PSLE Learner’s Guide · Volume 0922
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The sum of the first n positive even numbers is n(n + 1); count the terms correctly and subtract earlier totals when the list starts after 2.
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. Recognise the exact sequence
- 2. Find the term count from the last term
- 3. Derive the formula by factoring two
- 4. Use the formula with labelled quantities
- 5. Reverse a stated total
- 6. Handle a later starting point
- 7. Compare with the odd-number identity
- 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. Recognise the exact sequence
The direct formula applies to 2, 4, 6 and so on without gaps. A sequence beginning at 8 is still consecutive even numbers, but it requires subtracting the earlier total 2+4+6.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the consecutive even sequence and its endpoints.
- Check term count n and last term 2n.
- Check the total n times n plus one.
- Check starts-at-two condition reverse reasoning subtraction estimation and unit labels.
CHAPTER 2 OF 12 · Back to contents
2. Find the term count from the last term
The nth positive even number is 2n. If the last term is 40, then n=20. This count, not the last term itself, enters n(n+1).
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Derive the formula by factoring two
Write 2+4+…+2n as 2(1+2+…+n). Since 1+2+…+n=n(n+1)/2, multiplying by 2 gives n(n+1).
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Use the formula with labelled quantities
For the first 15 even numbers, n=15, the last term is 30 and the sum is 15×16=240. Keep the count, endpoint and total separate.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Reverse a stated total
If the first n even numbers sum to 210, solve n(n+1)=210. Consecutive factors 14 and 15 give n=14, so the last term is 28.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Handle a later starting point
To add 8+10+…+24, take the total from 2 to 24 and subtract the total from 2 to 6. This is 12×13−3×4.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Compare with the odd-number identity
The first n odd numbers sum to n², while the first n even numbers sum to n(n+1). The even total exceeds the odd total by n because each corresponding even term is one larger.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. 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
Verify consecutiveness, find n, use n(n+1), subtract an earlier total if necessary and check that the answer lies near number-of-terms times average term.
Build the explanation by naming the consecutive even sequence and its endpoints and term count n and last term 2n. A remembered keyword is not enough: connect the observation to the mechanism and identify what would have to change for the conclusion to change.
Repair the nearest misconception by checking the total n times n plus one and returning to starts-at-two condition reverse reasoning subtraction estimation and unit labels. 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 2+4+6+8+10.
Controlled answer. 30. There are five terms, so 5×6=30.
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 by direct addition. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “30” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Find the sum of the first 12 even numbers.
Controlled answer. 156. Use 12×13.
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. State the last term. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “156” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: What is the 25th positive even number?
Controlled answer. 50. Use 2n.
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 sum through 50. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “50” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Find 2+4+…+50.
Controlled answer. 650. There are 25 terms, so 25×26=650.
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. Estimate with an average. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “650” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: How many terms are in 2+4+…+64?
Controlled answer. 32. Solve 2n=64.
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 total. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “32” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: A first-even-number sum is 132. How many terms?
Controlled answer. 11. Eleven times twelve is 132.
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 term. 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: A first-even-number sum has 18 terms. What is the last term?
Controlled answer. 36. The nth even term is 2n.
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 total. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “36” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Find 8+10+12+14.
Controlled answer. 44. Use 7×8−3×4=44 or add directly.
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 subtraction boundary. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “44” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Find 22+24+…+40.
Controlled answer. 310. Use 20×21−10×11=310.
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. Count the included terms. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “310” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: Why does n(n+1) appear?
Controlled answer. Factoring two converts the sum to twice a triangular-number sum. The factor two cancels the division by two.
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 the algebra. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Factoring two converts the sum to twice a triangular-number 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 the consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Repair: 4+6+8+10=4×5.
Controlled answer. The sum is 28, not 20. The direct formula assumes the sequence starts at 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. Use 5×6−1×2. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The sum is 28, not 20” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Repair: the tenth even number is 10.
Controlled answer. The tenth positive even number is 20. Use 2n.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. List the first few terms. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “The tenth positive even number is 20” 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 consecutive even sequence and its endpoints, term count n and last term 2n, the total n times n plus one, starts-at-two condition reverse reasoning subtraction estimation and unit labels. 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 100 be the sum of the first n even numbers?
Explained answer. No. No consecutive whole numbers n and n+1 multiply to 100.
Delayed transfer. Locate 9×10 and 10×11. 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: Can 240 be such a sum?
Explained answer. Yes, with 15 terms. 15×16=240.
Delayed transfer. Find the last term. 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: Find the difference between the first 30-even sum and first 29-even sum.
Explained answer. 60. The difference is the 30th even number.
Delayed transfer. Verify using products. 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: How much larger is the first 20-even sum than the first 20-odd sum?
Explained answer. 20. Corresponding even terms are each one larger.
Delayed transfer. Compare formulas. 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 is the average of the first n even numbers?
Explained answer. n+1. The endpoints are 2 and 2n, whose average is n+1.
Delayed transfer. Multiply by n terms. 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: Why label n?
Explained answer. It is the number of terms, not the last term or total. Confusing these creates oversized products.
Delayed transfer. Make a three-label table. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 7: What is the nearest wrong shortcut?
Explained answer. Multiplying the last term by the next number. The formula uses term count n.
Delayed transfer. Use last term 12 as a counterexample. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 8: State the full method.
Explained answer. Confirm consecutive evens, find n from the count or last term 2n, use n(n+1), and subtract earlier even totals for a later start. Check with count times average.
Delayed transfer. Create a reverse problem. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
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.