G2 Mathematics K210 reasoning often forms chains. If A is greater than B and B is greater than C, then A must be greater than C. If two expressions equal the same quantity, they equal each other. These transitive structures can solve problems and expose contradictions quickly.
This Learner’s Guide is designed to upgrade capability rather than merely add facts. It makes the mechanism explicit, shows how to diagnose failure, defines the smallest useful repair, keeps competing explanations visible, states model limits, and ends in transfer to a changed problem surface.
Mechanism
Identify the relation that should remain true and the condition that controls how the answer changes. Write the causal, logical or quantitative handoff explicitly enough that another learner could predict the next state before seeing the result.
Diagnosis and smallest repair
When the answer fails, locate the first broken handoff. Repair that node only: source, role, denominator, sign, condition, mechanism, boundary or measurement. Recheck one downstream consequence before changing anything else.
1. A>B and B>C
For A>B and B>C, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
2. A<B and B<C
For A<B and B<C, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
3. A=B and B=C
For A=B and B=C, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
4. same total
For same total, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
5. same ratio
For same ratio, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
6. same unit rate
For same unit rate, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
7. same percentage base
For same percentage base, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
8. same gradient
For same gradient, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
9. same midpoint coordinate
For same midpoint coordinate, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
10. same distance
For same distance, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
11. same angle
For same angle, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
12. same length
For same length, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
13. same area
For same area, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
14. same probability
For same probability, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
15. same mean
For same mean, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
16. same sequence term
For same sequence term, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
17. same formula output
For same formula output, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
18. equal expressions
For equal expressions, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
19. equivalent fractions
For equivalent fractions, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
20. equivalent ratios
For equivalent ratios, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
21. equivalent percentages
For equivalent percentages, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
22. unit conversions
For unit conversions, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
23. currency conversions
For currency conversions, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
24. ordering values
For ordering values, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
25. ordering fractions
For ordering fractions, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
26. ordering percentages
For ordering percentages, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
27. ordering rates
For ordering rates, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
28. ordering densities
For ordering densities, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
29. ordering costs
For ordering costs, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
30. ordering times
For ordering times, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
31. ordering speeds
For ordering speeds, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
32. ordering probabilities
For ordering probabilities, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
33. comparison chain
For comparison chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
34. equality chain
For equality chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
35. substitution chain
For substitution chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
36. dependency chain
For dependency chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
37. contradiction cycle
For contradiction cycle, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
38. A>B but C>A with B>C inconsistency
For A>B but C>A with B>C inconsistency, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
39. strict versus non-strict inequality
For strict versus non-strict inequality, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
40. greater-or-equal chain
For greater-or-equal chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
41. equality inside inequality chain
For equality inside inequality chain, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
42. sign reversal caution
For sign reversal caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
43. reciprocal reverses order for positive values
For reciprocal reverses order for positive values, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
44. negative multiplication reverses inequality
For negative multiplication reverses inequality, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
45. squaring caution
For squaring caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
46. absolute value caution
For absolute value caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
47. domain caution
For domain caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
48. rounded equality caution
For rounded equality caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
49. approximate equality caution
For approximate equality caution, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
50. transitivity in geometry
For transitivity in geometry, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
51. transitivity in algebra
For transitivity in algebra, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
52. transitivity in data comparison
For transitivity in data comparison, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
53. transitivity MCQ elimination
For transitivity MCQ elimination, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
54. transitivity final rule
For transitivity final rule, state the observation or given information first. Then state the inference, calculation or mechanism separately. This keeps what is seen apart from what is concluded.
Construct a competing explanation or near-miss that fits part of the evidence. Identify one discriminating test, counterexample, changed condition or independent check that the two alternatives predict differently. Support raises confidence; it does not turn a model into unlimited proof.
For transfer, change the surface while preserving the deep relation. Alter names, numbers, source, graph, location, time, boundary or apparatus. Rebuild the reasoning rather than recalling the example, and state where the method would stop being reliable.
Delayed transfer and measurement
Return later to a mixed problem where the framework is not named. Measure success by independent reconstruction: can the learner identify the mechanism, diagnose a near-miss, choose the smallest repair, distinguish observation from inference, test a competing explanation and state the boundary without prompts?
Internal learning links
Use the Mathematics Hub, Vol 0138 Symmetry Reasoning, Vol 0118 Conservation and Balance Checks, the Examination Craft hub and the PSLE Learner’s Guide.
Official-source discipline
For the current 2027 Singapore-Cambridge Secondary Education Certificate G2 school-candidate framework, use the official SEAB G2 syllabus directory and its linked subject syllabus. The current directory lists K200 English Language, K210 Mathematics and K223–K225 Science combinations. This article’s framework is an eduKateSengkang learning method, not an additional SEAB syllabus topic or marking rule.
Final rule
A strong learner can reconstruct the relationship, locate failure, repair minimally, test an alternative and transfer the method to an unseen surface. That capability—not the length of the answer—is the standard.
