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 assumes that doing the same addition to the numerator and denominator preserves a fraction, confusing additive change with multiplying both terms by the same non-zero factor. 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 problem beneath the mistake
Multiplying 2/3 by 2/2 gives 4/6, which has the same value because the whole fraction is multiplied by 1.
Adding 2 to both terms gives 4/5, not 4/6. The new fraction is larger than 2/3, so equal additions did not preserve value.
The words “do the same thing to top and bottom” are incomplete. The operation matters. Same multiplication preserves a ratio; same addition usually does not.
In 2/3, adding 1 doubles the numerator but increases the denominator by only one third. The relative changes are unequal.
Cross-products provide a decisive check. If a/b and c/d are equal, then ad equals bc when denominators are non-zero.
For 2/3 and 3/4, the cross-products are 2×4=8 and 3×3=9. They are not equivalent.
A positive proper fraction has numerator smaller than denominator. Adding the same positive number keeps the gap fixed while the denominator grows, so the fraction moves closer to 1.
The gap from one is (denominator−numerator)/denominator. The top-bottom gap stays fixed after equal addition, but it becomes a smaller part of the larger denominator.
If numerator and denominator are already equal, the fraction is 1 and equal additions keep it at 1, provided the new denominator is not zero.
If the added number is zero, nothing changes. These special cases do not make equal addition a general equivalence method.
Negative numbers and zero denominators require extra care. At PSLE level, inspect the stated positive fractions and never allow a denominator of zero.
The safe routine is identify the operation, predict direction, compare by cross-products, check denominator validity and state whether value changed.
Core decision: Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same non-zero factor. Adding the same non-zero number to unequal numerator and denominator usually changes the value because the two relative changes are different. For positive proper fractions, adding a positive number to both terms moves the fraction closer to 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. Name the operation
Multiplication and addition affect ratios differently.
Diagnostic question: Were both terms scaled or shifted? Reliable response: Write ×k or +k beside the fraction.
2. Test relative change
An equal addition is not an equal percentage change when terms differ.
Diagnostic question: What fraction of each original term was added? Reliable response: Compare k/a with k/b.
3. Use cross-products
Two fractions a/b and c/d are equal exactly when ad=bc for valid denominators.
Diagnostic question: Do the diagonal products match? Reliable response: Calculate both before simplifying claims.
4. Track the gap from one
For a positive proper fraction, 1−a/b=(b−a)/b.
Diagnostic question: What happens to the numerator-denominator gap and denominator? Reliable response: The gap stays, denominator grows, so distance from one shrinks.
5. Identify special cases
Adding zero changes nothing, and equal terms represent one.
Diagnostic question: Is k zero or numerator equal denominator? Reliable response: State the exception instead of overgeneralising.
6. Protect denominator validity
A fraction is undefined when its denominator is zero.
Diagnostic question: Could b+k equal zero? Reliable response: Reject any transformed expression with zero denominator.
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: Simple proper fraction
Task. Add 1 to numerator and denominator of 1/2.
Reasoning. The new fraction is 2/3. Cross-products are 1×3=3 and 2×2=4. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. The value changes from 1/2 to 2/3.
Independent check. Decimal check: 0.5 versus about 0.667. 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: Compare with scaling
Task. Start with 3/5. Multiply both terms by 2, then separately add 2.
Reasoning. Scaling gives 6/10 and preserves value. Addition gives 5/7 and changes it. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 3/5=6/10 but 3/5≠5/7.
Independent check. Cross-products confirm both statements. 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: Closer to one
Task. Add 4 to both terms of 2/7.
Reasoning. The gap remains 5, but denominator grows from 7 to 11. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 6/11 is larger than 2/7 and closer to 1.
Independent check. 2×11=22 while 6×7=42, so new 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 4: Improper fraction
Task. Add 2 to 7/4.
Reasoning. Both terms increase, but the fraction moves toward 1 from above. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 9/6=3/2, which is less than 7/4.
Independent check. 1.5 is closer to 1 than 1.75. 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: Fraction equal to one
Task. Add 5 to 3/3.
Reasoning. Numerator and denominator remain equal. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 8/8=1, so value remains one.
Independent check. Both simplify to one. 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: Zero addition
Task. Add 0 to 5/9.
Reasoning. No term changes. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. The fraction remains 5/9.
Independent check. Identity check. 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: Negative shift
Task. Subtract 1 from both terms of 3/5.
Reasoning. This is adding -1. The new fraction is 2/4=1/2. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. The value changes from 3/5 to 1/2.
Independent check. Cross-products: 3×4=12, 2×5=10. 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: Invalid denominator
Task. Add -4 to both terms of 7/4.
Reasoning. The new denominator would be zero. The controlling idea is whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. The transformed expression is undefined.
Independent check. A denominator cannot be zero. 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: Multiply by two
Case. a/b becomes 2a/2b.
Analysis. This multiplies the fraction by 2/2=1. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Explain equivalence. 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 two
Case. a/b becomes (a+2)/(b+2).
Analysis. This is a shift, not multiplication by one. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Use cross-products. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 3: Proper fraction
Case. 0
Analysis. Equal positive addition increases the fraction toward one. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Prove with one example. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 4: Improper fraction
Case. a>b>0 and k>0.
Analysis. Equal positive addition decreases the fraction toward one. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare distances from one. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 5: Unit fraction
Case. 1/n becomes (1+k)/(n+k).
Analysis. It is no longer usually a unit fraction and value changes. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Check n=4,k=1. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 6: Equal terms
Case. a/a with valid a equals one.
Analysis. Adding the same k keeps equal terms if denominator stays valid. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. State restriction. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 7: Zero shift
Case. k=0.
Analysis. The fraction is unchanged trivially. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Separate identity case. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 8: Negative shift
Case. k<0.
Analysis. Direction depends on signs and validity. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Check denominator. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 9: Fixed gap
Case. b−a stays the same after adding k to both.
Analysis. A fixed gap does not mean a fixed ratio. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare 2/5 and 12/15. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 10: Changing denominator
Case. The same gap is divided by a different denominator.
Analysis. Distance from one changes. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Write 1−fraction. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 11: Cross-product proof
Case. Compare a/b and (a+k)/(b+k).
Analysis. Equality requires a(b+k)=b(a+k), so k(a−b)=0. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Interpret factors. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 12: Percentage view
Case. Adding k gives percentage changes k/a and k/b.
Analysis. They differ when a and b differ. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Calculate both. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 13: Simplification after addition
Case. The new fraction may simplify.
Analysis. Simplifying does not restore the original automatically. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare values. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 14: Word problem ratio
Case. Boys:girls is 2:3; add two boys and two girls.
Analysis. The new ratio 4:5 differs from 2:3. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Use counts. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 15: Mixture caution
Case. Equal added amounts change concentrations.
Analysis. Ratios depend on totals, not only differences. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Name components. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 16: Probability ratio
Case. Favourable/total cannot be changed by adding one to both without changing experiment meaning.
Analysis. Numerator is contained within denominator. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Model actual outcomes. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 17: Equivalent target
Case. To create an equivalent fraction, choose a scale factor.
Analysis. Multiply both terms by that factor. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Find 3/4 with denominator 20. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 18: Comparison target
Case. Equal addition can help compare movement toward one.
Analysis. Do not label it equivalence. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. State direction only. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 19: Decimal check
Case. Convert simple fractions to decimals.
Analysis. This provides a different-route check. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Use exact comparison where possible. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 20: Complete statement
Case. “Same to top and bottom” is unsafe.
Analysis. Name multiplication by the same non-zero factor. Notice that the answer depends on whether the operation is multiplication or addition, the original numerator-denominator relationship, and a comparison by cross-products or distance from one; changing a decorative detail would not be enough to change the reasoning.
Challenge. Rewrite the rule precisely. 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: Same operation always preserves value
What the learner may think. Doing the same addition to both terms must be fair.
Why it fails. Ratios depend on relative, not absolute, change. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Compare percentage increases. Use this short test immediately: Were terms scaled by one factor?
Misconception 2: Fixed difference means fixed fraction
What the learner may think. The top-bottom gap stays, so the ratio stays.
Why it fails. A fraction uses division, and denominator changes. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Use distance from one. Use this short test immediately: What fraction of the denominator is the gap?
Misconception 3: Simplifying proves equivalence
What the learner may think. The new fraction simplifies, so it equals the old one.
Why it fails. Simplification only preserves the new fraction’s value. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Compare new and old by cross-products. Use this short test immediately: Do diagonals match?
Misconception 4: Proper fraction gets smaller
What the learner may think. A larger denominator always makes the fraction smaller.
Why it fails. The numerator also increases; for equal positive addition it moves toward one. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Use cross-products or gap. Use this short test immediately: Which relative increase is larger?
Misconception 5: Every exception is a method
What the learner may think. 1/1 stays one, so equal addition is valid generally.
Why it fails. Equal terms are a special case. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. State k=0 or a=b exceptions. Use this short test immediately: Do terms differ?
Misconception 6: Denominator check can wait
What the learner may think. Algebra may create a denominator of zero.
Why it fails. Undefined expressions cannot be used. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Check b+k first. Use this short test immediately: Is the denominator non-zero?
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. Does adding 1 to both terms of 2/3 preserve value?
Explained answer. No; it gives 3/4.
Why this answer earns trust. Cross-products 8 and 9 differ. Transfer variation: Compare decimals.
Practice 2: Equivalent fraction
Question. Create an equivalent fraction to 2/3 with denominator 12.
Explained answer. 8/12.
Why this answer earns trust. Multiply both terms by 4. Transfer variation: State factor.
Practice 3: Direction
Question. Add 5 to both terms of 3/8. Does value rise or fall?
Explained answer. It rises toward one.
Why this answer earns trust. The proper fraction has fixed gap and larger denominator. Transfer variation: Verify 8/13 versus 3/8.
Practice 4: Improper
Question. Add 3 to 8/5. Does value rise or fall?
Explained answer. It falls toward one.
Why this answer earns trust. The excess over denominator stays 3 while denominator grows. Transfer variation: Compare 11/8 and 8/5.
Practice 5: Exception
Question. When can equal addition preserve a/b?
Explained answer. When the addition is zero or numerator equals denominator, subject to valid denominators.
Why this answer earns trust. The equality condition is k(a−b)=0. Transfer variation: Test each case.
Practice 6: Ratio story
Question. Boys:girls=4:6. Add two to each group. Is ratio unchanged?
Explained answer. No; 6:8=3:4, while 4:6=2:3.
Why this answer earns trust. Absolute additions differ proportionally. Transfer variation: Use cross-products.
Practice 7: Invalid
Question. Transform 5/2 by adding -2 to both terms.
Explained answer. The result is invalid because denominator becomes zero.
Why this answer earns trust. Fractions cannot have zero denominator. Transfer variation: Identify domain first.
Practice 8: Repair rule
Question. Improve “do the same thing to top and bottom”.
Explained answer. Multiply or divide numerator and denominator by the same non-zero factor to preserve value.
Why this answer earns trust. The operation and restriction are explicit. Transfer variation: Give a counterexample to addition.
Practice 9: Proof
Question. Why does positive equal addition move a proper fraction toward one?
Explained answer. The numerator-denominator gap stays fixed while the positive denominator grows, so the gap is a smaller fraction of the denominator.
Why this answer earns trust. This uses 1−a/b. Transfer variation: Apply to 9/10.
Practice 10: Routine
Question. State the checking routine.
Explained answer. Name operation, predict direction, cross-multiply, inspect special cases, protect denominator and conclude.
Why this answer earns trust. It separates equivalence from movement. Transfer variation: Apply to 7/9 plus 2.
Delayed transfer: prove that the learning lasts
Transfer test 1: Operation sort
Sort fraction changes into scaling, equal addition, unequal addition and simplification. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. Only justified scaling is labelled an equivalence transformation. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 2: Ratio investigation
Test positive proper and improper fractions with several equal additions. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. The direction toward one is predicted before calculation. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 3: Delayed proof
After two days, derive the equality condition for a/b and (a+k)/(b+k). Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. The learner interprets k(a−b)=0 and checks denominators. 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
Ask whether the terms were multiplied or merely increased; avoid the vague phrase “same thing”.
Use counters for ratios such as two red and three blue, then add equal counts and compare the changed composition.
Require one cross-product check so a familiar-looking fraction is not accepted by appearance.
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 multiplying both terms preserve a fraction?
It multiplies the whole value by k/k=1 when k is non-zero.
Why does adding the same number usually change it?
The unequal terms receive different relative or percentage increases.
What happens to a proper fraction after positive equal addition?
It increases toward one while remaining below one.
Are there exceptions?
Yes: adding zero, or starting with equal numerator and denominator, subject to a valid non-zero denominator.
What is the fastest test?
Cross-multiply the old and new fractions.
Can the denominator become zero?
Yes for some negative additions; that transformed fraction is undefined and must be rejected.
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 the point at which a learner must choose, not merely copy.