G2 Mathematics K210 problem solving becomes clearer when learners distinguish independent information from redundant information. Two equations can look different yet say the same thing. Several totals can be derived from one another. More data on the page does not always mean more constraints on the answer.
This one-hundred-and-sixty-sixth Learner’s Guide develops independent-versus-redundant constraint reasoning. It is especially useful for algebra, ratio, data, geometry, rates and word problems where learners need to know whether the available information is sufficient, excessive, contradictory or merely repetitive.
Mechanism: independent information shrinks the solution set
A new condition adds information when it removes possibilities that were still allowed before. If every solution satisfying the old conditions automatically satisfies the new one, the new condition is redundant. This is the practical idea of independence without requiring advanced linear algebra.
Diagnosis
Count the unknowns, translate each condition, and simplify. Ask of every new condition: does it rule out any solution that survived the previous conditions? If not, use it as a check rather than pretending it supplies a new equation.
Smallest repair
If a system is underdetermined, do not invent a condition. Look for a missed independent fact in the question or state that more information is required in the reasoning framework. If a condition is redundant, keep it for verification. If conditions conflict, locate the transcription, rounding, ownership or model error.
1. Same equation twice
Information structure: Two written equations are identical after simplification.
Use: They provide one independent condition, not two.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
2. Scaled duplicate equation
Information structure: One equation is a constant multiple of another.
Use: No new solution information is added.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
3. Reworded duplicate relation
Information structure: Two sentences describe the same mathematical relationship.
Use: Translate both before counting them as separate constraints.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
4. Independent linear conditions
Information structure: Two genuinely different relations can determine two unknowns.
Use: Check they are not multiples/rephrasings.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
5. Too few conditions
Information structure: Two unknowns with one independent equation usually leave a family of solutions.
Use: Do not invent a second assumption.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
6. Extra consistent condition
Information structure: A third condition agrees with the solution from two independent conditions.
Use: Use it as verification rather than another unknown-maker.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
7. Extra inconsistent condition
Information structure: A third condition conflicts with the others.
Use: Diagnose data error, rounding, model mismatch or wrong ownership.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
8. Total plus part
Information structure: Knowing total and one part independently determines the other.
Use: The two pieces are not redundant because each constrains a different component.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
9. Two parts plus total
Information structure: If total equals known sum of the two parts, one relation may be a consequence of the other two.
Use: Identify which condition is verification rather than new information.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
10. Ratio plus total
Information structure: A ratio and total are independent enough to determine parts.
Use: Check both are needed; ratio alone fixes only relative size.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
11. Percentage plus total
Information structure: Percentage and whole determine the part.
Use: The part value then becomes derived, not independent.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
12. Part plus percentage
Information structure: Can determine the whole if base is clear.
Use: Do not count the reconstructed whole as a new condition later.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
13. Mean and total
Information structure: Mean×count=total means mean and total are linked once count is known.
Use: Do not treat all three as independent.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
14. Mean and count
Information structure: Together determine total.
Use: Total is derived.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
15. Frequency sum
Information structure: Category frequencies sum to sample size.
Use: If sample size is already stated, the sum is a consistency check.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
16. Relative frequencies
Information structure: Complete category proportions sum to 1.
Use: One proportion may be derived from the others.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
17. Probability complement
Information structure: P(A)+P(not A)=1.
Use: Knowing one determines the other; they are not independent.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
18. Two-way table row total
Information structure: Row entries determine row total.
Use: If both entries known, total is derived.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
19. Two-way table column total
Information structure: Likewise for columns.
Use: Avoid double-counting totals as extra independent data.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
20. Grand total
Information structure: Can be derived from row totals or column totals.
Use: Use to check table consistency.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
21. Triangle angle sum
Information structure: Two angles determine the third.
Use: The 180° total is a rule, not an extra measured condition.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
22. Quadrilateral angles
Information structure: Three interior angles determine the fourth.
Use: Again, total is structural.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
23. Isosceles triangle
Information structure: Equal sides imply equal base angles.
Use: Once one relation is used, the mirrored angle may be derived.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
24. Parallel lines
Information structure: Angle relations may create several equal/supplementary statements from one geometric condition.
Use: Do not count every angle equality as independent.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
25. Similarity
Information structure: Corresponding side ratios are all equal.
Use: Several ratio equations may encode the same scale factor.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
26. Scale factor
Information structure: One corresponding side pair can recover scale factor if similarity is already established.
Use: A second pair verifies the factor.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
27. Pythagoras condition
Information structure: Right angle plus two sides can determine third.
Use: The theorem is a relation, not another independent measurement.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
28. Coordinate midpoint
Information structure: Midpoint coordinates are derived from endpoints.
Use: Do not count them as independent if computed from same points.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
29. Gradient
Information structure: Gradient from two points is derived.
Use: Using the same two points to claim an extra independent line condition is redundant.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
30. Line equation
Information structure: One point plus gradient determines a line.
Use: A second point on the same line can verify.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
31. Two points
Information structure: Two distinct points determine a straight line.
Use: Gradient and intercept recovered from them are derived parameters.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
32. Graph intersection
Information structure: Two curves/lines provide independent relations when their equations differ.
Use: Intersection satisfies both.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
33. Direct proportion
Information structure: One nonzero point can recover constant if direct proportionality is given.
Use: Additional points test the model.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
34. Inverse proportion
Information structure: One nonzero pair can recover constant if inverse proportionality is given.
Use: Additional pairs verify product constancy.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
35. Arithmetic sequence
Information structure: First term and common difference determine all terms.
Use: Later terms are derived conditions unless used to recover/check parameters.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
36. Geometric sequence
Information structure: First term and common ratio determine the sequence.
Use: Later terms are consequences under the model.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
37. Two sequence terms
Information structure: May determine parameters if positions are known, but information content depends on model.
Use: Count unknowns and independent relations.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
38. Currency conversion pair
Information structure: One exact conversion rate determines reverse direction by reciprocal.
Use: Forward and backward rates are linked, not independent.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
39. Unit conversion
Information structure: Centimetres and metres describe same length.
Use: Two unit values are representations of one quantity, not two measurements.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
40. Speed-distance-time
Information structure: Any two determine the third.
Use: The third is derived from the relationship.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
41. Density-mass-volume
Information structure: Any two determine the third under the model.
Use: Do not count all three as independent if one was calculated.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
42. Flow-rate amount-time
Information structure: Any two determine the third.
Use: Track which were measured/given versus derived.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
43. Work-rate task-time
Information structure: Same structure.
Use: Derived quantities are not new evidence.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
44. Percentage multiplier
Information structure: Final=original×multiplier links three quantities.
Use: Once two known, the third follows.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
45. Profit relation
Information structure: Profit=revenue−cost.
Use: Knowing all three may provide a consistency check, not three independent facts.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
46. Capacity used+remaining
Information structure: Used and remaining sum to capacity.
Use: One component may be derived.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
47. Budget categories
Information structure: If categories exhaust budget, final category can be residual.
Use: Avoid treating residual as separately observed.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
48. Case split exhaustive
Information structure: Two cases can be independent branches but together exhaust the domain.
Use: A branch condition is not redundant if it changes the formula.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
49. Overlapping cases
Information structure: Two cases partly cover same values.
Use: Information structure is flawed because branches are not independent/exhaustive.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
50. Symmetric cases
Information structure: Cases may be equivalent by symmetry.
Use: Solve one and map the other instead of counting two independent difficulties.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
51. MCQ elimination constraints
Information structure: Units and bounds can be independent filters.
Use: If both eliminate the same option for the same underlying reason, information gain may be partly redundant.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
52. Bound plus exact value
Information structure: An exact value automatically satisfies a correct bound if inside it.
Use: The bound then serves as verification.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
53. Round-trip conversion
Information structure: Forward and reverse conversions are linked.
Use: Agreement checks execution; it does not create a second independent physical fact.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
54. Residual check
Information structure: Observed−predicted=0 when model fits exactly.
Use: Residual is derived from observed and predicted values.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
55. Normalisation
Information structure: Normalised rate is derived from raw count and base.
Use: Do not treat raw and normalised versions as independent measurements.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
56. Benchmark anchor
Information structure: A benchmark constrains plausibility but may not narrow to one exact solution.
Use: It is weaker information than an independent exact equation.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
57. Measurement repeated identically
Information structure: Repeat trials can improve confidence/precision but do not create a new model relation.
Use: Distinguish repeated evidence from independent constraints.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
58. Different method same quantity
Information structure: A genuinely independent measurement route can add verification value.
Use: Independence is about failure modes as well as algebra.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
59. Independent representation
Information structure: Graph and equation from the same data are not fully independent evidence.
Use: They are different views of one dataset.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
60. Independent dataset
Information structure: A second dataset under comparable conditions can test transfer.
Use: This adds stronger external verification.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
61. Hidden assumption as fake condition
Information structure: Learner invents equality/constancy to get enough equations.
Use: Do not count assumptions as givens.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
62. Parameter count
Information structure: Count free unknowns before solving.
Use: You generally need enough independent constraints to determine them.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
63. Constraint rank intuition
Information structure: Ask whether the new condition removes any solutions still allowed by earlier conditions.
Use: If it removes none, it is redundant.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
64. Redundancy as check
Information structure: Redundant information is not useless.
Use: It can verify consistency and expose transcription errors.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
65. Independence final rule
Information structure: A condition is independent when it genuinely narrows the remaining solution set.
Use: Do not count equations by appearance; count new information.
For practice, list the candidate solutions allowed before the condition and after it. If the set does not shrink, label the condition redundant. If it shrinks, explain what new restriction was added.
For transfer, change the numbers or context while preserving the same information structure. The learner should identify independence from relationships rather than from how many equations are printed.
Competing explanations for inconsistency
When conditions disagree, possible explanations include arithmetic error, copied data, rounding, wrong unit, wrong ownership, an invalid assumption or a model that should not be constant across all cases. Use the simplest discriminating check before discarding data.
Observation versus inference
The printed equations, totals and data points are observations from the task. The claim that they are independent or redundant is an inference about their relationships. Simplification and substitution make that inference testable.
Support versus proof
A second condition that agrees with the first supports consistency but may not add independence. Agreement is valuable, but it should not be mistaken for extra degrees of freedom in solving.
Model limits
This framework assumes the conditions belong to the same mathematical model and context. If the rule changes at a breakpoint, different intervals can require different parameters. What looks contradictory under one model may be coherent under a piecewise model explicitly supported by the task.
Delayed transfer
Return later with mixed algebra, geometry, probability and data questions. Ask the learner to count unknowns, identify independent conditions, mark redundant checks, diagnose inconsistency and state what extra information would be needed if the problem is underdetermined.
Internal learning links
Use the Mathematics Hub, Vol 0162 Parameter Recovery, Vol 0118 Conservation and Balance Checks, Examination Craft and the PSLE Learner’s Guide.
MOE and SEAB current framework
MOE states that Full Subject-Based Banding is fully implemented and that, from 2027, the Singapore-Cambridge Secondary Education Certificate (SEC) replaces the former N- and O-Level examinations, with graduating students sitting subjects at their respective G1, G2 or G3 levels. See the official MOE Full SBB / SEC announcement. For current 2027 G2 Mathematics subject code K210 and syllabus links, use the official SEAB G2 syllabus directory.
Final rule
Do not count equations; count new information. Independent conditions narrow the solution set. Redundant conditions verify it. Contradictory conditions demand diagnosis before calculation continues.
G2 SEC Learner’s Guide: Open the Vol 0132–0175 hub and subject index.
