PSLE Learner’s Guide · Volume 0938
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To count multiples of k from a to b inclusive, count multiples up to b and subtract multiples up to a−1.
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. Count from one to an endpoint
- 2. Protect the lower boundary
- 3. Use the interval formula
- 4. Check with first and last multiples
- 5. Handle endpoints deliberately
- 6. Avoid counting gaps as items
- 7. Use complements and divisibility
- 8. Control the examination solution
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Count from one to an endpoint
The number of positive multiples of k not exceeding b is the whole-number quotient when b is divided by k. For k=6 and b=40, the multiples are 6 through 36, so the count is 6.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the divisor k and inclusive lower and upper endpoints.
- Check whole-number quotients for multiples up to each boundary.
- Check count through b minus count through a minus one.
- Check zero negative ranges endpoint multiples units and a short direct list.
CHAPTER 2 OF 12 · Back to contents
2. Protect the lower boundary
Subtract multiples up to a−1, not up to a. This keeps a in the interval when a itself is a multiple.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Use the interval formula
For positive integers with 1≤a≤b, count = floor(b÷k)−floor((a−1)÷k), where floor means keep only the whole-number quotient.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Check with first and last multiples
Find the smallest multiple at least a and largest multiple at most b. Their difference divided by k plus one should match the quotient method.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Handle endpoints deliberately
If the lower endpoint is not a multiple, moving from a−1 to a changes no count. If it is a multiple, the subtraction boundary is decisive.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Avoid counting gaps as items
After locating first and last multiples, difference÷step gives gaps. Add one to include both endpoint multiples.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Use complements and divisibility
A total interval count minus multiples can count non-multiples. Keep the same inclusive universe and verify that wanted plus unwanted equals all integers.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control the examination solution
Write the interval and divisor, compute both whole-number quotients, subtract in the correct order, list a small sample and test whether endpoint multiples were included.
Build the explanation by naming the divisor k and inclusive lower and upper endpoints and whole-number quotients for multiples up to each boundary. 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 count through b minus count through a minus one and returning to zero negative ranges endpoint multiples units and a short direct list. 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: How many multiples of 5 are from 1 to 30 inclusive?
Controlled answer. 6. 30÷5 gives six whole multiples.
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 5 through 30. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: How many multiples of 5 are from 6 to 30 inclusive?
Controlled answer. 5. Count through 30 is 6; count through 5 is 1; subtract to get 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 the first is 10. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “5” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: How many multiples of 5 are from 5 to 30 inclusive?
Controlled answer. 6. Use count through 30 minus count through 4, giving 6−0.
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. Notice why a−1 matters. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: How many multiples of 7 are from 20 to 60 inclusive?
Controlled answer. 6. Whole quotients are 8 and 2, so 8−2=6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. List 21,28,35,42,49,56. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: How many multiples of 4 are from 13 to 39 inclusive?
Controlled answer. 6. Whole quotients are 9 and 3, so 9−3=6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Check first 16 and last 36. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: How many multiples of 9 are from 1 to 100 inclusive?
Controlled answer. 11. The whole-number quotient of 100÷9 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. Find the last multiple 99. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “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 the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: How many multiples of 12 are from 25 to 95 inclusive?
Controlled answer. 5. Whole quotients are 7 and 2, so 7−2=5; the multiples are 36,48,60,72,84.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Verify by listing. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “5” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: How many multiples of 8 are from 18 to 70 inclusive?
Controlled answer. 6. Whole quotients are 8 and 2, so 8−2=6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. List 24 through 64. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: How many integers from 1 to 50 are not multiples of 6?
Controlled answer. 42. There are 50 integers and 8 multiples of 6.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Check wanted plus unwanted. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “42” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: How many multiples of 3 are from 10 to 10 inclusive?
Controlled answer. 0. Ten is not 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. Test a one-number interval containing 12. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “0” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Repair: Count multiples from 20 to 60 by 60÷7−20÷7.
Controlled answer. Use whole-number quotients through 60 and through 19. Subtracting through 20 can wrongly remove an included lower endpoint in other cases.
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. Write a−1. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Use whole-number quotients through 60 and through 19” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Repair: multiples from 16 to 36 by 4 give (36−16)÷4=5.
Controlled answer. There are 6 multiples. Five is the number of gaps; add one.
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 both endpoints. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “There are 6 multiples” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the divisor k and inclusive lower and upper endpoints, whole-number quotients for multiples up to each boundary, count through b minus count through a minus one, zero negative ranges endpoint multiples units and a short direct list. 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 does the whole-number quotient count?
Explained answer. Complete groups of k from 1 up to the endpoint. Each complete group ends at one multiple.
Delayed transfer. Connect quotient to list position. 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: Why use a−1?
Explained answer. It counts everything strictly below the inclusive lower endpoint. Then an endpoint multiple is retained.
Delayed transfer. Test a=5 and k=5. 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: What if there is no multiple in the interval?
Explained answer. The count is zero. The first possible multiple lies beyond the upper endpoint.
Delayed transfer. Check a one-number interval. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 4: Can the answer exceed the number of integers in the interval?
Explained answer. No. Multiples form a subset of the interval.
Delayed transfer. Use this as a bound check. 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: How can first and last multiples verify the answer?
Explained answer. Use (last−first)÷k+1 when at least one exists. This is an independent inclusive count.
Delayed transfer. Compare methods. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 6: What is the nearest wrong shortcut?
Explained answer. Subtracting raw decimal quotients or forgetting the lower-bound adjustment. Only complete multiples count.
Delayed transfer. Keep whole-number quotients. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 7: How should non-multiples be counted?
Explained answer. Subtract the multiple count from the total number of integers in the same interval. The partition must be complete and disjoint.
Delayed transfer. Find the total as b−a+1. 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. For positive integers, count complete multiples through the upper endpoint, subtract complete multiples through one less than the lower endpoint, then verify with first and last multiples and inclusive bounds. Correct any mismatch before proceeding.
Delayed transfer. Apply to a new divisor. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
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.