PSLE Learner’s Guide · Volume 0950
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From each vertex, exclude itself and its two neighbours, then divide by two because every diagonal was named from both endpoints.
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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.
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CHAPTER 1 OF 12 · Back to contents
1. Define a diagonal precisely
A diagonal joins two non-adjacent vertices of a polygon. A side joins adjacent vertices, so sides must be excluded even though they also connect vertex pairs.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the number of polygon vertices.
- Check eligible non-neighbour endpoints from each vertex.
- Check the duplicate count from reversing each endpoint pair.
- Check polygon type sides versus diagonals small-case checks and whole-number verification.
CHAPTER 2 OF 12 · Back to contents
2. Count from one vertex
In an n-sided polygon, a chosen vertex cannot join to itself and its two neighbouring vertices by a diagonal. It therefore has n−3 eligible diagonal endpoints.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Count from every vertex
Repeating the one-vertex count gives n(n−3) endpoint descriptions. This is not yet the number of distinct diagonals because each segment is described from both ends.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Divide by two
The diagonal AC is the same segment as CA. Every distinct diagonal appears exactly twice in the endpoint count, so the total is n(n−3)/2.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Check small polygons
A triangle has zero diagonals, a quadrilateral has two and a pentagon has five. These familiar cases expose formulas that accidentally include sides or omit the division.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Separate sides and diagonals
The number of sides is n, while the number of diagonals grows faster. If a question asks for all segments joining vertex pairs, add sides and diagonals or use n(n−1)/2 directly.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Use algebra in reverse
When the diagonal count is given, solve n(n−3)/2=D and keep only a whole number n of at least three. Test the value in the original relationship.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Control examination work
State the one-vertex exclusions, multiply by the number of vertices, divide for reversed pairs and verify the result against a sketch or a nearby known polygon.
Build the explanation by naming the number of polygon vertices and eligible non-neighbour endpoints from each vertex. 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 duplicate count from reversing each endpoint pair and returning to polygon type sides versus diagonals small-case checks and whole-number verification. 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 diagonals has a triangle?
Controlled answer. 0. 3(3−3)/2=0, and every pair of vertices is already a side.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Explain why a triangle cannot contain a diagonal. 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: How many diagonals has a quadrilateral?
Controlled answer. 2. 4(1)/2=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. Draw both without counting a side. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “2” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: How many diagonals has a pentagon?
Controlled answer. 5. 5(2)/2=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. Count two from each vertex and identify the duplicates. 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: How many diagonals has a hexagon?
Controlled answer. 9. 6(3)/2=9.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Check from a labelled sketch. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “9” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Why is n−3 used?
Controlled answer. Exclude the chosen vertex and its two adjacent vertices. Those three cannot be the other endpoint of a diagonal from the chosen vertex.
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. Apply the exclusions to an octagon vertex. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Exclude the chosen vertex and its two adjacent vertices” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Why divide by two?
Controlled answer. Each diagonal is counted once from each endpoint. AB and BA name the same segment.
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. Mark both descriptions of one diagonal. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Each diagonal is counted once from each endpoint” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: Repair: an octagon has 8×5=40 diagonals.
Controlled answer. An octagon has 8×5÷2=20 diagonals. Forty counts every diagonal twice.
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 two orientations of one pair. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “An octagon has 8×5÷2=20 diagonals” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Repair: a pentagon has 5×4÷2=10 diagonals.
Controlled answer. Subtract the five sides, giving 5 diagonals, or use 5×2÷2. The original count includes every vertex pair, adjacent or not.
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. Connect the two methods. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Subtract the five sides, giving 5 diagonals, or use 5×2÷2” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: How many diagonals has a decagon?
Controlled answer. 35. 10(7)/2=35.
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 first using a nearby case. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “35” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: A polygon has 14 diagonals. How many sides?
Controlled answer. 7. 7(4)/2=14, so it is a heptagon.
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 possible whole numbers systematically. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “7” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: A polygon has 20 diagonals. How many sides?
Controlled answer. 8. 8(5)/2=20.
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. Substitute to verify. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “8” 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 number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Can a polygon have exactly 7 diagonals?
Controlled answer. No. No whole number n≥3 satisfies n(n−3)/2=7.
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 counts on either side. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “No” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck the number of polygon vertices, eligible non-neighbour endpoints from each vertex, the duplicate count from reversing each endpoint pair, polygon type sides versus diagonals small-case checks and whole-number verification. 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: How many segments join every pair of vertices of a hexagon?
Explained answer. 15. There are 6×5÷2 unordered pairs, comprising 6 sides and 9 diagonals.
Delayed transfer. Reconcile with the diagonal formula. 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: A regular polygon and an irregular polygon both have nine vertices. Compare diagonal counts.
Explained answer. They have the same number, 27. Diagonal count depends on vertex adjacency, not equal side lengths or angles.
Delayed transfer. State a condition needed for a simple polygon. 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: Does a diagonal have to lie completely inside every non-convex polygon?
Explained answer. No. In a concave polygon, a segment between non-adjacent vertices can pass partly outside, but it remains a diagonal by endpoints.
Delayed transfer. Keep the counting definition endpoint-based. 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 many diagonals emerge from one vertex of a 12-gon?
Explained answer. 9. Exclude itself and two neighbours from 12 vertices.
Delayed transfer. Turn the local count into a total. 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: The diagonal count rises from a hexagon to a heptagon by how much?
Explained answer. 5. The totals are 9 and 14; the new vertex creates connections while one former side becomes a diagonal.
Delayed transfer. Explain the increment structurally. 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 integer check should follow the formula?
Explained answer. The result must be a whole non-negative number. Distinct segments cannot be fractional or negative.
Delayed transfer. Explain why n(n−3) is always even. 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. Counting all vertex pairs as diagonals. Adjacent pairs are sides and must be excluded.
Delayed transfer. Compare pair count and diagonal count. 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. Count n−3 eligible endpoints per vertex, multiply by n, divide by two for reversed pairs and check a small case. Keep sides and unordered pairs distinct.
Delayed transfer. Solve a delayed forward and 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.