PSLE Learner’s Guide · Volume 0822
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Write a two-digit number as 10a + b and its reversal as 10b + a; their sum is 11(a + b), so it is divisible by 11.
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 original number
- 2. Represent the reversal
- 3. Add the expressions
- 4. Conclude divisibility correctly
- 5. Use the result as a fast check
- 6. Handle repeated and zero digits
- 7. Do not reverse the theorem
- 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 original number
Let a be the tens digit and b the ones digit. The original two-digit number is 10a + b, with a from 1 to 9 and b from 0 to 9.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check tens digit and ones digit.
- Check place-value expressions.
- Check common factor of 11.
- Check leading-zero and converse limits.
CHAPTER 2 OF 12 · Back to contents
2. Represent the reversal
Reversing the digits gives 10b + a. If b is zero, the written reversal may appear as a one-digit number, but 10 × 0 + a still represents its value correctly.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Add the expressions
The sum is (10a + b) + (10b + a) = 11a + 11b = 11(a + b). Factoring shows the complete sum contains 11.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Conclude divisibility correctly
Because a + b is a whole number, 11(a + b) is a multiple of 11. This proves the claim for every valid two-digit starting number, not only the examples checked.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Use the result as a fast check
Add a two-digit number and its reversal, then divide the sum by 11 or compare it with 11 times the digit sum. A mismatch reveals an addition or reversal error.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Handle repeated and zero digits
Numbers such as 44 reverse to themselves and give 88. A number such as 70 reverses in value to 7 and gives 77. Both follow the same place-value proof.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse limits. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Do not reverse the theorem
Every such sum is divisible by 11, but not every multiple of 11 must come from a unique two-digit reversal pair. Different pairs can share the same digit sum.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse 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 the digits, write both place-value expressions, factor 11 and state the domain. Avoid claiming that the original number or each digit must be divisible by 11.
Build the explanation by naming tens digit and ones digit and place-value expressions. 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 common factor of 11 and returning to leading-zero and converse 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: Find 23 + 32 and test divisibility by 11.
Controlled answer. 55, which is 5 × 11. The digit sum is 2 + 3 = 5.
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 with 11(a+b). Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “55, which is 5 × 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Find 47 + 74.
Controlled answer. 121, which is 11 × 11. The digit sum is 11.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Verify the addition column by column. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “121, which is 11 × 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Find 70 plus its reversal.
Controlled answer. 77. The reversal has value 7, or 07 in two 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. Explain the leading zero. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “77” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: What happens for 44?
Controlled answer. 44 + 44 = 88, a multiple of 11. Reversing equal digits gives the same 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. Keep the general proof unchanged. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “44 + 44 = 88, a multiple of 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Write 10a + b plus 10b + a in factored form.
Controlled answer. 11(a + b). Like terms give 11a + 11b.
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 common factor. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “11(a + 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Why must a be non-zero?
Controlled answer. It is the tens digit of a two-digit number. If a were zero, the original would not be a standard two-digit 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. Keep b allowed to be zero. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “It is the tens digit of a two-digit number” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: Repair: each digit must be divisible by 11.
Controlled answer. The sum of the two numbers is divisible by 11. Digits are between 0 and 9; the factor comes from place value.
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 algebra. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The sum of the two numbers is divisible by 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Repair: the original number must be a multiple of 11.
Controlled answer. Only the original plus its reversal is guaranteed to be a multiple of 11. Twenty-three is a counterexample to the stronger claim.
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 23 and 55. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Only the original plus its reversal is guaranteed to be a multiple of 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Does 12 + 21 fit the rule?
Controlled answer. Yes, the sum is 33. Thirty-three equals 11 times the digit sum 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. Use the formula. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes, the sum is 33” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: Does 98 + 89 fit the rule?
Controlled answer. Yes, the sum is 187 = 17 × 11. The digit sum is 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. Avoid multiplying the original digits. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Yes, the sum is 187 = 17 × 11” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Why is checking many examples not a complete proof?
Controlled answer. Examples do not cover every possible digit pair. The place-value argument works for all allowed a and b.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. State the domain. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Examples do not cover every possible digit pair” 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse limits. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Can two different pairs give the same sum?
Controlled answer. Yes. Any pairs with the same digit sum produce the same 11(a+b).
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 14 and 23. 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 tens digit and ones digit, place-value expressions, common factor of 11, leading-zero and converse 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: What is the smallest possible sum?
Explained answer. 11. Ten plus its reversal 1 gives 11.
Delayed transfer. Use a=1, b=0. 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 largest possible sum?
Explained answer. 198. Ninety-nine reverses to itself and 99 + 99 = 198.
Delayed transfer. Use a=9, b=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 3: Is every multiple of 11 from 11 to 198 obtainable?
Explained answer. Only those whose quotient can be a digit sum from 1 to 18. The digit sum sets the quotient.
Delayed transfer. Construct digits for a chosen 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 4: Is the representation 10b+a still valid when b=0?
Explained answer. Yes. It becomes a, matching the numerical value of the reversal.
Delayed transfer. Distinguish written digits from value. 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 quick check follows from the theorem?
Explained answer. The sum should equal 11 times the sum of the original digits. This checks both reversal and addition.
Delayed transfer. Apply it to 62 + 26. 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 found?
Explained answer. Set the given total equal to 11(a+b) and solve for the digit sum. Digit limits then restrict possibilities.
Delayed transfer. List all valid pairs without duplicates. 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. Assuming divisibility by 11 comes from matching digits alone. It comes from the place values 10a+b and 10b+a.
Delayed transfer. Factor the algebra. 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. Let the digits be a and b, write the numbers as 10a+b and 10b+a, add to get 11(a+b), and conclude the sum is divisible by 11. Handle a zero ones digit by place value and do not reverse the theorem.
Delayed transfer. Create and verify a fresh 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.
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.