A number is increased by 38.
Then 25 is subtracted.
The final result is 94.
What was the starting number?
If we try to move forward, the first quantity is missing.
The final quantity is known.
This is a strong signal that reversing the sequence may be useful.
Working backwards means beginning from a known later state and undoing reversible changes in reverse order until the unknown earlier state is recovered.
Start from 94.
The last forward action was subtract 25.
Undo it by adding 25:
94 + 25 = 119.
The earlier forward action was add 38.
Undo it by subtracting 38:
119 − 38 = 81.
Check forward:
81 + 38 = 119.
119 − 25 = 94.
The recovered start is correct.
Working backwards is a general problem-solving heuristic rather than a separately named Primary 4 syllabus topic. It fits naturally with the Singapore Mathematics emphasis on problem solving, representations and using known relationships to find unknown quantities. It is especially useful as Primary 4 problems become more structurally varied and learners must choose a route rather than simply recognise an operation from a keyword.
The quick answer: reverse both order and operation
If the forward sequence is:
start → add 38 → subtract 25 → finish,
then the backward sequence is:
finish → add 25 → subtract 38 → start.
Two things change:
- the order of the steps reverses;
- each reversible operation is replaced by its inverse.
Inverse pairs include:
- addition ↔ subtraction;
- multiplication ↔ division;
- doubling ↔ halving, when exact;
- adding a known fraction ↔ subtracting the same fraction;
- increasing a quantity by a fixed amount ↔ decreasing by that amount.
The word “inverse” is important because working backwards is not simply “do the opposite-looking thing”.
The backward operation must restore the previous state exactly.
Why the order must reverse
Suppose a number is doubled, then 7 is added.
The final value is 31.
Forward:
x → ×2 → +7 → 31.
To undo, begin with the last operation.
31 − 7 = 24.
Then:
24 ÷ 2 = 12.
Starting number = 12.
If the learner divides by 2 first:
31 ÷ 2 = 15.5, then subtracts 7,
the result does not recover the original state because the transformations were undone in the wrong order.
Undo the last change first because it sits directly on top of everything that came before it.
A state diagram makes the structure visible
Write the process as boxes and arrows.
Unknown start → +38 → intermediate state → −25 → 94.
Now draw backward arrows beneath:
94 → +25 → intermediate state → −38 → unknown start.
This representation keeps operation order separate from arithmetic execution.
It also makes checking easier because forward and backward paths can be compared directly.
Working backwards from money remaining
A child buys a book for $12.80 and a pen for $3.40. She has $8.80 left. How much money did she have at first?
Forward story:
start − 12.80 − 3.40 = 8.80.
Work backwards:
8.80 + 3.40 = 12.20.
12.20 + 12.80 = 25.00.
Starting amount:
$25.00.
Check forward:
$25.00 − $12.80 − $3.40 = $8.80.
Working backwards from a fraction remainder
A tank contains some water.
12 litres are added.
Then 1/4 of the new amount is removed.
45 litres remain.
This is more subtle because “remove one quarter” is a multiplicative change, not a fixed subtraction.
If one quarter was removed, three quarters remained.
So 45 litres represents 3/4 of the amount just before removal.
Find one quarter:
45 ÷ 3 = 15 litres.
Find four quarters:
15 × 4 = 60 litres.
Before the 12 litres were added:
60 − 12 = 48 litres.
Check forward:
48 + 12 = 60.
Remove 1/4 of 60 = 15.
60 − 15 = 45.
Working backwards through multiplication and division
A number is multiplied by 6, then divided by 4, giving 72.
Forward:
x × 6 ÷ 4 = 72.
Backward:
72 × 4 = 288.
288 ÷ 6 = 48.
Start = 48.
Check:
48 × 6 = 288.
288 ÷ 4 = 72.
The operation must actually be reversible with the available information
Not every process can be reversed uniquely.
Suppose a number is rounded to the nearest 10 and the result is 50.
What was the original number?
There is no unique answer.
Many whole numbers round to 50.
The forward operation lost information.
Working backwards can recover only a range, not one exact start.
Working backwards succeeds cleanly only when the forward steps preserve enough information to be inverted under the problem’s conditions.
This is an important limit. The heuristic is not magic.
Another non-unique case: absolute difference
Suppose a hidden number is compared with 10 and the absolute difference is 3.
The hidden number could be 7 or 13.
The final information does not tell us which side of 10 the starting value occupied.
Working backwards reveals multiple valid possibilities rather than one answer.
When working forwards is easier
Do not use working backwards merely because it is the name of a strategy.
If the start is known and the finish is unknown, forward calculation may be direct.
If the problem contains a simple part-whole relationship, a bar model may reveal the unknown more quickly.
If several cases are possible, systematic listing may be better.
If a repeated pattern is involved, pattern generalisation may be more useful.
Problem-solving strategies should be selected by structure, not performed as compulsory rituals.
Worked example: a multi-step classroom problem
A class had some markers.
The teacher bought 24 more.
Then the markers were packed equally into 6 boxes.
Each box contained 15 markers.
How many markers were there at first?
Final total after packing:
6 × 15 = 90 markers.
Those 90 markers were the amount after 24 had been added.
So:
90 − 24 = 66 markers.
This example shows that working backwards can begin by reconstructing a hidden final total from a multiplicative condition.
Worked example: repeated change
A game score is doubled, then increased by 9, then reduced by 5. The final score is 40.
Backward:
40 + 5 = 45.
45 − 9 = 36.
36 ÷ 2 = 18.
Start = 18.
Check forward:
18×2 = 36; 36+9 = 45; 45−5 = 40.
Working backwards is closely related to solving equations
The story:
double a number, then add 7, result 31
can be written:
2x + 7 = 31.
Solving algebraically:
2x = 24.
x = 12.
This is the same inverse-operation logic formalised symbolically.
Working backwards in Primary Mathematics therefore prepares learners for algebraic equation solving later.
Use a table when several states must be tracked
For a longer process, write:
- State 0: unknown start;
- State 1: after first change;
- State 2: after second change;
- State 3: known final state.
Then record the backward inverse beside each arrow.
This reduces memory load because the learner no longer has to hold the entire sequence mentally while calculating.
Reasonableness still matters
Suppose a problem says a starting amount increased, then decreased slightly, and ended at 94.
A recovered start of 8,100 should trigger suspicion unless the changes were correspondingly enormous.
Working backwards produces an exact arithmetic route, but estimation and contextual reasonableness remain useful independent checks.
Common misconception 1: reverse the operations but keep the same order
If the forward sequence is ×2 then +7, the backward sequence is −7 then ÷2.
Repair: write the forward arrows first, then travel through them from right to left.
Common misconception 2: subtract whenever the problem asks for the starting amount
The inverse depends on the actual forward operation.
If a quantity was divided by 5, working backwards multiplies by 5.
Common misconception 3: every operation has one unique reverse
Rounding and other information-losing operations may not have unique inverses.
Repair: ask whether the final state contains enough information to reconstruct one earlier state.
Common misconception 4: working backwards means guessing a start and checking
Guess-and-check is a different strategy.
Working backwards uses inverse relationships to reconstruct states systematically.
Common misconception 5: a successful backward path needs no forward check
The cleanest verification is to run the recovered starting value through the original forward process.
If it does not reproduce the stated final value, the reversal contains an error.
A diagnostic ladder for working backwards
- Can the learner identify the starting state, changes and final state?
- Can the learner name inverse pairs such as +/− and ×/÷?
- Can the learner reverse the order of two steps correctly?
- Can the learner draw a state-and-arrow diagram?
- Can the learner work backwards through mixed addition and multiplication?
- Can the learner handle a fraction-of-a-quantity step by reconstructing the whole?
- Can the learner recognise a non-invertible or non-unique situation?
- Can the learner decide when working forwards would be simpler?
- Can the learner check by replaying the original process forwards?
- Can the learner connect the same reasoning to a simple equation?
A five-minute home activity
Create a “number machine”.
Choose a secret starting number.
Apply two or three reversible operations, for example:
- ×3;
- +8;
- −5.
Reveal only the final number and the operation sequence.
Ask the learner to recover the start.
Then switch roles.
Finally include one information-losing operation such as rounding and ask whether the original number can still be recovered uniquely.
The contrast teaches both the power and limit of the strategy.
What parents should listen for
- “The last forward step was subtract 25, so I undo it first by adding 25.”
- “I have to reverse the order as well as the operation.”
- “Removing one quarter left three quarters, so the final amount represents three equal parts.”
- “Rounding lost information, so there may be several possible starting numbers.”
- “I checked by running my answer forwards through the original steps.”
What teachers and tutors should avoid
- Avoid presenting working backwards as a keyword-triggered recipe.
- Avoid reversing operations without reversing their order.
- Avoid using only arithmetic chains with one obvious route. Include money, measurement and fraction states.
- Avoid implying every process can be uniquely reversed.
- Avoid accepting a backward answer without forward verification.
- Avoid treating the strategy as an isolated trick. Connect it to inverse operations and later equations.
How this fits Singapore Primary 4 Mathematics
“Working backwards” is best understood as a problem-solving heuristic rather than a standalone content heading in the current Primary 4 syllabus. It draws on Primary Mathematics content that is explicitly taught: inverse operation relationships, fractions, decimals, measurement, multi-step word problems and mathematical problem solving.
Evidence-informed mathematics guidance from EEF and the Institute of Education Sciences supports giving learners opportunities to analyse and compare multiple problem-solving strategies rather than treating one method as universally correct. Working backwards belongs in that strategic repertoire.
The deeper lesson: some problems become easier when time is reversed
Forward processes accumulate change.
Backward reasoning peels those changes away.
The learner is not reversing time literally.
The learner is reversing a chain of mathematical transformations.
When the destination is known and the starting point is hidden, the shortest route may begin at the destination.
Final thought
Working backwards is powerful because it changes the learner’s question.
Instead of asking:
“How do I get from an unknown start to the final answer?”
the learner asks:
“What must have been true immediately before this final state?”
One predecessor at a time, the hidden beginning becomes recoverable.