PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 11 · GUIDE 42
Many Primary 5 word problems become manageable once the learner identifies what does not change. Sometimes the total stays constant. Sometimes a difference stays constant. Sometimes one subject remains unchanged while another changes. These are three distinct invariant families and they should not be mixed.
The labels used here are heuristic descriptions rather than official syllabus chapter names. Their purpose is to make invariant selection explicit and teachable.
Return to the Primary 5 Mathematics Learning Hub. Related guides: Unchanged-Quantity Reasoning · Internal & External Transfer Problems.
1. Constant total problems
When quantity only moves within a closed system, the total remains fixed. If A+B=140 before a transfer, A+B=140 afterward.
2. Constant difference problems
If equal amounts are added to or removed from both quantities, their difference stays unchanged. If A−B=30, then (A+12)−(B+12)=30.
3. Constant single-subject problems
If B remains unchanged while A changes, B becomes the anchor connecting before and after states.
4. Do not assume the wrong invariant
Internal transfer preserves total but changes difference. Equal addition preserves difference but changes total. One-subject change preserves only the untouched subject.
5. Constant total example
A=90, B=50. Some transfer internally until A has 10 more. Total=140. Final values are 75 and 65. Transfer=15.
6. Constant difference example
A=80, B=50. Both receive the same amount. Their difference remains 30 no matter how much equal addition occurs.
7. Constant single-subject example
B has $120 and stays unchanged. A spends $40 and afterward has twice B. After A=$240; before A=$280.
8. Constant total with fractions
A+B=300. A is 3/5 of total and B is 2/5. A transfer changes their split, but total stays 300.
9. Constant difference across time
If two ages differ by 4 years, the difference remains 4 years as both increase by one year annually. This is a classic constant-difference structure.
10. Constant single subject with percentage
B remains $200. A changes until it becomes 125% of B. Then A=250. If A had decreased by $30 to reach that state, A originally had $280.
11. Bar models for constant total
The combined bar length stays fixed while internal segments shift between groups.
12. Bar models for constant difference
Two bars may both lengthen or shorten by the same amount while the extra segment remains unchanged.
13. Bar models for constant single subject
Keep the unchanged bar fixed and compare the changing subject against it before and after.
14. Invariants reduce unknowns
An invariant acts like an equation already given by the story. The learner does not need to model every changing quantity independently.
15. Invariants can combine
A problem can preserve total during one stage and then preserve one subject during another. Mark stages clearly so the correct invariant is used at the correct time.
16. External events break constant-total reasoning
If 20 units enter from outside, the total increases by 20. If 20 leave, total falls by 20. The invariant must be updated or abandoned.
17. Unequal additions break constant difference
If A gains 10 and B gains 4, the difference changes by 6. Constant difference requires equal changes to both quantities.
18. Constant single-subject problems require evidence
Do not freeze a quantity merely because the story focuses on another one. Confirm that no stated action changes the anchor subject.
19. Error map
| Error | Cause | Repair question |
|---|---|---|
| Uses constant total after outside addition | System changed | Did anything enter or leave? |
| Uses constant difference after unequal changes | Equal-change condition absent | Did both quantities change by the same amount? |
| Freezes a subject that changed | Anchor misidentified | Was this quantity truly untouched? |
| Uses same invariant across all stages | State transitions ignored | Which invariant applies at this stage? |
20. Practice laboratory
- A+B=200. An internal transfer occurs. What stays constant?
- A−B=35. Both gain 12. What stays constant?
- B=$150 unchanged. A spends $30 and ends at twice B. Find A before.
- A=90, B=50, internal transfer until A has 10 more. Find transfer.
- A=80, B=50, both gain 15. Find new values and difference.
21. Answers
1. Total 200.
2. Difference 35.
3. After A=300; before A=$330.
4. Transfer 15.
5. 95 and 65; difference 30.
22. Full invariant problem
A and B have 360 counters altogether. A has 80 more than B. Then both receive 20 counters from outside. After that, 30 counters move from A to B. Find the final values.
Initial: total 360, difference 80 → values 220 and 140. Equal outside additions preserve difference but raise total to 400 → 240 and 160. Internal transfer of 30 preserves total but reduces difference by 60 → final 210 and 190.
The invariant changes by stage: first total+difference, then constant difference, then constant total.
23. Final checkpoint
A strong Primary 5 learner can distinguish constant-total, constant-difference and constant-single-subject structures, justify the invariant from the story, change invariants when the system changes and use the correct anchor to connect before and after states.
Continue to Model Method Language Traps.
