PSLE Learner’s Guide · Volume 0830
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For a three-digit number 100a + 10b + c, subtracting a + b + c gives 99a + 9b = 9(11a + b), a multiple of 9.
Series routes: Use the complete PSLE Learner’s Guide volume directory, the original PSLE Learning Guide and the Mathematics learning hub.
This guide uses original Primary 6 practice and does not predict a live question. Consult the SEAB PSLE formats examined in 2026 page for official information.
Open the full chapter index
- 1. Represent the number by place value
- 2. Represent the digit sum
- 3. Subtract the expressions
- 4. Factor out 9
- 5. Extend the mechanism
- 6. Use the theorem as a check
- 7. Handle zero and repeated digits
- 8. Control examination reasoning
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Represent the number by place value
For a three-digit whole number, let a, b and c be its hundreds, tens and ones digits. Its value is 100a + 10b + c.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check place-value digit representation.
- Check sum of the digits.
- Check difference after subtraction.
- Check common factor of 9 and theorem limits.
CHAPTER 2 OF 12 · Back to contents
2. Represent the digit sum
The digit sum is a + b + c. It uses the digit values themselves, not their place-value contributions.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Subtract the expressions
The difference is 100a + 10b + c minus a + b + c, which simplifies to 99a + 9b because the c terms cancel.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Factor out 9
Write 99a + 9b as 9(11a + b). Since the bracket is a whole number, the difference is divisible by 9.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Extend the mechanism
For more digits, every place-value coefficient beyond the ones place differs from 1 by a multiple of 9: 10 minus 1, 100 minus 1 and 1000 minus 1.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Use the theorem as a check
A number and its digit sum leave the same remainder when divided by 9. This explains the familiar digit-sum divisibility check rather than treating it as magic.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Handle zero and repeated digits
Zeros contribute nothing to the digit sum but still hold places. Repeated digits are separate place contributions and do not change the proof.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control examination reasoning
Define digits, write the place-value expression, subtract with brackets, factor 9 and state the conclusion. Do not claim the number or digit sum must itself be divisible by 9.
Build the explanation by naming place-value digit representation and sum of the digits. 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 difference after subtraction and returning to common factor of 9 and theorem limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 9 OF 12 · Back to contents
9. Worked examples
Worked case 1: For 472, find the digit sum.
Controlled answer. 13. Add 4+7+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. Keep place values out of the digit sum. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “13” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Find 472 minus its digit sum.
Controlled answer. 459. Compute 472-13.
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. Check 459 ÷ 9. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “459” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Is 459 divisible by 9?
Controlled answer. Yes, 459 = 9 × 51. The theorem predicts this result.
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 multiplication. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes, 459 = 9 × 51” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Write a three-digit number using digits a, b and c.
Controlled answer. 100a + 10b + c. The coefficients are place values.
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 a is non-zero. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “100a + 10b + c” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Subtract the digit sum symbolically.
Controlled answer. 99a + 9b. The c terms cancel.
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. Show the brackets. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “99a + 9b” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Factor the result.
Controlled answer. 9(11a + b). Both remaining coefficients contain 9.
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. Expand to check. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “9(11a + b)” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: Why does c disappear?
Controlled answer. The ones contribution c is exactly the same as its contribution to the digit sum. Subtracting gives c-c=0.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Do not say the ones digit is irrelevant to the original number. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The ones contribution c is exactly the same as its contribution to the digit sum” 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Test 100.
Controlled answer. 100-(1+0+0)=99, divisible by 9. Zeros hold the tens and ones places.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Keep the number three-digit. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “100-(1+0+0)=99, divisible by 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Test 999.
Controlled answer. 999-27=972, divisible by 9. Nine times 108 is 972.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Use the algebra with a=b=c=9. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “999-27=972, divisible by 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: Repair: every number is divisible by 9.
Controlled answer. A number minus its digit sum is divisible by 9. The theorem concerns a difference, not every starting number.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Use 472 as a counterexample. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “A number minus its digit sum is divisible by 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Repair: the digit sum must be divisible by 9.
Controlled answer. The number and digit sum have the same remainder modulo 9. Either both are divisible by 9 or neither may be.
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. Compare 472 and 13. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The number and digit sum have the same remainder modulo 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Why do a number and its digit sum have the same remainder on division by 9?
Controlled answer. Their difference is a multiple of 9. Subtracting a multiple of 9 does not change the remainder.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Give a numerical example. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Their difference is a multiple of 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 place-value digit representation, sum of the digits, difference after subtraction, common factor of 9 and theorem limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
CHAPTER 10 OF 12 · Back to contents
10. Independent practice
Practice 1: Can the method test divisibility by 9?
Explained answer. Yes. If the digit sum is divisible by 9, the original number is too because their difference is a multiple of 9.
Delayed transfer. State both directions carefully. 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 same proof automatically work for divisibility by 8?
Explained answer. No. Place-value powers do not all differ from 1 by multiples of 8.
Delayed transfer. Use the appropriate divisibility rule. 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: How does the result extend to four digits?
Explained answer. 1000a+100b+10c+d minus a+b+c+d equals 999a+99b+9c. Every term has factor 9.
Delayed transfer. Factor it fully. 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 are examples not a complete proof?
Explained answer. They cover only selected digit choices. The variable place-value argument covers every allowed number.
Delayed transfer. State digit limits. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 5: What if the number has one digit?
Explained answer. The difference is zero, which is divisible by 9. The number equals its digit sum.
Delayed transfer. Include zero as a multiple of 9. 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 can a missing digit be checked?
Explained answer. Use the digit sum and the required remainder modulo 9. The theorem links the original number to its digit sum.
Delayed transfer. Keep the digit from 0 to 9. 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 nearest wrong shortcut?
Explained answer. Applying the digit-sum test without explaining why it works. The place-value difference supplies the mechanism.
Delayed transfer. Show one algebraic line. 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. Represent the number by place value, subtract the sum of its digits, simplify and factor 9; then conclude that the difference is a multiple of 9. Do not claim the starting number is always divisible by 9.
Delayed transfer. Prove the four-digit case independently. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
CHAPTER 11 OF 12 · Back to contents
11. Delayed transfer
On day one, solve clear examples. On day three, mix examples, near-nonexamples and incomplete evidence. On day seven, solve without headings or a word bank. Delay reveals whether the method can be reconstructed independently.
Keep an error log for the first failed stage: reading, representation, rule, execution or checking. The earliest failure identifies the repair.
Interleave this skill with neighbouring ideas that share vocabulary. Classify each item only after naming the distinguishing condition.
In timed work, mark the relationship, apply the rule, write the conclusion and audit one condition. If checking changes the answer, record the overlooked feature.
Create a teach-back card with a near-nonexample on the front and the repaired mechanism on the back. Explain why the tempting answer fails.
Finish with one basic example, one boundary case and one transfer case. Supply explanations that a classmate can verify.
For a second delayed test, reverse the direction: start from a conclusion and invent evidence that would justify it, then evidence that would be insufficient. This exposes the boundary between relevance and proof.
CHAPTER 12 OF 12 · Back to contents
12. Family guidance and FAQs
Guidance for parents and tutors
Ask for the decisive condition before confirming the answer. Use prompts such as “What exactly happened?”, “Which stage or meaning is this?” and “What would prove it?”
Fade support from labelled examples to mixed questions and delayed open response. Independent retrieval after time and topic change is the target.
Praise precise claim control. Narrowing an answer when evidence is incomplete is scientific and academic strength.
Frequently asked questions
Should I memorise the rule?
Memorise the compact rule, then practise examples and near-nonexamples so the conditions remain visible.
What if two answers seem possible?
Return to the exact relationship and choose the option satisfying every stated condition without adding facts.
Why delay practice?
Immediate repetition may depend on the recent explanation; delay tests reconstruction.
How should an error be corrected?
Name the first failed stage and practise it with a changed example.
When is the skill ready?
When the learner can explain, apply and transfer it after a delay.
What is the final audit?
Check relationship, mechanism, claim strength and whole-question fit.
Next routes
Continue at the Mathematics learning hub or return to the series directory.