PSLE Learner’s Guide · Volume 0882
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Count squares size by size: a k by k square has (m − k + 1)(n − k + 1) positions in an m by n grid.
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 grid by cells
- 2. List every possible size
- 3. Count positions for one size
- 4. Sum across sizes
- 5. Use the square-grid shortcut carefully
- 6. Separate squares from rectangles
- 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 grid by cells
An m by n grid has m rows and n columns of unit cells. A square counted in the grid must follow its lines and have equal whole-number side lengths.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check m rows and n columns of unit cells.
- Check possible side length k up to the shorter dimension.
- Check horizontal and vertical starting positions.
- Check size table sum and rectangle-total check.
CHAPTER 2 OF 12 · Back to contents
2. List every possible size
The smallest squares are 1 by 1. Continue with 2 by 2, 3 by 3 and so on until the side length reaches min(m,n), the shorter grid dimension.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Count positions for one size
A k by k square can start in m − k + 1 row positions and n − k + 1 column positions. Multiplying gives (m − k + 1)(n − k + 1) placements.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. Sum across sizes
Add the placement counts for every valid k. The total is the sum from k = 1 to min(m,n) of (m − k + 1)(n − k + 1).
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Use the square-grid shortcut carefully
For an n by n grid, the total becomes 1² + 2² + … + n², which equals n(n + 1)(2n + 1) ÷ 6. The shortcut does not transfer unchanged to a non-square grid.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Separate squares from rectangles
Every square is a rectangle, but not every rectangle is a square. A rectangle total can check that the square count is not larger, but it cannot replace the equal-side test.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. 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
Count each square by its top-left starting position and side length. That pair identifies it uniquely, so the same boundary is not counted from multiple corners.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. 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
Write a size table with side length, row positions, column positions and product. Stop at the shorter dimension, sum once and compare with a small drawing or the all-rectangles total.
Build the explanation by naming m rows and n columns of unit cells and possible side length k up to the shorter dimension. 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 horizontal and vertical starting positions and returning to size table sum and rectangle-total check. 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 1 by 1 squares are in a 2 by 3 grid?
Controlled answer. 6. Each unit cell is one smallest square.
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 next possible size. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: How many 2 by 2 squares are in a 2 by 3 grid?
Controlled answer. 2. There is one vertical start and two horizontal starts.
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. Add the size counts. 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: How many squares are in a 2 by 3 grid?
Controlled answer. 8. Six unit squares plus two 2 by 2 squares give eight.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Compare with the rectangle total. 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: How many squares are in a 2 by 2 grid?
Controlled answer. 5. There are four 1 by 1 squares and one 2 by 2 square.
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 and label them. 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: How many squares are in a 3 by 3 grid?
Controlled answer. 14. Nine 1 by 1, four 2 by 2 and one 3 by 3 give fourteen.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Use the sum-of-squares shortcut. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “14” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: How many 2 by 2 squares are in a 3 by 4 grid?
Controlled answer. 6. The start counts are 2 rows and 3 columns.
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 1 by 1 and 3 by 3 sizes. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “6” and invent a fresh prompt that would justify it for the same reason. Then change one decisive condition to create a near-nonexample. This reverse task checks whether the learner owns the boundary rather than merely recognising the original sentence.
Explanation standard. A complete explanation names the observation, connects it to the rule and limits the conclusion to what follows. It should remain valid if decorative details are removed. If the explanation depends on the story topic instead of the decisive relation, rebuild it.
Audit. Recheck m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: How many squares are in a 3 by 4 grid?
Controlled answer. 20. There are 12 unit, 6 two-by-two and 2 three-by-three squares.
Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.
Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.
Transfer. Verify the maximum size. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Why can a 4 by 2 square not fit in the grid?
Controlled answer. Its sides are unequal, so it is a rectangle rather than a square. A square must have equal side lengths.
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 it only in the rectangle total. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Its sides are unequal, so it is a rectangle rather than a square” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Why does a k by k square have m − k + 1 vertical positions?
Controlled answer. Its k rows can begin at each row start that leaves k rows inside the grid. The final valid start is row m − k + 1.
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 m = 5 and k = 3. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “Its k rows can begin at each row start that leaves k rows inside the grid” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: What is the largest square in a 4 by 7 grid?
Controlled answer. 4 by 4. The shorter dimension limits equal side 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. Count its horizontal placements. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “4 by 4” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: How many 4 by 4 squares fit in a 4 by 7 grid?
Controlled answer. 4. There is one row start and four column starts.
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. Locate the four starts. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “4” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Repair: a 3 by 4 grid has only 12 squares.
Controlled answer. It has 20 squares. Twelve counts only unit squares and misses larger sizes.
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 missing sizes. Solve the altered case from the mechanism rather than copying the previous wording.
Reverse construction. Begin with the controlled answer “It has 20 squares” 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 m rows and n columns of unit cells, possible side length k up to the shorter dimension, horizontal and vertical starting positions, size table sum and rectangle-total check. 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: Repair: use 1²+2²+3²+4² for a 3 by 4 grid.
Explained answer. Use size-specific placement counts for the rectangular grid. The square-grid shortcut assumes equal dimensions.
Delayed transfer. Build the 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 2: Can the square total exceed the rectangle total?
Explained answer. No. Squares are a subset of rectangles.
Delayed transfer. Use this as a reasonableness 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 3: How many squares are in a 1 by 5 grid?
Explained answer. 5. Only 1 by 1 squares fit because the shorter dimension is one.
Delayed transfer. Explain why no larger square fits. 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 squares are in a 4 by 4 grid?
Explained answer. 30. The counts are 16 + 9 + 4 + 1.
Delayed transfer. Check with the first-four-squares 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 5: What uniquely identifies a counted square?
Explained answer. Its side length and starting row and column. These details prevent duplicate counting.
Delayed transfer. Describe one 2 by 2 square. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 6: What is the nearest wrong shortcut?
Explained answer. Multiplying rows by columns and stopping. That counts only unit squares.
Delayed transfer. Add the larger-size placements. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 7: How should a size table be audited?
Explained answer. Check every k from one to the shorter dimension and verify both start counts stay positive. A skipped size or off-by-one count changes the total.
Delayed transfer. Audit a 5 by 3 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 8: State the full method.
Explained answer. For each side length k up to min(m,n), multiply m − k + 1 by n − k + 1 and add the results. Count each square once by size and start position.
Delayed transfer. Apply to a new rectangular grid. 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.