Units and Parts is a Primary Mathematics method for problems where quantities are related by equal abstract parts but the value of one part is not known at first. Ratios, fractions and multiplicative comparisons often become easier when the student temporarily stops thinking in dollars, marbles or pupils and thinks in units.

Students searching for units and parts method, unitary method PSLE, ratio units, fraction units or Primary 5/6 problem sums often know how to divide once the units are written. The hard part is deciding what one unit represents and ensuring units from different states are genuinely comparable.

This page is the canonical Units and Parts owner under the PSLE Mathematics Heuristics hub.

Quick answer: what is a unit?

A unit is one equal part in a mathematical relationship. In a ratio 3:5, the quantities are represented as 3 equal units and 5 equal units. The units are not actual values until the problem gives enough information to convert them.

Structure first, value second.

The four-step unit method

1. Write the unit relationship.
2. Count the relevant units.
3. Match those units to a known actual value.
4. Find one unit, then scale to the required quantity.

Example 1: total known

Problem: The ratio of boys to girls is 3:5. There are 64 pupils altogether. How many girls are there?

Total units: 3 + 5 = 8 units.
1 unit: 64 ÷ 8 = 8 pupils.
Girls: 5 × 8 = 40 pupils.

Example 2: one quantity known

Problem: The ratio of apples to oranges is 4:7. There are 28 apples. How many oranges are there?

4 units: 28.
1 unit: 28 ÷ 4 = 7.
7 units: 49 oranges.

Example 3: difference known

Problem: The ratio of Aisha’s savings to Ben’s savings is 7:4. Aisha has $45 more than Ben. How much does Ben have?

Difference in units: 7 − 4 = 3 units.
3 units: $45.
1 unit: $15.
Ben: 4 × $15 = $60.

This is a critical pattern: when the actual difference is known, match it to the difference in units, not the total units.

Example 4: fraction as units

Problem: 5/8 of a tank contains 35 litres. What is the full capacity?

5 parts: 35 L.
1 part: 7 L.
8 parts: 56 L.

Example 5: remaining fraction

Problem: 3/7 of a sum was spent. $80 remained. What was the original sum?

If 3/7 was spent, 4/7 remained.
4 units: $80.
1 unit: $20.
7 units: $140.

The challenge is identifying which fraction the known value belongs to.

Units are labels, not numbers

Writing “3 units = 45” is safe. Writing “3 = 45” is not. The word unit protects the student from treating the ratio number itself as the actual quantity.

Units also need identity. “1 unit of Aisha’s ratio” and “1 unit after the change” are not automatically the same. Before-and-after ratio problems require an invariant to connect states.

When units can be compared directly

• same ratio state
• same whole split into equal parts
• same unit size explicitly preserved
• quantities scaled from one common actual value

When units cannot be compared directly

• ratio changes after transfer
• whole changes between fractions
• one side gains while another loses and the unit size changes
• different diagrams use independently defined units

This is where constants become essential. Use Constant Total, Constant Difference and Constant Part.

The unit bridge

A unit bridge connects an abstract relationship to an actual value.

Example: ratio 5:8, difference = 21.
Difference = 3 units = 21.
Therefore 1 unit = 7.

Always state the bridge explicitly during learning. It reduces accidental division by the wrong number of units.

Common unit-method errors

• Adds ratio units when difference is known.
• Subtracts ratio units when total is known.
• Uses spent fraction instead of remaining fraction.
• Assumes unit size remains unchanged across a changed ratio.
• Finds one unit correctly but scales to the wrong side.
• Forgets the units represent equal parts, not actual counts.

A mixed decision table

• Total known: add units.
• Difference known: subtract units.
• One side known: match that side’s units.
• Fraction known: match numerator parts to actual value.
• Remainder known: identify remaining fraction first.
• Before/after ratio: identify constant before equating unit values.

Transfer drill

Take one ratio 3:5 and create four different problems: total known, difference known, smaller quantity known, larger quantity known. The ratio is identical but the unit bridge changes. This teaches structure instead of a fixed operation.

Frequently asked questions

What is the difference between units and actual values?

Units describe proportional structure. Actual values are found only after the problem gives a bridge between units and real quantities.

When do I add ratio numbers?

When the total of both quantities is known.

When do I subtract ratio numbers?

When the difference between the quantities is known.

Can one unit change size?

Yes across different ratio states. Do not equate units across before/after situations unless a constant justifies the connection.

Is this the same as the Model Method?

Units and Parts often uses bar models, but the core reasoning is the equal-unit relationship.

Where does this sit in Atlas?

This is the canonical Units and Parts owner under PSLE Mathematics Heuristics.

The final Units rule

Ask what the known number represents in units. Total? Difference? One side? Remaining fraction? Once the unit bridge is correct, the arithmetic is usually straightforward.

Future Master Publish Deepening | Units and Parts

Use this guide to build a method you can apply independently. The operating rule is to treat ratio or fraction parts as equal relational units, then convert those units into actual values only when a total or difference fixes one unit. The route below turns public explanation into independent capability: recognition, representation, execution, fresh practice, verification, specialist handoff and return. One key boundary is that unit values must not be carried across changing states without proof.

1. Reader job

The focus at this stage is to move from a word problem to a defensible representation. For Units and Parts, the diagnostic question is: What relationship is hidden and what exactly is the target? The tutor should not treat a correct final number as enough evidence. Better evidence is that the student can state the target and relationship before calculating. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to classify three short problems without solving them. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

2. Structure recognition

The focus at this stage is to separate mathematical structure from surface vocabulary. For Units and Parts, the diagnostic question is: Which quantities are compared, combined, repeated, or changed? The tutor should not treat a correct final number as enough evidence. Better evidence is that method choice survives a change of names and story context. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to rewrite one familiar problem with different language. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

3. Boundary test

The focus at this stage is to recognise when the method should not be used. For Units and Parts, the diagnostic question is: What defining condition is missing from this problem? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can reject the method for a structural reason. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to mix one valid and two false-positive examples. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

4. Dependency check

The focus at this stage is to find the earliest weak prerequisite. For Units and Parts, the diagnostic question is: Which lower skill fails before the method itself fails? The tutor should not treat a correct final number as enough evidence. Better evidence is that a smaller diagnostic task reproduces the error. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to repair the dependency before returning to the full problem. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

5. Target control

The focus at this stage is to keep the requested quantity visible. For Units and Parts, the diagnostic question is: What must the final answer describe and in what unit? The tutor should not treat a correct final number as enough evidence. Better evidence is that intermediate calculations remain connected to the target. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to write the target above the working. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

6. Representation choice

The focus at this stage is to choose the cheapest valid representation. For Units and Parts, the diagnostic question is: Does a bar, table, diagram, statement, or equation reduce uncertainty? The tutor should not treat a correct final number as enough evidence. Better evidence is that the representation preserves all important conditions. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to compare two possible representations. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

7. Label discipline

The focus at this stage is to keep every part of the representation meaningful. For Units and Parts, the diagnostic question is: What does each bar, unit, segment, or symbol represent? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can explain every label without guessing. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to cover the question and read the representation back as a story. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

8. Invariant search

The focus at this stage is to identify what remains fixed across changes. For Units and Parts, the diagnostic question is: What stays the same while other quantities move? The tutor should not treat a correct final number as enough evidence. Better evidence is that the chosen bridge remains true in both states. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to circle the invariant before solving. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

9. Worked reasoning

The focus at this stage is to make each transformation inspectable. For Units and Parts, the diagnostic question is: Why is this step allowed by the relationship? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can justify major steps in ordinary language. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to pause after each step and name its purpose. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

10. Alternative route

The focus at this stage is to compare valid methods rather than worship one. For Units and Parts, the diagnostic question is: Could another representation reach the same result more safely? The tutor should not treat a correct final number as enough evidence. Better evidence is that two routes agree and their trade-offs are understood. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to solve once visually and once arithmetically. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

11. Error taxonomy

The focus at this stage is to classify the first wrong decision. For Units and Parts, the diagnostic question is: Was the error recognition, representation, execution, or checking? The tutor should not treat a correct final number as enough evidence. Better evidence is that the same error category predicts mistakes on another problem. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to record mechanism rather than page number. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

12. Error ledger

The focus at this stage is to turn correction into future prevention. For Units and Parts, the diagnostic question is: What rule would prevent this error next time? The tutor should not treat a correct final number as enough evidence. Better evidence is that the error disappears on delayed retest. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to write one corrected rule and one fresh test. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

13. Near transfer

The focus at this stage is to remove exact-number memory. For Units and Parts, the diagnostic question is: Can the same reasoning survive different numbers? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner rebuilds rather than copies the prior layout. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to change all numbers while preserving structure. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

14. Language transfer

The focus at this stage is to remove phrase dependence. For Units and Parts, the diagnostic question is: Can the relationship survive different wording and sentence order? The tutor should not treat a correct final number as enough evidence. Better evidence is that recognition remains accurate with new vocabulary. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to paraphrase the problem before solving. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

15. Representation transfer

The focus at this stage is to move between visual and symbolic forms. For Units and Parts, the diagnostic question is: Can the same relationship be expressed another way? The tutor should not treat a correct final number as enough evidence. Better evidence is that the student translates without changing meaning. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to convert a model to an equation or table. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

16. Method discrimination

The focus at this stage is to choose between neighbouring strategies. For Units and Parts, the diagnostic question is: Why does this method fit better than the closest alternative? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can explain both selection and rejection. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to mix two similar problem families. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

17. Counterexample

The focus at this stage is to learn the edge of the heuristic. For Units and Parts, the diagnostic question is: What looks familiar here but is structurally different? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner rejects keyword traps. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to design one problem containing a misleading cue. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

18. Reverse engineering

The focus at this stage is to reconstruct language from mathematics. For Units and Parts, the diagnostic question is: What story could generate this completed representation? The tutor should not treat a correct final number as enough evidence. Better evidence is that the invented story preserves every relationship. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to hide the original question and rebuild it. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

19. Problem design

The focus at this stage is to control the structure from the inside. For Units and Parts, the diagnostic question is: Can the learner write a valid new problem for this method? The tutor should not treat a correct final number as enough evidence. Better evidence is that another student can solve it unambiguously. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to author, exchange, solve, and revise a question. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

20. Contradiction test

The focus at this stage is to recognise impossible or insufficient data. For Units and Parts, the diagnostic question is: Do all stated conditions fit one solution? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can demonstrate inconsistency instead of forcing an answer. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to alter one condition until the problem breaks. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

21. Spaced retrieval

The focus at this stage is to make the method survive time. For Units and Parts, the diagnostic question is: Can the learner recover the rule after several days? The tutor should not treat a correct final number as enough evidence. Better evidence is that recognition returns without reopening notes. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to schedule one-day, three-day, and seven-day retests. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

22. Timed transfer

The focus at this stage is to add pressure only after reliability. For Units and Parts, the diagnostic question is: Where is time being spent: recognition, setup, calculation, or checking? The tutor should not treat a correct final number as enough evidence. Better evidence is that speed improves without loss of structure. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to time a short mixed set and tag the delay source. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

23. Verification

The focus at this stage is to test the result against the original story. For Units and Parts, the diagnostic question is: Which independent condition can confirm or reject the answer? The tutor should not treat a correct final number as enough evidence. Better evidence is that the answer reproduces the given relationships. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to reconstruct the total, ratio, difference, or state. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

24. Return route

The focus at this stage is to reconnect the method to the larger heuristic map. For Units and Parts, the diagnostic question is: What different method should be compared next? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can name one similarity and one decisive difference. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to return to the PSLE heuristic map and choose a neighbour. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

Worked example and verification

Problem: The ratio of fiction to non-fiction books is 4:7. There are 33 more non-fiction books. Find the total. The difference is 3 units. Three units represent 33 books, so one unit is 11. Fiction is 44 and non-fiction is 77. Total = 121.

Check: 77−44=33 and 44:77 simplifies to 4:7. The answer is accepted because it satisfies the original relationships, not merely because the arithmetic produced a plausible number.

Specialist handoff and return

For deeper specialist Mathematics treatment, continue to How Representation Works in PSLE Mathematics Problem Solving, How PSLE Mathematics Works, and How Method Marks and Working Steps Work. Then return to PSLE Mathematics Heuristics and compare this method with another family.

The route receipt is an unseen problem solved without the method name, with a defensible representation, visible working and a verification step. The learner should also identify one problem where Units and Parts would be a poor choice. That comparison shows that the method has become part of a larger decision system rather than a memorised trick.