How to Perform in PSLE | Learner’s Guide Vol 0762 | Mathematics: Test Divisibility by 75 With the Conditions for 3 and 25

CHAPTER 1 OF 12 · Back to contents

1. Factor the target

Since 75 equals 3 × 25 and 3 and 25 are coprime, a whole number is divisible by 75 exactly when it passes both factor tests.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. State why the tempting alternative is too strong, incomplete or about a different stage.

• Check coprime factor conditions.
• Check digit-sum test for 3.
• Check final-two-digit test for 25.
• Check complete candidate verification.

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CHAPTER 2 OF 12 · Back to contents

2. Test divisibility by 3

Add all digits. The number is divisible by 3 when that digit sum is divisible by 3.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Test divisibility by 25

Inspect the final two digits. They must be 00, 25, 50 or 75 for a whole number to be divisible by 25.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Require both conditions

Passing the digit-sum test alone or having a valid ending alone is insufficient. The two conditions form an AND statement.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Reject efficiently

Check the last two digits first when an ending is obviously invalid; otherwise use whichever condition gives the fastest certain rejection.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Construct missing digits systematically

Restrict the ending to 00, 25, 50 or 75, then apply the digit-sum condition to every remaining candidate.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Verify with division

After a candidate passes the tests, divide by 75 when a quotient is useful or when checking a constructed list.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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. Control examination work

State both tests, show the relevant digits and sum, keep all candidates, then confirm that the list is complete.

Build the explanation by naming coprime factor conditions and digit-sum test for 3. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.

Repair the nearest misconception by checking final-two-digit test for 25 and returning to complete candidate verification. 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: Is 2,325 divisible by 75?

Controlled answer. Yes. Its ending 25 is divisible by 25 and its digit sum 12 is divisible by 3; 2,325 ÷ 75 = 31.

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. Change the hundreds digit and retest. 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 2: Is 1,250 divisible by 75?

Controlled answer. No. The ending 50 passes 25, but the digit sum 8 fails 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. Find the nearest larger multiple of 75. 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 3: Is 4,500 divisible by 75?

Controlled answer. Yes. The ending 00 passes 25 and the digit sum 9 passes 3; the quotient 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. Explain why the two tests are sufficient. 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 4: Is 975 divisible by 75?

Controlled answer. Yes. The ending 75 passes 25 and the digit sum 21 passes 3; the quotient is 13.

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. Create another three-digit example ending 75. 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 5: Repair: 1,125 is not divisible by 75 because it does not end in 75.

Controlled answer. 1,125 is divisible by 75. A multiple of 25 may end in 00, 25, 50 or 75; its digit sum 9 also passes 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. State its quotient. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “1,125 is divisible by 75” 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 6: Repair: Every number ending 50 is divisible by 75.

Controlled answer. A number ending 50 must also have a digit sum divisible by 3. The ending proves only divisibility by 25.

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 one example and one counterexample. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “A number ending 50 must also have a digit sum divisible by 3” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.

Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.

Audit. Recheck coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 7: The number 12□5 is divisible by 75. Find the digit.

Controlled answer. 7. Only endings 25 and 75 can pass 25; □ = 2 gives digit sum 10, while □ = 7 gives digit sum 15 and 1,275 ÷ 75 = 17.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. Explain why 2 fails despite ending 25. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “7” 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 8: The number 4□50 is divisible by 75. Find all digits.

Controlled answer. 0, 3, 6 or 9. The ending 50 passes 25; the digit sum 9 + □ must be divisible by 3.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. Verify the four quotients. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “0, 3, 6 or 9” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.

Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.

Audit. Recheck coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 9: The number 3□00 is divisible by 75. Find all digits.

Controlled answer. 0, 3, 6 or 9. The ending 00 passes 25; the digit sum 3 + □ must be divisible by 3.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. List the four quotients. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “0, 3, 6 or 9” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.

Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.

Audit. Recheck coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 10: What are the only possible final two digits?

Controlled answer. 00, 25, 50 or 75. These are the two-digit endings divisible by 25.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. Explain why 05 is not included. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “00, 25, 50 or 75” 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 11: Why is divisibility by 5 insufficient?

Controlled answer. A number can end in 0 or 5 without its final two digits being divisible by 25. For example, 30 passes 5 but fails 25.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. Name the stronger ending test. Solve the altered case from the mechanism rather than copying the previous wording.

Reverse construction. Begin with the controlled answer “A number can end in 0 or 5 without its final two digits being divisible by 25” 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 12: Does divisibility by 15 and 5 prove divisibility by 75?

Controlled answer. No. The least common multiple of 15 and 5 is only 15; the extra factor 5 is not secured.

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 a counterexample. 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 coprime factor conditions, digit-sum test for 3, final-two-digit test for 25, complete candidate verification. 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: Does divisibility by 3 and 25 prove divisibility by 75?

Explained answer. Yes. The factors are coprime and their product is 75.

Delayed transfer. Express this with a least common multiple. 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: What is the fastest rejection for 98,765?

Explained answer. Reject it because the ending 65 is not divisible by 25. There is no need to complete every test after a decisive failure.

Delayed transfer. Check the digit sum as a separate exercise. Cover the answer later and rebuild the chain.

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

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

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

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

Practice 3: Can a multiple of 75 end in 00?

Explained answer. Yes. Numbers such as 300 and 1,500 pass both 25 and 3.

Delayed transfer. State the smallest positive example ending 00. 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 a multiple of 75 end in 25?

Explained answer. Yes. Examples include 225 and 525 when their digit sums also pass 3.

Delayed transfer. Construct a four-digit 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 5: Repair: 750 passes because 7 + 5 + 0 is divisible by 3.

Explained answer. 750 passes because its digit sum is divisible by 3 and its ending 50 is divisible by 25. Both required conditions must be stated.

Delayed transfer. Find the quotient. Cover the answer later and rebuild the chain.

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

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

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

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

Practice 6: How do you prove a candidate list is complete?

Explained answer. Enumerate all valid endings and then apply the digit-sum restriction to every remaining digit choice. Systematic filtering prevents omissions.

Delayed transfer. Build a small table. 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 check catches an arithmetic slip?

Explained answer. Divide each final candidate by 75 and confirm a whole-number quotient. The division verifies the constructed result.

Delayed transfer. Estimate the quotient first. 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. Require a digit sum divisible by 3 and final two digits 00, 25, 50 or 75; filter missing digits systematically and verify candidates. These factor conditions are jointly equivalent to divisibility by 75.

Delayed transfer. Create a missing-digit question with two solutions. 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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