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 the PSLE Mathematics hub provides subject-specific continuation.
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This volume solves one precise learner problem: the learner assumes that adding the same number to numerator and denominator keeps a fraction unchanged. The aim is not to memorise a slogan. It is to understand the mechanism well enough to recognise the idea when the wording, numbers, diagram or context changes. A Primary 6 learner should be able to explain the decision, perform it under time pressure, and detect a tempting answer that only looks familiar.
- Understand the mechanism before memorising a form.
- Study the worked examples and compare near-misses.
- Repair common misconceptions with targeted tests.
- Complete graded independent practice before reading the answers.
- Use delayed transfer and examination control to make the skill durable.
- Use the parent and tutor notes only after the learner has attempted the work.
The current official SEAB page for PSLE formats examined in 2026 is the primary route for format-specific information. This article teaches a reasoning habit rather than predicting a particular question. If a school instruction or an official paper direction differs from a practice routine here, follow the instruction printed for that task.
The problem beneath the mistake
Equal changes do not always preserve a ratio. The fraction 2/5 is not kept equal by adding 3 to both terms: 5/8 is larger than 2/5. The operation looks balanced because both written numbers change equally, but numerator and denominator have different jobs.
A fraction can be read as a quotient, a part-whole comparison, or a position on the number line. Adding the same positive amount to both terms creates a new quotient and a new whole. It does not add the same amount to the fraction’s value.
For a proper fraction, the denominator exceeds the numerator. Adding the same positive number preserves the absolute gap between them while making the denominator larger. That fixed gap becomes a smaller fraction of the new denominator, so the value moves closer to 1.
Core decision: For a positive proper fraction a/b with a < b, adding the same positive number k to numerator and denominator gives (a+k)/(b+k), which is greater than a/b but still less than 1.
A useful rule has three parts: a trigger, an operation and a check. The trigger tells you when the rule applies. The operation tells you what to do. The check tells you whether the result still matches the meaning of the question. Learners who remember only the operation often apply it in the wrong place; learners who remember only an example often freeze when the surface details change.
Mechanism: why the rule works
1. Numerator and denominator are not separate counters
The numerator tells how many equal parts are considered; the denominator tells how many equal parts make the whole. Changing the denominator changes the size of each part, so equal written increments do not cancel.
Diagnostic question: Did the size of the whole or each part change? Reliable response: Treat the new pair as a new ratio.
2. Compare the missing part to 1
For a/b, the distance from 1 is (b-a)/b. After adding k to both terms, the distance is still based on gap b-a, but becomes (b-a)/(b+k), which is smaller.
Diagnostic question: What is the gap between denominator and numerator? Reliable response: Keep the gap, enlarge the denominator, and infer a smaller distance from 1.
3. Cross multiplication proves the direction
Compare (a+k)/(b+k) with a/b. Cross products are b(a+k) and a(b+k). Their difference is k(b-a), which is positive when k>0 and b>a.
Diagnostic question: Are k and b-a positive? Reliable response: Then the new fraction is larger.
4. The new fraction remains proper
Adding the same number preserves the difference b-a, so a+k remains less than b+k. The value increases without reaching or exceeding 1.
Diagnostic question: Does numerator remain smaller? Reliable response: Yes, because the original positive gap remains.
5. Scaling is the operation that preserves a fraction
Multiplying numerator and denominator by the same non-zero number creates an equivalent fraction because the quotient is scaled in both positions. Adding is not scaling.
Diagnostic question: Was the same factor used or the same addend? Reliable response: Same factor preserves; same positive addend changes a proper fraction.
6. The rule has conditions
The direction stated here assumes a positive proper fraction and a positive number added. Improper fractions behave in the opposite direction toward 1, and negative changes require domain checks.
Diagnostic question: Is the starting fraction proper and is k positive? Reliable response: Verify conditions before quoting the rule.
Worked examples: make the decision visible
Do not race through these examples. Before reading each explanation, name the evidence that matters and the information that does not. Then compare the correct route with the tempting route. That comparison is where the transferable learning sits.
Worked example 1: Two fifths plus three
Task. Compare 2/5 with (2+3)/(5+3).
Reasoning. The new fraction is 5/8. The gap from 1 remains 3 units, but changes from 3/5 to 3/8. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 5/8 is greater than 2/5 and closer to 1.
Independent check. Cross multiply: 5×5=25 and 2×8=16, so 5/8 is larger. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 2: Seven tenths plus two
Task. Compare 7/10 with 9/12.
Reasoning. Both have a gap of 3 between denominator and numerator. The missing fraction shrinks from 3/10 to 3/12. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 9/12=3/4 is greater than 7/10.
Independent check. Decimal estimates 0.75 and 0.70 confirm the direction. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 3: Near one
Task. Add 5 to both terms of 19/20.
Reasoning. The gap stays 1, while the denominator grows from 20 to 25. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 24/25 is larger than 19/20 and remains below 1.
Independent check. Distances from 1 are 1/20 and 1/25. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 4: Very small fraction
Task. Add 10 to both terms of 1/100.
Reasoning. The new value 11/110 simplifies to 1/10. Equal addition has changed the ratio substantially because the numerator was initially tiny. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 1/10 is greater than 1/100.
Independent check. A place-value comparison confirms ten hundredths exceeds one hundredth. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 5: Contrast with multiplication
Task. Compare adding 3 to 2/5 with multiplying both terms by 3.
Reasoning. Addition gives 5/8, while multiplication gives 6/15=2/5. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. Only multiplication by the same non-zero factor preserves the fraction.
Independent check. Simplify 6/15 and cross multiply 2/5 with 5/8. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 6: Model with coloured squares
Task. A model changes from 3 shaded out of 8 to 7 shaded out of 12 by adding four shaded and four total squares.
Reasoning. The four added squares are all shaded, not a mini-copy of the original 3-out-of-8 composition. This raises the proportion. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 7/12 is greater than 3/8.
Independent check. Cross products: 7×8=56 and 3×12=36. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 7: General numeric proof
Task. Explain why (n+1)/(n+2) is greater than n/(n+1) for positive n.
Reasoning. Both fractions are one unit below their denominators. Distances from 1 are 1/(n+2) and 1/(n+1), and the first distance is smaller. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. (n+1)/(n+2) is closer to 1 and therefore greater.
Independent check. Cross multiplication differs by 1. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 8: Improper fraction contrast
Task. Add 2 to both terms of 7/4.
Reasoning. The new fraction 9/6=3/2 is smaller than 7/4. An improper fraction begins above 1, and the same addition moves it toward 1 from above. The controlling idea is the changing gap relative to the changing denominator. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. The proper-fraction direction cannot be used without checking conditions.
Independent check. Compare 1.5 and 1.75. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Deepening lab: compare cases, conditions and limits
A method becomes independent when you can say where it applies, where it stops and which change would alter the conclusion. The cases below deliberately vary one structural feature at a time. For each case, pause before the analysis and predict whether the core rule remains available. Then name the exact condition that controls the decision. This is harder than repeating a worked solution, but it is also closer to what an unfamiliar examination item demands.
Deepening case 1: Add one represented as a new success
Case. A team wins 6 of 10 games, then wins one additional game. Its record becomes 7 wins in 11 games.
Analysis. Adding one to numerator and denominator models one new event that belongs to the numerator category. The success rate rises because the new event is a success. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare this with adding one new loss. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 2: Add several successes
Case. A reader completes 4 of 9 planned tasks, then completes the next 5 tasks without adding unfinished ones.
Analysis. The record becomes 9/14. Each added task is counted as completed and total, so the completion fraction moves upward toward 1. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Use cross multiplication and a decimal estimate. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 3: Keep the fraction unchanged by copying the mix
Case. A bag has 2 red counters out of 5. Another identical batch of 2 red out of 5 is added.
Analysis. The combined fraction is 4/10=2/5 because both terms were multiplied by 2. Preserving composition requires proportional addition, not equal numerical addition. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Explain why adding 2 red and 2 total would describe a different batch. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 4: Approach without reaching one
Case. Starting from 1/4, add 100 to both terms.
Analysis. The result 101/104 is very close to 1 but remains below it because the gap of 3 persists. A large addend shrinks the relative gap without removing it. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Find the exact distance from 1 before and after. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 5: Start at one
Case. The fraction 7/7 receives 5 in both numerator and denominator.
Analysis. The result 12/12 remains 1. The gap begins at zero and stays zero, so there is no movement from either side. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Place this boundary case between proper and improper examples. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 6: Improper fraction as a rate
Case. A machine produces 8 items in 5 minutes, and a comparison forms 11/8 after adding 3 to both terms.
Analysis. The numerical ratio decreases toward 1 because the starting numerator exceeds the denominator. The real-world units also require care: adding items and minutes is not automatically a meaningful process. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Separate the algebraic fact from the modelling question. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 7: Subtracting the same positive number
Case. A learner subtracts 1 from both terms of 5/8 to get 4/7.
Analysis. This is the reverse of adding one to 4/7, so 4/7 is smaller than 5/8. Denominators must remain positive, and the operation still does not preserve value. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Prove the comparison by cross multiplication. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 8: Fixed gap family
Case. Consider 2/5, 7/10, 12/15 and 102/105.
Analysis. Every fraction has gap 3. As denominators grow, the missing fraction 3/d shrinks, so the values rise toward 1 in a predictable order. Notice that the answer depends on the changing gap relative to the changing denominator; changing a decorative detail would not be enough to change the reasoning.
Challenge. Order the family without converting to decimals. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Finish the lab with a blank-page recall. Write the rule, its conditions, one confirming example, one near-miss and one independent check from memory. Compare that page with the article only after finishing. If the rule is correct but the condition is missing, the learning is not yet safe: an overgeneralised rule can produce confident errors. If the condition is correct but no example can be generated, practise construction rather than rereading.
Common misconceptions and targeted repairs
Misconception 1: Equal addition cancels
What the learner may think. Adding the same number to top and bottom changes nothing.
Why it fails. Cancellation applies to common factors in multiplication, not common addends across a division bar. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Contrast (a×k)/(b×k) with (a+k)/(b+k). Use this short test immediately: Does 2/5 equal 5/8?
Misconception 2: The gap alone determines value
What the learner may think. Both fractions have the same numerator-denominator gap, so they are equal.
Why it fails. The gap must be considered relative to the denominator. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Write distance from 1 as gap/denominator. Use this short test immediately: Compare 1/2 and 9/10; both gaps are 1.
Misconception 3: A larger denominator means a smaller fraction
What the learner may think. The denominator increased, so the fraction decreased.
Why it fails. That shortcut applies only under controlled conditions such as equal positive numerators. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Track both terms and compare ratios. Use this short test immediately: Compare 2/5 with 5/8 by cross multiplication.
Misconception 4: Adding k adds k to the value
What the learner may think. 2/5 plus 3 becomes 3.4 or 5/8 by direct value addition.
Why it fails. Changing terms is not the same as adding to the quotient. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Rewrite the complete new fraction before evaluating. Use this short test immediately: What operation is actually shown by (2+3)/(5+3)?
Misconception 5: The rule works for every fraction
What the learner may think. Same addition always increases a fraction.
Why it fails. Improper fractions above 1 move downward toward 1. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Classify proper, equal-to-one or improper first. Use this short test immediately: Compare 7/4 with 9/6.
Misconception 6: The fraction reaches 1 after enough additions
What the learner may think. Repeated equal additions will eventually make terms equal.
Why it fails. The positive difference b-a stays constant. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Track the invariant gap. Use this short test immediately: Can adding the same k make unequal numbers equal?
Graded independent practice
Attempt each item without looking ahead. Write one line that identifies the controlling evidence before writing the final answer. Items 1–3 check recognition, 4–7 require explanation, and the later items change the surface form so that memory alone is insufficient.
Practice 1: Recognition
Question. Which is larger: 3/7 or 4/8?
Explained answer. 4/8 is larger.
Why this answer earns trust. Both are proper and one was added to both terms; cross products are 28 and 24. Transfer variation: Compare 3/7 with 13/17.
Practice 2: Distance from 1
Question. Compare 8/11 and 13/16 using the gap method.
Explained answer. 13/16 is larger.
Why this answer earns trust. Both gaps are 3, but 3/16 is a smaller missing part than 3/11. Transfer variation: Create another pair with gap 3.
Practice 3: Equivalence
Question. Which operation preserves 4/9: add 5 to both terms or multiply both by 5?
Explained answer. Multiplying gives 20/45=4/9; adding gives 9/14, a larger fraction.
Why this answer earns trust. Common factors cancel; common addends do not. Transfer variation: Explain without using decimals.
Practice 4: Proof
Question. Show that (a+k)/(b+k)>a/b when 0
Explained answer. Cross multiply: b(a+k)-a(b+k)=k(b-a)>0.
Why this answer earns trust. Both factors in the final product are positive. Transfer variation: State what fails if a>b.
Practice 5: Model
Question. A jar changes from 5 red counters out of 12 to 9 red out of 16. Did the red fraction rise?
Explained answer. Yes. 9/16>5/12.
Why this answer earns trust. Four added counters were all red, so the composition rose. Transfer variation: How many non-red counters remain in each jar?
Practice 6: Near one
Question. Which is closer to 1: 49/50 or 99/100?
Explained answer. 99/100 is closer.
Why this answer earns trust. The gaps are both 1, but 1/100 is smaller than 1/50. Transfer variation: Order 9/10, 99/100 and 999/1000.
Practice 7: Condition check
Question. Compare 11/7 and 14/10 after adding 3 to both terms.
Explained answer. 14/10 is smaller than 11/7.
Why this answer earns trust. The starting fraction is improper, so it moves toward 1 from above. Transfer variation: What happens if the fraction starts exactly at 1?
Practice 8: Error repair
Question. A learner says 6/13=10/17 because 4 was added to both terms. Correct the reasoning.
Explained answer. The fractions are not equal; 10/17 is greater because the proper fraction moved closer to 1.
Why this answer earns trust. Equality requires common scaling, not common addition. Transfer variation: Find an equivalent fraction to 6/13.
Practice 9: Generalisation
Question. For fractions with the same positive gap g, which is larger: (d-g)/d or (D-g)/D when D>d?
Explained answer. (D-g)/D is larger.
Why this answer earns trust. Its distance g/D from 1 is smaller than g/d. Transfer variation: Use g=4 with two numeric examples.
Practice 10: Independent explanation
Question. Explain the result to a learner using one model and one calculation.
Explained answer. Model the added items as all belonging to the numerator category, then verify with cross multiplication or distances from 1.
Why this answer earns trust. Two representations make the mechanism visible and checkable. Transfer variation: Choose 2/9 and add 5 to both terms.
Delayed transfer: prove that the learning lasts
Transfer test 1: Ratio transfer
Interpret adding the same number to wins and total games as adding only wins in the new games. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. Recognise why the win rate rises when every added game is a win. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 2: Algebra transfer
Prove the direction for symbols a, b and k after a two-day delay. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. State conditions and obtain k(b-a)>0 without copying. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 3: Boundary transfer
Compare proper, unit and improper starting fractions under the same addition. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. Describe movement toward 1 from below, no change at 1, and movement toward 1 from above. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Examination control
1. Read the job before the story. Underline the command and the noun it governs. “Explain the change,” “state the word,” “find the fraction” and “identify the evidence” are different jobs even when the surrounding context is similar.
2. Separate given information from remembered information. Use only the facts supplied for the current task plus secure subject knowledge. Do not let a previous question, a familiar worksheet or an expected pattern silently become new evidence.
3. Write enough working to expose the decision. A marker or teacher should be able to see the relationship that justifies the answer. Extra sentences are not automatically stronger; the strongest working is concise, relevant and checkable.
4. Reserve a different-route check. For language, read the sentence for meaning and grammar. For mathematics, estimate or reverse the operation. For science, check the claim against the stated observation and a conservation or fair-test principle.
5. Stop when the requested job is complete. Unrequested speculation can contradict a correct answer. Add detail only when it explains the mechanism or satisfies another explicit part of the question.
Under time pressure, use a compact four-part margin routine: trigger → evidence → operation → check. Trigger names the feature that activates the method. Evidence identifies the exact words, values or observations allowed. Operation performs the language, mathematical or scientific step. Check asks whether the final statement answers the requested job, preserves units or referents, and avoids a stronger claim than the evidence supports.
A compact error log
After practice, do not record only “careless mistake.” Name the broken decision. Useful labels include trigger missed, evidence imported, relationship reversed, referent lost, unit dropped, operation correct but job wrong, and check repeated the same route. Add one prevention cue beside the label. A short accurate log is more useful than a long emotional account because it tells the learner what to do on the next item.
Review the log after two days and again after a week. Reattempt only one or two representative items first. If the method transfers, move to a mixed set. If it fails, repair the exact link between trigger and operation. This spacing matters because a solution can feel obvious while it is still visible and disappear when the page is closed.
Notes for parents and tutors
Use counters or shaded grids carefully. When adding the same number to numerator and denominator, the added items are all counted in the numerator category as well as in the total. This is why the proportion changes; it is not a neutral enlargement of the original group.
A tutor should require conditions before the rule: positive proper fraction and positive addend. Then include an improper-fraction counterexample so the learner does not overgeneralise.
Ask for two checks: cross multiplication and distance from 1. If both agree, the learner has an arithmetic check and a structural explanation.
The adult’s most useful question is not “Why were you careless?” but “Which decision did the evidence require?” Ask the learner to point to the controlling word, value, relationship or observation. If the learner can explain that link, reduce support. If the learner cannot, model one example and then change the surface details for the next attempt.
Frequently asked questions
Why does adding the same number not cancel?
A fraction bar represents division. Common multiplication factors can be divided out; common addends cannot generally be cancelled across a quotient.
Does the numerator-denominator gap change?
No. Adding the same number preserves the absolute gap, but the gap becomes a smaller fraction of the larger denominator.
Will the new proper fraction exceed 1?
No. The numerator remains smaller because the original positive gap is preserved.
What if the original fraction equals 1?
If numerator and denominator are equal, adding the same number keeps them equal, so the value remains 1.
What if the original fraction is improper?
Adding the same positive number moves it closer to 1 from above, so its value decreases. Check the starting class before using the direction rule.
What is the fastest reliable comparison?
For this special structure, compare distances from 1. For general fractions, cross multiplication is dependable when denominators are positive.
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 the PSLE Mathematics hub for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is the point at which a learner must choose, not merely copy.