the GCD is 7 and the LCM is 42. Find the other number.

Controlled answer. 21. The missing number is 7 × 42 ÷ 14.

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 all divisibility conditions. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 6: Check GCD 4 and LCM 48 for the numbers 12 and 16.

Controlled answer. Valid. Four times 48 equals 192, which is 12 times 16.

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 by prime factorisation. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 7: A learner claims GCD 6 and LCM 72 for 18 and 24. Is the pair valid?

Controlled answer. Yes. Six times 72 equals 432, the product of 18 and 24.

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. Confirm 6 divides both and both divide 72. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 8: A learner claims GCD 5 and LCM 60 for 15 and 20. Repair it.

Controlled answer. The LCM should be 60 and the GCD should be 5, so the claim is valid. Five times 60 equals 300, the product of the numbers.

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 by listing multiples and factors. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 9: A learner claims GCD 4 and LCM 40 for 12 and 20. Repair it.

Controlled answer. The LCM should be 60. Four times 40 is 160, not the product 240; the correct LCM is 60.

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 by listing multiples or prime factors. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 10: The product of two numbers is 540 and their GCD is 6. Find their LCM.

Controlled answer. 90. LCM = 540 ÷ 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. State what remains unknown about the individual pair. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 11: The product is 840 and the LCM is 120. Find the GCD.

Controlled answer. 7. GCD = 840 ÷ 120.

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 that a possible pair must both be multiples of 7. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 12: If two numbers are coprime, what does the identity become?

Controlled answer. Their LCM equals their product. The GCD is 1.

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 with 8 and 15. 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 positive whole numbers and whether the question really concerns exactly two numbers, their prime-factor exponents and the minimum or maximum chosen for GCD and LCM, the identity GCD times LCM equals the product and the divisibility checks on any recovered value, factor pair common factor common multiple prime exponent cancellation missing number integer condition verification and problem scope. Decide whether the evidence proves the conclusion or only supports a limited statement.

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CHAPTER 10 OF 12 · Back to contents

10. Independent practice

Practice 1: If one positive number divides the other, what are GCD and LCM?

Explained answer. The smaller is the GCD and the larger is the LCM. Their product still equals GCD times LCM.

Delayed transfer. Test with 6 and 24. 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: Does the identity identify a unique pair from GCD 2 and LCM 60?

Explained answer. No. Several pairs can share those values, although every valid pair has product 120.

Delayed transfer. Find two valid pairs. 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: Why must a recovered number be checked?

Explained answer. Algebraic division alone may produce a number that fails the stated GCD or LCM conditions if the data are inconsistent. The original relationships are constraints, not decoration.

Delayed transfer. Recompute both quantities. 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: Can the missing number be a decimal?

Explained answer. Not in this identity’s stated whole-number setting. The rule here assumes positive whole numbers.

Delayed transfer. Check the question domain. 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 minimum exponent plus maximum exponent work?

Explained answer. For each prime, the smaller and larger exponents together equal the sum of the two original exponents. That is exactly the exponent in the product a × b.

Delayed transfer. Demonstrate with 12 and 18. 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 GCD of three numbers times LCM of three numbers always equal their product?

Explained answer. No. The simple identity does not extend directly to three numbers.

Delayed transfer. Test 2, 3 and 6. Cover the answer later and rebuild the chain.

Control question. Which exact condition does the nearest wrong answer miss? State it before checking.

Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.

Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.

Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.

Practice 7: What is the fastest verification?

Explained answer. Multiply the proposed GCD and LCM and compare with the product of the two numbers. Unequal products prove an error.

Delayed transfer. Then check factor and multiple relationships. 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. Confirm there are exactly two positive whole numbers, find or identify their GCD and LCM, apply GCD times LCM equals the product, rearrange only for the requested quantity, and verify any result by divisibility prime factors and recalculating the pair. The method combines a structural identity with independent checks.

Delayed transfer. Apply after a delayed missing-number 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.

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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.

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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.

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