PRIMARY 4 MATHEMATICS LEARNING GUIDE · BATCH 11 · GUIDE 41
The Unitary Method solves a large family of Primary 4 Maths word problems by finding the value of one equal unit first. A question may tell us the cost of several identical items, the size of several equal groups, a fractional part of a collection, or the combined length of equal pieces. Instead of trying to jump directly from the given amount to the final unknown, we reduce the relationship to one unit, understand what that unit represents, and then build the required amount from it.
In Singapore Mathematics, this “find one first” structure appears across equal groups, fractions, money, measurement and bar-model reasoning. Different schools and publishers may use different labels, so this guide treats Unitary Method as a problem-solving heuristic rather than claiming it is a separate official Primary 4 syllabus chapter. The official curriculum boundary is the MOE Primary Mathematics Syllabus, updated October 2025.
The central route is simple enough to remember but deep enough to transfer: identify equal units → find one unit → find the required number of units → return the answer to the story. The difficult part is not always the division. It is deciding what “one unit” means.
Series route: return to the Primary 4 Mathematics Learning Hub. For fractions whose reference whole changes during a problem, use Fraction Word Problems, Reference Wholes and Mixed Numbers.
Navigate: the unitary engine · equal groups · fractions · money and measurement · reverse problems · diagnosis · practice · answers.
1. The Unitary Method asks one precise question: what is one unit worth?
Suppose six identical boxes contain 144 counters altogether. How many counters are in one box?
Six equal units total 144. One unit is 144 ÷ 6 = 24 counters.
If the question then asks for the number of counters in nine such boxes, scale from one unit: 24 × 9 = 216 counters.
The method is not “divide, then multiply” as a blind rule. Division is used because several equal units are combined in the known total. Multiplication is used later because the required amount contains several copies of the recovered unit.
Always label the unit. “One unit = 24” is weaker than “one box = 24 counters”. The label tells us what can safely be multiplied later.
2. Separate the number of units from the value of one unit
In “six boxes contain 144 counters”, the number six counts boxes. The number 144 counts counters.
Dividing 144 counters by six boxes produces 24 counters per box.
A common error is to treat both numbers as though they count the same object. Writing “144 ÷ 6 = 24 boxes” loses the meaning of the quotient.
Before calculating, write:
6 boxes → 144 counters
1 box → ? counters
This two-line representation is often enough to prevent unit confusion.
3. From several equal groups to a different number of groups
Eight identical packets contain 360 stickers altogether. How many stickers are in five such packets?
First find one packet: 360 ÷ 8 = 45 stickers.
Then find five packets: 45 × 5 = 225 stickers.
Check by scale. Five packets should contain less than eight packets because the packet size is unchanged. 225 is less than 360, so the direction is sensible.
We can also compare directly: five packets are 5/8 of the eight-packet collection. But the Unitary Method keeps the equal-group structure especially visible.
4. The one-unit step can be mental even when the final answer is written
Four identical books cost $28 altogether. Seven identical books cost how much?
One book costs $7. Seven books cost $49.
A full written solution can still be compact:
1 book: $28 ÷ 4 = $7.
7 books: $7 × 7 = $49.
The method should not create unnecessary writing for easy facts. Its purpose is to make the relationship explicit where needed.
5. Unitary reasoning is stronger when the unit is derived, not guessed
A learner may look at 8 packets → 360 stickers and guess that one packet contains 40 because 8 × 40 = 320 is nearby.
That guess can be checked, but the exact unit must satisfy the total: 360 ÷ 8 = 45.
Once one unit is known exactly, every later scaled amount follows consistently.
This is one reason the method is reliable: it converts a multi-unit relationship into a single reusable building block.
6. The method works in both scaling directions
If twelve identical ribbons have total length 18 m, one ribbon is 18 ÷ 12 = 1.5 m.
Three ribbons have total length 1.5 × 3 = 4.5 m.
Twenty ribbons have total length 1.5 × 20 = 30 m.
After one unit is known, we can scale down or up.
Do not assume “scale up” always means the final number is larger than the original total. If the required unit count is smaller, the final total will be smaller.
7. A known fractional part can be treated as several equal units
Three eighths of a collection are 27 counters. Find the whole collection.
Think of the whole as eight equal units. Three units correspond to 27 counters.
One unit = 27 ÷ 3 = 9 counters.
Eight units = 9 × 8 = 72 counters.
This is a Unitary Method interpretation of reverse fraction-of-a-set reasoning.
The key question is not “What do I divide by?” but “How many equal fraction units does the known amount represent?”
8. A fraction unit is defined by the denominator
If 5/6 of a collection is 40, the 40 represents five equal sixth-units.
One sixth = 40 ÷ 5 = 8.
The whole six sixths = 8 × 6 = 48.
Do not divide 40 by six first. Six is the number of equal parts in the whole, not the number of those parts represented by the known 40.
The numerator tells us how many units are currently known; the denominator tells us how many units make the whole.
9. Finding a different fraction after recovering the whole
Three eighths of a collection are 27. Find five eighths.
One eighth = 27 ÷ 3 = 9.
Five eighths = 9 × 5 = 45.
There is no need to recover all eight units first unless the whole is also required.
The one-unit value can be reused directly for any number of equal eighth-units.
10. Be careful when the reference whole changes
A problem says: “One third of 72 counters are used. Then one quarter of the remainder are used.”
The first unitary step uses the original 72. One third = 24, leaving 48.
The second one-quarter refers to the new whole, 48. One quarter = 12.
The final amount is 36.
Do not carry the original unit value into the second stage automatically. The reference whole changed.
This is where the dedicated Fraction Word Problems guide becomes the stronger owner.
11. Money problems often hide a unit price
Four identical pens cost $7.20. Find the cost of seven pens.
One pen = $7.20 ÷ 4 = $1.80.
Seven pens = $1.80 × 7 = $12.60.
Check: seven pens should cost more than four pens because the unit price is positive and unchanged.
Keep cents aligned. $1.80 × 7 = $12.60, not $12.6 if the task expects standard money notation.
12. Measurement problems use the same structure
Five identical ropes have a total length of 12.5 m. Find the total length of eight ropes.
One rope = 12.5 ÷ 5 = 2.5 m.
Eight ropes = 2.5 × 8 = 20 m.
The arithmetic is ordinary division and multiplication. The Unitary Method contributes the representation: several equal measured objects → one object → required number of objects.
Preserve the measurement unit through both stages.
13. Unit conversions may come before the one-unit step
Four identical bottles contain 3 L 200 mL altogether. How much do seven bottles contain?
Convert 3 L 200 mL to 3,200 mL.
One bottle = 3,200 ÷ 4 = 800 mL.
Seven bottles = 800 × 7 = 5,600 mL = 5 L 600 mL.
Trying to divide 3 L and 200 mL separately without control can create regrouping errors.
A common unit makes the one-unit relationship easier to execute.
14. A unitary route can expose whether a proposed answer is reasonable
Six identical bags weigh 18 kg. A learner says ten bags weigh 25 kg.
One bag must weigh 18 ÷ 6 = 3 kg.
Ten bags therefore weigh 30 kg.
The proposed 25 kg would imply a smaller unit weight even though the bags are stated to be identical.
The one-unit value functions as an invariant check.
15. Reverse problems can begin with the final number of units
Seven identical notebooks cost $42. How many notebooks can be bought for $60 at the same unit price, assuming whole notebooks and exact pricing?
One notebook costs $42 ÷ 7 = $6.
Number of notebooks for $60 = $60 ÷ $6 = 10 notebooks.
The second step is division rather than multiplication because the required unknown is the number of units, not their combined value.
Unitary Method does not force a fixed divide-then-multiply pattern. After finding one unit, choose the operation that matches the new unknown.
16. Whole-object conditions matter
Five identical toys cost $18. What is the cost of one toy?
The mathematical unit price is $18 ÷ 5 = $3.60, which is meaningful for money.
But if the question instead says five sealed packets contain 18 indivisible counters equally, the condition is impossible because 18 is not divisible by five.
The same numerical division can have different contextual validity.
Before accepting a decimal one-unit answer, ask whether the unit can meaningfully be fractional.
17. Some questions do not contain enough information for a unitary solution
“Several identical packets contain 120 counters altogether. How many counters are in one packet?”
The number of packets is missing.
Many unit values are possible: two packets would hold 60 each; three would hold 40; five would hold 24.
The Unitary Method cannot manufacture the missing unit count.
A correct response is that the value of one packet cannot be determined uniquely from the information given.
18. Compare Unitary Method with bar-model reasoning
Three eighths of a collection are 27.
Bar model: draw eight equal units and bracket three of them as 27.
Unitary Method: three units → 27; one unit → 9; eight units → 72.
These are not competing methods. The bar model shows the relationship visually; the unitary calculation extracts the one-unit value.
Good learners should see how the representations correspond.
19. Compare Unitary Method with direct multiplication
Four books cost $28. Seven books cost?
A learner who immediately recognises one book costs $7 may not need to write the division.
Another learner might use proportional scaling mentally.
The Unitary Method is especially valuable when the one-unit value is not obvious or when the question reverses the usual unknown.
Do not turn a useful structure into compulsory extra writing on every simple example.
20. Estimation protects the one-unit step
Eight packets cost $39.20. One packet should cost a little under $5 because $40 ÷ 8 = $5.
Exact unit price = $39.20 ÷ 8 = $4.90.
If a learner obtains $49, the estimate exposes a place-value error immediately.
Estimate the unit before scaling it. A wrong one-unit value contaminates every later step.
21. Common Unitary Method errors
| Error | Likely cause | Repair question |
|---|---|---|
| Divides by denominator instead of known numerator in reverse fraction problem | Known unit count misidentified | How many fraction units does the given amount represent? |
| Reports one-unit value as final answer | Target forgotten | How many units does the question actually ask for? |
| Multiplies when required unknown is number of items | Value per unit confused with unit count | Are we finding total value or number of units? |
| Carries old fraction unit into a new remainder | Reference whole changed unnoticed | What is the whole at this stage? |
| Drops units | Number detached from quantity | One what equals how many what? |
| Accepts fractional indivisible object count | Context ignored | Can this object be split? |
22. A five-question diagnostic
- Six identical boxes contain 144 counters. Find one box.
- Eight packets contain 360 stickers. Find five packets.
- Three eighths of a collection are 27. Find the whole.
- Four pens cost $7.20. Find seven pens.
- Seven notebooks cost $42. How many can be bought for $60?
If Question 1 fails, equal-group division may be unstable. If Question 1 succeeds but Question 3 fails, reverse fraction units may be the weak dependency. If the first four succeed but Question 5 fails, the learner may be treating Unitary Method as a fixed “divide then multiply” recipe rather than following the new unknown.
23. Practice laboratory
- Six boxes contain 144 counters. Find the number in one box.
- Eight packets contain 360 stickers. Find the number in five packets.
- Twelve ribbons have total length 18 m. Find the total length of three ribbons.
- Four books cost $28. Find the cost of seven books.
- Three eighths of a collection are 27 counters. Find the whole collection.
- Five sixths of a collection are 40 counters. Find the whole.
- Three eighths of a collection are 27. Find five eighths.
- Four pens cost $7.20. Find seven pens.
- Five ropes have total length 12.5 m. Find eight ropes.
- Four bottles contain 3 L 200 mL altogether. Find seven bottles.
- Six bags weigh 18 kg. Find ten bags.
- Seven notebooks cost $42. How many notebooks can be bought for $60?
- Nine equal groups contain 225 items. How many are in four groups?
- Four sevenths of a collection are 32. Find three sevenths.
- Seven tenths of a length are 28 m. Find the whole length.
- Five identical toys cost $18. Find one toy and eight toys.
- A learner says eight packets costing $39.20 means one packet costs $49. Explain the error and give the correct value.
- Several identical packets contain 120 counters, but the number of packets is unknown. Can one packet be found uniquely?
- Five sealed packets contain 18 indivisible counters equally. Is the condition possible?
- One third of 72 counters is used, then one quarter of the remainder. Find the final amount and explain why the one-unit value changes between stages.
24. Explained answers
1. 144 ÷ 6 = 24 counters per box.
2. One packet = 360 ÷ 8 = 45. Five packets = 45 × 5 = 225 stickers.
3. One ribbon = 18 ÷ 12 = 1.5 m. Three = 4.5 m.
4. One book = $7. Seven = $49.
5. Three eighth-units = 27. One = 9. Eight = 72.
6. Five sixth-units = 40. One = 8. Six = 48.
7. One eighth = 9. Five eighths = 45.
8. One pen = $1.80. Seven = $12.60.
9. One rope = 2.5 m. Eight = 20 m.
10. 3 L 200 mL = 3,200 mL. One bottle = 800 mL. Seven = 5,600 mL = 5 L 600 mL.
11. One bag = 3 kg. Ten = 30 kg.
12. One notebook = $6. $60 ÷ $6 = 10 notebooks.
13. One group = 225 ÷ 9 = 25. Four groups = 100 items.
14. Four units = 32, so one = 8. Three units = 24.
15. Seven units = 28 m, so one = 4 m. Ten units = 40 m.
16. One toy = $18 ÷ 5 = $3.60. Eight toys = $28.80.
17. $39.20 ÷ 8 = $4.90. $49 is ten times too large; an estimate near $5 exposes the place-value mistake.
18. No. The number of equal units is missing, so several unit values are possible.
19. No if the counters are indivisible and every packet must have a whole-number equal count. Eighteen is not divisible by five.
20. First whole = 72: one third = 24, leaving 48. New whole = 48: one quarter = 12, leaving 36. The one-unit value changes because the reference whole changes.
25. Teaching routine: label the unit before calculating it
Use a two-column prompt: number of units and value.
Ask the learner to fill the known row, then add a row for one unit.
Once one unit is found, ask what the target row should contain. Do not automatically tell the learner whether to multiply or divide.
Rotate the unknown across three versions of the same relationship: total value unknown, value per unit unknown, number of units unknown.
This rotation prevents the Unitary Method from becoming a memorised operation pair.
26. Handover to shortage and surplus
The Unitary Method works when equal units can be reduced to one. The next heuristic handles a different structure: the unit count itself is unknown, but two distribution scenarios create a measurable excess or shortage.
Continue to Shortage and Surplus: Excess, Deficit, Fixed Groups, Left Over and Short Of.
Final checkpoint: can the learner identify what one unit means, derive its value, scale to a new target, change operation when the unknown changes, and preserve the reference whole and physical unit?
Source and editorial note
The curriculum boundary is referenced to the MOE Primary Mathematics Syllabus, updated October 2025. “Unitary Method” is used here as an instructional heuristic label; the examples and learning sequence are independently written by eduKate Publishing.
Editorial control: Wintour House V1.0 · CivDJ · eduKate Publishing.
