PSLE Learner’s Guide · Volume 1026
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For positive fractions that refer to the same whole and have equal non-zero numerators, the fraction with the smaller positive denominator is larger because each equal part is larger.
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. Start from one whole
- 2. Hold the numerator fixed
- 3. Use unit fractions first
- 4. State the conditions
- 5. Do not compare denominator digits directly in every case
- 6. Use visual models carefully
- 7. Confirm by cross multiplication
- 8. Audit the answer
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Start from one whole
A denominator tells how many equal parts the same whole is divided into. Dividing into fewer equal parts makes each part larger.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
- Check the shared whole numerator denominator sign and whether the fractions are positive.
- Check the relationship between the number of equal parts and the size of each part.
- Check whether equal numerators and valid positive denominators justify the inverse comparison.
- Check fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation.
CHAPTER 2 OF 12 · Back to contents
2. Hold the numerator fixed
When two positive fractions select the same number of parts, the size of each part decides the comparison. The smaller denominator supplies larger parts.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 3 OF 12 · Back to contents
3. Use unit fractions first
One fifth is larger than one eighth because each fifth is a larger share of the same whole. Multiplying both unit fractions by the same positive numerator preserves the order.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 4 OF 12 · Back to contents
4. State the conditions
The quick rule assumes positive denominators, positive equal numerators and the same whole. Negative signs or differently sized wholes require separate reasoning.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 5 OF 12 · Back to contents
5. Do not compare denominator digits directly in every case
A larger denominator does not always mean a smaller fraction when numerators differ. The equal-numerator condition makes this shortcut legal.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 6 OF 12 · Back to contents
6. Use visual models carefully
Draw equal-size bars or circles before partitioning them. Unequal drawings can make the wrong fraction look larger.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 7 OF 12 · Back to contents
7. Confirm by cross multiplication
For a over b and a over c with positive a, b and c, compare a times c with a times b. The algebra confirms the same inverse denominator order.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. State why the tempting alternative is too strong, incomplete or about a different stage.
CHAPTER 8 OF 12 · Back to contents
8. Audit the answer
Check the shared whole, signs, equal numerators and positive denominators, state the larger fraction with a reason and verify it lies in the expected number-line position.
Build the explanation by naming the shared whole numerator denominator sign and whether the fractions are positive and the relationship between the number of equal parts and the size of each part. 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 whether equal numerators and valid positive denominators justify the inverse comparison and returning to fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. 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: Which is larger, 3/5 or 3/8?
Controlled answer. 3/5. Both select three parts, and fifths are larger than eighths.
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. The smaller denominator gives the larger positive fraction. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 2: Which is smaller, 7/9 or 7/12?
Controlled answer. 7/12. Twelfths are smaller than ninths while seven parts are selected in each.
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. Answer the requested direction. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 3: Compare 1/4 and 1/6.
Controlled answer. 1/4 is larger. A fourth is a larger equal part than a sixth of the same whole.
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. Begin with unit fractions. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 4: Repair: 5/11 is larger than 5/7 because 11 is larger.
Controlled answer. 5/7 is larger than 5/11. With equal positive numerators, the larger denominator creates smaller parts.
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. Reverse the denominator comparison. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 5: Repair: A larger denominator always gives a smaller fraction.
Controlled answer. A larger denominator gives a smaller positive fraction only when the numerator is held equal and the whole is the same. Different numerators can change the order.
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 shortcut’s conditions. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 6: Which is nearer 1, 9/10 or 9/13?
Controlled answer. 9/10. Tenths are larger than thirteenths, so nine tenths lies farther to the right.
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 number line as a check. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 7: Compare 12/7 and 12/5.
Controlled answer. 12/5 is larger. Both are positive and have numerator 12; fifths are larger than sevenths.
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. The rule also works for improper fractions. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 8: Why must the whole be the same?
Controlled answer. A fraction describes a share of a particular whole. Three fifths of a small cake can be less food than three eighths of a much larger cake.
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. Separate fraction size from physical amount. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 9: Can denominator zero be used?
Controlled answer. No. Division by zero is undefined.
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. The comparison rule requires positive non-zero denominators. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 10: What if both numerators are zero?
Controlled answer. Both fractions equal zero when denominators are non-zero. The fractions are equal, so the non-zero numerator condition matters for strict inequality.
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 boundary cases. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 11: Does the rule apply directly to negative fractions?
Controlled answer. No. Order with negative values requires tracking the sign; the value nearer zero is greater.
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 copy a positive-fraction shortcut across signs. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. Decide whether the evidence proves the conclusion or only supports a limited statement.
Worked case 12: Use cross multiplication for 4/7 and 4/9.
Controlled answer. 4/7 is larger. 4 times 9 is 36 and 4 times 7 is 28, confirming 4/7 exceeds 4/9.
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. Keep cross products attached to the opposite denominators. Solve the altered case from the mechanism rather than copying the previous wording.
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 shared whole numerator denominator sign and whether the fractions are positive, the relationship between the number of equal parts and the size of each part, whether equal numerators and valid positive denominators justify the inverse comparison, fraction numerator denominator same whole equal parts unit fraction positive proper improper compare order number line model cross multiplication condition boundary and explanation. 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: Draw a fair model for 2/3 and 2/5.
Explained answer. Use two equal-size bars, split one into thirds and one into fifths, then shade two parts in each. The thirds are larger equal parts.
Delayed transfer. Equal-size wholes make the picture valid. 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: Which is larger, 6/100 or 6/10?
Explained answer. 6/10. Tenths are much larger than hundredths.
Delayed transfer. Place values and fraction parts agree. 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: If the denominator decreases while a positive numerator stays fixed, what happens?
Explained answer. The fraction increases. Each part becomes larger while the selected part count stays constant.
Delayed transfer. The denominator must remain positive. 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: If the denominator doubles while the positive numerator stays fixed, what happens?
Explained answer. The fraction is halved. Each original part is divided into two smaller parts.
Delayed transfer. Connect inverse change to the same numerator. 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: Why is 3/4 not automatically comparable to 5/9 by this shortcut?
Explained answer. The numerators differ. Both part size and selected part count change.
Delayed transfer. Use a common denominator, decimals or cross multiplication. 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 fastest equal-numerator check?
Explained answer. Confirm positive signs and the same whole, then choose the fraction with the smaller denominator as larger. Say why the parts are larger.
Delayed transfer. Do not omit conditions. 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 can estimation check 8/9 versus 8/15?
Explained answer. Eight ninths is close to one, while eight fifteenths is only a little above one half. The estimate supports 8/9 as larger.
Delayed transfer. Use benchmarks after the exact rule. Cover the answer later and rebuild the chain.
Control question. Which exact condition does the nearest wrong answer miss? State it before checking.
Extension. Change one decisive feature while keeping the story similar. Explain why the conclusion now stays or changes.
Independent proof. Restate the answer without copying its wording. Your version must identify the evidence, the governing relationship and the boundary of the claim. Then write one fact that would be relevant but still insufficient on its own. This separates explanation from repetition.
Delayed comparison. Save this item beside the near-nonexample you created. On the next attempt, compare them before answering and point to the single feature that controls the difference. If more than one feature changed, revise the pair so the comparison stays fair.
Practice 8: State the full method.
Explained answer. Confirm the fractions refer to the same whole, check that denominators are positive and non-zero, verify equal positive numerators, reason that fewer equal parts make each part larger, reverse the denominator order to compare the fractions, and confirm with a model benchmark or cross multiplication without extending the shortcut to unequal numerators or negative values. The method makes the inverse relationship visible and conditional.
Delayed transfer. Apply it after a delayed mixed-fraction comparison. 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.