eduKate Singapore · Primary 6 · Learner’s Guide Vol 0082
Choose the part that helps you now
1. Understand the job · 2. Study the worked examples · 3. Diagnose and repair · 4. Practise independently · 5. Check, control and continue
Full chapter index · Practice · Subject hub · PSLE Learning Guide
Chapter index
Chapters 1–4
2. Why the difference changes by twice the transfer
Chapters 5–8
5. Worked example: use equal final amounts
6. A fraction transfer must belong to a particular amount
7. Work through several transfers in order
8. Design independent checks instead of repeating the same arithmetic
Chapters 9–12
9. Independent practice: fixed transfers
10. Independent practice: fractions and outside movements
When one person gives another person some counters, two quantities change. The giver has fewer counters and the receiver has more. If your working records only one side, you can produce accurate arithmetic from an incomplete model.
The strongest first check is to write who gives, who receives, how much moves and when the movement happens. Then record the before-and-after quantities for both sides. A transfer between two people keeps their combined total unchanged only when nothing enters or leaves the pair.
This Primary 6 workshop teaches you to audit a transfer solution rather than guess a familiar method. You will use short records, worked examples, error repairs and delayed practice to catch missing changes, wrong directions and percentages attached to the wrong moment. The examples are original teaching tasks, not predictions of examination questions.
Suppose Asha has 38 counters and Ben has 22. Asha gives Ben 6 counters. Asha then has 32 and Ben has 28. Their combined total remains 60. You can describe the movement with two ordinary statements: Asha loses 6; Ben gains 6.
Writing only “38 − 6 = 32” answers a question about Asha’s new amount, but it is not a complete record if a later question uses Ben’s amount too. Writing “22 + 6 = 28” alone has the same limitation on the other side. Both calculations are correct; either can become misleading when you forget the unrecorded change.
Use the names before using signs. “Asha to Ben” makes the direction visible. Then attach subtraction to Asha and addition to Ben. A subtraction sign does not mean “the smaller person”, and an addition sign does not mean “the person mentioned second”. The direction of the transfer decides the operation.
A simple spoken check is: “After giving, does the giver have less? After receiving, does the receiver have more?” If your record fails this check, return to the sentence. Do not repair the error by swapping random signs until the final answer looks sensible.
The combined-total check is also useful, but it has conditions. If Asha gives 6 counters to Ben, the pair’s total is unchanged. If Asha gives 6 counters to a third person outside the pair, Asha and Ben’s combined total decreases. If Ben buys 6 new counters from a shop, the pair’s total increases.
Therefore, do not write “the total stays the same” just because a question uses the word “gives”. First identify the group whose total you are tracking. A transfer within that group preserves the total. A movement across its boundary changes the total.
Your first audit
Start with 45 stickers for Lina and 15 for Omar. Lina gives Omar 8. A learner records 37 and 15. Which entry is missing? Omar should have 23 because he receives the same 8 that Lina loses. The total 52 in the learner’s record exposes the missing gain: the original total was 60.
Suppose the giver begins with more. Each item transferred makes the giver’s amount smaller by one and the receiver’s amount larger by one. The gap therefore closes by two for each item moved.
With 38 and 22 counters, the initial difference is 16. After a transfer of 6 from the larger amount to the smaller amount, the difference is 32 − 28 = 4. The gap has decreased by 12, which is twice 6.
You can see this without memorising a rule. Think of the difference as the giver’s amount minus the receiver’s amount. Reducing the first amount by 6 reduces that difference by 6. Increasing the second amount by 6 reduces it by another 6. Both changes matter.
This explains a common equalisation task. If one pupil has 42 beads and another has 18, the gap is 24. To make their amounts equal by moving beads from the first to the second, transfer 12. The resulting amounts are 30 and 30.
Transferring 24 would not make them equal. It would reverse the original amounts: 18 and 42. A learner who says “the difference is 24, so give 24” has treated the transfer as if it changed only the giver. The repair is to record both new amounts, not simply remember “divide by two”.
The twice-the-transfer relationship applies to the signed difference when the same quantity is subtracted from one amount and added to the other. Once the amounts pass equality, the identity of the larger amount changes. If the task asks for an ordinary positive difference, check which amount is larger at the end.
For example, 30 and 10 become 18 and 22 after a transfer of 12 from the first to the second. The original gap was 20. The signed difference becomes 20 − 24 = −4, meaning the second now has 4 more. In Primary 6 working, you can simply compute 22 − 18 = 4 and label who has more.
Do not apply the same gap rule when both people give items away, when one amount increases from an outside source, or when different quantities move at the two ends. The relationship comes from a specific mechanism. Record that mechanism before choosing the shortcut.
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A good record makes the relationship easier to inspect. It is not decoration. You can write two lines, a small table or a bar model, depending on the question. The minimum information is the identity of each quantity, the time state and the change.
For a transfer of 9 marbles from Kai to Mei, write: Kai after = Kai before − 9; Mei after = Mei before + 9. If the initial quantities are known, substitute them. If they are unknown, keep the relationships visible while using the other information.
Suppose Kai has 7 more marbles than Mei after the transfer. That is an after-state relationship. It belongs with the after amounts, not the before amounts. The initial gap must have been 7 + 18 = 25 because the transfer closed the gap by 18.
A before-and-after table prevents three confusions at once. The column labels keep the time states separate. The row labels keep the people separate. The changes show the transfer direction. If a table does not do at least one of these jobs, it may be unnecessary.
Do not use the same letter or phrase for different quantities without saying so. “Kai’s amount” could mean before or after. “One unit” could refer to an initial ratio or a final ratio. A vague label can make valid calculations appear to fit the wrong relationship.
Consider an initial ratio of 5:3 that becomes 1:1 after a transfer. The original eight ratio units describe the initial quantities. The final equal amounts need not be called “one original unit” each. If you use ratio units, keep their meaning fixed or explicitly build a new representation.
This workshop is about auditing the movement. For broader strategies involving changing ratios and before-and-after problems, use the existing PSLE before-and-after Mathematics guide. Its wider problem-solving role remains separate from the two-sided check practised here.
Keep the record small
If the question asks only for Ben’s new amount in a straightforward transfer, a labelled addition may be enough. If several transfers or ratio changes occur, a fuller record becomes worthwhile. Choose the smallest representation that keeps every relevant change visible.
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Original task: “Rina had 18 more cards than Sam. She gave Sam 5 cards. Sam then had 24 cards. How many cards did Rina have at first?”
Begin with the final information that is easiest to reverse. Sam received 5 and ended with 24, so he began with 24 − 5 = 19. Rina originally had 18 more than Sam, so she began with 19 + 18 = 37.
Now audit both sides. Rina after = 37 − 5 = 32. Sam after = 19 + 5 = 24. The final gap is 8. The initial gap was 18, and the transfer reduced it by 10. That matches twice the transferred 5.
The combined total gives another check. Before: 37 + 19 = 56. After: 32 + 24 = 56. The movement stayed within the pair, so the totals should agree. This check tests whether you lost or created cards while recording the transfer.
A wrong solution might begin with 24 + 18 = 42 and state that Rina originally had 42. The arithmetic is correct, but it compares Rina’s before amount with Sam’s after amount. The phrase “18 more” described their original quantities. It cannot be attached to Sam’s final 24 without adjustment.
Another wrong solution might use 24 + 5 = 29 for Sam’s original amount. That reverses the receiving action in the wrong direction. To work backwards from a receiver’s final amount, subtract the received amount. The receiver had less before the receipt.
A third wrong solution might correctly find 37 but show only an unexplained chain of numbers. During practice, label the intermediate quantity: “Sam at first = 19 cards.” The label helps both you and another reader see why the next addition uses 18.
A changed-detail check
Change the task so that Sam gives Rina 5 cards and Sam then has 24. Sam originally had 29. Rina originally had 29 + 18 = 47. The numbers look similar, but the transfer direction changes the reverse operation.
This changed version is useful after a delay. If you automatically subtract 5 whenever you see 24, you have remembered the old surface pattern rather than the action. Name the giver and receiver again before calculating.
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Original task: “Nadia and Theo had 84 tokens altogether. Nadia gave Theo 10 tokens. They then had the same number of tokens. How many tokens did each have at first?”
Since the transfer stayed within the pair, their total remained 84. At the end, equal amounts must be 84 ÷ 2 = 42 each. Nadia gave 10, so she originally had 42 + 10 = 52. Theo received 10, so he originally had 42 − 10 = 32.
Check the story forwards. Nadia: 52 − 10 = 42. Theo: 32 + 10 = 42. Check the original total: 52 + 32 = 84. Check the original gap: 52 − 32 = 20, which is twice the transfer of 10.
Notice that the final equal state makes the problem simple. You do not need to start with an unknown ratio or trial different starting pairs. The known total and equal final quantities give you a useful anchor.
A common error is to divide the total equally and call 42 the starting amount for each. That ignores the transfer. The 42 belongs to the after state. The task asks for the before state. Correct the label before changing the calculation.
Another error is to add 10 to both final amounts. That would produce 52 and 52, with a starting total of 104. Both people cannot originally have had 10 more if one gave and the other received the same 10.
If a task uses money, water or mass instead of tokens, the movement still has two ends, but units need attention. Ten dollars moved between wallets is not ten cents. A volume poured from one container to another is preserved only if the task gives no spilling, leakage or other loss.
Keep such conditions within the question. Do not invent a loss because real water might spill. Conversely, if the question explicitly says that 3 mL was spilled, do not preserve the two-container total as if everything reached the receiver.
Explain the method in one sentence
“At the end, each had half the unchanged total; I reversed the giving for Nadia and the receiving for Theo.” This sentence describes the reasoning. It is more useful than “I used the transfer method”, because it names the relationships that make the method valid.
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Now the transferred quantity is described as a fraction. You must identify whose amount and which time state the fraction refers to.
Original task: “Ella had 48 stickers and Farid had 24. Ella gave Farid one quarter of her stickers. How many stickers did each have afterwards?”
One quarter of Ella’s original 48 is 12. Ella has 48 − 12 = 36. Farid has 24 + 12 = 36. The total remains 72. The fraction determined the amount moved; it did not mean that Farid should receive one quarter of his own stickers.
A wrong record might say Farid receives one quarter of 24, or 6. That changes the reference whole. Another wrong record might subtract 12 from Ella but add 6 to Farid. This creates a missing 6 stickers. The combined-total check exposes the inconsistency.
When the wording says “one quarter of her remaining stickers”, the reference whole is different. Suppose Ella first gives away 8 of her 48 stickers to someone else, then gives Farid one quarter of what remains. The remaining amount is 40, and the second transfer is 10.
Record the first event before the second. Ella has 40 after the outside gift, then 30 after giving Farid 10. Farid becomes 34. The pair’s total starts at 72 and ends at 64 because 8 stickers left the pair. The internal transfer of 10 did not change the new combined total of 64.
This example shows why “total unchanged” cannot be applied across the whole story without checking every movement. It also shows why fractions cannot float free of their reference quantity.
Use an “of what, when?” check before calculating a fraction transfer. Write “one quarter of Ella’s amount before this transfer” or “one quarter of her remaining amount after the first gift”. Once the meaning is settled, perform the arithmetic.
For deeper topic instruction, return through the Mathematics Learning Hub and the existing fraction and ratio guides linked from the PSLE Learning Guide. Here, your performance job is to ensure the same calculated transfer appears at both ends of the record.
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With more than one movement, the after state of one event becomes the before state of the next. A correct final answer depends on updating the quantities at each stage.
Original task: “A box held 60 red counters and another box held 30 blue counters. Twelve counters were moved from the first box to the second. Then 8 counters were moved back from the second box to the first. How many counters were in each box at the end?”
The colours are not needed if the question asks only for total counters per box and does not specify which colours moved back. First transfer: first box 48, second box 42. Second transfer: first box 56, second box 34. The final combined total is 90.
You can also find the net movement: 12 out and 8 back means a net transfer of 4 from the first box to the second. Then 60 − 4 = 56 and 30 + 4 = 34. This shortcut is valid because the quantities are fixed numbers moving between the same two boxes with no other changes.
Do not use a net shortcut blindly when the second transfer is a fraction of the receiver’s current amount. The current amount depends on the first transfer. Calculate the first state before finding the second transferred quantity.
For example, first box 60 and second box 30; transfer 12 to the second; then return one third of the second box’s counters. The second box now has 42, so one third is 14. Final amounts are 62 and 28. The second transfer is not one third of the original 30.
A useful record has a row for each event and an entry for both quantities after that event. You do not need an elaborate diagram. What matters is that the value used for a later fraction or comparison belongs to the right stage.
Spot the stage error
A learner calculates the return as one third of 30, or 10, and obtains 58 and 32. Their final total remains 90, so the combined-total check passes. Yet the answer is wrong because the fraction used the wrong stage.
This illustrates an important limit: one successful check does not prove the whole solution. The total check catches lost or created items. The stage check catches a fraction or relationship attached to the wrong moment. Use the check that matches the possible failure.
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Chapter 8 of 12
8. Design independent checks instead of repeating the same arithmetic
Recalculating the same operations can catch an arithmetic slip, but it may preserve a wrong model. A stronger audit tests a different relationship from the one used to solve.
If you solved using the final equal amounts, check the original total and the transfer forwards. If you solved using a gap, check both final quantities. If you solved backwards, run the story forwards from your proposed starting values.
Suppose you found starting amounts of 52 and 32 in the equalisation example. Running the transfer forwards gives 42 and 42. That tests the given final condition. Adding the starting amounts gives 84. That tests the given total. These checks address separate constraints.
The combined-total check is powerful when all movements stay inside the tracked group. Before total should equal after total. If it does not, look for a missing gain, missing loss or accidental double transfer.
The gap check is useful when a fixed quantity moves from the larger amount to the smaller. The signed gap changes by twice the transfer. But you must still label which person has more if the transfer crosses equality.
The direction check asks whether the giver decreases and the receiver increases. It catches a reversed arrow before you do a long calculation. The time-state check asks whether each ratio, difference or fraction uses quantities from the same specified stage.
The physical-feasibility check asks whether a person gives more than they have, whether counters remain whole when required, or whether an amount becomes negative without a meaningful reason. Such failures often reveal an interpretation error.
None of these checks is a guarantee in isolation. A wrong solution can preserve the total while using the wrong transfer amount. It can produce whole numbers while reversing the people. Choose two checks that test the most likely weaknesses in your particular working.
A check with a purpose
Instead of writing “check answer”, write “run the transfer forwards to recover the final equality” during practice. The named purpose guides your attention. Later, the shorter mental version is enough. Checking should produce evidence about correctness, not merely a feeling that you have looked at the page again.
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Attempt these tasks before reading the answer discussion. For each, identify the giver and receiver, name the state asked for, and show at least one check that is different from repeating the solving steps.
Task A: “Yun had 46 cards and Zane had 18. Yun gave Zane 7 cards. How many did each have afterwards?” Task B: “Isha gave Leon 9 beads. They then had 35 beads each. How many did each have before the transfer?”
Task C: “Ari had 14 more stamps than Bo. Ari gave Bo 4 stamps. Bo then had 23 stamps. How many did Ari have at first?” Task D: “Two jars contained 96 buttons altogether. After 8 buttons were moved from Jar P to Jar Q, Jar Q contained 10 more buttons than Jar P. How many were in each jar at first?”
For A, Yun has 39 and Zane has 25. Their total is 64 before and after. The initial gap of 28 becomes 14, a decrease of 14 from the transfer of 7.
For B, Isha began with 44 and Leon with 26. Reversing Isha’s giving adds 9 to her final 35; reversing Leon’s receipt subtracts 9. The original gap is 18, which closes when 9 moves across.
For C, Bo originally had 23 − 4 = 19. Ari originally had 19 + 14 = 33. Afterwards, Ari has 29 and Bo has 23. The final gap of 6 is consistent with an original gap of 14 reduced by 8.
For D, the final total remains 96. The final amounts are 43 in P and 53 in Q because they sum to 96 and differ by 10. Reverse the transfer: P originally had 51 and Q originally had 45.
The last task is especially useful because P begins with more but ends with less. If you assume the original larger amount must remain larger, you may attach the final difference to the wrong person. Running the transfer forwards gives 51 − 8 = 43 and 45 + 8 = 53.
Repair, then retest
If you made an error, name it precisely. “I reversed the receiver incorrectly” is a useful diagnosis. “I forgot the other side” is another. Then change a number or transfer direction and attempt a fresh task. Copying the provided answer does not test whether the repaired decision can survive new wording.
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These tasks require you to decide whether the total is preserved across the whole sequence and which quantity supplies a fraction.
Task E: “Maya had 72 stickers and Noah had 18. Maya gave Noah one sixth of her stickers. Find their final amounts.” Task F: “P had 50 counters and Q had 30. P gave 10 counters to a third pupil, then gave Q one quarter of P’s remaining counters. Find P’s and Q’s final amounts.”
Task G: “Container A held 900 mL of water and Container B held 300 mL. A pupil poured 200 mL out of A; 180 mL entered B and 20 mL was spilled. Find the amount in each container afterwards.” Task H: “Sara had 40 cards and Tim had 20. Sara gave Tim 8 cards. Tim then returned one quarter of his current cards to Sara. Find the final amounts.”
For E, the transfer is 72 ÷ 6 = 12. Maya ends with 60 and Noah with 30. Their combined total stays 90. If you used one sixth of Noah’s 18, you attached the fraction to the wrong person.
For F, P first has 40. One quarter of 40 is 10. P ends with 30 and Q with 40. Their combined total is 70, compared with the original 80, because the first 10 left the pair. The second transfer preserved the new total.
For G, A ends with 700 mL and B with 480 mL. The final combined volume is 1180 mL, 20 mL less than the original 1200 mL. Subtract 200 from A but add only 180 to B because the problem explicitly states the spill.
For H, after the first transfer Sara has 32 and Tim has 28. One quarter of Tim’s current 28 is 7. Sara ends with 39 and Tim with 21. The total remains 60. Using one quarter of Tim’s original 20 would produce a different, incorrect return.
Explain why the total changed
For F and G, do not merely note that the totals differ. Identify the movement across the tracked group’s boundary: counters given to a third pupil, or water spilled outside the containers. This explanation shows that you understand why the total check should allow a change.
For E and H, all described movement stays within the pair. The unchanged total is a valid audit. Keep that condition attached to the check so that you can recognise when a future problem requires something different.
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Return to this skill after a delay with a new setting. Use library books, two savings envelopes or water containers instead of stickers. The object changes, but the accounting should still identify both ends and any movement outside the pair.
Delayed task: “Two classes had 120 reading cards altogether. Class A sent 15 cards to Class B. Class B then had twice as many cards as Class A. Find the number each class had originally.”
At the end, one share plus two shares makes three equal shares of 120. Class A has 40 and Class B has 80. Reverse the transfer: A originally had 55 and B had 65. Running it forwards gives 40 and 80, which satisfies the final ratio.
This example shows that the giver need not originally have more. Class A had fewer than B and still gave cards. The words “gives” and “receives” identify changes, not initial ranking. Do not use size to guess direction.
In examination work, start with a small labelled record if the story contains a transfer. If the problem is simple, the record may be only two expressions. If it contains successive events, preserve the intermediate states long enough to use the right quantities.
When you cannot start, look for a known final total, final equality or final ratio that can anchor the reverse work. If that route does not fit, use the original relationships. Do not keep adding formulas without identifying what each result represents.
The official SEAB 2026 PSLE Mathematics format is the current-year reference checked for this guide. Standard Mathematics uses two written papers; calculators are not allowed in Paper 1 and are allowed in Paper 2. Follow the official approved-calculator information and your school’s instructions.
The transfer audit here is a teaching routine, not a required named method. Use clear working that fits the question. A calculator can verify arithmetic where permitted, but it cannot decide which person gives, which state a ratio describes or whether a quantity has left the tracked group.
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Adults can check the learner’s understanding without giving a full method. Ask, “Who loses the items? Who gains them? Which total are we tracking?” Then let the learner record the changes. If the record is correct but arithmetic fails, support the calculation separately.
Use a pair of similar tasks with reversed direction. This reveals whether the learner understands the movement or has memorised an operation sequence. Add a fractional second transfer only after the fixed-transfer record is stable.
Do I always divide the difference by two?
Only when the conditions make that relationship valid, such as finding the fixed transfer needed to equalise two amounts by moving items from the larger to the smaller. Explain the two-sided gap change first. A remembered shortcut should not replace the story.
Is an unchanged total enough to prove my answer?
No. It shows that your record did not lose or create quantity within the tracked group. You could still use the wrong fraction, stage or direction. Check another given condition, preferably by running the original story forwards.
What if the question gives a final ratio?
Attach the ratio to the final quantities. If the total is unchanged and known, use it to find the final amounts, then reverse the transfer. Do not attach the final ratio to the original amounts unless the question explicitly says they are the same.
Should I draw a bar model or make a table?
Choose the representation that makes the needed relationship clearest. A table is useful for several stages; a bar model can show a gap or ratio; two labelled calculations can handle a simple movement. You are trying to preserve meaning, not display every possible method.
How do I know this skill is becoming independent?
You can identify the transfer direction, update both sides, use the correct time state and perform a relevant check on a new task without a hint. A correct answer immediately after seeing a model is useful practice, but delayed success gives stronger evidence of transfer.
Return to the PSLE Learning Guide or the Mathematics Learning Hub for the next topic. Keep this question available whenever quantities move: have I recorded the same movement at both ends, with any loss or outside movement accounted for?
Contents · Previous chapter · Choose your next PSLE route
Continue through these four performance workshops
Vol 0081: English: Keep Each Comprehension Answer Inside the Requested Time Window
Vol 0082: Mathematics: Audit a Transfer by Recording Both the Giver and the Receiver
Vol 0083: Science: Judge What Repeated Results Let You Conclude
Vol 0084: English, Mathematics and Science: Test Whether Your Answer Can Be Understood Without You