PRIMARY 4 MATHEMATICS LEARNING GUIDE · BATCH 11 · GUIDE 44
A Constant Difference problem changes two quantities by the same amount, so the gap between them stays unchanged. If one child has 20 more counters than another and both receive 15 more counters, the first child still has 20 more. If both spend 7 counters from their collections, the same 20-counter gap remains. The values move; the difference does not.
This invariant is easy to state but frequently confused with Constant Total. Equal changes to both quantities usually change their combined total, while preserving their difference. An internal transfer between the quantities does the opposite: it can preserve total while changing the difference.
“Constant Difference” is used here as a Singapore problem-solving heuristic label rather than a separate official syllabus chapter. The official curriculum reference is the MOE Primary Mathematics Syllabus, updated October 2025. All examples and practice tasks are independently written by eduKate Publishing.
Series route: return to the Primary 4 Mathematics Learning Hub. For the contrasting invariant, use Constant Total.
Navigate: the invariant · number-line view · age problems · when difference changes · constant total contrast · practice · answers.
1. Equal additions preserve the difference
A has 65 counters and B has 38.
Difference = 65 − 38 = 27 counters.
Both receive 12 more counters.
New amounts: A=77, B=50.
New difference = 77 − 50 = 27 counters.
The two values changed by the same amount in the same direction, so their gap stayed constant.
2. Equal subtractions also preserve the difference
A has 65 and B38. Both use 10 counters.
New amounts:55 and28.
Difference=55−28=27.
Subtracting the same amount shifts both quantities down together.
As long as both changes are equal and subtraction remains meaningful in the context, the difference is preserved.
3. A simple explanation without algebra
Imagine two runners standing 27 metres apart on the same straight path.
If both walk forward 12 metres, the space between them is still27 metres.
If both walk backward5 metres, the space remains27 metres.
Their positions change together, so the separation stays fixed.
This spatial image mirrors Constant Difference reasoning on a number line.
4. Translate Constant Difference onto a number line
Start with38 and65.
The gap is27.
Add12 to both:50 and77.
Both points shift12 units to the right.
The segment between them remains27 units long.
This representation makes the invariant visible: translation of both points by the same amount preserves distance between them.
5. Constant Difference can help us avoid unnecessary recalculation
A has 43 more cards than B.
Both receive 25 cards.
Question: how many more cards does A have now?
No new subtraction is necessary. The difference remains 43 cards.
The heuristic saves work because the equal change tells us the comparison gap directly.
6. The total usually changes even though the difference does not
A=65, B=38. Original total=103.
Add12 to each. New total=77+50=127.
Total increased by24 because two quantities each increased by12.
Difference remained27.
This shows why Constant Difference and Constant Total must not be confused.
7. Equal changes can be hidden inside a story
Two children each receive a birthday gift of $15.
Before the gift, Amir has $42 more than Bea.
After both receive the same $15, Amir still has $42 more.
The word “same” is the important structural clue.
Do not focus only on the new amount added; focus on whether both quantities changed equally.
8. The invariant works through several equal-change stages
A is 18 counters ahead of B.
Both receive10.
Then both use7.
Then both receive3.
Every stage applies an equal change to both quantities.
The difference remains 18 counters throughout.
Several equal shifts can be compressed into one net shift without changing the gap.
9. Age difference is a natural Constant Difference example
An older sibling is 4 years older than a younger sibling.
Five years later, both are five years older.
The age difference is still 4 years.
Their combined age, however, increases by10 years.
This everyday example shows Constant Difference especially clearly.
10. Age problems can be solved by carrying the difference across time
Today, Mei is 13 and Jia is 9.
Difference=4 years.
When Jia is 15, six years have passed. Mei will be19.
We can check using the invariant:19−15=4.
The difference does not need to be recomputed from scratch at every year.
11. Equal increases to lengths preserve the length difference
Ribbon A is 80 cm and Ribbon B is55 cm.
Both are extended by20 cm.
New lengths:100 cm and75 cm.
Difference remains 25 cm.
The same structure can appear in measurement, money, counts and ages.
12. Equal decreases preserve the difference only while the context remains valid
A has12 counters and B5, difference7.
Subtract4 from both:8 and1, difference7.
Subtract6 from both would produce6 and−1 in abstract integer arithmetic, but negative counter counts are not meaningful in an ordinary Primary 4 collection story.
The invariant is mathematically valid in broader number systems, but the physical context can restrict permissible changes.
At Primary 4, keep the story quantities meaningful.
13. A known final amount plus the constant difference can recover the other amount
A has 27 more counters than B.
After both receive12 counters, B has50.
Because the difference remains27, A now has50+27=77.
Undo the equal addition if original amounts are required:
A originally65, B originally38.
The invariant can be used at the final stage before reversing the common change.
14. A final total plus the constant difference can recover both final amounts
A has 20 more than B. Both receive the same amount. Their final total is 140.
The difference is still20.
Remove the excess:140−20=120.
Split equally:60 and60.
So final B=60 and A=80.
If the common addition is known, subtract it from both to recover the original amounts.
15. The common change may be unknown and unnecessary
A has35 more than B.
Both receive some equal unknown amount.
Question: what is their new difference?
We do not need to know the common amount.
The answer remains 35.
Invariants can remove irrelevant unknowns from a problem.
16. Unequal changes do not preserve the difference
A=65, B=38.
A receives12 while B receives5.
New amounts77 and43.
New difference34.
Original difference27.
The difference increased by7 because A received7 more than B.
Constant Difference requires equal changes, not merely changes in the same direction.
17. Internal transfer changes the difference by twice the transfer
A=80, B=70. Difference10.
A gives3 to B.
New amounts77 and73. Difference4.
The gap decreased by6, which is twice the transfer.
Why? A moved3 down while B moved3 up, closing the gap from both sides.
Total remains constant, but difference changes.
18. Transfer in the opposite direction increases the difference by twice the transfer
A=80, B=70. B gives3 to A.
New amounts83 and67.
Difference16.
The gap increased by6.
Again, total remains150, while difference changes by twice3.
The direction of transfer determines whether the gap closes or widens.
19. Multiplying both quantities by the same factor does not preserve the difference
A=8, B=5. Difference3.
Double both:16 and10. Difference6.
The difference also doubles.
Equal multiplication preserves a multiplicative relationship, not an additive difference.
This is an important boundary: “same operation” is not enough. Constant Difference specifically concerns equal additive changes.
20. Same percentage increase does not necessarily mean same amount increase
This is a later-mathematics warning rather than a Primary 4 percentage lesson.
If two unequal quantities both grow by the same percentage, the numerical additions are generally different.
Therefore their difference need not remain constant.
The Primary 4 heuristic should be stated precisely: adding or subtracting the same amount to both quantities preserves their difference.
21. One quantity can cross the other only when changes are unequal or transferred
If A begins larger than B and both change by the same amount, A remains larger by the same gap.
Equality cannot suddenly appear.
For the order to reverse, the two quantities must experience different changes.
This provides a fast plausibility check.
If a solution says equal additions made the smaller quantity become larger, something is wrong.
22. Constant Difference versus Constant Total
| Structure | Total | Difference |
|---|---|---|
| Add same amount to both | Changes | Stays constant |
| Subtract same amount from both | Changes | Stays constant |
| Transfer from A to B | Stays constant | Changes |
| Transfer from B to A | Stays constant | Changes |
This comparison is one of the most useful heuristic distinctions in the Batch 11 lane.
23. Bar models can show the preserved gap
Draw A’s bar longer than B’s by a fixed segment of27.
When both receive12, extend both bars by an equal12-unit segment.
The unmatched27-unit segment remains unchanged.
When both lose10, remove equal10-unit segments from both bars. The unmatched segment remains27.
The diagram makes the invariant visible.
24. Number sequences can preserve a constant difference
Sequence A:10,15,20,25,…
Sequence B:3,8,13,18,…
Each sequence increases by5.
The difference between corresponding terms remains7.
This is Constant Difference across repeated equal changes.
If one sequence later increases by6 instead, the gap changes.
25. Constant Difference can simplify future-state questions
Today, one jar contains 18 more marbles than another.
Every week, 4 marbles are added to each jar.
After 10 weeks, the difference is still 18 marbles.
We do not need to calculate either final amount if the question asks only for the gap.
The invariant removes unnecessary work.
26. Constant Difference does not determine actual amounts by itself
“A has 20 more than B.”
Possible pairs include30 and10,50 and30, and100 and80.
The difference gives one relationship but not the individual values.
We need another condition such as a total, one actual amount, or a final amount after an equal change.
An invariant is powerful but does not eliminate the need for sufficient information.
27. A total plus difference determines the pair
A has20 more than B and together they have140.
Remove the excess20:120 remains as two equal parts.
B=60; A=80.
If both then receive15, new amounts95 and75, still difference20.
This connects total-and-difference models with Constant Difference.
28. A final amount can be used without calculating the other original amount first
A has27 more than B.
Both receive12.
Afterwards B has50.
Because the difference is constant, after the change A=77.
If the question asks for original A only, subtract12 from77 to get65.
There is no need to reconstruct every intermediate number unless required.
29. Diagnostic error table
| Error | Likely cause | Repair question |
|---|---|---|
| Adds common change to the difference | Difference confused with individual values | Did both endpoints move equally? |
| Calls transfer a constant-difference case | Total and difference invariants confused | Did one lose while the other gained? |
| Assumes same multiplication preserves difference | Additive and multiplicative change confused | Was the same amount added, or were values scaled? |
| Cannot use final amount with known gap | Invariant not carried across stages | What is the gap after the equal change? |
| Invents actual amounts from difference alone | Insufficient information ignored | What second condition fixes the pair? |
30. Practice laboratory
- A has65 counters and B38. Find the difference.
- Both receive12. Find the new difference.
- Both use10. Find the new difference.
- A has43 more cards than B. Both receive25. Find the new difference.
- A is18 counters ahead of B. Both receive10, use7, then receive3. Find the final difference.
- An older sibling is4 years older. What is the age difference five years later?
- Mei is13 and Jia9. How old is Mei when Jia is15?
- Ribbon A is80 cm and B55 cm. Both are extended20 cm. Find the new difference.
- A has27 more counters than B. Both receive12. Afterwards B has50. Find final A and original amounts.
- A has20 more than B. After both receive the same amount, their total is140. Find final amounts.
- If the common addition in Question10 is15, find original amounts.
- A has35 more than B. Both receive an unknown equal amount. Find the new difference.
- A=65,B38. A receives12 while B receives5. Find new difference.
- A=80,B70. A gives3 to B. Find new difference and change in gap.
- A=80,B70. B gives3 to A. Find new difference and change in gap.
- A=8,B5. Double both. Is the difference constant?
- A and B differ by20 and total140. Find them.
- Sequence A is10,15,20,25 and B is3,8,13,18. Find the corresponding difference.
- Two jars differ by18 marbles and4 marbles are added to each every week. Find the difference after10 weeks.
- A has20 more than B, but no other information is given. Can their actual amounts be found uniquely?
31. Explained answers
1. 65−38=27.
2. Equal addition preserves the gap: 27.
3. Equal subtraction preserves the gap: 27.
4. 43 cards.
5. Every stage changes both equally, so 18.
6. 4 years.
7. Six years later Jia15; Mei19. Difference remains4.
8. Original difference25 cm; new difference 25 cm.
9. Final B50, so final A77. Undo +12: original A65,B38.
10. Difference20,total140→A80,B60.
11. Subtract15 from each: A65,B45.
12. 35; the common amount is irrelevant to the gap.
13. New A77,B43,difference=34.
14. New77 and73,difference4; gap decreased by6, twice the transfer.
15. New83 and67,difference16; gap increased by6.
16. No. Difference changes from3 to6.
17. Remove excess20:120; half60. A80,B60.
18. Difference is consistently7.
19. 18 marbles.
20. No. Many pairs have difference20.
32. Teaching routine: move both endpoints visibly
Place two counters or marks on a number line.
Measure the gap.
Move both by the same amount and re-measure.
Then try an internal transfer: move one down and the other up by the same amount. Observe that the total behaves differently from the gap.
This physical contrast makes the two invariants difficult to confuse.
33. Batch 11 World Return
Batch 11 adds four Singapore-facing heuristic owners without replacing the existing Primary 4 content guides.
Unitary Method reduces equal relationships to one unit. Shortage & Surplus compares two distribution scenarios. Constant Total tracks redistribution inside a closed system. Constant Difference tracks equal additive changes that preserve a comparison gap.
Final checkpoint: can the learner identify which quantity stays invariant, distinguish equal change from internal transfer, and choose the heuristic because of the relationship rather than because of a memorised keyword?
Source and editorial note
The curriculum boundary is referenced to the MOE Primary Mathematics Syllabus, updated October 2025. “Constant Difference” is used here as an instructional heuristic label; all examples and routines are independently written by eduKate Publishing.
Editorial control: Wintour House V1.0 · CivDJ · eduKate Publishing.
