PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 10 · GUIDE 37
The units-and-parts method turns complicated comparisons into equal-sized building blocks. Instead of carrying several fractions, totals and differences at once, the learner identifies a common unit, expresses each quantity as a number of those units, and solves the problem through scaling.
This is a problem-solving method, not a separate MOE syllabus topic. It is especially useful when fractions, multiplicative comparison, equal quantities and bar models interact.
Return to the Primary 5 Mathematics Learning Hub. Related foundations: Comparison Bar Models · Equal-Fraction & Unchanged-Quantity Reasoning.
1. What is a unit?
A unit is one equal part of a comparison model. If A is represented by 3 equal units and B by 5 equal units, the unit is not necessarily “1 item”. It is one common chunk shared by the model.
2. Units compress comparison structure
If A has 3 units and B has 5 units and their difference is 40, then 2 units = 40. One unit = 20. Therefore A = 60 and B = 100.
The entire problem is solved by identifying the unit gap.
3. Parts and units are related but not identical words
“Parts” describe how a whole is partitioned. “Units” are convenient equal blocks used to compare quantities. In a bar model they often coincide, but learners should still ask what the unit represents.
4. Fractions can become unit counts
If A is 3/5 of B, represent A as 3 units and B as 5 units. If A = 72, one unit = 24, so B = 120.
This converts fraction language into equal-unit reasoning.
5. Common units connect equal fractional quantities
If 2/3 of A equals 3/5 of B, choose a common equal amount of 6 units.
2/3 of A = 6 units → A = 9 units.
3/5 of B = 6 units → B = 10 units.
The common quantity becomes the bridge.
6. Why six units?
Six is convenient because it is divisible by both fractional numerators 2 and 3. Any common multiple would work, but the smallest convenient one reduces arithmetic.
7. Equivalent parts preserve proportion
3 units out of 5 and 6 units out of 10 represent the same multiplicative relationship. Doubling every unit count changes the representation, not the relationship.
8. Scaling all parts equally preserves the comparison
If A:B is represented by 4 units and 7 units, multiplying both by 3 gives 12 and 21. The scale changes; the structural comparison does not.
9. Use units for total-and-comparison problems
A is 3 units and B is 5 units. Together they total 240.
8 units = 240. One unit = 30. A = 90 and B = 150.
10. Use units for difference-and-comparison problems
A is 4 units and B is 7 units. Their difference is 45.
3 units = 45. One unit = 15. A = 60 and B = 105.
11. Use units when a fraction refers to another quantity
If C is 4/7 of D, represent C as 4 units and D as 7. If C + D = 220, then 11 units = 220 and one unit = 20. C = 80 and D = 140.
12. Units can align changing wholes
If 3/4 of an earlier quantity equals 5/6 of a later quantity, the equal physical amount can be expressed using a common unit count. This connects two different wholes without confusing their denominators.
13. Do not force common denominators when common units are clearer
For equation-style fraction addition, common denominators are essential. For equal-quantity comparison, common-unit modelling may be more intuitive because it focuses on the quantities that are equal.
14. Unit bars can reveal the whole immediately
If 5 equal units represent 125, one unit = 25. A 7-unit quantity is then 175. The bar model externalises the multiplicative scaling.
15. Common units support percentage thinking too
If 40% of A equals 60% of B, write 2/5 of A = 3/5 of B. Choose a common amount of 6 units: A = 15 units, B = 10 units. Thus A is larger.
Percentage can be converted into fraction structure when that makes the units easier to see.
16. Units method versus algebra
2/3 A = 3/5 B can be solved algebraically, but common units often reduce symbol load for Primary 5 learners. Later algebra will compress the same structure more formally.
17. Units method versus direct division
If A = 72 and A is 3/5 of B, direct division 72 ÷ 3 × 5 is efficient. A units model explains why that works: 3 units = 72, so one unit = 24, then 5 units = 120.
18. Use the units method only when it reduces uncertainty
Do not draw a seven-part model for a one-step calculation already understood. Use units when the comparison itself is the difficult part.
19. Error map
| Error | Cause | Repair question |
|---|---|---|
| Different-sized units drawn | Equal-part assumption lost | Are all units meant to represent the same amount? |
| Uses denominator as quantity automatically | Whole/reference confused | What does one unit represent here? |
| Scales only one side | Comparison not preserved | Did both quantities scale by the same factor? |
| Common-unit method used without an equality | Bridge not justified | Which quantities are actually equal? |
20. Practice laboratory
- A is 3/5 of B. A = 84. Find B.
- A and B are 4 units and 7 units. Their total is 220. Find both.
- A and B are 5 units and 8 units. Their difference is 54. Find both.
- 2/3 of A equals 3/5 of B. If A+B=190, find A and B.
- 40% of A equals 60% of B. Compare A and B using common units.
- C is 4/7 of D and together they total 242. Find C and D.
21. Answers
1. 3 units = 84; one = 28; B = 140.
2. 11 units = 220; one = 20; values 80 and 140.
3. 3 units = 54; one = 18; values 90 and 144.
4. Common amount 6 units → A 9 units, B 10 units; 19 units = 190; A=90, B=100.
5. A = 15 units, B = 10 units, so A is 1.5 times B.
6. 11 units = 242; one = 22; C=88, D=154.
22. Full units-and-parts problem
Three fifths of A equals four sevenths of B. A and B total 345. Find both.
Make the equal fractional amounts 12 units.
3/5 of A = 12 → A = 20 units.
4/7 of B = 12 → B = 21 units.
Total = 41 units = 345, so one unit = 345/41. The result is not a whole number. If A and B must be whole objects, the data are incompatible with that requirement.
The method therefore solves and checks feasibility.
23. Final checkpoint
A strong Primary 5 learner can identify equal units, translate fractional comparison into unit counts, choose a common unit for equal fractional quantities, scale all parts consistently, use total or difference to recover one unit and reject unit models when the equality or whole-object conditions do not support them.
Continue to Primary 5 Mathematics Learning Guide | Equal Stage Method, Before–After Equalisation & Same-State Anchors.
