PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 11 · GUIDE 41
Transfer problems become much easier when the learner first decides whether the transfer stays inside the system or crosses its boundary. If counters move from Box A to Box B, the combined total of A and B is conserved. If counters leave A and go to an outside group, the total inside A+B changes. That single distinction controls many before–after problems.
This guide separates internal transfer from external transfer, then develops total conservation, difference change, equalisation, reverse transfer and multi-stage state control. These are problem-solving descriptions rather than standalone syllabus chapters.
Return to the Primary 5 Mathematics Learning Hub. Related guides: Before–After Models · Equal Stage Method · Unchanged-Quantity Reasoning.
1. Internal transfer preserves the combined total
Box A has 90 counters and Box B has 50. If 20 counters move from A to B, the new values are 70 and 70. The combined total stays 140.
2. External transfer changes the system total
If 20 counters leave A and are given to someone outside the A+B system, the total inside A+B falls by 20. Conservation only applies inside the chosen system boundary.
3. Define the system before calculating
Write the groups being tracked. If the problem concerns A and B only, ask whether anything enters or leaves A+B. If not, total is invariant. If yes, update total.
4. Internal transfer changes difference twice as fast
If x moves from larger A to smaller B, A decreases by x while B increases by x. The difference shrinks by 2x.
5. External transfer changes difference differently
If x leaves A for an outside group while B is unchanged, the difference between A and B changes by only x. One side moves; the other does not.
6. Same words can hide different mathematics
“A gives 20 to B” is an internal transfer for system A+B. “A gives 20 away” is external unless the receiving party is included in the system.
7. Equalisation is a special internal-transfer state
A=90, B=50. Total=140, so the equal stage is 70 each. Therefore 20 moves from A to B.
8. Final-difference transfer problems
A=90, B=50. Some move from A to B until A has 10 more. Total 140. Final pair must be 75 and 65. Transfer = 15.
9. Reverse transfer from final state
After an internal transfer, A=64 and B=56. If 12 moved from A to B, before the transfer A=76 and B=44.
10. Internal transfer with fractions
A and B total 300. Before transfer A is 3/5 of total=180 and B=120. After transfer they are equal at 150 each. Transfer = 30.
11. Internal transfer with percentage
A and B total $500. A holds 70%, B 30%. After transfer they are equal. Start 350 and 150; equal 250 each; transfer $100.
12. External addition changes the total first
A=80, B=60. Someone gives 20 to B from outside. New total=160 and new pair=80,80. Do not conserve the original total 140.
13. External removal changes the active state
A=100, B=60. A gives 20 outside. New total=140, values=80 and 60. If a later transfer occurs between A and B, conserve 140 from that point onward.
14. Multi-stage transfer problems need state labels
Use S0, S1, S2. Mark whether each stage is internal or external. This prevents using the wrong total or difference rule after a boundary-crossing event.
15. Internal transfer and constant total
Whenever only redistribution happens, total remains fixed. This lets learners solve final states from total and final difference without naming the transfer first.
16. Internal transfer and bar models
Before and after bars should keep the same combined length if nothing enters or leaves. The shifted segment appears on the other bar after transfer.
17. External transfer and bar models
If material leaves the system, the combined after-bars are shorter. If material enters, they are longer. The diagram should show the system boundary change.
18. Transfer with rate or money
If $48 moves from Account A to B, combined money stays fixed. If $48 is spent externally, the combined account total falls by $48. Same amount, different system action.
19. Error map
| Error | Cause | Repair question |
|---|---|---|
| Conserves total after money is spent outside | External transfer misread | Did the quantity remain inside the system? |
| Gap changes by x for A→B transfer | Only one side tracked | What happens to both A and B? |
| Uses original total after external addition | Active state not updated | What is the total now? |
| Mixes before and after relations | Stage control lost | Which statements belong to the same stage? |
20. Practice laboratory
- A=90, B=50. How much moves internally to equalise?
- A=90, B=50. How much moves internally until A has 10 more?
- A=100, B=60. A gives 20 outside. Find new total and difference.
- A=80, B=60. B receives 20 from outside. Find new total and difference.
- A+B=300, A=3/5 of total. How much moves to equalise?
- A+B=$500, A has 70%. How much transfers to equalise?
21. Answers
1. 20.
2. 15.
3. New total 140; values 80 and 60; difference 20.
4. New total 160; values 80 and 80; difference 0.
5. Start 180/120; equal 150/150; transfer 30.
6. Start 350/150; equal 250/250; transfer $100.
22. Full transfer problem
A and B have 420 counters altogether. A has twice as many as B. Then 30 counters are added to B from outside. After that, some counters move from A to B until A has 20 more than B. How many counters move internally?
Start: A=280, B=140. External addition gives A=280, B=170; active total=450. Final difference 20: remove difference → 430; equal part=215. Final A=235, B=215. Internal transfer = 280−235 = 45 counters.
23. Final checkpoint
A strong Primary 5 learner can define the system boundary, distinguish internal from external transfer, conserve totals only when justified, track how differences change, rebuild active totals after external events and solve hidden transfers from before–after states.
Continue to Constant Total, Constant Difference & Constant Single-Subject Problems.
