G2 Mathematics K210 problem solving sometimes fails because the learner forces one formula across situations that actually have different cases. A threshold, sign, domain restriction, rate band, probability condition or geometric configuration can change the mathematics.
This ninety-eighth Learner’s Guide develops case splitting: identify when one problem contains several legitimate mathematical states, solve each state under its own conditions, reject impossible branches and recombine the valid results. It complements Vol 0064 Assumption Audit and Vol 0094 Sensitivity Checks.
The case-splitting protocol
Ask: what condition changes the rule? Are the cases exhaustive? Do they overlap? Which branches violate domain or context? Only split when the mathematics genuinely changes; unnecessary casework creates complexity.
1. positive/negative root
Case trigger: an equation gives two algebraic roots.
Execution: test both against the original domain and context. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
2. length root
Case trigger: quadratic solution gives positive and negative lengths.
Execution: retain only physically valid positive length unless context says directed quantity. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
3. time root
Case trigger: equation gives two time values.
Execution: reject negative elapsed time and check both positive roots against scenario. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
4. integer/non-integer
Case trigger: algebra gives decimal but count must be whole.
Execution: apply integer feasibility after solving. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
5. minimum integer
Case trigger: required quantity must meet or exceed threshold.
Execution: use smallest integer satisfying inequality. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
6. maximum integer
Case trigger: capacity cannot be exceeded.
Execution: use largest integer satisfying condition. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
7. even/odd
Case trigger: condition changes by parity.
Execution: split only if the property genuinely depends on evenness. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
8. positive/zero/negative
Case trigger: expression behaves differently across sign regions.
Execution: analyse sign cases explicitly. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
9. denominator zero
Case trigger: rational expression excludes one value.
Execution: separate domain restriction before algebra. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
10. square root domain
Case trigger: radicand must be nonnegative in real-number context.
Execution: restrict cases before accepting solution. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
11. absolute difference
Case trigger: distance-like expression ignores sign.
Execution: consider ordering cases if removing absolute value. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
12. greater/less ordering
Case trigger: two quantities can occur in either order.
Execution: case split prevents hidden sign reversal. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
13. percentage increase/decrease
Case trigger: wording defines opposite multipliers.
Execution: identify which case applies before calculation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
14. profit/loss
Case trigger: same monetary quantities have different interpretation depending sign.
Execution: separate positive and negative difference. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
15. above/below baseline
Case trigger: change is measured relative to reference.
Execution: case direction controls wording and sometimes operation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
16. inside/outside interval
Case trigger: constraint changes formula or feasibility.
Execution: test interval membership. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
17. before/after threshold
Case trigger: piecewise rule changes at threshold.
Execution: use correct rule on each side. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
18. peak/off-peak rate
Case trigger: tariff depends on time category.
Execution: split usage by rate band. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
19. first block/remaining units
Case trigger: pricing changes after quota.
Execution: calculate each block separately. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
20. fixed fee plus variable fee
Case trigger: cost has common fixed part and usage-dependent part.
Execution: do not treat whole expression as one rate. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
21. discount threshold
Case trigger: discount applies only above qualifying amount.
Execution: test eligibility before applying multiplier. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
22. tax/service tier
Case trigger: rate changes by bracket or condition.
Execution: apply the live bracket rule. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
23. shipping free threshold
Case trigger: cost changes at purchase boundary.
Execution: test amount against threshold. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
24. age category
Case trigger: fee or rule depends on age band.
Execution: use correct interval including endpoints. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
25. ticket category
Case trigger: adult/child/student rates differ.
Execution: count each group separately. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
26. distance zones
Case trigger: fare depends on zone.
Execution: split by stated zone conditions. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
27. speed stages
Case trigger: journey has different speeds over segments.
Execution: compute segment times before total. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
28. stop/wait stage
Case trigger: journey includes zero-distance time.
Execution: separate moving and stationary intervals. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
29. outward/return trip
Case trigger: distances may match but speeds differ.
Execution: do not average speeds naively. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
30. uphill/downhill
Case trigger: rates differ by direction.
Execution: split segments. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
31. with/against current
Case trigger: effective speed changes by direction.
Execution: build separate expressions. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
32. work rates
Case trigger: workers join or leave during task.
Execution: split time intervals by active rate. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
33. machine rates
Case trigger: machines operate for different durations.
Execution: separate contribution intervals. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
34. fill/drain
Case trigger: inflow and outflow combine differently depending active valves.
Execution: split operating states. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
35. mixture before/after addition
Case trigger: concentration base changes after addition.
Execution: separate states. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
36. population before/after event
Case trigger: growth rule changes after migration/intervention.
Execution: split timeline. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
37. simple/compound stage
Case trigger: financial rule changes after a date.
Execution: apply each rule to correct period. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
38. currency conversion direction
Case trigger: buying and selling or quote direction differ.
Execution: choose correct reciprocal case. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
39. graph increasing/decreasing region
Case trigger: relationship changes direction around turning point.
Execution: analyse relevant interval. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
40. quadratic left/right of vertex
Case trigger: same y can correspond to two x values.
Execution: retain both until domain/context selects. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
41. line intersection
Case trigger: two objects meet at one or more valid points.
Execution: test coordinates against both equations. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
42. circle intersection
Case trigger: geometry can yield zero, one or two intersections.
Execution: case count follows relative positions. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
43. triangle existence
Case trigger: side lengths may or may not satisfy inequality.
Execution: test feasibility before angle work. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
44. acute/right/obtuse
Case trigger: angle category changes geometric properties.
Execution: classify from data rather than appearance. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
45. similar/congruent
Case trigger: same shape versus same size and shape.
Execution: choose relation from evidence. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
46. reflection/rotation
Case trigger: same image can look similar under different transformations.
Execution: test invariant and centre/line conditions. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
47. direct/inverse proportion
Case trigger: surface wording may fit different models.
Execution: test constant ratio versus constant product. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
48. linear/nonlinear
Case trigger: two points alone do not prove linear relation.
Execution: use stated model or more evidence. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
49. interpolation/extrapolation
Case trigger: target lies inside or outside data range.
Execution: case changes reliability of prediction. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
50. with replacement/without replacement
Case trigger: probability denominator changes differently.
Execution: identify sampling case before tree calculation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
51. independent/dependent events
Case trigger: later probability may or may not change.
Execution: use event structure. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
52. mutually exclusive/not exclusive
Case trigger: union formula changes overlap term.
Execution: test whether intersection can occur. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
53. ordered/unordered selection
Case trigger: arrangements and groups count differently.
Execution: decide whether order creates distinct outcome. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
54. repetition allowed/not allowed
Case trigger: counting universe changes.
Execution: apply condition before enumeration. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
55. at least one
Case trigger: complement case may be simpler.
Execution: compare direct case split with complement. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
56. exactly one
Case trigger: separate mutually exclusive component cases.
Execution: sum valid branches. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
57. one or both
Case trigger: include overlap correctly.
Execution: avoid double counting. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
58. success/failure branches
Case trigger: each trial creates cases.
Execution: keep path ownership clear. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
59. equal/unequal class widths
Case trigger: histogram interpretation changes.
Execution: use frequency density only when appropriate. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
60. grouped/ungrouped data
Case trigger: statistics may be exact or estimated.
Execution: identify data representation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
61. complete/incomplete table
Case trigger: missing frequency or total creates different unknown.
Execution: write governing total relation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
62. same/different denominator
Case trigger: fraction comparison method changes.
Execution: use common denominator or cross-product appropriately. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
63. positive/negative multiplier inequality
Case trigger: inequality direction changes only for negative factor.
Execution: split sign if multiplier sign is unknown. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
64. parameter positive/negative
Case trigger: formula behaviour depends on parameter sign.
Execution: do not assume sign without evidence. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
65. zero special case
Case trigger: division or cancellation may fail at zero.
Execution: separate zero before dividing by variable expression. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
66. factor zero
Case trigger: product equals zero creates alternative cases.
Execution: set each factor to zero. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
67. square equality
Case trigger: x²=a can create ± roots for positive a.
Execution: preserve both algebraic cases. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
68. absolute value equation
Case trigger: |x|=a gives two cases for positive a.
Execution: handle zero and negative a separately. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
69. absolute value inequality
Case trigger: inside/outside interval cases differ.
Execution: translate carefully. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
70. piecewise graph
Case trigger: formula changes by x interval.
Execution: evaluate target with correct branch. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
71. rounding boundary
Case trigger: displayed value represents interval.
Execution: split endpoint inclusion carefully. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
72. upper/lower bound product
Case trigger: extreme product uses different endpoint combinations.
Execution: positive assumption matters. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
73. quotient bounds
Case trigger: largest quotient uses high numerator/low denominator for positive quantities.
Execution: split sign cases if negatives possible. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
74. ratio feasibility
Case trigger: part values must satisfy totals and integer constraints.
Execution: filter algebraic candidates. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
75. geometry orientation
Case trigger: diagram can have mirror configurations.
Execution: check whether both satisfy given constraints. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
76. bearing direction
Case trigger: clockwise angle from north depends on direction.
Execution: distinguish reciprocal bearing. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
77. coordinate quadrant
Case trigger: signs depend on quadrant.
Execution: use location constraints. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
78. slope vertical line
Case trigger: gradient undefined rather than zero.
Execution: separate vertical case. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
79. horizontal line
Case trigger: gradient zero.
Execution: do not confuse with undefined. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
80. parallel/perpendicular
Case trigger: relationship test differs.
Execution: use equal gradients versus negative reciprocal where applicable. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
81. same point/different point
Case trigger: distance zero is special case.
Execution: check before direction calculations. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
82. rate zero
Case trigger: stationary interval changes average speed calculation.
Execution: include time even with zero distance. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
83. probability zero/one
Case trigger: impossible/certain events are boundary cases.
Execution: use range meaning. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
84. percentage 0/100
Case trigger: none/all are boundary cases.
Execution: interpret exactly. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
85. scale factor zero/negative
Case trigger: ordinary enlargement conventions may restrict factor.
Execution: follow syllabus/context definition. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
86. answer options
Case trigger: MCQ choices represent different cases.
Execution: use constraints to eliminate whole cases before exact calculation. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
87. case exhaustion
Case trigger: several cases are possible.
Execution: prove you covered all legitimate cases without duplication. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
88. case overlap
Case trigger: two branches describe same outcome.
Execution: avoid double counting. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
89. case impossibility
Case trigger: a branch violates a condition.
Execution: discard with explicit reason. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
90. case symmetry
Case trigger: two cases are equivalent by symmetry.
Execution: solve one and map the other when justified. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
91. case priority
Case trigger: one branch is much easier to test.
Execution: eliminate impossible branch first. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
92. case final rule
Case trigger: split only where a condition changes the mathematics.
Execution: then solve, filter and recombine without missing or double-counting cases. For practice, write the branch condition before the branch calculation. At the end, test every candidate against the original question rather than assuming algebraic validity means contextual validity.
A clean case solution has three visible layers: condition → method → filter. If two branches overlap, remove double counting. If a branch is impossible, state the condition it violates and close it.
When not to split
Do not create cases merely because the context has several nouns or stages. If one invariant formula applies throughout, a single model is cleaner. Case splitting is justified only when a sign, interval, rule, denominator, event structure or feasibility condition changes the operation.
Links
Use the Mathematics Hub, Vol 0086 Bounding Checks, Vol 0070 Dependency Graphs, the Examination Craft hub and the PSLE Learner’s Guide.
Official-source discipline
For the current 2027 SEC G2 school-candidate framework, use the official SEAB G2 syllabus directory and linked K210 Mathematics syllabus. Case splitting is an eduKateSengkang reasoning framework, not an additional SEAB syllabus topic.
Final rule
Split only when the condition changes the mathematics. Name each branch, solve it under its own rule, reject impossible cases and make sure the surviving branches cover the problem exactly once.
