PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 6 · GUIDE 24
Some problems are difficult because there are several possible cases and the learner must prove that none are missing. Systematic listing, tables and tree structures organise possibilities so that cases are complete, non-duplicated and easy to compare. Invariant thinking then asks what stays unchanged while the cases vary.
This guide develops non-routine reasoning through systematic lists, organised tables, case elimination, small-case analysis, pattern tracking, parity, fixed totals and invariants. These tools are not separate tricks; they are ways of controlling a search space.
For the official curriculum framework, see the MOE Primary Mathematics Syllabus.
Series route: return to the Primary 5 Mathematics Learning Hub. Earlier: Comparison Bar Models, Before–After Models and Assumption Method, Fixed Totals, Equalisation & Difference Reasoning.
1. A systematic list has an organising rule
Using digits 2, 4 and 7 exactly once, list all three-digit numbers.
Organise by first digit:
- 2: 247, 274
- 4: 427, 472
- 7: 724, 742
There are 6 numbers.
The structure proves completeness better than a random list.
2. Fix one choice, then vary the next
Systematic listing often begins by fixing the first decision. Once every possibility under that first choice is exhausted, move to the next.
This prevents duplication and omission.
3. Tables organise two changing quantities
Suppose a total of 20 items is split between A and B. A table can list possible A values and corresponding B = 20 − A.
| A | B |
|---|---|
| 0 | 20 |
| 1 | 19 |
| 2 | 18 |
The total 20 is invariant while the distribution changes.
4. Use tables when one case generates another predictably
If each adult ticket costs $12 and child ticket $8, with 20 tickets total, let adult count increase by 1. Child count decreases by 1, and revenue increases by $4.
A table makes that constant change visible.
5. Invariants reduce the number of cases
If the total number of objects is fixed at 20, you do not need two independent variables. Knowing one category count determines the other.
An invariant removes one degree of freedom.
6. Parity can eliminate impossible cases
An even number plus an even number is even. An odd plus odd is even. An odd plus even is odd.
If a problem requires an odd total but every possible contribution is even, the condition is impossible.
Parity is a simple invariant under many operations.
7. Example: possible sums from two number sets
Choose one number from {2, 5} and one from {10, 20, 30}. Systematically list:
- 2 + 10 = 12
- 2 + 20 = 22
- 2 + 30 = 32
- 5 + 10 = 15
- 5 + 20 = 25
- 5 + 30 = 35
There are 2 × 3 = 6 combinations.
8. Tree diagrams represent staged choices
If a student chooses one of 2 shirts and one of 3 shorts, a tree has two first branches, each with three second branches.
Total combinations = 2 × 3 = 6.
The multiplication principle emerges from the repeated branch structure.
9. Systematic listing and tree diagrams solve the same search problem
A list is compact when cases are few. A tree is useful when choices happen in stages. A table is useful when numerical quantities change together.
Choose the representation that makes completeness easiest to verify.
10. Small-case analysis can reveal a general pattern
For handshakes, 2 people give 1 handshake, 3 give 3, 4 give 6, 5 give 10.
Each new person adds handshakes with all previous people: +1, +2, +3, +4, …
For 8 people: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
11. Patterns need an explicit invariant or change rule
In the handshake pattern, existing handshakes remain while one new person contributes exactly one handshake with each existing person.
The invariant is that old pairs stay counted; the change is determined by the number of existing people.
12. Fixed-sum pairs can be listed efficiently
Find positive whole-number pairs with sum 10:
(1,9), (2,8), (3,7), (4,6), (5,5).
If order does not matter, stop at the midpoint to avoid duplicates like (9,1).
13. Ordered versus unordered cases
For outfits, red shirt + blue shorts differs from blue shirt + red shorts only if shirts and shorts come from different choice sets. In digit arrangements, 247 and 742 are different because position changes value.
Decide whether order matters before listing.
14. Case elimination using bounds
If 12 items each cost either $5 or $9, total cost must be at least $60 and at most $108.
A stated total of $120 is impossible without further calculation.
Bounds eliminate cases before detailed search.
15. Case elimination using divisibility
If a total is made entirely from packs of 6, the count must be a multiple of 6. A proposed total of 50 cannot arise from complete 6-item packs alone.
Divisibility is another fast structural filter.
16. Fixed total under transfer
If 10 counters move from A to B, the combined total is unchanged.
Tracking the total can be more powerful than tracking both changing amounts separately.
17. Difference is not invariant under transfer
When x moves from larger A to smaller B, the difference shrinks by 2x.
Know which quantity is invariant and which changes predictably.
18. Product invariance under doubling and halving
16 × 35 = 8 × 70. Doubling one factor and halving the other preserves the product.
This arithmetic invariant is useful in mental mathematics and pattern reasoning.
19. Difference invariance under equal shifts
70 − 45 = 25. Also 80 − 55 = 25.
Adding the same amount to both values preserves their difference.
20. Ratio invariance under equal scaling
If quantities are in a 2-to-3 comparison and both are doubled, the comparison becomes 4-to-6 but the multiplicative relationship remains the same.
This is a transition concept toward later formal ratio.
21. Systematic table for a two-price problem
Ten tickets cost either $6 or $10. Let the number of $10 tickets be x. Then $6 tickets = 10 − x.
| $10 tickets | $6 tickets | Revenue |
|---|---|---|
| 0 | 10 | 60 |
| 1 | 9 | 64 |
| 2 | 8 | 68 |
| 3 | 7 | 72 |
Revenue increases by $4 each row. Once the pattern is seen, there is no need to list every row individually.
22. A table can expose a linear relationship
If increasing x by 1 always changes y by the same amount, the table contains a constant-difference relationship.
Primary 5 learners need not use formal algebraic language to benefit from seeing this structure.
23. Search only as far as necessary
Systematic listing should reduce uncertainty, not create needless work. Once a constant pattern or decisive condition is found, switch to the more efficient calculation.
A scaffold should be abandoned when the structure is clear.
24. Prove completeness
After listing cases, ask: why can there be no others?
For the digits 2, 4, 7, there are exactly three choices for the first digit, then two for the second, then one for the third. That accounts for 3 × 2 × 1 = 6 arrangements.
A complete solution explains why the search space is exhausted.
25. Error map
| Visible error | Likely cause | Repair question |
|---|---|---|
| Cases repeated | No organising rule | Which first choice can you fix? |
| Cases missing | Search space not partitioned | Have all first-stage choices been exhausted? |
| Ordered pairs double-counted | Order relevance unclear | Does swapping positions create a new case? |
| Long table continued after pattern obvious | Scaffold not released | What constant change has emerged? |
| Invariant assumed incorrectly | Changing quantity misidentified | What exactly remains constant under this operation? |
26. Practice laboratory
- Using 1, 5, 8 exactly once, list all three-digit numbers.
- Choose one number from {2,5} and one from {10,20,30}. How many sums are possible?
- List unordered positive integer pairs summing to 12.
- Six people shake hands once with every other person. Find handshakes.
- A total of 20 items is split between A and B. If A is 7, find B and state invariant.
- 10 tickets cost either $6 or $10. Explain revenue change when one $6 ticket is replaced by a $10 ticket.
- Can 12 objects each weighing 2 or 5 kg total 31 kg? Explain using parity or cases.
- A difference of 30 exists between A and B. Both gain 12. What remains invariant?
- 16 × 35 is transformed to 8 × 70. What invariant is preserved?
- Explain why a random list is weaker evidence of completeness than a list organised by first choice.
27. Explained answers
1. 158, 185, 518, 581, 815, 851: 6.
2. 2 × 3 = 6 combinations.
3. (1,11), (2,10), (3,9), (4,8), (5,7), (6,6).
4. 1 + 2 + 3 + 4 + 5 = 15.
5. B = 13; invariant total = 20.
6. Revenue increases by $4.
7. No. Twelve 2-kg items give 24 kg. Replacing one by 5 kg adds 3 kg, so possible totals are 24,27,30,33,…; 31 is not possible.
8. Difference remains 30.
9. The product remains unchanged.
10. Organised listing partitions the search space and gives a reason no case is omitted or repeated.
28. Full mixed case problem
A snack set contains 8 items, each either $2 or $5, and costs $25. Find the number of $5 items.
Systematic table or assumption both work. Let x be $5 items. Then $2 items = 8 − x.
Total = 5x + 2(8 − x) = 25.
3x + 16 = 25, so x = 3.
Check: 3 × 5 + 5 × 2 = 15 + 10 = 25.
The invariant item count turns a two-category problem into one changing variable.
29. Final checkpoint
A strong Primary 5 case-reasoning learner can organise choices systematically, decide whether order matters, use tables to track linked quantities, exploit bounds, parity and divisibility, identify true invariants, stop listing once a stronger pattern emerges and explain why the final set of cases is complete.
Return to the Primary 5 Mathematics Learning Hub.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Partition the search space, preserve the invariant, eliminate impossible cases early, and stop enumerating the moment a stronger general relation can safely replace the list.
