How to compare two gas-sensor trials without comparing the wrong numbers
Two gas-sensor readings can be individually correct and still support a wrong comparison. Suppose one trial starts at 430 parts per million and ends at 690, while another starts at 500 and ends at 740. The second ends higher, but the first increases more. The difference between final values is not the difference between increases. In G3 Mathematics, that distinction is more important than the ability to press a subtraction key. It determines which quantity you calculate, how you explain it, and whether a practical recommendation matches the evidence.
This guide moves from a Biology demonstration of carbon dioxide to a mathematical dataset. You will read paired measurements, find absolute changes, calculate percentage changes with the correct baseline, compare groups with a fair summary, interpret graph axes, decide when a simple model is reasonable, and write answers that retain units and relevant limitations. You do not need specialist chemistry to answer the mathematics: the sensor values are an invented context for rates, percentages, statistics and sound reasoning.
Quick route: identify whether the question asks for final reading, increase or relative increase; keep each final reading paired with its own baseline; subtract in a consistent direction; divide by the starting value when finding percentage change; use the original readings for checking; compare like with like; label the graph; and explain any limitation before drawing a conclusion beyond the collected data.
What 2027 G3 SEC Mathematics actually requires
The SEAB 2027 G3 catalogue lists Mathematics as K310, with 4052 as the earlier reference code. The official 2027 K310 syllabus has two written papers. Paper 1 lasts 2 hours 15 minutes, carries 90 marks and is weighted 50 percent; Paper 2 also lasts 2 hours 15 minutes, carries 90 marks and is weighted 50 percent. Paper 2 has 9 to 10 questions, with the final question focused on applying mathematics to a real-world scenario. All questions in both papers are compulsory. The syllabus stresses essential working, correct interpretation, meaningful units and appropriate accuracy.
The official instructions specify non-exact numerical answers to three significant figures, or one decimal place for angles, unless a question specifies otherwise. In a teaching explanation, you may carry more digits internally to avoid rounding errors, then present the final answer to the accuracy requested. Sensor data here are not a new prescribed content topic. They provide a real-world setting for familiar mathematics. There is no separate official EMS paper; Mathematics is assessed as its own G3 subject.
Read the paired dataset before calculating
An invented school-science activity compares two labelled trial conditions. The instrument displays carbon dioxide concentration in parts per million, abbreviated ppm. For this mathematical exercise, each trial has a baseline reading taken before a specified change and a final reading taken afterward. These figures are illustrative, not measurements of a real classroom or a health assessment.
Condition A, trial 1: baseline 430 ppm, final 690 ppm. Condition A, trial 2: baseline 440 ppm, final 720 ppm. Condition A, trial 3: baseline 450 ppm, final 750 ppm. Condition B, trial 1: baseline 500 ppm, final 740 ppm. Condition B, trial 2: baseline 510 ppm, final 760 ppm. Condition B, trial 3: baseline 520 ppm, final 780 ppm. The labels are names for experimental conditions; they do not indicate who supplied gas, what process occurred or which condition is safer.
For each pair, calculate final minus baseline. Condition A increases are 260, 280 and 300 ppm. Condition B increases are 240, 250 and 260 ppm. The final readings for B are all higher than those for A, yet the mean increases are 280 ppm for A and 250 ppm for B. This is not contradictory. B begins from a higher baseline. The exact question determines whether final concentration or change from baseline is the relevant statistic.
Why subtraction is a model choice, not merely a button press
Subtracting the baseline answers a question about what changed between two points. It does not automatically isolate the effect of a named cause. For that stronger claim, you would also need a suitable design that controls other factors. If the prompt asks only for change, use final minus initial. If it asks which reading is largest at the end, compare the final values directly. Do not decide that ‘higher’ always refers to one or the other.
An easy checking identity is baseline + change = final. For A2, 440 + 280 = 720 ppm. For B3, 520 + 260 = 780 ppm. If the sum does not reproduce the stated final reading, revisit the subtraction. A negative result is also possible; it means the final value is lower than the baseline under the chosen direction of subtraction, not automatically that the instrument is broken.
Pairs must remain intact. If one student sorts all baselines from smallest to largest and independently sorts final readings, some pairs may cease to match their original trials. Subtracting these rearranged values produces arithmetic with no valid experimental meaning. The row identifier is part of the data, just as the unit is part of the value.
Percentage change needs the initial value as its denominator
Percentage change is (final minus initial) divided by initial, multiplied by 100 percent. For A1, the increase is 260 ppm and the baseline is 430 ppm. Hence 260/430 × 100 = approximately 60.5 percent. For B1, the increase is 240 ppm and the baseline is 500 ppm, giving 48.0 percent. Condition A1 has a slightly smaller final reading than B1 but a larger absolute and relative rise.
The denominator matters because a change of 100 ppm can mean very different relative changes. If the baseline were 400 ppm, it would be 25 percent; if the baseline were 800 ppm, it would be 12.5 percent. Neither percentage can be inferred from the change alone. When the initial value is zero, the ordinary percentage-change calculation is undefined; do not report an infinite or convenient 100 percent by habit.
A percentage-point change is not the same as a percentage change. If a survey share moved from 20 percent to 25 percent, that is five percentage points or a 25 percent relative increase. Our gas-sensor readings are expressed in ppm, not percentage share of pupils, so describe a 260 ppm increase as such. Use percent only when the correct ratio has been calculated.
Compare groups fairly
Condition A’s mean final reading is (690 + 720 + 750)/3 = 720 ppm. Condition B’s mean final reading is (740 + 760 + 780)/3 = 760 ppm. Comparing those means says that B finished higher in this small invented dataset. By contrast, A’s mean change is (260 + 280 + 300)/3 = 280 ppm and B’s mean change is (240 + 250 + 260)/3 = 250 ppm. Comparing those means says A rose more. Both conclusions can be true at the same time because they answer different questions.
The mean is not always the only relevant summary. The middle change in A is 280 ppm and in B is 250 ppm, so their medians agree with their mean increases in this example. A’s range of changes is 300 − 260 = 40 ppm; B’s is 260 − 240 = 20 ppm. In these three observations, B’s changes are more tightly clustered by the range measure. A few observations are not enough to claim that one method always gives more stable values in every setting.
If groups contain different numbers of trials, avoid averaging two group means without accounting for group sizes. For example, an average of 280 ppm from three trials and 250 ppm from seven trials does not combine to (280 + 250)/2 = 265 ppm. The overall mean is (3 × 280 + 7 × 250)/10 = 259 ppm. The weights are the numbers of observations in each group. The question may not require that combined mean, but you should recognise when one is needed.
Interpret graphs rather than believing their appearance
A bar chart of the final reading and a bar chart of the change are different graphs. Both need a title that identifies which statistic is plotted, category labels, a readable scale and the unit ppm. A bar chart that begins its vertical axis at 650 ppm may make a small difference in final values look dramatic. A student should inspect the scale before turning visual distance into an argument about a large practical effect.
A line graph can show how a reading changes over time, but only if the x-axis is genuinely time and the points are ordered correctly. Connecting three separate trial identifiers as though they were successive time points may imply a process that was not recorded. If the question gives paired baselines and finals without elapsed times, do not invent a time-gradient or quote ppm per minute. A rate needs a measured interval.
Suppose a poster labels the 720 ppm average as ‘720%’. This is a unit and meaning error, not just presentation. The value is a concentration reading measured in parts per million. The percentage rise for each trial is a calculated ratio using a particular starting value. Clear unit labels prevent those two ideas from merging.
The invented classroom threshold
Imagine the teacher defines an arbitrary practice threshold of 700 ppm solely so students can learn inequalities and classification. It is not an air-quality standard or medical limit. In Condition A, two final readings, 720 and 750 ppm, exceed 700 ppm; 690 ppm does not. In Condition B, all three final readings exceed 700 ppm. A statement that ‘all B finals are above the practice threshold’ is true for this data, while ‘all A finals are above it’ is false.
If the rule is ‘at least 700 ppm’, 700 exactly is included. If it is ‘greater than 700 ppm’, 700 is excluded. The wording of the inequality matters. Similarly, when the threshold applies to change rather than final value, the result changes completely: none of the six increases reaches 700 ppm. Always attach the threshold to the variable it is supposed to test.
A threshold count of 2/3 for A and 3/3 for B describes six invented final observations, not a general probability of future readings crossing the threshold. The fractions do not become population probabilities simply because they were expressed as percentages. To generalise to a wider setting, the question would need relevant assumptions about sampling and conditions.
A complete model response
Original question: ‘For A1 and B1, calculate the percentage increase in concentration and state which trial had the greater relative increase. Explain why comparing final readings alone would not answer the question.’ Model working: A1 change = 690 − 430 = 260 ppm; percentage increase = 260/430 × 100 = 60.5% to three significant figures. B1 change = 740 − 500 = 240 ppm; percentage increase = 240/500 × 100 = 48.0%. A1 has the greater percentage increase. The final values alone ignore the different initial readings and therefore cannot establish the relative change.’
The explanation earns its place because the numerical comparison is not meaningful without knowing what each percentage is relative to. It also states the matching denominator and checks a common distractor: the trial with the larger final value need not have the larger increase. In a timed paper, concise correct working is better than a long paragraph that leaves the actual question unanswered.
Twenty-six original worked transfer questions
Worked question 1: Find a change
Question. A1 has baseline 430 ppm and final 690 ppm. Solution. Subtract final minus baseline: 690 − 430 = 260 ppm. Write the unit. The result answers how much the reading increased, not what concentration was reached at the end. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 2: Reverse a baseline
Question. A reading finishes at 740 ppm after increasing by 240 ppm. Solution. Initial reading = 740 − 240 = 500 ppm. Check 500 + 240 = 740. The word ‘increasing’ fixes the sign and prevents addition of the change to the final value. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 3: Reverse a final
Question. A trial begins at 450 ppm and rises by 300 ppm. Solution. Final reading = 450 + 300 = 750 ppm. A clear statement makes it impossible to confuse the final concentration with the increase. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 4: Read a decrease
Question. An invented trial begins at 720 ppm and ends at 680 ppm. Solution. Final minus initial = −40 ppm, or a decrease of 40 ppm. Choose one consistent description. A negative change is not automatically evidence that the data were recorded in error. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 5: Calculate a percentage rise
Question. Trial B2 changes from 510 to 760 ppm. Solution. Increase = 250 ppm. Percentage increase = 250/510 × 100 = 49.0% to three significant figures. Do not divide by 760, because the question asks for change relative to the starting value. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 6: Compare endpoints
Question. Which of A3 and B3 finishes at the higher concentration? Solution. B3 finishes at 780 ppm and A3 at 750 ppm, so B3 is higher by 30 ppm. This compares final values only; it makes no claim about which experienced the larger rise. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 7: Compare gains
Question. Which of A3 and B3 experiences the larger absolute increase? Solution. A3 increases by 300 ppm, B3 by 260 ppm, so A3 has the larger rise by 40 ppm. It is possible for a trial to finish lower yet increase more because it began lower. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 8: Find a group mean
Question. What is the mean increase for Condition A? Solution. The changes are 260, 280 and 300 ppm. Mean = 840/3 = 280 ppm. Average values of the original final readings would answer a different question. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 9: Find a second mean
Question. What is the mean final reading for Condition B? Solution. The finals are 740, 760 and 780 ppm. Their sum is 2280 ppm; mean = 760 ppm. State ‘mean final reading’ so the result cannot be mistaken for mean change. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 10: Find the range
Question. Condition B has increases of 240, 250 and 260 ppm. Solution. Range = 260 − 240 = 20 ppm. Range measures spread between the extremes in this dataset; it does not identify the cause of the spread. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 11: Find the median
Question. Condition A increases are 300, 260 and 280 ppm in the recorded order. Solution. Sort as 260, 280, 300. The middle value is 280 ppm. Do not choose the second recorded value as a rule unless it happens also to be the ordered middle. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 12: Use a weighted mean
Question. Three trials average 280 ppm increase and seven other trials average 250 ppm. Solution. Combined mean = (3 × 280 + 7 × 250)/10 = 259 ppm. The arithmetic average of 280 and 250 ignores that the groups contain different numbers of observations. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 13: Avoid a false rate
Question. A final reading is 780 ppm and an initial reading is 520 ppm; no time interval is supplied. Solution. The increase is 260 ppm. You cannot calculate ppm per minute without a measured elapsed time. A rate requires a change and a corresponding duration. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 14: A true rate
Question. A sensor reading rises by 260 ppm over 20 minutes and a constant-average-rate model is requested. Solution. Average rate = 260 ÷ 20 = 13 ppm per minute. This is a mathematical average over the interval; it does not prove the sensor changed at exactly 13 ppm every minute. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 15: Relative versus absolute
Question. A rises by 260 ppm from 430; B rises by 240 ppm from 500. Solution. The absolute increases differ by 20 ppm. The percentage rises are approximately 60.5% and 48.0%, a difference of about 12.5 percentage points after rounding. Be clear about which comparison is requested. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 16: A zero baseline
Question. An invented device reads zero initially, then 100 units afterward. Solution. The absolute increase is 100 units, but ordinary percentage increase would require division by zero and is undefined. Report the absolute change and explain why a percentage cannot be calculated in the usual way. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 17: A swapped pair
Question. A student pairs B1’s final of 740 ppm with A1’s baseline of 430 ppm. Solution. That 310 ppm difference is not a valid within-trial change. Restore the original pairing: B1 goes with 500 ppm and A1 with 690 ppm. Identifiers preserve mathematical meaning. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 18: Wrong axis
Question. A poster displays final ppm values on a vertical axis titled ‘percentage change’. Solution. The label is wrong. Rename the axis ‘Final concentration (ppm)’ or recalculate percentage changes using individual baselines. An attractive visual cannot replace the underlying quantity. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 19: Truncated scale
Question. A bar chart begins at 700 ppm and visually magnifies the difference between 740 and 780. Solution. Read the numbers and scale rather than judging bar height alone. The actual difference is 40 ppm. If the chart communicates comparison poorly, change the axis presentation and clarify the practical question. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 20: Apply a threshold
Question. A practice rule says final concentration must be strictly above 700 ppm. Does A1 pass? Solution. A1 finishes at 690 ppm, so it does not meet the rule. A2 at 720 and A3 at 750 do. The threshold is a fictional teaching rule, not a safety guideline. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 21: Equality boundary
Question. A practice reading is exactly 700 ppm. The condition is ‘at least 700 ppm’. Solution. It qualifies because ‘at least’ means greater than or equal to. It would not qualify under ‘greater than 700’. Translating words to inequalities protects boundary cases. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 22: Count valid cases
Question. Two of three A finals exceed the invented 700 ppm threshold. Solution. The observed proportion is 2/3, about 66.7%. That percentage describes these three readings only. Do not infer a probability for every future reading unless the sampling assumptions support it. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 23: A calibration offset
Question. Every reading from one sensor is 35 ppm too high by a known constant offset. Solution. If both the baseline and final are from that same sensor under that same fixed offset, subtracting the two cancels the offset in the paired change. It does not necessarily fix all calibration errors or errors from different sensors. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 24: Sensor drift
Question. An instrument’s offset changes between the initial and final measurement. Solution. The error may no longer cancel in subtraction. The mathematical calculation of recorded change remains possible, but the interpretation as true change becomes less certain. A checking or calibration procedure is relevant. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 25: Missing baseline
Question. A row has final 760 ppm but its initial reading is absent. Solution. Its final value may be compared with other finals if appropriate, but its individual change and percentage change cannot be computed from the given data. Do not silently borrow a baseline from another trial. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
Worked question 26: A conclusion with limits
Question. A group’s mean increase is 280 ppm while another’s is 250 ppm. Solution. It is correct to say the first group has a higher mean observed increase by 30 ppm in this invented dataset. It is not correct to claim the condition always produces more carbon dioxide, let alone to make an air-health judgement, without suitable evidence. Transfer check. Name the target variable, use the matching row or population, retain its unit and see whether the answer can be checked by substituting it into the original relationship. If a conclusion mentions a cause or a wider population, ask whether the data actually establish it.
A structured strategy for the extended real-world question
Read the final question of a practice paper in two passes. On the first, identify every named quantity and its unit, the question’s decision target and the constraints. On the second, decide which calculations are needed in what order. With paired sensor readings you may need to calculate individual changes, compare means, count threshold cases and then choose a justified statement. Avoid jumping to the final recommendation from an eye-catching chart.
Keep a small working table with columns for trial identifier, initial reading, final reading, absolute change and percentage change. This prevents the denominator from wandering between rows. After calculation, compare like quantities. If a question asks for the most economical design or least error, remember that the information given may not be sufficient to make that decision, even if some values appear smaller. A mathematically sound limitation is better than an invented fact.
Finally use one sentence of context. For example: ‘In these three invented Condition A trials, the mean rise was 280 ppm, 30 ppm greater than the 250 ppm mean rise for Condition B.’ This is a complete answer to a comparison question. It avoids declaring a health standard, causal mechanism or general policy that the numbers have not established.
Ten-minute practice without notes
Create a new row with a 460 ppm baseline and a 736 ppm final reading. The change is 276 ppm; the percentage increase is 276/460 × 100 = 60%. Then imagine the final reading had been 700 ppm instead. Recalculate the change as 240 ppm and the percentage rise as approximately 52.2%. Explain in words why the denominator remains 460 in both cases. A student who understands the relationship can make the change without relearning the whole method.
Now sketch two bar charts, one showing Condition A’s three final readings and another showing its three changes. Give them different titles and clear units. Write two correct conclusions that the separate charts support. Then write an incorrect conclusion that confuses the two, and explain the mistake. This reversal exercise is a powerful way to detect when you are reading a diagram by appearance instead of by quantity.
Continue through the canonical subject routes
This mathematical application follows Vol 0563 Biology exhaled-air evidence, but does not repeat its gas-exchange teaching. Return to Vol 0560 valid denominator reasoning if percentage populations are uncertain. The existing Vol 0210 misleading-graphs guide is the owner for the general graph-reading method, and Vol 0214 sampling and classification explains the wider foundations. Use the SEAB G3 syllabus catalogue for official codes and the K310 2027 syllabus for format and mathematical conventions. Every sensor reading and threshold here is invented for practice.
