K310 rewards accurate technique, modelling and mathematical communication. A result can be numerically tidy and still be wrong because the model, unit, scale or interpretation failed. This workshop builds a disciplined plausibility and reverse-check system.
It extends the model sensitivity in Vol 0067, the feasibility work in Vol 0071 and the assumption audit in Vol 0073.
For 2027 school candidates, the official K310 Mathematics syllabus explicitly includes approximation and estimation, alongside reasoning, communication and application. Use the SEAB G3 syllabus directory for the current official documents. All cases below are original teaching examples.
Plausibility is a mathematical skill
A calculator can return a perfectly formatted number for an incorrectly modelled problem. Plausibility asks whether the result fits sign, scale, unit, range and context. It is not a vague feeling. It is a structured mathematical check.
Estimation comes before exactness
Before calculating, predict a rough range or order of magnitude. If the exact answer lies far outside it, inspect the setup. Approximation and estimation are explicit parts of the K310 syllabus and can be used as error control, not only as separate topics.
Units are part of the reverse check
A numerical answer without the expected unit may reveal the wrong relationship. Speed, area, rate, probability and money each carry different dimensional meaning. Unit checks often expose errors before the final arithmetic is repeated.
Reverse operations test structure
Addition can be checked by subtraction, multiplication by division, squaring by square root where the domain permits, and percentage increase by reversing to the original base. A reverse check should test the relationship, not merely repeat the same keystrokes.
Substitute back into the original condition
For an equation, one of the strongest checks is to place the solution into the original equation or context. A value that satisfies a rearranged line but not the original statement exposes an algebra or transcription error.
Use a second representation
A graph, table or diagram can verify an algebraic result. If two linear models supposedly intersect at x = 30, a quick table around 30 or a graph should tell the same story. Different representations catch different mistakes.
Reasonableness depends on context
A probability cannot exceed 1, a length cannot be negative in ordinary physical context, and a count of students must be a feasible whole number. Context supplies constraints that pure arithmetic may not enforce automatically.
Sign checks are cheap and powerful
If a quantity must increase, a negative change is suspicious. If a debt is reduced, the sign may have a clear interpretation. Predicting sign before calculation helps catch copied negatives and reversed subtraction.
Order of magnitude catches decimal slips
If a distance is about 5 km, an answer of 5000 km is almost certainly wrong. Standard form and estimation help detect misplaced powers of ten, unit prefixes and calculator-entry errors.
Significant figures do not rescue a wrong model
Rounding an incorrect answer neatly to three significant figures does not make it valid. Model first, calculate second, then apply the requested accuracy at the end.
Worked case 1: average speed
A vehicle travels 180 km in 3 hours. Before calculating, estimate about 60 km/h. The exact average speed is 60 km/h. An answer of 540 km/h signals a multiplication error; 0.0167 km/h signals a unit or division problem.
Worked case 1: reverse check
Multiply 60 km/h by 3 h to recover 180 km. The reverse check returns the original distance and unit relationship. Re-entering 180 ÷ 3 on the calculator is repetition, not an independent check.
Worked case 2: percentage increase
A price rises from $80 by 15%. Estimate a little above $90. Exact new price is 80 × 1.15 = $92. An answer of $68 has the wrong direction; $120 is too large for a 15% increase.
Worked case 2: reverse percentage check
Divide the new price $92 by 1.15 and recover $80. This checks the multiplicative model directly.
Worked case 3: reverse percentage
After a 20% discount, a price is $96. The original is 96 ÷ 0.8 = $120. Multiplying by 1.2 gives $115.20 and is not the correct reverse because the final price is 80% of the original base.
Worked case 4: geometry area
A rectangle 12 m by 7 m has area 84 m². If an answer says 38 m², compare with a rough benchmark: 10 × 7 = 70. The result should be near 80, not below 40.
Worked case 4: perimeter confusion
2(12 + 7) = 38 m is the perimeter. A plausible-looking 38 arose from solving a different quantity correctly. Unit and target checks distinguish area from perimeter.
Worked case 5: Pythagoras
A right triangle with legs 6 and 8 has hypotenuse 10. Before calculating, the hypotenuse must exceed 8. An answer of 7.2 fails a structural plausibility check immediately.
Worked case 5: square check
Verify 6² + 8² = 36 + 64 = 100 = 10². This tests the theorem relationship rather than merely trusting the square-root output.
Worked case 6: probability
A calculated probability of 1.24 is impossible. The error may be in the sample space, addition of overlapping events or arithmetic. The bound [0,1] is a context-free mathematical check available before detailed diagnosis.
Worked case 7: mean
For values 12, 14, 16 and 18, the mean must lie between 12 and 18. A result of 20 cannot be correct. Range bounds provide a simple reasonableness test.
Worked case 8: median
If the ordered data are 3, 5, 9, 12, 20, the median is 9. Averaging all values gives a mean, not a median. Plausibility also means checking the statistic requested.
Worked case 9: gradient
A line rises from (2,5) to (6,13). Gradient = 8/4 = 2. A negative gradient conflicts with the visual rise. Direction can check arithmetic.
Worked case 9: reverse using the gradient
From x = 2 to x = 6, a gradient of 2 predicts a rise of 2 × 4 = 8, giving y = 13. The point is recovered.
Worked case 10: linear equation
Solve 3x + 7 = 25 to get x = 6. Substitute: 18 + 7 = 25. This is faster and safer than rereading each algebra step when the answer is short.
Worked case 11: simultaneous interpretation
If x and y represent tickets sold, negative solutions may be mathematically possible in an unrestricted system but infeasible in context. Interpret after solving.
Worked case 12: direct proportion
If y is directly proportional to x, doubling x should double y. A calculated value that halves instead suggests the wrong proportional model.
Worked case 13: inverse proportion
If y is inversely proportional to x, doubling x should halve y. Direction-of-change checks can distinguish direct from inverse models before exact values are trusted.
Worked case 14: scale drawing
A map scale of 1:50,000 means 1 cm represents 50,000 cm, or 500 m. A 6 cm map distance is 3 km. An answer of 300 km reveals a unit-scale error.
Worked case 15: area scale
If linear scale factor is 3, area scale factor is 9. An answer that multiplies area by 3 has ignored dimensionality. The expected growth should be much larger than the linear change.
Worked case 16: volume scale
If all linear dimensions double, volume multiplies by 8. A result of only twice the volume fails a dimensional check.
Worked case 17: standard form
3.2 × 10⁵ is 320,000. If a calculation of similar quantities returns 3.2 × 10⁻⁵, inspect the exponent sign. Order-of-magnitude awareness can catch the inversion.
Worked case 18: money
A bill for 8 items at about $12 each should be near $96. If the calculator says $960, a misplaced decimal or extra zero is more likely than an extraordinary price change.
Worked case 19: time conversion
2.5 hours is 150 minutes, not 250 minutes. Decimal hours are base ten; minutes are base sixty. Reverse conversion by dividing 150 by 60 to recover 2.5 hours.
Worked case 20: rate units
A rate of 5 litres per minute for 12 minutes gives 60 litres. Dividing 5 by 12 produces litres per minute squared, which is not the requested total volume.
Calculator sanity starts before pressing keys
Write the relationship or at least state it mentally before entering numbers. Calculator-first behaviour can hide a wrong model behind a precise display.
Bracket checks matter
Long expressions require deliberate brackets. If the calculator answer is implausible, inspect grouping before assuming the formula is wrong.
Mode checks matter only when relevant
Degree mode is important for trigonometry in degrees. Rechecking mode after every arithmetic question wastes attention. Use tool checks where the mathematics requires them.
Round at the end unless instructed otherwise
Premature rounding can drift a multi-step result. Keep enough precision through working, then apply the required decimal places or significant figures to the final answer.
Approximate answers should match requested precision
A question asking for three significant figures should not receive an unnecessarily long calculator display. Reporting precision is part of communication.
Exact form can be more informative
If a question requires an exact answer, keep surds or fractions as appropriate rather than converting automatically to decimals. Exactness is a response-form decision.
Graph plausibility
If a quadratic graph should open upwards because its x² coefficient is positive, a sketch opening downwards signals a structural error before any roots are checked.
Function plausibility
If f(x)=2x+3, increasing x by 1 increases f(x) by 2. A table that changes by 5 each step cannot represent that function.
Inequality plausibility
After solving an inequality, test one value from the proposed solution region. This catches reversed signs after multiplication or division by a negative number.
Constraint plausibility
If a capacity is 40, any proposed solution using 45 people in one unit is infeasible even if the algebraic cost is smallest. Feasibility can override optimisation.
Break-even plausibility
At a break-even threshold, both models should give the same value. Substitute the threshold into both expressions. If the totals differ, the equation or arithmetic is wrong.
Sensitivity plausibility
Before changing a model input, predict the direction of the result. If a lower variable cost makes a plan become worse earlier, inspect the calculation. Direction is a conceptual check.
Worked case 21: average versus total
A class records 150 visits over 5 sessions. The average is 30 visits per session. If a later sentence says 30 students attended in total, the number has been translated into the wrong quantity.
Worked case 22: weighted average
If a component worth 70% improves, the overall score should move more than an equal change in a component worth 30%, all else equal. This direction can check a weighted formula.
Worked case 23: bounds from obvious context
If three positive lengths are each less than 10 cm, their sum must be less than 30 cm. A result of 45 cm violates a simple bound even before exact values are reconsidered.
Worked case 24: angle sum
Triangle angles must sum to 180°. If two angles are 70° and 80°, the third is 30°. A result of 210° signals that the target or subtraction was misunderstood.
Worked case 25: frequency total
If a frequency table represents 120 observations, the frequencies must sum to 120. A total of 118 indicates an omission or copying error before any mean is trusted.
Reverse-check ladder
Use the cheapest check first: sign, unit, rough size, feasible range, substitution, inverse operation, alternate representation. Do not perform a long second solution when a five-second unit check already exposes the error.
Independent task A
Estimate, then calculate 49.8 × 19.7. State a reasonable rough range before using the calculator and explain how it helps detect a decimal slip.
Independent task B
Solve 5x − 9 = 31 and verify by substitution.
Independent task C
A price after a 25% discount is $72. Find the original price and reverse-check your result.
Independent task D
A cylinder calculation gives a negative volume. Explain what this tells you before you inspect any formula.
Independent task E
A probability tree calculation gives 0.08. Give two independent checks you could perform without simply repeating the same multiplication.
Independent task F
A graph of cost against quantity has positive gradient and positive intercept. What should happen to cost as quantity increases, and how can this prediction help you check a table?
Worked feedback A
49.8 is about 50 and 19.7 about 20, so the product should be near 1,000. Exact calculation is 981.06. A result near 98 or 9,810 would suggest a decimal-position error.
Worked feedback B
5x − 9 = 31 gives 5x = 40 and x = 8. Substitution gives 40 − 9 = 31, confirming the solution.
Worked feedback C
$72 is 75% of the original, so original = 72 ÷ 0.75 = $96. Reverse-check: 25% of 96 is 24, and 96 − 24 = 72.
Worked feedback D
Volume cannot be negative in the physical context. The sign failure tells you the model or arithmetic must be inspected before reporting the result.
Worked feedback E
Check that each branch probability lies between 0 and 1 and that relevant branch totals are consistent. You can also reconstruct the event through a table or complementary probability where appropriate.
Worked feedback F
Cost should rise as quantity increases. A table showing falling costs while the model claims a positive gradient indicates a copying or formula inconsistency.
Repair route
If plausibility checks are weak, begin by predicting only sign, unit and rough size before every calculation. These three habits are fast enough to become automatic.
Stabilisation route
Mix questions where some calculator answers are correct and others contain planted sign, unit or magnitude errors. Ask the learner to identify which deserve recalculation and why.
Extension route
Require two independent checks on high-value problems: for example substitution plus graph, or estimation plus inverse operation. The goal is not redundant work but complementary evidence.
What progress should look like
Progress is visible when implausible answers are rejected before feedback, reverse checks target the original relationship, and the learner stops treating the calculator display as proof.
Final operating rule
Before accepting a numerical answer, ask: does the sign make sense, does the unit fit, is the size plausible, is the value feasible, and can I verify it through the original condition or another representation?
K310 plausibility and reverse-check checklist
- predict sign and rough size
- keep units visible
- check feasible range
- use required accuracy
- substitute solutions back
- reverse the operation where useful
- compare another representation
- recalculate only after diagnosing the likely fault
Continue the G3 Mathematics route
Return to the G3 SEC Learner’s Guide hub for the full Mathematics route and numbered series.
Advanced plausibility and verification laboratory
Independent verification should not copy the same error
If a solution used the same formula and the same calculator entry twice, the second result is not independent evidence. A useful check changes the route: substitute back, estimate, use a graph, use a complementary probability or reconstruct the quantity from another relationship.
Check the target before checking the arithmetic
Many wrong answers are correct calculations of the wrong quantity. Before recalculating, restate what the question asked. Area versus perimeter, total versus rate, mean versus median and original versus discounted price are classic target errors.
Use the answer space only as a presentation cue
Do not assume a long answer is required because there is much space, or that a one-line answer is enough because the box is small. The required mathematical communication comes from the command, marks and structure.
Plausibility under multi-step chains
For long questions, check important intermediate values before reusing them. A milestone check is cheaper than discovering at the end that every later part inherited one wrong quantity.
Milestone check: units
If an intermediate result changes unit unexpectedly, stop. A length should not become an area unless the operation explains the change. Units can reveal an error before the next subpart is attempted.
Milestone check: direction
If increasing an input should logically raise the output under the model, but the intermediate value falls, inspect the relationship. Direction-of-change reasoning is a fast conceptual check.
Milestone check: order of magnitude
A power of ten error can contaminate many later parts. Compare each large or small result with a rough benchmark before carrying it forward.
Milestone check: domain
If a logarithm, square root, probability or physical count has a restricted domain, verify the intermediate value remains feasible. A later calculation cannot repair an impossible earlier state.
Worked case 26: currency-free ratio
A recipe model uses 3 parts concentrate to 7 parts water. For 50 parts total, concentrate should be 15 and water 35. A result of 21 and 29 fails the 3:7 ratio even though the total is 50.
Worked case 26: reverse ratio check
Check 15:35 simplifies to 3:7. This tests the original relationship directly and is stronger than merely checking that 15 + 35 = 50.
Worked case 27: inverse proportion direction
If y = k/x and x doubles, y halves. If your table doubles both values, you have modelled direct proportion instead. The direction check can catch the error before solving for k.
Worked case 28: simultaneous equations
After solving x and y, substitute both values into both original equations. Satisfying only one equation is not enough. This is especially important when elimination or substitution involved several sign changes.
Worked case 29: quadratic roots
If x = 2 and x = 5 are proposed roots of x² − 7x + 10 = 0, verify by factorisation (x − 2)(x − 5). The coefficient and constant term provide a structural cross-check.
Worked case 30: graph intercept
For y = 3x + 4, the y-intercept must be 4. A plotted graph crossing at −4 signals a sign or plotting error even if the gradient looks correct.
Worked case 31: cumulative frequency intuition
A cumulative frequency graph cannot decrease as the variable increases. A downward segment indicates plotting or reading error because cumulative totals do not lose observations.
Worked case 32: histogram or frequency density context
Where a task uses unequal class widths and frequency density, bar height is not raw frequency. Check area interpretation if the syllabus context requires it. Do not carry a simple bar-chart assumption into a different representation.
Worked case 33: angle reasonableness
An obtuse angle must exceed 90° and be below 180°. If a trigonometric calculation returns 42° when the diagram and conditions require obtuse, inspect whether the supplementary angle is needed.
Worked case 34: circle context
A circumference must be greater than the diameter for a positive circle. An answer smaller than the diameter indicates a formula or calculator error.
Worked case 35: percentage composition
If percentages describe an entire set, their total should be 100% unless categories overlap or the question says otherwise. A total of 134% may be valid only if multiple selections are allowed; context controls the check.
Worked case 36: compound growth
Two successive 10% increases produce 1.1² = 1.21, or 21% overall. Adding percentages gives 20% and misses compounding. Reverse-check by dividing the final amount by the original.
Worked case 37: percentage decrease then increase
A 20% decrease followed by 20% increase does not restore the original. The multipliers 0.8 × 1.2 = 0.96 show a 4% net decrease. This is a strong test of base awareness.
Worked case 38: map scale
If 2 cm represents 1 km, then 7 cm represents 3.5 km. Reverse-check: 3.5 km divided by 0.5 km per cm returns 7 cm.
Worked case 39: density
If mass is 500 g and volume 250 cm³, density is 2 g/cm³. A result of 0.5 g/cm³ comes from reversing numerator and denominator. Unit wording tells you which way the ratio should be formed.
Worked case 40: simple interest intuition
If simple interest is $120 per year, three years should produce $360 interest, not a compounding pattern. The formula must match the financial model specified by the question.
Alternative method check: mental arithmetic
For simple products or percentages, a mental estimate can verify calculator output quickly. 19% of 200 is about 40; an answer of 380 is obviously suspect.
Alternative method check: table
A table around a threshold can verify algebra and reveal integer behaviour. This is especially useful after break-even or inequality problems.
Alternative method check: graph
A graph can confirm number of roots, intersection direction and broad solution region. It is useful when the algebraic result conflicts with visual structure.
Alternative method check: factorisation
A quadratic solved by formula can sometimes be checked by factorising. Different algebraic routes reduce the chance of repeating the same slip.
Alternative method check: substitution
A formula rearrangement can be checked by substituting the final value back into the original equation. This is often the fastest end-to-end verification.
Alternative method check: complement
Probability of an event can sometimes be checked using 1 minus the complement. If two independent methods disagree, inspect the event definition.
Calculator diagnostic: repeated wrong answer
If the same implausible number appears repeatedly, stop pressing equals. The mistake may be in the formula, unit conversion or brackets. Repetition only confirms the entered expression, not the intended Mathematics.
Calculator diagnostic: display mode
Scientific notation, degree/radian mode or fraction/decimal display can affect interpretation. Check mode only when relevant to the task rather than as a ritual.
Calculator diagnostic: stored memory
A stored value or Ans entry can create unexpected output if used unintentionally. For high-stakes calculations, know what expression the display actually contains.
Calculator diagnostic: fraction structure
A long numerator or denominator needs brackets. Write the mathematical fraction first, then make the calculator entry mirror that structure.
Boundary case check
Test simple values such as zero, one, or a maximum capacity when they are allowed. Boundary cases often reveal whether a formula or inequality has been interpreted correctly.
Zero case
If a cost model has a fixed fee, cost at zero units should equal that fixed fee. If your expression gives zero, you have probably dropped the intercept.
One-unit case
For a per-unit rule, substituting one unit can reveal whether the rate and fixed part were combined correctly. Simple cases expose algebra hidden by large numbers.
Maximum-capacity case
At the stated capacity, the solution should still be feasible. One unit above it should violate the constraint. This tests an inequality or integer decision.
Symmetry check
In a symmetric geometry problem, corresponding lengths or angles should behave consistently. A result that breaks a stated symmetry deserves reinspection.
Monotonicity check
If a model is known to increase over the relevant range, a table that decreases is inconsistent. This can catch copied signs or wrong coefficients.
Range check for trigonometric ratios
Sine and cosine values lie between −1 and 1. A calculated sine of 1.3 signals a setup or arithmetic error before inverse trigonometry is attempted.
Range check for probabilities
Probabilities lie from 0 to 1. Percent probabilities lie from 0% to 100%. Values outside the range demand immediate diagnosis.
Range check for proportions
A part-to-whole proportion usually lies between 0 and 1. If it exceeds 1, confirm whether the quantity is actually a ratio that can be greater than 1.
Range check for averages
A weighted or unweighted average should normally lie between the minimum and maximum component values when weights are positive and sum appropriately. If not, inspect the weights or formula.
Plausibility in exact algebra
Even symbolic answers can be checked. If solving for x in terms of a positive parameter should produce a positive expression, a negative result may expose a sign error.
Plausibility in geometry proofs
A proof step should follow from stated or previously proven conditions. The check is logical rather than numerical: does the theorem actually apply here?
Plausibility in statistics
A conclusion about consistency should match the chosen measure of spread. A smaller range can support one claim about spread but not necessarily every statement about the distribution.
Plausibility in probability
Event definitions must align with the sample space. A calculation can be arithmetically correct while the counted outcomes answer a different event.
Plausibility in modelling
Real-world outputs should respect feasibility. A negative number of buses, fractional people, or capacity beyond a stated maximum must be interpreted, rounded or rejected according to context.
Exam-time hierarchy
Use checks in order of cost. First target and unit. Then sign and rough size. Then feasibility. Then substitution or alternate representation if the question is high-value or uncertain. Do not spend three minutes on a check that a five-second unit inspection can settle.
When not to recheck
If the answer is simple, consistent with estimate, unit, range and original condition, move on. Checking has diminishing returns. Overchecking secure work can create time pressure elsewhere.
When to recheck deeply
Use deeper verification for marked uncertainty, long chains, model-heavy questions, suspicious calculator output and answers near a decision threshold. Risk-based checking is more efficient than equal checking.
Error-log category: implausible magnitude
Record the factor-of-ten or scale mistake and the benchmark that would have caught it. Practise another question with a different surface context.
Error-log category: wrong unit
Record the relationship between requested and produced units. The repair is to keep units attached during setup, not merely append them at the end.
Error-log category: wrong target
Record what you calculated and what was asked. Practise identifying the target before any operation.
Error-log category: wrong domain
Record the contextual constraint that invalidated the raw solution. Practise another integer, probability or geometry constraint question.
Error-log category: repeated method
If your check simply repeated the original calculation, add one alternate verification method in the next practice set.
Final plausibility standard
The learner is ready when an impossible or suspicious answer triggers diagnosis before feedback. The goal is not distrust of every result; it is the ability to distinguish a result worth accepting from one that deserves another look.
G3 Learner’s Guide navigation: ← Vol 0074 · Master Hub · Vols 0001–0077 · Vol 0076 → · Mathematics route.
