Series route: This guide belongs to the complete PSLE Learner’s Guide volume directory. The original PSLE Learning Guide supplies the wider English, Mathematics and Science route, while PSLE Mathematics provide verified subject-specific continuation.
Find your route through this guide
This volume solves one precise learner problem: the learner sees a ratio such as 2:3 and treats its one-part difference as one object, forgetting that each part may represent any common scale. The goal is durable control, not a remembered slogan. A Primary 6 learner should be able to identify the trigger, explain the mechanism, perform the decision under time pressure and reject an attractive answer that exceeds the evidence.
- Start with the exact mistake and the evidence boundary.
- Build the mechanism before memorising a shortcut.
- Study worked contrasts and name why the near-miss fails.
- Repair misconceptions with one targeted check each.
- Attempt graded independent practice before reading the explanations.
- Return after a delay and prove that the method transfers.
- Use the parent and tutor notes after the learner has attempted the work.
The problem beneath the mistake
A ratio of 2:3 means two equal parts compared with three equal parts. It does not say that the actual quantities are 2 and 3.
The quantities could be 8 and 12, 20 and 30, or 200 and 300. The common multiplier gives the value of one part.
The ratio-number difference is 1 part, but the actual difference equals one part multiplied by the part value.
If 2:3 quantities differ by 7, then one part is 7 and the quantities are 14 and 21.
If they differ by 35, one part is 35 and the quantities are 70 and 105. The same ratio describes both situations.
Totals use the sum of ratio parts. For 2:3, five parts represent the combined quantity.
A ratio is dimensionless until context supplies objects, money, mass, volume or another quantity. Units belong to the actual part value.
Simplifying a ratio changes the written part counts but not the comparison. A 4:6 ratio becomes 2:3, so the numerical difference changes from 2 parts to 1 part while the actual difference stays the same.
Part values can also expose impossible interpretations. If a whole-number count problem leads to one part equalling 2.5 pupils, the arithmetic may be formally possible but the context signals that the chosen ratio, total or difference has been matched incorrectly. The learner should return to the sentence and identify which actual quantity the given number represents.
Units must be carried through the model rather than attached at random at the end. If a three-part difference equals 18 litres, one part is 6 litres; if it equals $18, one part is $6. The ratio calculation has the same structure, but the answer belongs to a different measured quantity.
This is why “part” is a model unit, not automatically an object. The learner must convert between parts and actual quantities deliberately.
The reliable routine is ratio parts, target parts, part value, actual quantity and check.
Core decision: In a ratio a:b, a and b count equal model parts. An actual difference equals |a−b| multiplied by the value of one part. Find the part value from the given total, difference or one quantity before reporting objects or units.
Ratios describe multiplicative structure. The same comparison can be scaled up or down, so ratio numbers are coefficients rather than fixed actual counts.
Mechanism: why the rule works
1. Separate parts from objects
Write a boxes for the first quantity and b equal boxes for the second.
The boxes represent equal parts whose actual value is not yet known.
Diagnostic check. Has the question supplied one part’s value?
2. Count target parts
Use a+b for a total, |a−b| for a difference and the relevant coefficient for one quantity.
Different givens attach to different part counts.
Diagnostic check. Which combination of parts matches the given quantity?
3. Find one part
Divide the known actual quantity by its matching number of parts.
This step converts the model scale into real units.
Diagnostic check. Does the division produce a sensible part value?
4. Scale quantities
Multiply each ratio number by one part.
Both quantities must use the same part value.
Diagnostic check. Do the scaled quantities still simplify to the original ratio?
5. Attach units
One part may represent pupils, dollars, litres or grams.
A bare part count is not the requested actual answer.
Diagnostic check. What does one part measure in this problem?
6. Handle simplified ratios
Simplification merges model parts but increases the value represented by each new part.
Changing the written difference does not change the real difference.
Diagnostic check. What happened to the part value after simplification?
7. Verify all constraints
Check ratio, total or difference, and units.
A calculation can match one condition while failing another.
Diagnostic check. Do both quantities satisfy every given statement?
Worked examples: make the decision visible
Pause before each answer. State the controlling evidence in one line, then complete the operation. The explanations deliberately contrast a valid route with a plausible but invalid one, because transfer grows when the learner can say why a tempting route does not apply.
Worked example 1: Difference given
Task. A:B=2:3 and B−A=7.
Reasoning. The difference is one part, so one part is 7.
Answer. A=14 and B=21.
Independent check. 21−14=7 and 14:21=2:3.
Contrast-and-transfer pass. Keep the governing principle from “Difference given”, but change one surface detail in the task: A:B=2:3 and B−A=7. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—A=14 and B=21.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 2: Larger difference
Task. A:B=2:3 and the difference is 35.
Reasoning. One part now represents 35, not one object.
Answer. A=70 and B=105.
Independent check. The ratio and difference both match.
Contrast-and-transfer pass. Keep the governing principle from “Larger difference”, but change one surface detail in the task: A:B=2:3 and the difference is 35. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—A=70 and B=105.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 3: Two-part difference
Task. Red:Blue=3:5 and Blue exceeds Red by 18.
Reasoning. The difference is two parts, so one part is 9.
Answer. Red=27 and Blue=45.
Independent check. 45−27=18.
Contrast-and-transfer pass. Keep the governing principle from “Two-part difference”, but change one surface detail in the task: Red:Blue=3:5 and Blue exceeds Red by 18. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Red=27 and Blue=45.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 4: Total given
Task. Girls:Boys=4:3 and total is 35.
Reasoning. Seven parts represent 35, so one part is 5.
Answer. Girls=20 and Boys=15.
Independent check. The total is 35.
Contrast-and-transfer pass. Keep the governing principle from “Total given”, but change one surface detail in the task: Girls:Boys=4:3 and total is 35. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Girls=20 and Boys=15.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 5: One quantity given
Task. Pens:Pencils=5:2 and pens are 30.
Reasoning. Five parts equal 30, so one part is 6.
Answer. Pencils=12.
Independent check. 30:12 simplifies to 5:2.
Contrast-and-transfer pass. Keep the governing principle from “One quantity given”, but change one surface detail in the task: Pens:Pencils=5:2 and pens are 30. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Pencils=12.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 6: Money unit
Task. Savings:Spending=7:4 and difference is $45.
Reasoning. Three parts equal $45, so one part is $15.
Answer. Savings=$105 and spending=$60.
Independent check. Units remain dollars.
Contrast-and-transfer pass. Keep the governing principle from “Money unit”, but change one surface detail in the task: Savings:Spending=7:4 and difference is $45. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Savings=$105 and spending=$60.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 7: Simplification trap
Task. Compare 4:6 and 2:3 for quantities 20 and 30.
Reasoning. In 4:6, one part is 5; in 2:3, one part is 10.
Answer. Both models represent the same quantities and actual difference 10.
Independent check. Written part difference changed, part value compensated.
Contrast-and-transfer pass. Keep the governing principle from “Simplification trap”, but change one surface detail in the task: Compare 4:6 and 2:3 for quantities 20 and 30. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Both models represent the same quantities and actual difference 10.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 8: Wrong one-object claim
Task. A learner says 8:9 means the quantities differ by one item.
Reasoning. The ratio gives one-part difference but no part value.
Answer. The actual difference is unknown until scale is supplied.
Independent check. Examples 8 and 9 versus 80 and 90 disprove the claim.
Contrast-and-transfer pass. Keep the governing principle from “Wrong one-object claim”, but change one surface detail in the task: A learner says 8:9 means the quantities differ by one item. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The actual difference is unknown until scale is supplied.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 9: Fractional part value
Task. A:B=2:5 and total mass is 21 kg.
Reasoning. Seven parts equal 21 kg, so one part is 3 kg.
Answer. Masses are 6 kg and 15 kg.
Independent check. Units and total check.
Contrast-and-transfer pass. Keep the governing principle from “Fractional part value”, but change one surface detail in the task: A:B=2:5 and total mass is 21 kg. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Masses are 6 kg and 15 kg.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 10: Reverse design
Task. Create quantities in ratio 3:4 with difference 12.
Reasoning. One part must be 12.
Answer. Use 36 and 48.
Independent check. The difference and simplified ratio verify.
Contrast-and-transfer pass. Keep the governing principle from “Reverse design”, but change one surface detail in the task: Create quantities in ratio 3:4 with difference 12. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Use 36 and 48.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
The method lab: change one condition at a time
Build ratio strips for 2:3 at three scales and compare written parts with actual quantities.
Mix problems where the given is total, difference or one quantity; label target parts before calculating.
Simplify 6:9 to 2:3 and explain why the new part is three times as large.
After a delay, create a ratio problem whose one-part difference is not one object.
Common misconceptions and targeted repairs
Misconception 1: One part means one object
What the learner may think. The learner confuses model units with actual counts.
Why it fails. Each part may represent many or fractional units.
Targeted repair. Find the scale first.
Misconception 2: Always divide by total parts
What the learner may think. The learner uses a+b even when the given is a difference.
Why it fails. The matching part count depends on the given.
Targeted repair. Label total, difference or one side.
Misconception 3: Simplifying changes quantities
What the learner may think. The learner thinks 4:6 and 2:3 describe different actual sets.
Why it fails. Equivalent ratios preserve comparison.
Targeted repair. Track the changed part value.
Misconception 4: Use different part values
What the learner may think. The learner scales the two sides separately.
Why it fails. Ratio parts are equal model units.
Targeted repair. Write one shared multiplier.
Misconception 5: Drop the unit
What the learner may think. The learner reports “15 parts” instead of $15 or 15 pupils.
Why it fails. Actual quantities require context units.
Targeted repair. Attach the given unit to one part.
Misconception 6: Check only ratio
What the learner may think. The learner finds equivalent quantities that miss the stated total or difference.
Why it fails. All constraints matter.
Targeted repair. Verify ratio and the actual given.
Graded independent practice with explained answers
Attempt each item without looking ahead. Write the evidence or relationship before the final response. Early items check recognition; middle items require explanation; later items change the surface form so that copying a worked example is insufficient.
Practice 1: Meaning
Question. What does 2:3 directly state?
Explained answer. Two equal parts compared with three equal parts.
Why this earns trust. It does not fix actual counts.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Two equal parts compared with three equal parts.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Difference parts
Question. How many difference parts in 4:7?
Explained answer. Three parts.
Why this earns trust. Use 7−4.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Three parts.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Total parts
Question. How many total parts in 4:7?
Explained answer. Eleven parts.
Why this earns trust. Use 4+7.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Eleven parts.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Part value
Question. A 3:5 ratio has difference 16. Find one part.
Explained answer. Two parts equal 16, so one part is 8.
Why this earns trust. The given matches difference parts.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Two parts equal 16, so one part is 8.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Quantities
Question. Continue the previous problem.
Explained answer. The quantities are 24 and 40.
Why this earns trust. Multiply 3 and 5 by 8.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The quantities are 24 and 40.” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Total
Question. A:B=2:6 and total is 40.
Explained answer. One part is 5; quantities are 10 and 30.
Why this earns trust. Eight parts match the total.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “One part is 5; quantities are 10 and 30.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: One side
Question. Cats:Dogs=3:4 and cats=18.
Explained answer. One part is 6; dogs=24.
Why this earns trust. Three parts match 18.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “One part is 6; dogs=24.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Scale
Question. Can 2:3 represent 200 and 300?
Explained answer. Yes. One part is 100.
Why this earns trust. The comparison is multiplicative.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Yes. One part is 100.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Unknown difference
Question. Does 9:10 prove a difference of one object?
Explained answer. No. It proves one-part difference only.
Why this earns trust. Scale remains unknown.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No. It proves one-part difference only.” from memory, but to recreate a justified answer from the new evidence.
Practice 10: Simplify
Question. Why can one part change after simplifying?
Explained answer. Fewer larger model parts represent the same quantities.
Why this earns trust. The actual comparison is invariant.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Fewer larger model parts represent the same quantities.” from memory, but to recreate a justified answer from the new evidence.
Practice 11: Unit
Question. If one part is $12, what is three parts?
Explained answer. $36.
Why this earns trust. The unit scales with the part value.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “$36.” from memory, but to recreate a justified answer from the new evidence.
Practice 12: Routine
Question. State the five checks.
Explained answer. Parts, target parts, part value, actual quantities and verification.
Why this earns trust. The full translation is present.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Parts, target parts, part value, actual quantities and verification.” from memory, but to recreate a justified answer from the new evidence.
A four-session mastery cycle
Session 1: construct. Read the mechanism once, cover it and write a three-line model containing the trigger, the operation and the check. Add one example and one near-miss. Compare only after the model is complete. If the operation is right but the trigger is missing, the knowledge is unsafe because it may be applied in the wrong place.
Session 2: discriminate. Mix six items: two that clearly require this method, two that tempt the method but violate a condition, and two that require another skill. For every item, write “use” or “do not use” before answering. Accuracy in method selection matters more than speed at this stage.
Session 3: explain. Give a complete answer to an imagined classmate. Include the smallest sufficient evidence, the mechanism and the independent check. Remove any sentence that merely repeats the question or adds an unsupported claim. A concise causal explanation is stronger than a longer collection of familiar words.
Session 4: perform. Attempt a timed mixed set without notes. Mark uncertain decisions, but continue. During review, classify each error as trigger, evidence, operation, scope, unit or check. Reattempt one changed version after a delay. Mastery is shown when the learner chooses and adapts the method without needing the original example beside it.
Delayed transfer: prove that the learning lasts
Transfer test 1: Scale family
Represent one ratio at four different scales.
Success criterion. Every pair simplifies correctly.
Transfer test 2: Given-type sort
Classify totals, differences and one-side givens.
Success criterion. The correct matching part count is selected.
Transfer test 3: Simplification transfer
Explain equivalent ratios with changed part values.
Success criterion. Actual quantities remain invariant.
Transfer test 4: Delayed modelling
Solve a new ratio word problem after a week.
Success criterion. Parts and units are distinguished without prompting.
Examination control
1. Draw equal parts. Ratio numbers count model units.
2. Match the given. Use total, difference or one side correctly.
3. Find one part. Convert the model to actual scale.
4. Attach units. Report real quantities.
5. Verify twice. Check ratio and stated total or difference.
Under time pressure, compress the method to ratio parts → target parts → one part → quantities → check. The compressed routine is useful only after the full reasoning is understood. If it produces uncertainty, expand the step instead of guessing.
A compact error log
Label errors as parts treated as objects, wrong target parts, different multipliers, simplification misunderstood, unit dropped or condition unchecked.
Retest with 2:3 at scales 1, 7 and 100; the learner must distinguish part difference from actual difference.
Notes for parents and tutors
Use identical counters grouped into equal boxes to show that a part can contain several objects.
Ask “How many parts does the given describe?” before any division.
Keep units visible beside the one-part value.
Make the learner verify the original sentence, not just the ratio.
Frequently asked questions
Is one ratio part one object?
Not unless the problem states that one part equals one object.
What is a part?
An equal model unit shared by both quantities.
How do I use a difference?
Divide the actual difference by the difference in ratio parts.
How do I use a total?
Divide the total by the sum of ratio parts.
Can one part be fractional?
Yes when the context permits fractional quantities.
Why does simplification change part value?
Fewer model parts must represent the same actual amounts.
What must I check?
The ratio, the given total or difference and units.
What is the main trap?
Treating ratio coefficients as actual object counts.
Next routes
Return to the complete volume directory to locate the series, use the original PSLE Learning Guide for the foundation route, and continue through PSLE Mathematics for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is where the learner must choose a method rather than imitate one.