G2 Mathematics K210 working becomes more reliable when every symbol, coefficient, sign and denominator has a clear origin. This is term provenance: knowing exactly where each part of an equation came from, what quantity it represents and which condition gives it the right to appear.
This one-hundred-and-seventieth Learner’s Guide develops term-provenance control across algebra, rates, ratios, percentages, geometry, probability, graphs and data. It is distinct from ordinary equation formation because the focus is auditability: every term should be traceable back to the problem or to a valid derived quantity.
Mechanism: equations are compressed ownership maps
An equation is not just numbers and symbols. Each term carries ownership: quantity, unit, sign, condition and source. When provenance is preserved, equations are easier to verify and repair. When provenance is lost, a plausible-looking formula can hide a wrong denominator, swapped variable or invented constant.
Diagnosis
If a calculation goes wrong, ask each term five questions: what quantity is this, where was it given/derived, what unit should it carry, why is its sign correct, and under what condition is it valid? The first term that cannot answer those questions is the main diagnostic target.
Smallest repair
Relabel the broken term, correct its ownership or replace the invalid relation, then propagate forward only through expressions that depend on it. Do not rebuild independent terms whose provenance remains sound.
1. Variable definition
Provenance: Every symbol should point to a named quantity.
Control: Write the noun and unit beside the symbol before forming equations.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
2. Constant term
Provenance: A fixed fee, starting value or intercept must come from a stated or derived fixed quantity.
Control: Do not let a remembered formula invent it.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
3. Coefficient
Provenance: A multiplier should trace to a rate, scale, count or relationship.
Control: Name what one unit of the coefficient means.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
4. Numerator
Provenance: Trace it to the quantity being counted or accumulated.
Control: Check ownership before division.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
5. Denominator
Provenance: Trace it to the base of comparison, time, area, volume, attempts or original value.
Control: Most rate/percentage errors are provenance errors.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
6. Positive sign
Provenance: A plus sign should represent addition, inflow, gain or same-direction contribution.
Control: Do not use + merely because numbers are combined.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
7. Negative sign
Provenance: A minus sign should represent subtraction, loss, opposite direction or removal.
Control: Attach the sign to a physical/mathematical meaning.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
8. Equality sign
Provenance: Both sides must represent the same quantity under the model.
Control: Do not use ‘=’ as ‘the next step is’.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
9. Inequality sign
Provenance: Trace >, <, ≥ or ≤ to the wording: more than, less than, at least, at most.
Control: Endpoint inclusion is part of provenance.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
10. Fraction bar
Provenance: The denominator defines what the numerator is relative to.
Control: Translate the fraction into words.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
11. Percentage multiplier
Provenance: 1+r or 1−r must trace to an increase/decrease rate.
Control: Keep r in decimal form and tied to the original base.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
12. Scale factor
Provenance: Trace to corresponding lengths under similarity.
Control: Do not import it into areas/volumes unchanged.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
13. Area factor
Provenance: Trace to square of length scale where similarity applies.
Control: Verify dimension.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
14. Volume factor
Provenance: Trace to cube of length scale where similarity applies.
Control: Verify dimension and context.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
15. Gradient numerator
Provenance: Rise/change in y must come from the y-axis quantity.
Control: Do not swap axes.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
16. Gradient denominator
Provenance: Run/change in x must come from the x-axis quantity.
Control: Units reveal orientation.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
17. Intercept
Provenance: Trace to the predicted value when x=0 under the model.
Control: Check whether x=0 is meaningful in context.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
18. Mean numerator
Provenance: Total of values or weighted contributions.
Control: Do not divide a count total by the wrong denominator.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
19. Mean denominator
Provenance: Number of observations or total weight.
Control: Trace whether frequencies/weights are involved.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
20. Weighted term
Provenance: Value×weight/frequency must keep the correct pairing.
Control: A row mismatch creates a hidden provenance error.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
21. Probability numerator
Provenance: Count or probability of favourable event.
Control: Ensure event matches the question.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
22. Probability denominator
Provenance: Total equally likely outcomes or total reference probability under the model.
Control: Define the universe.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
23. Complement term
Provenance: 1−P(A) uses 1 as the complete probability total.
Control: Do not use complement unless events exhaust the universe.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
24. Tree branch
Provenance: Each factor should trace to the condition active on that branch.
Control: Conditional probabilities need branch ownership.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
25. Sequence first term
Provenance: Trace to position n=1 or the stated initial position.
Control: Do not confuse with first difference.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
26. Sequence difference
Provenance: Trace to equal additive change between consecutive terms.
Control: Check number of intervals when using spaced terms.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
27. Sequence ratio
Provenance: Trace to multiplicative change between consecutive terms.
Control: Check sign and zero issues.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
28. Speed numerator
Provenance: Distance travelled.
Control: Attach distance unit.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
29. Speed denominator
Provenance: Time taken.
Control: Attach time unit and period.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
30. Density numerator
Provenance: Mass.
Control: Do not swap with volume.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
31. Density denominator
Provenance: Volume.
Control: Compound unit is a provenance check.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
32. Population density numerator
Provenance: Population count.
Control: Ensure group matches target population.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
33. Population density denominator
Provenance: Area.
Control: Keep spatial unit.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
34. Unit price numerator
Provenance: Cost.
Control: Trace currency ownership.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
35. Unit price denominator
Provenance: Number of items/units.
Control: Do not confuse with items per dollar.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
36. Fuel efficiency numerator
Provenance: Distance.
Control: Keep route/condition consistent.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
37. Fuel efficiency denominator
Provenance: Fuel amount.
Control: Direction differs from consumption.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
38. Flow-rate numerator
Provenance: Volume or amount transferred.
Control: Trace to the moving quantity.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
39. Flow-rate denominator
Provenance: Time interval.
Control: Do not use total experiment duration if flow only occurs part of the time.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
40. Work-rate numerator
Provenance: Task/work amount.
Control: Define one whole job if used.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
41. Work-rate denominator
Provenance: Time.
Control: Combined-worker equations depend on compatible work units.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
42. Currency conversion factor
Provenance: Trace to quote direction and units.
Control: Write currencies as a fraction to avoid inversion.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
43. Tax term
Provenance: Trace to taxable base, not automatically final price.
Control: Base definition matters.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
44. Discount term
Provenance: Trace to marked/original price.
Control: Do not apply discount to already-discounted base unless stated.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
45. Profit term
Provenance: Trace to revenue−cost.
Control: Do not confuse revenue with profit.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
46. Margin denominator
Provenance: Revenue/selling price.
Control: Distinct from markup denominator.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
47. Markup denominator
Provenance: Cost.
Control: Distinct from margin.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
48. Capacity term
Provenance: Trace to maximum allowed amount.
Control: Used and remaining should reconstruct capacity.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
49. Occupancy term
Provenance: Trace to occupied units and available units.
Control: A percentage without these labels is fragile.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
50. Bound endpoint
Provenance: Trace to rounding precision or measurement interval.
Control: Upper/lower bounds are not arbitrary.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
51. Case formula
Provenance: Trace to the condition that activates it.
Control: A formula without its branch label loses provenance.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
52. Absolute value branch
Provenance: Trace to sign/interval condition.
Control: The two cases must cover all allowed values.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
53. Root
Provenance: Trace to the equation and domain that produced it.
Control: A candidate root is not yet a valid problem answer.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
54. Geometry length
Provenance: Trace to a labelled segment, not visual appearance.
Control: Diagram ownership matters.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
55. Angle
Provenance: Trace to the exact vertex/order or theorem.
Control: Avoid swapping angles with similar appearance.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
56. Corresponding side
Provenance: Trace vertex-to-vertex mapping before ratios.
Control: Similarity errors often start here.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
57. Table entry
Provenance: Trace row, column, heading and unit.
Control: Copying the right number from the wrong cell is still wrong.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
58. Graph point
Provenance: Trace series, x-coordinate, y-coordinate and scale.
Control: Each coordinate has provenance.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
59. Derived intermediate
Provenance: Write a short label for what the number means.
Control: Unlabelled intermediate values are easy to reuse incorrectly.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
60. Calculator memory
Provenance: A stored value should retain its meaning and unit.
Control: Do not let memory registers become anonymous numbers.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
61. Equation transformation
Provenance: Each new term should descend from an earlier term through a legal operation.
Control: If a term appears from nowhere, provenance is broken.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
62. Cancellation
Provenance: The factors cancelled must actually be common multiplicative factors.
Control: Do not cancel terms across addition.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
63. Cross multiplication
Provenance: Trace each product back to equality of ratios/fractions.
Control: Check denominators nonzero.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
64. Substitution
Provenance: Replacement value must belong to the same variable.
Control: Name the source equation/value.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
65. Rearrangement
Provenance: The target variable changes position but meaning/units remain.
Control: Use dimensional consistency as a check.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
66. Decomposition
Provenance: Each component term should represent one part of the whole.
Control: Recombine and verify total.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
67. Recombination
Provenance: Signs and weights must match how components enter the whole.
Control: A correct component can be mis-owned at recombination.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
68. Term provenance final rule
Provenance: Every symbol, coefficient, sign and denominator should answer the question ‘where did this come from and what does it mean?’
Control: If the answer is unclear, stop before calculating further.
For practice, annotate one equation by writing a short noun phrase under each term. Then remove the labels and ask the learner to reconstruct them. If a term cannot be named in words, its role is not yet secure.
For transfer, change the context and variable letters while preserving the same mathematical relationship. The learner should recover term meaning from units and structure rather than memorise the old symbols.
Competing explanations for a wrong answer
The same wrong result can come from arithmetic, model choice or provenance failure. A dimensional check can expose swapped numerator/denominator; reconstruction can expose wrong ownership; substitution can expose a copied intermediate. Use the check that targets the earliest uncertain term.
Observation versus inference
Given quantities and labelled data are observed task information. The equation assembled from them is a model inference. Keeping provenance explicit prevents the model from silently adding conditions the question never gave.
Support versus proof
A term having the right unit supports its plausibility but does not prove its ownership. Two different quantities can share units. Use semantic meaning, source and context as well as dimensions.
Model limits
Some problems allow several valid formulations. Provenance is not about forcing one canonical equation; it is about ensuring that whichever formulation is used, every term remains interpretable and conditionally valid.
Delayed transfer
Return later with unfamiliar word problems and deliberately anonymous algebra. Ask the learner to label each term, identify one invented or mis-owned term, repair it and verify the corrected equation on another case.
Internal learning links
Use the Mathematics Hub, Vol 0162 Parameter Recovery, Vol 0078 Dimensional Consistency, 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
Before trusting an equation, make every term answer: where did I come from, what do I mean, what unit do I carry, and under what condition am I valid? Anonymous terms are where hidden errors survive.
G2 SEC Learner’s Guide: Open the Vol 0132–0175 hub and subject index.
