G2 Mathematics K210 problem solving becomes safer when units are treated as part of the equation rather than decoration added at the end. A calculator can produce a plausible number from an inverted rate, a missing square, a wrong conversion or a formula applied to the wrong quantity. Units often expose the mistake before the arithmetic is finished.
This seventy-eighth Learner’s Guide develops dimensional consistency: use quantity type and units as an equation sanity check. The method complements Vol 0074 Invariant Hunting and Vol 0055 Error Signatures. The question is simple: if the formula is structurally correct, do the units and quantity types make sense?
The dimensional check
- Name the target quantity before calculating.
- Write or imagine the units of every major term.
- Only add or equate like quantity types.
- Use multiplication and division to predict the final compound unit.
- If the unit is wrong, inspect formula orientation, powers and conversions before trusting the number.
1. speed
Relationship: distance divided by time. The expected result carries km/h, m/s or another length-per-time unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: time divided by distance produces a reciprocal quantity and should not be labelled as speed. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
2. time per distance
Relationship: time divided by distance. The expected result carries h/km or s/m. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: this is meaningful but different from speed; the reciprocal relationship should be explicit. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
3. area
Relationship: length multiplied by length. The expected result carries cm², m² or another square unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: a linear unit at the end signals a missing dimension or wrong formula. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
4. volume
Relationship: three-dimensional length product. The expected result carries cm³, m³ or another cubic unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: square units suggest one dimension has disappeared. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
5. density
Relationship: mass divided by volume. The expected result carries kg/m³, g/cm³ or another mass-per-volume unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: mass-per-area or volume-per-mass answers a different question. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
6. unit price
Relationship: money divided by quantity. The expected result carries $/item or equivalent. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: items per dollar is reciprocal and reverses interpretation. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
7. population density
Relationship: people divided by area. The expected result carries people/km² or people/m². Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: area per person is a different planning measure. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
8. gradient
Relationship: change in vertical quantity divided by change in horizontal quantity. The expected result carries vertical-unit per horizontal-unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: swapped axes create the reciprocal gradient and may reverse meaning. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
9. percentage
Relationship: part or change divided by a reference whole, then scaled by 100. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: remaining physical units indicate numerator and denominator were not comparable. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
10. probability
Relationship: favourable share over total valid outcomes. The expected result carries dimensionless number from 0 to 1. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: a physical unit or value above 1 reveals structural error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
11. mean
Relationship: sum of like quantities divided by count. The expected result carries same unit as the original data. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: squared or reciprocal units show the denominator is wrong. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
12. frequency density
Relationship: frequency divided by class width where applicable. The expected result carries frequency per unit of variable. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using raw frequency alone does not represent density when widths differ. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
13. scale factor
Relationship: corresponding length divided by corresponding length. The expected result carries dimensionless. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: retaining centimetres means unlike quantities were compared or units were not aligned. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
14. similarity area factor
Relationship: area ratio. The expected result carries dimensionless and related to square of length scale factor. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using the linear factor directly for area loses a dimension. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
15. similarity volume factor
Relationship: volume ratio. The expected result carries dimensionless and related to cube of length scale factor. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using linear or square factor for volume gives a dimensional mismatch. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
16. Pythagoras
Relationship: squared length plus squared length equals squared length. The expected result carries all terms carry length squared before square root. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: adding a length to a squared length is structurally invalid. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
17. distance formula
Relationship: square root of sum of squared coordinate differences. The expected result carries length in coordinate unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: omitting the square root leaves squared distance. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
18. perimeter
Relationship: sum of lengths. The expected result carries length. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: square units indicate area and perimeter have been confused. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
19. circle circumference
Relationship: constant times radius or diameter. The expected result carries length. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: square units indicate the area formula has been used. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
20. circle area
Relationship: constant times radius squared. The expected result carries area. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: linear units indicate the radius was not squared dimensionally. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
21. rate of change
Relationship: change in output divided by change in input. The expected result carries output-unit per input-unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the unit should reveal what the gradient means in context. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
22. cost rate
Relationship: money divided by time, distance or item count. The expected result carries money per chosen base. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the denominator must match the word after ‘per’. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
23. work productivity
Relationship: output divided by labour time. The expected result carries items/hour or equivalent. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: hours/item is a reciprocal productivity measure. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
24. fuel efficiency
Relationship: distance divided by fuel or fuel divided by distance depending definition. The expected result carries km/L or L/100 km. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the two conventions are reciprocal in direction and cannot be compared casually. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
25. currency rate
Relationship: one currency divided by another. The expected result carries SGD/USD or USD/SGD. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the quoted direction determines which currency belongs in denominator. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
26. percentage change
Relationship: change divided by original quantity. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: dividing by final quantity answers a different relative comparison. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
27. relative error
Relationship: error divided by reference value where defined. The expected result carries dimensionless fraction or percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using a quantity with different units in denominator is invalid. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
28. probability path
Relationship: product of stage probabilities. The expected result carries dimensionless. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: any physical unit in the final path probability indicates a category error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
29. expected count
Relationship: probability multiplied by number of trials. The expected result carries count. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: leaving the result dimensionless ignores the trial-count factor. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
30. weighted mean
Relationship: weighted total divided by total weight or frequency. The expected result carries same unit as measured values. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: dividing by number of categories changes the represented population. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
31. compound units
Relationship: nested rates such as cost per kilometre per passenger. The expected result carries money/(distance·passenger) or equivalent. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the written unit should mirror the exact sequence of normalisations. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
32. unit conversion before area
Relationship: convert each length consistently before multiplying. The expected result carries area in square target unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: converting a final area with only a linear factor creates a factor-of-scale error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
33. unit conversion before volume
Relationship: convert length dimensions consistently before cubic calculation. The expected result carries volume in cubic target unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using a single linear conversion factor on volume is dimensionally wrong. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
34. hours to seconds in speed
Relationship: time conversion changes only the time unit. The expected result carries length per new time unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: changing distance as well without need may introduce a second error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
35. percentage with mixed units
Relationship: numerator and denominator represent the same quantity in different units. The expected result carries dimensionless after unit alignment. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: taking the ratio before conversion can create a factor-of-100 or factor-of-1000 error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
36. ratio with unlike quantities
Relationship: two parts must represent commensurable quantities if treated as a pure ratio. The expected result carries dimensionless when like quantities are compared. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: mixing dollars and kilograms produces a rate, not an ordinary ratio. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
37. equation addition
Relationship: terms added or subtracted should represent the same quantity type. The expected result carries matching units across terms. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: adding metres to square metres is structurally impossible regardless of numbers. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
38. equation equality
Relationship: both sides of an equation should represent the same quantity. The expected result carries matching overall units. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: mismatched units indicate the equation cannot be correct as written. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
39. formula substitution
Relationship: a symbol must receive a value of the right quantity type. The expected result carries units consistent with the symbol definition. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: putting area into a length variable can produce plausible arithmetic but invalid mathematics. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
40. graph intercept
Relationship: intercept carries the vertical-axis unit. The expected result carries same unit as y-variable. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: treating it as a gradient confuses unit type. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
41. graph slope
Relationship: slope carries vertical-unit per horizontal-unit. The expected result carries compound unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: a unitless slope is only appropriate when the axis quantities make it so. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
42. probability frequency conversion
Relationship: probability multiplied by total trials yields expected frequency. The expected result carries count. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: dividing instead produces probability per trial count, not expected occurrences. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
43. distance-time graph slope
Relationship: distance change over time change. The expected result carries speed unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: time over distance is the reciprocal and changes interpretation. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
44. speed-time graph area
Relationship: speed multiplied by time. The expected result carries distance. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using slope when area is needed produces acceleration rather than distance. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
45. speed-time graph slope
Relationship: speed change over time change. The expected result carries acceleration-type unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using area instead produces distance and answers a different question. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
46. price-per-mass
Relationship: money divided by mass. The expected result carries $/kg or equivalent. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: kg/$ is reciprocal purchasing yield. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
47. mass-per-item
Relationship: mass divided by item count. The expected result carries g/item or kg/item. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: item/g answers packing density in the reciprocal sense. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
48. time-per-task
Relationship: time divided by tasks. The expected result carries minutes/task. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: tasks/minute is productivity rather than duration per task. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
49. tasks-per-time
Relationship: tasks divided by time. The expected result carries tasks/hour. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: hours/task is reciprocal and should not be interpreted as the same performance metric. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
50. area-per-person
Relationship: area divided by people. The expected result carries m²/person. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: people/m² is occupancy density, not personal allocation. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
51. percentage-point change
Relationship: difference between two percentages. The expected result carries percentage points. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: treating it as percent change requires another denominator step. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
52. index number
Relationship: current quantity divided by base quantity then scaled. The expected result carries dimensionless index. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: raw units should cancel when the base is the same quantity type. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
53. average rate over intervals
Relationship: total change divided by total interval. The expected result carries quantity per interval-unit. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: simple averaging of rates can ignore unequal interval lengths. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
54. conversion factor
Relationship: a valid conversion factor represents one, expressed in different units. The expected result carries dimensionless multiplier. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: a factor that changes the physical quantity rather than only its unit is not a pure conversion. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
55. formula rearrangement
Relationship: rearranging should preserve dimensional equality. The expected result carries same units implied before and after rearrangement. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: if rearrangement makes a variable carry the wrong dimension, an algebraic error occurred. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
56. square root
Relationship: square root halves unit powers appropriately. The expected result carries sqrt(m²)=m. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: leaving squared units after a square root is a presentation and reasoning error. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
57. squaring
Relationship: squaring a quantity squares its units. The expected result carries (m)²=m². Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: keeping linear units after squaring hides a dimensional change. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
58. product of ratios
Relationship: dimensionless ratios remain dimensionless when multiplied. The expected result carries dimensionless. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: unexpected units show one ratio was actually a rate or unlike quantities were compared. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
59. normalised data
Relationship: raw measurement divided by an appropriate base. The expected result carries often dimensionless or per-unit base. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: normalising by the wrong base changes the meaning even if units cancel. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
60. real-world capacity
Relationship: amount divided by capacity or capacity divided by amount depending question. The expected result carries dimensionless utilisation or amount-per-capacity. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: the chosen direction determines whether 1 means full use or reciprocal slack. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
61. runway
Relationship: cash divided by cash-per-time burn rate. The expected result carries time. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: burn rate divided by cash produces reciprocal time and cannot answer months remaining. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
62. work completion
Relationship: work remaining divided by work rate. The expected result carries time. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: multiplying by work rate produces squared work over time, not completion time. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
63. flow model
Relationship: volume divided by time. The expected result carries volume/time. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: time/volume is reciprocal duration per unit volume. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
64. data rate
Relationship: data amount divided by time. The expected result carries MB/s or equivalent. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: seconds/MB is reciprocal transfer time. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
65. growth per year
Relationship: change divided by years. The expected result carries quantity/year. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using final quantity over years is average level per year, not change rate. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
66. return per dollar
Relationship: gain divided by invested amount. The expected result carries dimensionless return or percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: dollars per gain reverses the metric. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
67. margin
Relationship: profit divided by revenue when margin is requested. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: profit over cost is markup, a different denominator. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
68. markup
Relationship: profit divided by cost. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: profit over revenue is margin, not markup. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
69. occupancy
Relationship: occupied capacity divided by total capacity. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: capacity per occupied unit is reciprocal and not occupancy rate. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
70. utilisation
Relationship: used resource divided by available resource. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: available/used can exceed 1 and answers a different efficiency question. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
71. success rate
Relationship: successes divided by opportunities. The expected result carries dimensionless proportion. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: opportunities/success is trials per success, a reciprocal measure. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
72. error rate
Relationship: errors divided by attempts or observations. The expected result carries dimensionless proportion. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: attempts/error is attempts per error, useful perhaps, but not the stated error rate. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
73. survey response rate
Relationship: responses divided by invitations or eligible population as defined. The expected result carries dimensionless percentage. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: approval among respondents uses a different denominator and should not be confused with response rate. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
74. mean waiting time
Relationship: total waiting time divided by number of waits. The expected result carries time. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: dividing by operating hours creates waiting-time-per-hour, not mean customer wait. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
75. change per unit area
Relationship: change quantity divided by area. The expected result carries quantity/area. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: using length as denominator changes the dimensional meaning. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
76. change per unit volume
Relationship: change quantity divided by volume. The expected result carries quantity/volume. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: area or length denominators create different densities. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
77. dimensionless comparison
Relationship: same physical quantity divided by same physical quantity. The expected result carries unitless ratio. Writing this unit before or during the calculation gives the learner an independent structural check on the chosen formula.
Error signal: if units fail to cancel, the comparison is not dimensionless and may not be the intended ratio. A numerically tidy answer cannot rescue a dimensional mismatch. If the unit type is wrong, return to the relation, numerator/denominator direction, exponent or conversion rather than merely repeating calculator arithmetic.
For transfer practice, change the numerical values and units while preserving the same deep relationship. The learner should be able to predict the target unit before substitution and reject a formula whose dimensions cannot possibly produce it.
When units are dimensionless
Percentages, probabilities, ratios, scale factors and many index values can be dimensionless because like units cancel. Dimensionless does not mean structure-free. The learner still has to verify the correct base, sample space or corresponding quantity. A wrong denominator can cancel units perfectly and still answer the wrong question.
When school Mathematics uses units informally
Not every K210 algebraic exercise requires formal dimensional-analysis notation. The purpose is not to introduce university physics language into every question. The practical habit is lighter: know what each quantity represents, keep unlike quantities from being added, and use the target unit to test the direction of rates, conversions and powers.
Links into the Mathematics system
Use the Mathematics Hub for underlying number, algebra, geometry, statistics and probability concepts; Vol 0072 Verification Asymmetry for independent checking; Vol 0064 Assumption Audit for modelling conditions; the Examination Craft hub for timed review; and the PSLE Learner’s Guide for earlier unit and ratio foundations.
Readiness criteria
- You can predict the target unit before calculation.
- You recognise reciprocal rates by their reversed units.
- You square or cube units when quantities are squared or cubed.
- You do not add unlike quantity types.
- You use unit mismatch as a reason to inspect structure, not merely arithmetic.
- You distinguish a dimensionless ratio from a rate with unlike units.
- You can transfer the same unit logic across unfamiliar real-world contexts.
Official-source discipline
For current K210 requirements, including SI-unit use and compound-unit notation, use the official SEAB 2027 G2 syllabus directory and linked Mathematics syllabus. If SEAB updates the syllabus, the current official document takes priority.
Final rule
Before trusting the number, ask what kind of quantity the answer is. Units are a compressed statement of structure. When the unit is impossible, the mathematics has already told you where to look: formula direction, denominator, power, conversion or quantity meaning.
