PSLE Learner’s Guide · Volume 0920
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Per links one quantity to a unit of another quantity, while each distributes a statement across individual members; translate the relationship before calculating or explaining.
Series routes: Use the complete PSLE Learner’s Guide volume directory, the original PSLE Learning Guide and the English learning hub, Mathematics learning hub, Science 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. Read per as a two-quantity relationship
- 2. Read each as distribution across members
- 3. Preserve the denominator unit
- 4. Reconstruct totals deliberately
- 5. Do not assume each means equal in every context
- 6. Use per carefully in Science
- 7. Use units as a reasoning check
- 8. Control the examination interpretation
- 9. Worked examples
- 10. Independent practice
- 11. Delayed transfer
- 12. Family guidance and FAQs
CHAPTER 1 OF 12 · Back to contents
1. Read per as a two-quantity relationship
Kilometres per hour links distance to time; dollars per kilogram links cost to mass. The word per often signals division, but the useful meaning is how much of one quantity corresponds to one unit of another.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
- Check two related quantities or a stated set.
- Check unit rate versus individual distribution.
- Check units multiplication or division and total reconstruction.
- Check scope denominator averaging controlled conditions and unsupported equality.
CHAPTER 2 OF 12 · Back to contents
2. Read each as distribution across members
Each pupil receives two cards means the same allocation applies to every pupil considered individually. To find the total, multiply the number of pupils by two cards per pupil, provided no exception is stated.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 3 OF 12 · Back to contents
3. Preserve the denominator unit
A rate of 60 kilometres per hour is not merely 60. The per-hour denominator tells how the number scales with time and makes comparison possible.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 4 OF 12 · Back to contents
4. Reconstruct totals deliberately
If a pump transfers 4 litres per minute for 6 minutes at a constant rate, the total is 24 litres. Multiply rate by the matching number of units and cancel the denominator conceptually.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 5 OF 12 · Back to contents
5. Do not assume each means equal in every context
Each sample was observed does not say the samples had equal mass or identical results. Each distributes the action, while equality must come from other wording or controlled design.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 6 OF 12 · Back to contents
6. Use per carefully in Science
Heartbeats per minute, distance per unit time and mass per unit volume are rates or ratios. A measured rate describes a relationship under stated conditions; it does not automatically reveal the mechanism causing it.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 7 OF 12 · Back to contents
7. Use units as a reasoning check
When dollars per item is multiplied by items, dollars remain. When kilometres are divided by kilometres per hour, hours remain. Unit tracking exposes many reversed operations.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 8 OF 12 · Back to contents
8. Control the examination interpretation
Name the numerator quantity, denominator unit, stated members and constant-condition assumption. Decide whether the question asks for a unit rate, a total, a count of members or a comparison.
Use a four-step explanation. First identify two related quantities or a stated set. Next establish unit rate versus individual distribution. Then connect the evidence to units multiplication or division and total reconstruction. Finally audit scope denominator averaging controlled conditions and unsupported equality. This order prevents a familiar word or number from replacing the actual relationship.
The nearest misconception usually preserves one visible feature but drops a controlling condition. Repair it by naming what the tempting answer notices, what it ignores and why the ignored condition changes the conclusion. A correct answer should survive a small change of story while still responding to the decisive structure.
For independent control, rewrite the example with one decisive condition changed and predict whether the result stays or changes. Explain the boundary in a complete sentence. This contrast is more useful than repeating the original wording because it tests whether the mechanism can be transferred.
CHAPTER 9 OF 12 · Back to contents
9. Worked examples
Worked case 1: A car travels 180 km in 3 h. Find kilometres per hour.
Controlled answer. 60 km/h. Divide distance by time.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Reconstruct 180 km from the rate. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “60 km/h”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 2: Five pupils receive 3 pencils each. How many pencils?
Controlled answer. 15 pencils. Multiply five members by three per member.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. State the no-exception assumption. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “15 pencils”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 3: A tap releases 6 L per minute for 8 minutes. Find the total.
Controlled answer. 48 L. Rate times time gives volume.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Name the constant-rate assumption. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “48 L”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 4: A 2 kg bag costs $14. Find cost per kilogram.
Controlled answer. $7/kg. Divide total cost by mass.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Compare with a 3 kg bag. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “$7/kg”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 5: Each container was measured. Were the readings equal?
Controlled answer. Not necessarily. Each distributes the measuring action, not the result.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Name wording that would state equality. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “Not necessarily”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 6: Each plant received 50 mL of water. What is equal?
Controlled answer. The water allocation per plant. Other conditions may still differ.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. List two conditions to control. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “The water allocation per plant”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 7: A runner covers 400 m in 80 s. Find metres per second.
Controlled answer. 5 m/s. Divide distance by time.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Use the rate for 30 seconds if constant. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “5 m/s”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 8: Twelve stickers are shared equally among four children. How many each?
Controlled answer. 3 stickers. Equal sharing makes the per-child amount 12÷4.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Rebuild the total. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “3 stickers”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 9: A packet contains 24 cards at $0.50 each. Find cost.
Controlled answer. $12. Multiply count by cost per card.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Track dollar units. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “$12”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 10: What does 90 beats per minute compare?
Controlled answer. Number of beats with one minute of time. It is a rate, not a total over every duration.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Predict beats in half a minute only under a constant-rate assumption. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “Number of beats with one minute of time”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 11: Why can two rates with different units not be compared directly?
Controlled answer. They describe different relationships. Convert to common compatible units first.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Compare km/h with m/s after conversion. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “They describe different relationships”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
Worked case 12: Repair: 5 L per minute means exactly 5 L in every future minute.
Controlled answer. It states a rate under the relevant conditions; the rate may change if conditions change. Do not extend beyond the evidence.
Mechanism. Begin with two related quantities or a stated set, then use unit rate versus individual distribution to decide the operation or language choice. The answer is justified only when units multiplication or division and total reconstruction is shown explicitly. Keep every unit, referent and condition attached to the quantity or clause it controls.
Misconception repair. A tempting answer may copy the nearest noun, visible number or remembered slogan. That shortcut is unsafe because it does not test scope denominator averaging controlled conditions and unsupported equality. State the missing condition and show how it affects the result instead of merely labelling the first answer wrong.
Transfer. Name a condition that could alter flow. Solve the altered case from the relationship, not from surface resemblance. If the conclusion changes, identify the exact changed condition; if it stays, explain why the mechanism is invariant.
Examination control. Write the decisive evidence before the conclusion. Reread the full prompt, check that the response answers the requested quantity or meaning, and verify that no unstated fact has been added. A concise answer is strong only when its reasoning remains inspectable.
Reverse construction. Invent a new prompt whose correct answer is “It states a rate under the relevant conditions; the rate may change if conditions change”, then create a near-nonexample by changing one condition. Explain the pair without reusing the original story. This reverse task checks genuine ownership of the method.
CHAPTER 10 OF 12 · Back to contents
10. Independent practice
Practice 1: Repair: Each object fell, so all fell at the same speed.
Explained answer. Each states that every object fell, not that their speeds were equal. Equality needs measurements or explicit wording.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Separate action from rate. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 2: What does $3 per 100 g mean?
Explained answer. Three dollars for each 100-gram unit. The denominator is 100 g, not 1 g.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Find cost per gram if needed. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 3: A machine makes 12 parts per hour for 5 hours. What total follows if the rate is constant?
Explained answer. 60 parts. Multiply 12 parts/hour by 5 hours.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Discuss whole-part constraints. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 4: Why are units part of the answer?
Explained answer. They identify what the number measures and which quantity is in the denominator. A bare number can reverse the relationship.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Label numerator and denominator. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 5: When does each support multiplication?
Explained answer. When the same amount applies to every counted member and the total is required. Multiply amount per member by member count.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Give a counterexample with exceptions. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 6: When does per support division?
Explained answer. When a total numerator quantity is spread over a known denominator quantity to find a unit rate. Divide compatible totals.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Check the resulting unit. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 7: What is the nearest wrong shortcut?
Explained answer. Treating every per as a command to divide the numbers in their written order. First identify quantities, target and units.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Use unit cancellation to choose the operation. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
Practice 8: State the complete method.
Explained answer. Translate per as a relationship between two quantities and each as distribution across members; identify units and scope; choose multiplication or division; then reconstruct the original total to check. Do not infer equality or constancy beyond the wording.
Reasoning route. Identify two related quantities or a stated set, determine unit rate versus individual distribution, apply units multiplication or division and total reconstruction and audit scope denominator averaging controlled conditions and unsupported equality. Do not reveal the answer first when practising; reconstruct this chain from the prompt.
Delayed transfer. Create one English, one Mathematics and one Science example. Return after two days and solve without the headings. Then compare the new reasoning with this explanation and mark the first step that was missing or uncertain.
Boundary test. Change one decisive condition while leaving the remaining story intact. Predict the effect before calculating or editing. A useful near-nonexample differs in only one important way, so the reason for the changed answer remains visible.
Independent proof. Restate the conclusion in your own words and give one relevant fact that would still be insufficient by itself. This separates proof from association and prevents a topic label from standing in for a mechanism.
Self-check. Confirm the target, evidence, operation or grammar, units or referents, and claim strength. If the answer relies on a hidden assumption, state it. If the prompt does not justify that assumption, narrow the claim.
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11. Delayed transfer
Use three passes. On day one, solve clear examples and explain each step. On day three, mix valid examples, near-nonexamples and incomplete evidence. On day seven, answer without a word bank or chapter heading. The delay tests reconstruction rather than short-term recognition.
Keep an error log that records the first failed stage: reading, representation, rule, execution or checking. Repair that stage with a changed example before completing another full set. Ten repeated answers are less useful than one accurately diagnosed error.
Interleave this skill with neighbouring ideas that use similar words or diagrams. Before solving, classify the relationship and state the condition that distinguishes it from the neighbour. Classification should follow evidence, not topic appearance.
In timed work, mark the controlling words or quantities, write one intermediate relationship, complete the answer and audit one boundary condition. If checking changes the answer, record the overlooked feature so the same error becomes easier to recognise.
For a second delayed test, begin with a conclusion and construct sufficient evidence for it. Then construct evidence that is relevant but insufficient. Reversing the direction makes the boundary between support and proof visible.
Finish with a basic case, a boundary case and a transfer case from a different context. Explain all three to a classmate who is allowed to ask, “Which exact condition makes that true?”
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12. Family guidance and FAQs
Guidance for parents and tutors
Ask for the decisive condition before confirming an answer. Helpful prompts include “What exactly is related?”, “Which word or quantity controls the result?”, “What stayed fixed?” and “What would make the answer change?”
Fade support from labelled examples to mixed questions and delayed open response. Praise precise claim control, visible units and the willingness to narrow a conclusion when evidence is incomplete.
Frequently asked questions
Should I memorise the compact rule?
Yes, but pair it with examples and near-nonexamples. Memory supplies speed; contrast practice protects the conditions.
What if two answers seem possible?
Return to the exact relationship and choose the answer that satisfies every stated condition without inventing facts.
Why write an intermediate step?
It makes the controlling relationship visible and gives you something specific to audit.
Why delay practice?
Immediate repetition may rely on the recent explanation. Delay reveals whether the method can be rebuilt independently.
How should an error be corrected?
Name the earliest failed stage, repair it with one changed example and then retry the original question.
When is the skill ready?
When the learner can explain, apply and transfer it after a delay while preserving the relevant conditions.
Next routes
Continue at the English, Mathematics and Science learning hub or return to the series directory.