G2 Mathematics K210 often changes behaviour at a boundary. A price rule changes after a quota; an integer answer jumps at a threshold; an inequality changes feasibility at an endpoint; a graph switches formula at a breakpoint. Near these boundaries, ordinary smooth intuition can fail.
This one-hundred-and-twenty-second Learner’s Guide develops breakpoint reasoning. It extends Vol 0098 Case Splitting and Vol 0094 Sensitivity Checks.
The breakpoint protocol
Find the value where the rule changes. Test just below, exactly at and just above it. Record which formula, inequality, rounding rule or feasibility condition applies in each region. Endpoint ownership is part of the mathematics.
1. free shipping threshold
For free shipping threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
2. bulk discount threshold
For bulk discount threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
3. tax bracket boundary
For tax bracket boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
4. ticket age boundary
For ticket age boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
5. child/adult fare boundary
For child/adult fare boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
6. minimum order
For minimum order, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
7. maximum capacity
For maximum capacity, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
8. container ceiling
For container ceiling, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
9. group floor
For group floor, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
10. rounding half-unit
For rounding half-unit, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
11. integer crossing
For integer crossing, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
12. percentage eligibility
For percentage eligibility, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
13. interest tier
For interest tier, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
14. rate band
For rate band, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
15. peak/off-peak time
For peak/off-peak time, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
16. distance zone
For distance zone, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
17. first block/next block
For first block/next block, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
18. data-plan quota
For data-plan quota, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
19. overtime threshold
For overtime threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
20. commission tier
For commission tier, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
21. bonus threshold
For bonus threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
22. pass/fail cutoff in abstract practice
For pass/fail cutoff in abstract practice, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
23. graph piecewise boundary
For graph piecewise boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
24. absolute-value zero point
For absolute-value zero point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
25. denominator zero
For denominator zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
26. square-root domain edge
For square-root domain edge, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
27. probability zero
For probability zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
28. probability one
For probability one, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
29. triangle inequality boundary
For triangle inequality boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
30. degenerate geometry boundary
For degenerate geometry boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
31. acute/right/obtuse boundary
For acute/right/obtuse boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
32. parallel/perpendicular condition
For parallel/perpendicular condition, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
33. circle tangent boundary
For circle tangent boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
34. line intersection count boundary
For line intersection count boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
35. quadratic discriminating visual boundary where appropriate
For quadratic discriminating visual boundary where appropriate, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
36. sequence sign change
For sequence sign change, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
37. ratio equal-one boundary
For ratio equal-one boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
38. growth/decline multiplier one
For growth/decline multiplier one, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
39. profit/loss zero
For profit/loss zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
40. positive/negative change zero
For positive/negative change zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
41. above/below baseline zero
For above/below baseline zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
42. utilisation 100%
For utilisation 100%, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
43. discount 100% theoretical boundary
For discount 100% theoretical boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
44. percentage share 0/100
For percentage share 0/100, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
45. frequency class boundary
For frequency class boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
46. histogram class endpoint
For histogram class endpoint, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
47. median position boundary
For median position boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
48. quartile-style boundary where context permits
For quartile-style boundary where context permits, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
49. measurement rounding boundary
For measurement rounding boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
50. lower bound inclusion
For lower bound inclusion, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
51. upper bound exclusion
For upper bound exclusion, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
52. strict inequality
For strict inequality, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
53. inclusive inequality
For inclusive inequality, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
54. open interval
For open interval, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
55. closed interval
For closed interval, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
56. case-switch point
For case-switch point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
57. piecewise formula
For piecewise formula, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
58. saturation threshold
For saturation threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
59. capacity overflow
For capacity overflow, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
60. budget exceedance
For budget exceedance, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
61. minimum feasible integer
For minimum feasible integer, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
62. maximum feasible integer
For maximum feasible integer, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
63. break-even point
For break-even point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
64. equal-cost point
For equal-cost point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
65. equal-time point
For equal-time point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
66. equal-speed comparison
For equal-speed comparison, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
67. equal-rate point
For equal-rate point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
68. same-density boundary
For same-density boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
69. same-mean comparison
For same-mean comparison, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
70. same-probability point
For same-probability point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
71. MCQ option boundary
For MCQ option boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
72. estimate crossing
For estimate crossing, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
73. unit-conversion threshold
For unit-conversion threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
74. currency affordability threshold
For currency affordability threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
75. packing leftover threshold
For packing leftover threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
76. work-completion threshold
For work-completion threshold, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
77. flow fill point
For flow fill point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
78. runway zero point
For runway zero point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
79. coordinate quadrant axis
For coordinate quadrant axis, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
80. gradient sign zero
For gradient sign zero, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
81. turning point
For turning point, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
82. intercept crossing
For intercept crossing, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
83. root crossing
For root crossing, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
84. translation boundary
For translation boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
85. scale factor one
For scale factor one, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
86. probability complement half
For probability complement half, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
87. ratio half/one benchmarks
For ratio half/one benchmarks, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
88. percentage 50% benchmark
For percentage 50% benchmark, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
89. angle 90 benchmark
For angle 90 benchmark, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
90. angle 180 boundary
For angle 180 boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
91. full-turn 360 boundary
For full-turn 360 boundary, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
92. breakpoint sensitivity
For breakpoint sensitivity, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
93. breakpoint case split
For breakpoint case split, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
94. breakpoint verification
For breakpoint verification, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
95. breakpoint final rule
For breakpoint final rule, identify the breakpoint and write three test cases: one below, one at, one above. Explain whether the output changes continuously, jumps, changes formula or merely changes interpretation.
For practice, draw a number line or small table showing region → rule. This prevents learners from applying one familiar formula across a boundary where the conditions have changed.
Verification should target the boundary explicitly. If equality is allowed, test the endpoint in the inclusive case; if it is excluded, make that visible. A correct interior solution can still fail at the breakpoint.
Breakpoints explain abrupt answer changes
Integer ceilings, floors, eligibility rules and piecewise prices can jump even when the input changes only slightly. This is not an arithmetic error. It is the expected behaviour of a thresholded model.
Links
Use the Mathematics Hub, Vol 0118 Conservation and Balance Checks, Vol 0086 Bounding Checks, 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. Breakpoint reasoning is an eduKateSengkang problem-solving framework, not an additional SEAB syllabus topic.
Final rule
Whenever a rule has a threshold, test below, at and above. Make endpoint inclusion explicit. The mathematics on one side of a breakpoint is not automatically the mathematics on the other.
