Many PSLE Mathematics word problems describe change: counters are transferred, money is spent, quantities are increased, groups are rearranged, dimensions change, or a percentage is applied. Learners often focus only on what changed. The more powerful question is frequently: what stayed the same?
This Learner’s Guide develops one advanced but practical habit: before comparing a before-state and an after-state, identify the invariant—the quantity, relationship or condition that did not change. The invariant becomes an anchor. It reduces the number of unknowns and helps the learner choose a representation that survives the change.
The guide builds on Vol 0003: Represent Before You Calculate, Vol 0007: Recover When the First Method Fails, and Vol 0011: Name the Intermediate Quantity Before You Use It.
BEFORE → WHAT CHANGED? → WHAT STAYED THE SAME? → REPRESENT THE INVARIANT → CONNECT TO AFTER → SOLVE.
The quick answer: an invariant is a stable anchor
An invariant is something that remains unchanged while other quantities change. It may be a total, a difference, a ratio, a unit rate, an area, a perimeter, a number of objects, a fixed time, or a shared condition. The question itself determines what is stable.
Do not assume a familiar invariant. A total is not always conserved. A ratio is not always constant. The learner must read the change carefully and prove what stays the same.
Three questions before calculation
- What is different after the change?
- What must be the same before and after?
- Which stable quantity gives the cleanest bridge between the two states?
Common invariant patterns
- Transfer between groups: combined total may stay the same.
- Add or subtract the same amount from both: difference stays the same.
- Scale both parts by the same factor: ratio stays equivalent.
- Rearrange without adding or removing: total quantity stays the same.
- Same unit rate: per-unit relationship stays the same.
- Fixed geometry condition: perimeter, area or volume may be stated as constant.
- Same baseline: a common starting amount can anchor percentage or difference comparisons.
Sixty-four invariant cases
Counters transferred
Situation: Counters move from Box B to Box A; none enter or leave.
What stays the same: combined total.
Mathematical move: Represent the same total before and after.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Marbles transferred
Situation: Some marbles move from Red to Blue.
What stays the same: total marbles.
Mathematical move: Individual counts change, system total does not.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Money transferred
Situation: $20 moves from Aisha to Ben.
What stays the same: combined money.
Mathematical move: One decreases exactly as the other increases.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Books rearranged
Situation: Books move between two shelves.
What stays the same: total books.
Mathematical move: Shelf counts change; combined count stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Pupils switch groups
Situation: Pupils move from Group A to Group B.
What stays the same: class total.
Mathematical move: Use one fixed total across both ratios.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Stickers gifted equally
Situation: Both children receive 12 stickers.
What stays the same: difference.
Mathematical move: Same addition preserves the gap.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Points deducted equally
Situation: Both teams lose 5 points.
What stays the same: difference.
Mathematical move: Same subtraction preserves the gap.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Equal spending
Situation: Two people each spend $10.
What stays the same: difference in money.
Mathematical move: Difference remains; ratio usually changes.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Equal distance added
Situation: Two runners each run 2 km more.
What stays the same: difference in distance.
Mathematical move: Same addition preserves difference.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Equal shortening
Situation: Two ropes are each shortened by 3 cm.
What stays the same: difference in lengths.
Mathematical move: Same subtraction preserves difference.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Recipe doubled
Situation: Every ingredient is doubled.
What stays the same: ingredient ratio.
Mathematical move: Scaling preserves proportion.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Recipe halved
Situation: Every ingredient is halved.
What stays the same: ingredient ratio.
Mathematical move: Common division preserves proportion.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Map scale
Situation: All distances use one scale.
What stays the same: scale factor.
Mathematical move: Map and actual values change together.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Constant speed
Situation: A car travels at 60 km/h for different times.
What stays the same: rate.
Mathematical move: Distance varies, rate stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Constant unit price
Situation: Each notebook costs the same.
What stays the same: price per notebook.
Mathematical move: Total cost scales with quantity.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Constant box capacity
Situation: Each carton holds 24 cans.
What stays the same: cans per carton.
Mathematical move: Number of cartons changes, capacity stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Constant flow rate
Situation: Tap flows at 5 L/min.
What stays the same: litres per minute.
Mathematical move: Volume scales with time.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Constant work rate
Situation: Machine makes 8 items per minute.
What stays the same: items per minute.
Mathematical move: Output changes with duration.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same perimeter rectangle
Situation: Perimeter fixed at 40 cm while length changes.
What stays the same: perimeter.
Mathematical move: Width must compensate.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same area rectangle
Situation: Area fixed at 60 cm² while length changes.
What stays the same: area.
Mathematical move: Other dimension adjusts inversely.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same volume cuboid
Situation: Volume fixed while base area changes.
What stays the same: volume.
Mathematical move: Height adjusts to preserve volume.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same total score
Situation: Marks redistributed between sections.
What stays the same: overall score.
Mathematical move: Section values change, total stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same total mass
Situation: Material split into two portions without loss.
What stays the same: total mass.
Mathematical move: Parts sum to original mass.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same ribbon length
Situation: Ribbon cut into two pieces.
What stays the same: total length.
Mathematical move: Piece lengths sum to original.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same object count
Situation: Objects regrouped into different sets.
What stays the same: total count.
Mathematical move: Grouping changes, count stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same journey distance
Situation: Two travel plans cover same route.
What stays the same: distance.
Mathematical move: Speed and time may change.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same work target
Situation: Same number of items must be completed.
What stays the same: total items.
Mathematical move: Time changes with rate.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same tank capacity
Situation: Tank capacity fixed while fill level changes.
What stays the same: capacity.
Mathematical move: Current volume varies, maximum stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same whole
Situation: Different fractions describe same collection.
What stays the same: whole.
Mathematical move: Compare parts against one fixed whole.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same original price
Situation: Several changes refer to original price.
What stays the same: 100% base.
Mathematical move: Keep original amount as reference.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same population
Situation: Categories change but no people enter or leave.
What stays the same: total population.
Mathematical move: Category counts sum to same whole.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same class size
Situation: Boy-girl distribution changes without membership change.
What stays the same: class total.
Mathematical move: Use fixed total to bridge ratios.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same days
Situation: Two plans compare output over same number of days.
What stays the same: time period.
Mathematical move: Output rate is the changing factor.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same budget
Situation: Different allocations use the same budget.
What stays the same: total budget.
Mathematical move: Category spending changes.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same lap length
Situation: Runner completes different numbers of laps.
What stays the same: distance per lap.
Mathematical move: Total distance changes, lap length stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Equivalent fractions
Situation: Numerator and denominator scaled equally.
What stays the same: fraction value.
Mathematical move: Numerical parts change, value stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same tax rate
Situation: Tax percentage fixed across purchases.
What stays the same: percentage rate.
Mathematical move: Tax amount changes.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same conversion factor
Situation: 100 cm equals 1 m.
What stays the same: conversion relationship.
Mathematical move: Numerical representation changes, physical length stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same time interval
Situation: Machines compared over 10 minutes.
What stays the same: time window.
Mathematical move: Output comparison uses common duration.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same starting amount
Situation: Two percentage changes begin from same baseline.
What stays the same: baseline.
Mathematical move: Compare changes from common start.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same target sum
Situation: Different parts must total 100.
What stays the same: target total.
Mathematical move: Unknowns constrained by fixed sum.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Transfer then add
Situation: First, counters transfer; later, 10 new counters are added.
What stays the same: total only during transfer stage.
Mathematical move: Do not carry the invariant past the addition.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Transfer then remove
Situation: Items move between groups, then some leave system.
What stays the same: total before removal.
Mathematical move: Invariant changes at stage boundary.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Equal addition then unequal subtraction
Situation: Both receive 10, then only one spends 4.
What stays the same: difference only during equal-add stage.
Mathematical move: Track when invariant stops.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same ratio then extra addition
Situation: Both quantities scale, then one gets extra units.
What stays the same: ratio only before extra addition.
Mathematical move: Do not preserve ratio after asymmetric change.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Fixed difference then scaling
Situation: Two counts differ by 8, then both double.
What stays the same: original difference only before scaling.
Mathematical move: After scaling, the difference doubles.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same total after swap
Situation: Two pupils exchange equal-value tokens.
What stays the same: combined total.
Mathematical move: Individual holdings may change, total remains.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same average with redistribution
Situation: Total score and number of pupils fixed while scores redistribute.
What stays the same: overall average.
Mathematical move: Distribution changes, mean stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same number of groups
Situation: Items per group change while group count fixed.
What stays the same: group count.
Mathematical move: Total changes with items per group.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same items per group
Situation: Number of groups changes.
What stays the same: items per group.
Mathematical move: Total changes with group count.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same fare per trip
Situation: Trips increase but fare per trip fixed.
What stays the same: unit fare.
Mathematical move: Total transport cost scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same distance per trip
Situation: Number of identical trips changes.
What stays the same: trip distance.
Mathematical move: Total distance scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same daily allowance
Situation: Number of days changes.
What stays the same: allowance per day.
Mathematical move: Total allowance scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same packet mass
Situation: Packets added or removed.
What stays the same: mass per packet.
Mathematical move: Total mass scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same page count per book
Situation: Number of books changes.
What stays the same: pages per book.
Mathematical move: Total pages scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same water per bottle
Situation: Bottle count changes.
What stays the same: volume per bottle.
Mathematical move: Total volume scales.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Triangle angle sum
Situation: Triangle shape changes.
What stays the same: 180° interior-angle sum.
Mathematical move: Individual angles vary, sum stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Straight line angles
Situation: Adjacent angles change along a straight line.
What stays the same: 180° sum.
Mathematical move: One increase forces the other decrease.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Angles around a point
Situation: Individual angles vary.
What stays the same: 360° full-turn sum.
Mathematical move: Parts change, total stays.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Right-angle parts
Situation: Two component angles form a right angle.
What stays the same: 90° sum.
Mathematical move: Use fixed right-angle total.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Simple probability total
Situation: All mutually exclusive outcomes are considered.
What stays the same: probability total of 1.
Mathematical move: Individual probabilities sum to whole.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same denominator whole
Situation: Several category fractions share one whole.
What stays the same: denominator base.
Mathematical move: Numerators vary against same total.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same route length
Situation: Walking and cycling compare time on same route.
What stays the same: distance.
Mathematical move: Rate difference explains time difference.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same clock duration
Situation: Two taps run for same time.
What stays the same: time.
Mathematical move: Volume difference reflects rate difference.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same number of items
Situation: Two pricing schemes compare cost for same quantity.
What stays the same: quantity.
Mathematical move: Cost difference isolates pricing.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
Same amount invested
Situation: Two returns compare from same principal.
What stays the same: principal.
Mathematical move: Return amounts are comparable against one base.
Test the proposed invariant with simple numbers before committing. Then solve the original problem using the stable relationship as the bridge between states.
The wrong-invariant trap
A learner may preserve a ratio after equal addition, preserve a total after new items are introduced, or preserve a difference after unequal changes. The repair is to state the operation on each quantity and test whether the proposed relationship survives.
With A=10 and B=6, adding 3 to both keeps the difference 4 but changes the ratio. Multiplying both by 2 preserves the ratio 5:3 but changes the difference from 4 to 8.
Stage boundaries
One problem can contain several phases. A total may stay fixed during a transfer and then change when new items are added. Label the stage where the invariant is valid. Do not carry a stable relationship beyond the event that breaks it.
Before–after diagrams
Draw two simple states labelled BEFORE and AFTER. Mark the invariant with the same symbol. Mark changes with arrows. This externalises the structure and reduces the temptation to apply one state’s relationship to the other.
Invariant decision lab
Equal addition test
Test: Start with 12 and 20; add 5 to both.
Result: Difference stays 8; ratio changes.
Lesson: Use this to distinguish additive from multiplicative invariants.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Equal multiplication test
Test: Start with 12 and 20; multiply both by 3.
Result: Ratio stays 3:5; difference becomes 24.
Lesson: Scaling preserves ratio, not absolute gap.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Transfer test
Test: Start with groups 10 and 30; move 5 from second to first.
Result: Total remains 40; difference changes.
Lesson: Conservation anchors transfer problems.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Addition-to-system test
Test: Start with groups totalling 40; add 8 new items.
Result: Total becomes 48.
Lesson: Do not preserve a total when the system gains objects.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Removal-from-system test
Test: Start with 48 items; 6 are discarded.
Result: Total becomes 42.
Lesson: Conservation stops when items leave.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Fixed-perimeter test
Test: Rectangle 6 by 4 has perimeter 20; change length to 7 while perimeter stays 20.
Result: Width must become 3.
Lesson: Use fixed perimeter, not fixed area.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Fixed-area test
Test: Rectangle area 24; length changes from 6 to 8.
Result: Width changes from 4 to 3.
Lesson: Use product invariant.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Constant-rate test
Test: Rate is 5 L/min for 4 and 7 minutes.
Result: Rate stays 5; totals become 20 and 35.
Lesson: Unit rate connects durations.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Same-baseline test
Test: Two prices begin at $100; one rises 20%, one rises 30%.
Result: Starting base stays $100.
Lesson: Percentage changes are directly comparable.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Different-baseline warning
Test: One price starts at $100 and another at $200.
Result: Baseline differs.
Lesson: Equal percentages do not imply equal dollar changes.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Stage-change test
Test: Transfer preserves total, then 10 new items enter.
Result: Invariant changes after second event.
Lesson: Mark stage boundary explicitly.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Equivalent-fraction test
Test: 2/3 becomes 4/6.
Result: Value stays the same.
Lesson: Numerator and denominator scale together.
Use this laboratory style during practice: test the relationship with simple values, explain why it survives or fails, then return to the original word problem.
Under time pressure
Use one cue: what stayed? If a before–after problem feels crowded, stop calculating and answer that question. If no useful invariant exists, do not force the strategy; switch to another representation.
A seven-day invariant cycle
- Day 1: transfers and fixed totals.
- Day 2: equal changes and fixed differences.
- Day 3: scaling and fixed ratios.
- Day 4: constant rates and unit prices.
- Day 5: geometry invariants.
- Day 6: mixed before-after problems.
- Day 7: delayed transfer without topic labels.
What parents and tutors should ask
Ask: What changed? What did not change? Why must it remain the same? Can you prove it with simple numbers? At what stage does it stop being true? These questions reveal structure rather than operation guessing.
Common mistakes
- Preserving ratio after equal addition.
- Preserving total after new items enter or leave.
- Preserving difference after unequal changes.
- Carrying an invariant past a stage boundary.
- Ignoring units when comparing stable quantities.
- Forcing invariant reasoning where no useful invariant exists.
Frequently asked questions
Is total always the invariant?
No. It may be difference, ratio, rate, time, perimeter, area, volume, base amount or another condition.
Does adding the same number keep the ratio?
Usually no. It keeps the difference unchanged.
Does multiplying both numbers by the same factor keep the difference?
No. It preserves an equivalent ratio while scaling the difference.
How do I know whether an invariant is useful?
It should connect the before and after states more directly than raw calculation. If it does not simplify the structure, choose another method.
Independence indicators
- The learner asks what stays the same before calculating.
- Totals, differences and ratios are not confused.
- Before–after representations show the invariant.
- Invariants are tested rather than assumed.
- Stage boundaries are noticed.
- Method selection becomes faster in unfamiliar transfer problems.
Next route
Continue to Vol 0023: Science — Compare Change From the Same Starting Point.
Official PSLE reference
Use SEAB’s PSLE page and PSLE Formats Examined in 2026 for current examination-year information. Official examination documents and school instructions take priority over generic study advice.
Series route
Continue through the PSLE Learning Guide and Primary 6 Mathematics Learning Hub. These hubs remain the wider subject and PSLE owners for the learner route.
Series: How to Perform in PSLE | Learner’s Guide · Vol 0022 · Advanced Mathematics invariant reasoning
