PSLE Learner’s Guide · Volume 0894
Find your next learning step
For shortest grid routes, first lock the required horizontal and vertical moves; then build a route-count table in which each point receives the sum of counts from valid preceding points.
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. Define the route problem
- 2. Lock the shortest length
- 3. Seed the starting point
- 4. Add routes entering a point
- 5. Handle blocked edges correctly
- 6. Handle required checkpoints
- 7. Avoid double counting
- 8. Control examination work
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Define the route problem
Mark the start, finish, grid intersections and permitted directions. A route count is meaningless until movement along edges, through squares or between centres is stated clearly.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the start finish and allowed moves.
- Check shortest required step counts.
- Check the add-in rule at each point.
- Check blocked edges checkpoints and double-count control.
CHAPTER 2 OF 12 · Back to contents
2. Lock the shortest length
If the finish is four steps right and three steps up, every shortest route contains exactly four right moves and three up moves. Any left or down move creates a detour that must be excluded.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Seed the starting point
Write 1 at the start because there is one empty way to be there before moving. Along an unobstructed boundary, each reachable point initially has one shortest route.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Add routes entering a point
At an interior point reached only from the left or below, add the counts at those two predecessors. Every shortest route arrives through exactly one of them, so the cases are separate and complete.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Handle blocked edges correctly
A blocked edge removes only arrivals using that edge. Do not erase a point if it can still be reached through another valid predecessor.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Handle required checkpoints
For a route required to pass through point P, count shortest routes from start to P and from P to finish, then multiply because each first segment can pair with each second segment.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Avoid double counting
A full route is defined by its ordered moves. Two drawings that use the same sequence are one route, while different orderings are different even if they share many edges.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. 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
Show the step requirement, fill counts in a consistent direction, mark barriers before adding, audit the final total with a small case and state whether checkpoints are required or forbidden.
Build the explanation by naming the start finish and allowed moves and shortest required step counts. 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 add-in rule at each point and returning to blocked edges checkpoints and double-count control. 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 shortest routes go one step right and one step up?
Controlled answer. 2. The sequences are RU and UR.
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. Build the 1-by-1 addition table. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: How many shortest routes go two right and one up?
Controlled answer. 3. The single up move can occupy any of three positions.
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 RRU, RUR and URR. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “3” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: A finish is three right and two up. How many moves are in every shortest route?
Controlled answer. 5. Shortest routes use all three right and two up moves with no reverse step.
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. Reject a six-move proposal. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Why is the start labelled 1 rather than 0?
Controlled answer. There is one way to be at the start before taking a move. That seed lets later additions count complete sequences.
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. Trace the first boundary values. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “There is one way to be at the start before taking a move” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: A point has 4 valid routes from the left and 7 from below. How many reach it?
Controlled answer. 11. The last move comes from exactly one valid predecessor.
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 addition, not multiplication, is used. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: A blocked edge lies immediately left of a point, but 5 routes arrive from below. What is the point’s count?
Controlled answer. 5. The blocked edge removes the left contribution only.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Do not mark the point unreachable. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: A required checkpoint has 6 routes into it and 4 routes onward. How many full routes pass through it?
Controlled answer. 24. Each incoming route can pair with each onward route.
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 multiplication principle. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “24” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: If a point is forbidden, what value should it contribute?
Controlled answer. 0. No valid route may enter or leave through the forbidden point.
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. Distinguish a forbidden point from one blocked edge. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Repair: every 2-by-2 grid has four shortest routes.
Controlled answer. A 2-by-2 square grid has 6 corner-to-corner shortest edge routes. The move sequences contain two R and two U moves.
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. Generate the table values 1,2,3,6. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “A 2-by-2 square grid has 6 corner-to-corner shortest edge routes” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: Can a shortest route contain a left move when the finish lies to the right and only right/up travel is needed?
Controlled answer. No. A later right move would have to undo the detour, increasing length.
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 monotone-move condition. 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Why can boundary points have count 1?
Controlled answer. Only one all-horizontal or all-vertical shortest sequence reaches each unobstructed boundary point. There is no alternative predecessor on that boundary.
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. Change one boundary edge to blocked. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Only one all-horizontal or all-vertical shortest sequence reaches each unobstructed boundary point” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: A point can be entered from three valid directions in a different network. What is the rule?
Controlled answer. Add all three predecessor counts. Separate final-step cases still form a complete partition.
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 that no route is counted in two final-step cases. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Add all three predecessor counts” 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 start finish and allowed moves, shortest required step counts, the add-in rule at each point, blocked edges checkpoints and double-count control. 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: Does rotating the grid change an unobstructed corner-to-corner count?
Explained answer. No. Swapping horizontal and vertical labels preserves the move orderings.
Delayed transfer. Use symmetry as a 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 2: Two routes merge and then share every later edge. Are they still distinct?
Explained answer. Yes. Their earlier move sequences differ.
Delayed transfer. Keep both contributions at the merge point. 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: Why should barriers be marked before filling numbers?
Explained answer. A missed barrier contaminates every downstream count. The recurrence carries earlier errors forward.
Delayed transfer. Recompute only the affected region. 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: For routes through P, may the shortest start-to-P and P-to-finish segments be multiplied if their combined length exceeds the direct shortest length?
Explained answer. Not for a shortest-route question. Passing through P may force a detour and violate the original shortest length.
Delayed transfer. Compare combined length first. 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 nearest wrong shortcut?
Explained answer. Counting visible drawn lines instead of ordered complete routes. A route is a sequence of connected moves from start to finish.
Delayed transfer. Encode each path with R and U. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 6: How can a small case check a large table?
Explained answer. Recompute a 1-by-1 or 2-by-1 corner and verify the same add-in rule. Local consistency catches direction and seed errors.
Delayed transfer. Audit an interior cell. 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: When should addition be used and when multiplication?
Explained answer. Add mutually exclusive last-step cases; multiply independent choices for consecutive route segments. The operation follows the structure of choices.
Delayed transfer. Label one checkpoint example. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 8: State the full method.
Explained answer. Define allowed moves, lock the shortest step counts, seed the start with 1, add valid predecessor counts point by point, treat barriers as zero contributions and multiply segment counts only for a valid required checkpoint. Finish with a length and double-count audit.
Delayed transfer. Solve a blocked-grid case. 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.