How to Perform in PSLE | Learner’s Guide Vol 1034 | Mathematics: Adding the Same Positive Number to a Proper Fraction’s Numerator and Denominator Makes It Larger but Still Below 1

CHAPTER 1 OF 12 · Back to contents

1. State every condition

Let a and b be positive with a smaller than b, and let k be positive. The original fraction a over b is proper, while the new fraction is (a plus k) over (b plus k).

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

• Check the positive numerator denominator common addition and original proper-fraction condition.
• Check the unchanged absolute gap between denominator and numerator compared with a growing denominator.
• Check whether the new fraction is above the original value yet still below one.
• Check fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof.

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CHAPTER 2 OF 12 · Back to contents

2. Do not call the fractions equivalent

Equivalent fractions come from multiplying or dividing numerator and denominator by the same non-zero factor. Adding the same number generally changes the value.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 3 OF 12 · Back to contents

3. Track the gap from one

The gap between denominator and numerator is b minus a before and after the addition. The absolute gap stays fixed while the new denominator grows, so that gap becomes a smaller fraction of the whole denominator.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 4 OF 12 · Back to contents

4. Write the complement to one

One minus a over b equals (b minus a) over b. After adding k, one minus (a plus k) over (b plus k) equals the same gap b minus a over the larger denominator b plus k. The new fraction is therefore closer to one from below.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 5 OF 12 · Back to contents

5. Prove the increase by cross multiplication

Compare (a plus k) over (b plus k) with a over b. The difference in cross-products is bk minus ak, which equals k times (b minus a) and is positive.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 6 OF 12 · Back to contents

6. Protect the boundary cases

If k is zero, the value is unchanged. If the original numerator equals the denominator, both fractions equal one. For an improper fraction above one, adding the same positive number moves the value downward toward one instead.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 7 OF 12 · Back to contents

7. Estimate before calculating

A transformed positive proper fraction must lie between the original fraction and one. This interval catches the common but false claim that adding equal amounts leaves the value unchanged.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 8 OF 12 · Back to contents

8. Control examination work

Verify the conditions, write the transformed fraction, compare by complement-to-one or cross-products, explain the unchanged gap and identify any boundary case before generalising.

Build the explanation by naming the positive numerator denominator common addition and original proper-fraction condition and the unchanged absolute gap between denominator and numerator compared with a growing denominator. 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 the new fraction is above the original value yet still below one and returning to fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. State why the tempting alternative is too strong, incomplete or about a different stage.

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CHAPTER 9 OF 12 · Back to contents

9. Worked examples

Worked case 1: Compare three fifths and four sixths.

Controlled answer. Four sixths is larger. Adding one to numerator and denominator gives a value closer to one from below.

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. Simplify four sixths and compare exactly. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 2: Compare two sevenths and five tenths.

Controlled answer. Five tenths is larger. Adding three to both terms preserves the gap five but increases the denominator.

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 complements five sevenths and five tenths. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 3: Repair: adding the same number to numerator and denominator makes an equivalent fraction.

Controlled answer. The fraction generally changes value. Equivalent fractions require equal scaling, not equal addition.

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. Contrast two thirds, three fourths and four sixths. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 4: Why does the new fraction remain below one?

Controlled answer. Its numerator is still smaller than its denominator. Adding the same number preserves the positive gap b minus a.

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 gap before and after. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 5: Why does the fraction become larger?

Controlled answer. The fixed gap from one is spread over a larger denominator. Its complement to one becomes smaller.

Mechanism. Identify the exact relationship, apply the relevant rule and preserve every stated condition. A surface clue may be compatible with several explanations, so the answer must show what makes this case decisive.

Misconception repair. Do not turn partial evidence into complete proof. Name the closest wrong answer, identify its missing condition and explain why the observed evidence supports the narrower conclusion.

Transfer. Write both complements. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 6: Compare one half and eleven twelfths.

Controlled answer. Eleven twelfths is larger. Adding ten to both terms moves the proper fraction much closer to one.

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 claim it reaches one. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 7: What happens to five fifths after adding three to both terms?

Controlled answer. It remains one as eight eighths. The original gap is zero.

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. Identify why the strict proper-fraction condition fails. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 8: What happens when zero is added?

Controlled answer. The fraction is unchanged. k must be positive for a strict increase.

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 non-decrease from strict increase. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 9: Compare five thirds and six fourths.

Controlled answer. Six fourths is smaller than five thirds. The original fraction is improper and moves toward one from above.

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 the proper-fraction rule cannot be copied. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 10: Use cross-products to compare a over b with (a plus k) over (b plus k).

Controlled answer. The new fraction is larger when k times (b minus a) is positive. That follows from positive k and b greater than a.

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. Name both required 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 11: A learner compares four ninths with seven twelfths. Which is larger?

Controlled answer. Seven twelfths. Three was added to both terms of a positive proper fraction.

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. Confirm by cross multiplication. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

Worked case 12: Is the denominator increase alone enough to predict the direction?

Controlled answer. No. Both numerator and denominator change, so their relationship must be analysed.

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 fixed-gap mechanism. 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 positive numerator denominator common addition and original proper-fraction condition, the unchanged absolute gap between denominator and numerator compared with a growing denominator, whether the new fraction is above the original value yet still below one, fraction numerator denominator proper equivalent difference gap complement to one cross multiplication common addition positive number boundary zero improper mixed value comparison and proof. Decide whether the evidence proves the conclusion or only supports a limited statement.

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CHAPTER 10 OF 12 · Back to contents

10. Independent practice

Practice 1: What stays unchanged after equal addition?

Explained answer. The difference between denominator and numerator. The absolute gap stays b minus a.

Delayed transfer. Distinguish absolute gap from fractional value. 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: What changes about the complement to one?

Explained answer. It becomes smaller. The same gap is divided by a larger denominator.

Delayed transfer. Connect smaller complement to larger original fraction. 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: Can repeated equal additions make the value exceed one?

Explained answer. No, not when the original fraction is proper and additions remain equal. The numerator-denominator gap stays positive.

Delayed transfer. Describe the value approaching but not reaching one for finite additions. 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: Order two fifths, three sixths and four sevenths.

Explained answer. Two fifths, three sixths, four sevenths. Each step adds one to both terms and increases the value.

Delayed transfer. Confirm with common denominators or decimals. 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 multiplying both terms by two different?

Explained answer. Multiplying by the same non-zero factor preserves the ratio exactly. Equal addition changes the ratio by shrinking its relative gap from one.

Delayed transfer. Compare three fifths, six tenths and four sixths. 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 reasonableness range?

Explained answer. Original fraction less than new fraction less than one. The range follows from the positive proper-fraction and positive-addition conditions.

Delayed transfer. Use it to reject wrong options. 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 general proof finish?

Explained answer. State that k times (b minus a) is positive and that a plus k remains below b plus k. The first proves increase and the second proves the new value stays below one.

Delayed transfer. Do not omit the 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 8: State the full method.

Explained answer. Check that the original fraction is positive and proper and that the common addition is positive, preserve the numerator-denominator gap, compare complements to one or cross-products, place the result between the original value and one, and treat zero one or improper boundary cases separately. The method joins representation, proof and checking.

Delayed transfer. Apply it after a delayed mixed fraction-transformation set. 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.

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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.

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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.

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