Why a Refrigerator Is a Mathematics System That Never Truly Stops
A refrigerator does not manufacture cold. It uses electrical work to move heat from a cool interior to a warmer room while heat continually leaks back through walls, seals, door openings and warm contents. A thermostat or controller decides when to run, and the result is a repeating temperature and power pattern rather than one fixed number.
This is why mathematics matters in everyday life. Heat transfer is a rate, energy accumulates over time, thermostat limits form a band, compressor operation creates a duty cycle, and annual kilowatt-hours summarise many hours under a standard test. Mixing those quantities leads to poor explanations and unreliable savings claims.
Use published data and teacher-supplied temperature traces. Do not alter settings for experiments involving food, open panels, bypass switches or place sensors where they interfere with doors and seals. Electrical service and refrigerant work require qualified people.
Quick Reading Routes
- Heat entering the cabinet
- Thermostat bands and cycles
- Duty cycle and energy
- Student investigations
- Misconceptions and FAQs
Heat Keeps Entering the Cold Space
For a simple wall model, heat-transfer rate is Q̇=UAΔT. U combines the effect of materials and thicknesses, A is area, and ΔT is the temperature difference. With illustrative U=0.35 W/(m²·K), A=4.0 m² and ΔT=20 K, heat gain is 28 W.
The model is steady and simplified. Real cabinets have corners, door seals, thermal bridges and changing surface temperatures. A small high-conductance path can matter disproportionately. The calculation is useful because it names the variables, not because 28 W fits every appliance.
Door openings add air and moisture loads
If 0.20 kg of room air effectively enters and cools by 20 K, using cp≈1.0 kJ/(kg·K) gives about 4 kJ of sensible heat. Moisture that condenses or freezes adds latent heat. Air exchange depends on door geometry, opening duration, buoyancy and shelves.
Ten openings are not always exactly ten times one opening because temperatures and controller state change. Still, the linear first model helps compare behaviour while clearly naming what it omits.
Warm contents are thermal loads
Cooling 0.50 kg of a water-like drink by 18 K requires Q=mcΔT≈0.50×4.18×18=37.6 kJ, or about 10.4 Wh of heat removed from the drink. The refrigerator consumes more electrical energy than that because it must move heat against a temperature difference and has losses.
This calculation is not a food-safety timetable. Packaging, circulation, starting temperature, load placement and controller behaviour determine the real cooling curve.
Thermostat Bands Turn Temperature into Control
An on/off controller may start cooling at an upper threshold and stop at a lower threshold. If an illustrative sensor cuts in at 5.0 °C and cuts out at 2.0 °C, the 3.0 °C difference is the control band or hysteresis.
Without a band, tiny noise around one threshold could cause rapid switching. Hysteresis creates memory: the same temperature can produce different actions depending on whether the system is warming or cooling.
One sensor does not equal one cabinet temperature
Air near the door, rear wall and shelves can differ. A sensor sees its own location and response time. Food temperature changes more slowly than nearby air because of thermal mass. A graph should label what was measured rather than calling every line “fridge temperature.”
Suppose a supplied trace shows upper-shelf air from 2.5 to 5.2 °C, lower-shelf air from 1.8 to 4.4 °C and a sealed water bottle from 3.1 to 3.8 °C. The different ranges are expected from different heat capacities and airflow exposure; they do not alone prove a fault.
Cooling curves reveal dynamics
After a door closes, excess temperature may fall roughly exponentially for part of the recovery. A time constant describes how quickly a simple first-order model decays. Compressor switching, fan cycling and multiple thermal masses can produce residual patterns that show the model is incomplete.
The aim is not to force every trace into an exponential. It is to use a model, inspect its errors and choose a better description when needed.
Duty Cycle Connects Power with Time
Duty cycle is running time divided by total observation time. If a compressor runs 18 minutes in a 45-minute window, duty cycle is 0.40 or 40%.
Observation windows must include complete representative behaviour. Averaging 40% from one 45-minute window and 60% from a 15-minute window gives 50%, but the time-weighted duty cycle is (18+9)/(45+15)=45%. Equal-weight averaging would be wrong.
Energy requires every power state
Suppose running input is 120 W, non-running electronics use 3 W and duty cycle is 40% over 24 hours. Running time is 9.6 h and standby time 14.4 h. Energy is 120×9.6+3×14.4=1,195 Wh, or about 1.20 kWh.
Real units may have evaporator fans, anti-sweat heaters, defrost heaters, ice makers and variable-speed compressors. A two-state model must therefore be labelled as illustrative.
An annual label can be turned into an average. Three hundred kWh/year divided by 365 is about 0.822 kWh/day; dividing by 24 gives 34.2 W average input. That does not mean the compressor is a 34 W motor. It is time-averaged energy across all operating states under a test method.
What Official Test Results Mean
The U.S. Department of Energy’s current 10 CFR Part 430, Subpart B, Appendix A provides a uniform method for measuring annual energy use in kWh/year, refrigerated volume and adjusted volume for covered consumer refrigeration products. The eCFR page is authoritative but unofficial and its displayed update date should be checked when making a compliance claim.
A standardised test result supports comparison because conditions and calculations are controlled. It does not recreate one household’s room temperature, door openings, food load, settings or maintenance. The article uses the public legal scope and does not reproduce incorporated protected standards.
NIST’s research report on refrigerator energy test procedures is also a reminder that procedure choice shapes reported consumption. A number without its method is incomplete evidence.
Kettle and refrigerator equations point in opposite directions
The recently published electric kettle article models adding heat to water over minutes. A refrigerator removes heat from a compartment over repeated cycles. Both require energy, power, time and boundary definitions, but the direction and operating pattern differ.
Fair Investigations and Honest Savings Claims
A before-and-after energy comparison needs comparable conditions. Ambient temperature, contents, door openings, control setting and observation length should be logged. Changing a seal and changing household behaviour at the same time prevents causal attribution.
Longer records reduce some random variation but do not remove bias. A plug meter’s resolution, missing defrost cycles or an unusually empty week can still distort results. Report a range and the observation dates rather than a universal annual promise.
Common Misconceptions
“The compressor is either 100% efficient or wasting energy”
Refrigeration performance depends on temperatures, components and cycle conditions. Electrical input and heat moved are different quantities.
“A 40% duty cycle means 40% of rated annual energy”
Only under a matching power-state model. Fans, heaters and variable-speed operation add states.
“The thermostat holds one exact temperature everywhere”
It controls using a sensor and a band. Air and food temperatures vary across space and time.
“Opening the door adds only warm air”
Moisture, warm contents, mixing and recovery behaviour also contribute.
How Students Can Learn and Transfer the Mathematics
Primary learners can read thermometers, calculate elapsed time and compare fractions of an hour. Secondary learners can use Q=mcΔT, percentages, time-weighted averages and line graphs. Older learners can fit dynamic models, analyse residuals, use thermal resistances and compare test procedures.
Parents can ask, “What is the boundary of your energy calculation?” and “Did you average by time?” Those two questions catch many errors.
The same ideas appear in cold-chain logistics, building services, food engineering, energy analysis and control systems. Mathematics opens understanding and options; it does not promise a job or justify unsafe appliance changes.
Frequently Asked Questions
Does a refrigerator make cold?
It moves heat from the cool compartment to the warmer surroundings using work.
What is thermostat hysteresis?
It is the gap between switching thresholds, helping prevent rapid on/off chatter around one temperature.
What is duty cycle?
The fraction of an observation period spent in a defined running state.
Why can my measured energy differ from a label?
Test conditions, climate, settings, loads, door use and measurement period may differ.
Can students measure a working refrigerator?
Use only non-invasive, adult-approved observations that do not affect food safety, doors, seals or electrical parts. Teacher-supplied datasets are safer and easier to compare.
The Bigger Answer to “Why Mathematics?”
A refrigerator makes invisible heat flows legible. Rates accumulate into energy, thresholds become control decisions, cycles become duty fractions and standard methods turn long operation into comparable evidence. Mathematics makes the system understandable without pretending a simple model can certify a real appliance.
Continue through the eduKate Mathematics Learning Hub for further reading on mathematics in science, technology and daily decisions.
Worked Case Study: A Day of Power States
Suppose a teacher supplies a 24-hour state summary: compressor-high 3.2 h at 145 W, compressor-low 6.4 h at 72 W, defrost 0.35 h at 310 W, fans and controls without compressor 11.0 h at 7 W, and deep standby 3.05 h at 2 W. The durations sum to 24 hours.
State energies are 464, 460.8, 108.5, 77 and 6.1 Wh. Total daily energy is about 1,116 Wh or 1.116 kWh. Compressor-related states account for roughly 925 Wh, but defrost and background loads are not negligible.
If someone modelled the appliance as 145 W on for 40% of the day and zero otherwise, the estimate would be 1.392 kWh. That simplified model overstates this constructed dataset because much running occurs at lower power and non-running states are not zero.
The lesson is not that five states are always enough. It is that duty cycle requires a state definition. Variable-speed operation makes “on” ambiguous, so integrating the power trace or summing state energy is often clearer.
Worked Case Study: A Thermal-Resistance Network
Represent main wall conduction as resistance R_wall=1/(UA). With UA=1.4 W/K, R_wall≈0.714 K/W. Represent seals and bridges as a parallel path with UA=0.45 W/K, or R_seal≈2.22 K/W.
Parallel conductances add: total UA=1.4+0.45=1.85 W/K. At ΔT=20 K, total steady heat gain is 37 W, divided into 28 W through walls and 9 W through seals and bridges.
If seal-path conductance rises from 0.45 to 0.80 W/K in a hypothetical damaged condition, total heat gain becomes 44 W. The change is 7 W, or about 19% relative to 37 W. Over 24 hours that adds 168 Wh of heat load, not necessarily 168 Wh of electrical input because refrigeration performance must be considered.
This network demonstrates why a small area can matter if conductance is high. It also prevents a common mistake: adding thermal resistances directly when paths are parallel rather than in series.
Worked Case Study: Reading a Hysteresis Trace
Suppose compartment-air readings are sampled each minute. The compressor starts when temperature reaches 5.0 °C and stops at 2.0 °C. Warming takes 24 minutes; cooling takes 16 minutes. One complete cycle is 40 minutes and compressor duty within that stable cycle is 16/40=40%.
Now a door opening at minute 10 raises temperature quickly to 7.2 °C. The compressor starts earlier and runs 23 minutes before reaching 2.0 °C. That disturbed cycle has a higher duty fraction, but averaging it with stable cycles requires weighting by their durations.
If sensor uncertainty is ±0.2 °C, the apparent cut-in could be 4.8–5.2 °C and cut-out 1.8–2.2 °C. The band remains clearly nonzero, but reporting thresholds to hundredths would be unjustified.
The air trace may overshoot after switch-off because cold coil and air continue exchanging heat. A switching event and a temperature minimum need not occur at the same moment. A labelled graph is more informative than a table of extrema alone.
Worked Case Study: Comparing Two Weeks without Overclaiming
Week A uses 8.4 kWh with average room temperature 26.0 °C, 92 recorded door openings and a moderately full cabinet. Week B uses 7.9 kWh with room temperature 24.5 °C, 71 openings and a different food load. Energy fell by 0.5 kWh or about 6%.
It would be incorrect to attribute the reduction to one behaviour because several variables changed. A regression across many days might estimate associations, but correlation would still need design knowledge before causal interpretation.
A more controlled follow-up could keep setting and load stable, alternate one behaviour by day and record ambient temperature. Even then, weather and defrost timing add noise. Confidence intervals or a paired-difference plot would show whether the effect is distinguishable from ordinary variation.
Responsible mathematics sometimes rejects a tempting headline. “Week B used less energy under different conditions” is accurate. “This change saves 6%” is not supported by the two-week comparison.
Worked Case Study: Cooling a Mixed Grocery Load
Suppose a grocery load contains 1.5 kg of water-rich food with effective heat capacity 3.8 kJ/(kg·K), 0.8 kg of packaging and dry goods at 1.6 kJ/(kg·K), and 0.4 kg of glass containers at 0.84 kJ/(kg·K). All cool by 16 K in the model.
Sensible heat removed is 1.5×3.8×16=91.2 kJ, plus 0.8×1.6×16=20.5 kJ and 0.4×0.84×16=5.4 kJ. Total is about 117 kJ or 32.5 Wh of thermal energy.
If the refrigeration system requires an illustrative 18 Wh of electrical input per 32.5 Wh removed under that condition, the quotient of heat moved to electrical input is about 1.8. This is a coefficient-of-performance-style ratio, not an efficiency capped at 100% because the machine moves heat rather than converting all input into it.
The calculation excludes moisture, packaging geometry, airflow and baseline cabinet heat gain during cooling. It should not determine food-storage timing. It simply reveals why mass and material composition matter.
Worked Case Study: Door-Opening Event versus Continuous Leakage
Assume baseline heat gain is 35 W. Over one hour it adds 126 kJ. A single illustrative door event adds 8 kJ of sensible and latent load. Six such openings add 48 kJ, raising the hour’s total to 174 kJ.
In this example, openings contribute about 27.6% of the hour’s load. But if the room-to-cabinet temperature difference changes, baseline leakage changes; if openings are longer, event load changes. The percentage is not portable to another home.
Represent the model as Q_total=UAΔT·t+nQ_event+Q_contents. The terms answer different questions: continuous envelope gain, repeated events and one-time contents. Keeping them separate makes a sensitivity analysis straightforward.
If a proposed behaviour cuts openings from six to four, modelled load falls by 16 kJ, or about 4.4 Wh of heat. Electrical savings are not automatically 4.4 Wh; they depend on system performance. That last conversion is where many simple claims overreach.
Worked Case Study: Sampling Can Miss a Defrost Cycle
Suppose a one-hour observation contains compressor and fan energy of 70 Wh. Multiplying by 24 predicts 1.68 kWh/day. But the appliance runs a 300 W defrost heater for 20 minutes once during the day, adding 100 Wh. Background electronics add another 60 Wh/day.
The revised total is 1.84 kWh/day, about 9.5% above the short-window extrapolation. If the chosen hour happened to include defrost, multiplying it could instead overstate the day severely.
This is a sampling problem. The observation window must represent cycles with different frequencies. A full day may still miss a weekly adaptive event; several weeks may better capture household variation.
Stratified accounting is useful: ordinary compressor cycles, defrost, door recovery and idle states can be counted separately. The final estimate weights each by its actual frequency. The method is more work than multiplying one convenient hour, but it produces an auditable estimate.
Building a Safe Mathematics Project
Use a prepared dataset with timestamp, air temperature at several positions, sealed-water temperature, compressor-state indicator and electrical power. Begin by plotting, not averaging. Switching events, gradients and missing intervals should be visible.
Define one question such as, “How does door recovery change duty cycle over the next 60 minutes?” Mark the event, choose a baseline window and state how recovery ends. Calculate time-weighted energy rather than averaging power samples when intervals are unequal.
Check sensitivity to the recovery definition. A return within 0.5 °C of baseline may give one duration; a return within 0.2 °C gives another. Report both if the conclusion changes. Threshold choice is part of the method.
Include an ethical and safety boundary: do not compromise food temperatures, hold doors open, disable controls or access mains wiring. A dataset can teach full mathematical reasoning without risking food or equipment.
End with an evidence map linking every claim to a graph, table or calculation. A good report can say “the chosen recovery threshold changed the result by 12 minutes.” That transparency is more educational than pretending there is one perfect answer.
Final Transfer Challenge: Reconcile Heat, Power and Time
Give students a prepared 24-hour dataset containing cabinet temperatures, ambient temperature, compressor state, defrost state and electrical power. Their task is to make three accounts agree: thermal events, control states and energy use.
First, calculate energy from power over each time interval. With irregular sampling, use the interval duration explicitly; do not multiply a simple average by 24 hours unless the samples represent equal time. Sum compressor, defrost and background intervals separately before combining them.
Second, label thermal events such as a door opening or warm-load addition without assuming causation from timing alone. Compare each event with a matched baseline period at a similar ambient temperature and control state. A nearby compressor start may be part of a normal cycle rather than a response to the marked event.
Third, test the thermostat-band model. Estimate the temperatures at which cooling begins and ends for several cycles. Report a distribution, not a single perfect pair, because sensor location, delay, defrost and load can shift observed thresholds. A scatter plot of start and stop temperatures can reveal drift or distinct operating regimes.
Add a conservation-style reasonableness check. The electrical energy input cannot be equated directly with heat removed from the cabinet. Some input becomes heat at components, while the system moves heat across a temperature difference. State the system boundary before comparing quantities.
Finally, write two conclusions: one about the supplied dataset and one about what further evidence would be needed for a household claim. The first may quantify duty cycle and measured energy. The second may call for longer monitoring, calibrated sensors, controlled loads and seasonal conditions.
This challenge shows why mathematics matters beyond obtaining a daily total. It connects sampling, integration, control logic, uncertainty and physical modelling. It also makes the limits visible, which is essential when everyday measurements could otherwise be mistaken for food-safety or appliance-performance certification.
Add a missing-data drill before submission. Remove a ten-minute interval that crosses a compressor transition and ask students to bound, rather than invent, its energy. One lower estimate can use the observed idle power and one upper estimate the observed running power. The result is an interval whose width reveals how much the gap matters.
Finally, compare two normalisations: watt-hours per day and watt-hours per degree-hour of ambient-to-cabinet temperature difference. The second may help explore weather effects, but it is not automatically a universal performance metric because loads, humidity, cycling and appliance design remain different. Normalisation is useful when its denominator reflects a real mechanism and its limitations are stated.
These exercises replace false exactness with an honest range and make the evidence chain robust even when real datasets are imperfect.
A Topic-Specific Mathematics Laboratory
These investigations use diagrams, supplied data, unplugged models and published specifications. Choose a small set that answers one clear question; doing every activity is not the goal. Keep raw values, units, assumptions and limitations together so another student can reproduce the reasoning.
Investigation 1: Draw the thermal boundary
Sketch a cold compartment inside a warmer room. Mark heat entering through walls, door seals, openings and warm contents, plus heat moved out by the refrigeration system.
**Mathematics move.** Use arrows and watts for rates, joules for accumulated energy and degrees Celsius for temperature.
**Boundary and transfer.** The cabinet is not perfectly uniform; a one-node diagram is a deliberate simplification. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 2: Calculate wall conduction
For an illustrative panel with U=0.35 W/(m²·K), area 4.0 m² and 20 K temperature difference, Q̇=UAΔT=28 W.
**Mathematics move.** Repeat for 15 K and 25 K to show the linear assumption. Keep U as supplied, not invented for a real refrigerator.
**Boundary and transfer.** Thermal bridges, seals and changing temperatures can make the true heat gain larger. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 3: Model a door opening
Suppose 0.20 kg of room air effectively replaces cold air and cools by 20 K. With cp≈1.0 kJ/(kg·K), the sensible load is about 4 kJ.
**Mathematics move.** Compare ten openings with one opening and discuss why moisture condensation adds latent heat not included here.
**Boundary and transfer.** The exchanged mass is hard to measure and depends on geometry, duration and buoyancy. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 4: Cool a warm drink safely on paper
For 0.50 kg of water-based drink cooling by 18 K, Q≈0.50×4.18×18=37.6 kJ.
**Mathematics move.** Convert to watt-hours by dividing by 3.6, giving about 10.4 Wh of heat removed from the drink alone.
**Boundary and transfer.** This is not a food-safety timetable; packaging, airflow and refrigerator control affect the real process. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 5: Read a thermostat band
Use a supplied trace with compressor cut-in at 5.0 °C and cut-out at 2.0 °C.
**Mathematics move.** The 3.0 °C difference is the control band or hysteresis in this illustrative model. Mark each switch event on the graph.
**Boundary and transfer.** Air temperature at one sensor is not the same as every food temperature. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 6: Calculate duty cycle
If a compressor runs 18 minutes in a 45-minute observation, duty cycle is 18/45=0.40 or 40%.
**Mathematics move.** Compare several complete cycles and weight by total time rather than averaging percentages from unequal windows.
**Boundary and transfer.** Short observations can be misleading when defrost or recovery cycles occur. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 7: Turn duty cycle into energy
At 120 W while running and 3 W in standby, a 24-hour model with 40% duty uses 120×9.6+3×14.4=1,195 Wh≈1.20 kWh.
**Mathematics move.** Separate power states, multiply by hours and add. Compare with a single average-power calculation.
**Boundary and transfer.** Actual units may have fans, heaters, controls and variable-speed compressors, so two states can be too simple. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 8: Fit a cooling curve
Use teacher-supplied compartment temperatures after a door closes. Plot excess temperature above the eventual baseline on a logarithmic axis.
**Mathematics move.** If the plot is roughly linear, estimate a time constant; otherwise use a piecewise description.
**Boundary and transfer.** Compressor switching violates a simple passive exponential, so residuals matter. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 9: Locate temperature gradients
Place hypothetical sensors on upper shelf, lower shelf, door and rear wall and compare simultaneous readings.
**Mathematics move.** Calculate range and mean absolute deviation, then map values with colours without claiming continuous coverage.
**Boundary and transfer.** Sensor accuracy, airflow and loading affect the pattern; four points do not describe every location. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 10: Compare thermal mass
Model an empty compartment and one containing sealed water bottles using lumped heat capacity.
**Mathematics move.** More thermal mass changes the temperature response to a brief heat input, but also requires energy to cool initially.
**Boundary and transfer.** Do not recommend loading practices from this model; follow appliance and food-safety guidance. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 11: Audit a kWh-per-year label
Divide an illustrative 300 kWh/year by 365 to obtain 0.822 kWh/day, then by 24 to obtain 34.2 W average input.
**Mathematics move.** Explain why average input is not compressor running power. Use annual energy only within its stated test method.
**Boundary and transfer.** Climate, use and settings can make household consumption differ from a test result. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 12: Model a damaged seal as resistance
Represent wall, seal and air leakage as parallel heat paths. Increasing seal conductance raises total UA.
**Mathematics move.** Use a simple two-path example and show why the easier heat path can dominate even if its area is small.
**Boundary and transfer.** A mathematical symptom does not diagnose a real appliance; repair and electrical work require competent service. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 13: Study uncertainty in duty cycle
For manual observations rounded to the nearest 30 seconds, calculate best- and worst-case run time across six switch events.
**Mathematics move.** Report an interval for duty cycle and show how a longer observation reduces relative timing uncertainty.
**Boundary and transfer.** Controller logs can still have sampling and timestamp errors; more data does not erase systematic bias. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 14: Compare fixed-speed and variable-speed models
Draw a square-wave power trace for on/off control and a smoother trace for modulating control with the same daily energy.
**Mathematics move.** Compare peak power, average power and temperature variation separately. Do not equate a smoother trace with lower energy automatically.
**Boundary and transfer.** Real algorithms and hardware are proprietary and product-specific. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 15: Read the official test scope
Use the current eCFR Appendix A scope: annual energy in kWh/year, refrigerated volume and adjusted volume under a uniform test method.
**Mathematics move.** Explain why a standardised result supports comparison but does not reproduce one family’s room, door openings or food load exactly.
**Boundary and transfer.** Do not copy protected incorporated standards or claim regulatory compliance from a classroom calculation. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Investigation 16: Write the bounded conclusion
Combine heat-gain terms, thermostat events, duty cycle and daily energy in one systems diagram.
**Mathematics move.** Identify which conclusion survives different observation windows and which depends on the chosen thermal boundary.
**Boundary and transfer.** End with food-safety and electrical-safety boundaries and no promise of savings from an untested change. Finish by changing one input, recalculating, and explaining whether the conclusion stays stable.
Turning the Laboratory into a Strong Report
Select at least three different representations: a labelled mechanism diagram, a table with units and a graph or equation. Begin with the question, define the system boundary, show one worked calculation, compare a second case, and end with a limitation. A polished page is less valuable than a traceable chain from evidence to conclusion.
Ask a partner to reproduce one number using only the written method. If the result differs, inspect units, rounding, state definitions and selection rules before deciding anyone calculated badly. Reproducibility is a mathematical habit: it turns a private answer into evidence that can be checked and improved.