A graph can look persuasive before it is justified
A school group tests how an electromagnet lifts small steel washers when its current is adjusted under teacher supervision. Its worksheet records current in amperes and the number of washers held for a fixed two-second interval. The group sees an increasing trend and announces that doubling current always doubles the magnetic force. There is a useful pattern in the results, but the conclusion is stronger than the numbers permit. This Mathematics guide teaches students to read the data correctly, compare rates, recognise integer outcomes, make a decision under a stated threshold and avoid inventing a universal law.
The working order is uncomplicated: define the quantity, list the controlled conditions, calculate accurately, inspect the graph, test the proposed relationship and make a decision within the observed range. Students should separate a numerical result from an interpretation. A mean of 6.67 washers is an average of trial counts, not a claim that two-thirds of a washer physically hung from the coil in one trial.
This chapter follows Volume 0492 on current-produced magnetic fields. That article owns the Physics explanation: current direction, field patterns, compass evidence and the need to control the apparatus. This article owns a different task: a G3 Mathematics model of repeated discrete results, including graphs, ratios and a defensible threshold decision. It does not repeat the previous Physics lesson or substitute a fabricated law of electromagnetism.
Know the 2027 Mathematics paper before practising
The official SEAB 2027 G3 syllabus list, updated 26 October 2026, gives Mathematics code K310, with 4052 as the pre-2027 reference. The K310 syllabus places Number and Algebra, Geometry and Measurement, and Statistics and Probability within a larger emphasis on reasoning, communication and application. Modelling a school trial is an exercise in those skills rather than a promise that this specific experiment will appear in an examination.
K310 Paper 1 lasts two hours fifteen minutes, has about 26 compulsory short-answer questions, carries 90 marks and counts for 50%. Paper 2 also lasts two hours fifteen minutes and carries 90 marks, with nine to ten questions of varying length; the final question focuses on applying Mathematics to a real-world scenario. Paper 2 counts for the other 50%. The syllabus tells candidates to show essential working and normally give non-exact numerical answers to three significant figures, or angles in degrees to one decimal place, unless the question specifies otherwise.
Do not confuse the SEC Mathematics paper with the Physics practical. The group experiment here supplies data for mathematical reasoning. In an actual Mathematics question, use exactly the units, measurements, thresholds and definitions the question supplies. In an actual Physics investigation, the correct scientific controls and interpretation still matter. These are complementary but separate examinations.
Definition lock: the quantities are not interchangeable
The independent variable in the proposed investigation is the current, measured in amperes (A), assuming that the teacher deliberately sets it. The outcome here is the number of identical steel washers held for two seconds. It is a non-negative integer count, not magnetic flux density or magnetic force in newtons. The repeated trial count is a proxy for performance under the specified apparatus and loading procedure.
Current, magnetic field, lifting force and washer count may be connected, but they are not synonyms. Magnetic properties, contact area, washer size, core shape, heating and the success criterion can affect the count. A Mathematics question need not prove the entire Physics mechanism, but it must preserve the meaning of each column.
To claim direct proportion between two measured quantities, the student must have more than a generally increasing graph. In the ideal mathematical model y = kx, the ratio y/x is constant for every nonzero x and the graph is a straight line through the origin. A line with a nonzero intercept, y = mx + c, is linear but is not a direct-proportion model when c is not zero.
The fictional trial table
At currents 0.20 A, 0.30 A, 0.40 A, 0.50 A and 0.60 A, the repeated washer counts are respectively (2, 3, 2), (4, 4, 3), (5, 6, 5), (7, 7, 6) and (8, 8, 7). Every group uses the same coil, core, washer specification, power supply arrangements, contact geometry and two-second holding time. These numbers are invented teaching data; they are not readings from an actual electromagnet or a laboratory specification.
The means are 7/3 = 2.33, 11/3 = 3.67, 16/3 = 5.33, 20/3 = 6.67 and 23/3 = 7.67 washers, rounded to two decimal places only for display. A full calculation should retain the exact fraction or calculator result until the final step. The ranges of the three repeated values are all 1 washer, because the maximum and minimum differ by one in every row. An identical range does not mean identical relative uncertainty or identical means.
Mean washer count divided by current gives approximately 11.7, 12.2, 13.3, 13.3 and 12.8 washers per ampere. Those are not constant. The dataset therefore does not establish exact direct proportionality. Even if they had been very similar, an experiment with integer counts and limited currents would justify at most a tested-range approximation, not an unlimited physical law.
Suppose the school requires an acceptable design to hold at least seven washers in every repeat, using the smallest of the five tested currents. The row at 0.50 A has counts 7, 7 and 6, so it fails. The row at 0.60 A has counts 8, 8 and 7, so it qualifies. On the stated dataset, 0.60 A is the smallest tested setting meeting the condition. This is not permission for a student to set a real current without teacher approval; the numerical task is a decision under given conditions.
Mean, median and range answer different questions
The mean uses every observation and is useful as a centre, but it can produce a fractional count even when every trial was an integer. The median is the middle value after sorting three readings: at 0.50 A, the ordered counts 6, 7, 7 give a median of 7. The range is maximum minus minimum: 7 − 6 = 1. They are different summaries of the same data, not competing guesses about the correct result.
A threshold such as ‘at least seven in every repeat’ cannot be replaced by ‘mean at least seven’. The former requires the minimum of each trial group to reach seven. For 0.50 A, the median is seven but the minimum is six. One can meet the median criterion and fail the all-trials criterion. Always translate the wording into the right mathematical condition before calculating.
If the question instead says ‘the mean must be at least 6.5’, the 0.50 A row qualifies because 20/3 is about 6.667. A change in wording can change the decision without any change in the data. The student’s job is to honour the condition set, not to choose the easier statistic.
How to build a readable graph
Place current on the horizontal axis if it is the independent variable; place the trial outcome or its stated summary on the vertical axis. Write units in the axis labels: ‘Current, I (A)’ and ‘Mean washers held (washers)’. Plot the actual calculated mean values carefully and use a scale that covers all data. If the question supplies a graph grid, follow that grid and label the numeric intervals clearly.
Do not draw a single best-fit straight line and then claim it proves direct proportion. A trend can be represented by a line or curve for descriptive purposes, but the relationship is a model requiring checks. Ask whether the line passes reasonably near points, whether there is a meaningful intercept, whether scatter is compatible with the measuring process, and whether extrapolation is justified.
A residual is the difference between an observed outcome and a model prediction. In a simple classroom check, suppose someone proposes mean washers = 13 × current. At 0.20 A the prediction is 2.6, while the observed mean is 7/3, about 2.333, so the residual observed minus predicted is approximately −0.267. At 0.60 A the prediction is 7.8, while the observed mean is 23/3, about 7.667, so the residual is about −0.133. A small residual is not automatically proof of a universal model, and residuals at all tested currents should be considered.
Difference, ratio and percentage: choose the right language
The absolute increase from a mean of 7/3 washers at 0.20 A to 23/3 at 0.60 A is 16/3, or about 5.33 washers. The ratio of final to initial mean is (23/3)/(7/3) = 23/7, about 3.29. The current itself increased by a factor of three. This mismatch is a direct warning against stating that washer count is exactly proportional to current across these readings.
The relative increase in mean count is (23/7 − 1) × 100%, about 229%. The current’s relative increase is (0.60 − 0.20)/0.20 × 100% = 200%. Do not confuse a final value of 329% of the original with an increase of 229%. Always state the base from which the percentage is measured.
When discussing physical implications, keep the units distinct. The gradient of a graph with washer count on the vertical axis and current in amperes on the horizontal axis has units of washers per ampere. A gradient in washers per ampere is not the magnetic field strength. It is simply a rate of change for this particular chosen outcome.
An answer structure for longer real-world problems
Read the decision rule before completing a table. Mark words such as ‘all’, ‘at least’, ‘not more than’, ‘smallest’, ‘mean’ and ‘tested’. These are mathematical operators in ordinary English. If a rule requires every repeat to pass, calculate the row minimum. If it requires the lowest mean cost, calculate the mean and compare feasible options. If it limits current to a maximum, exclude any row above the limit before optimizing a second quantity.
Write a concise chain of working: define the variable, select the rows, calculate the correct summary, test the inequality, compare feasible settings and state the result with its restriction. A final sentence such as ‘Of the five supplied settings, 0.60 A is the smallest that meets at least seven washers in every recorded trial’ is stronger than ‘0.60 is best.’ It reveals the comparison set and the decision rule.
This approach continues the reasoning on the Mathematics learning index and connects with Volume 0324’s two-point graph model. The earlier volume owns basic two-point recovery of gradients and intercepts. This one teaches model rejection, repeated integer outcomes and decision thresholds in a magnetic-test dataset.
What a credible answer should not do
Do not treat a dotted extension of a graph as an observed result. Do not calculate a ratio with a zero denominator. Do not round means before comparing close thresholds. Do not turn counts into force units without a valid measurement. Do not assume that the average is the same as the minimum. Do not call a near-straight trend a physical law. Those errors are independent: fixing one does not automatically fix the others.
The best examination habit is to read the constraint twice and the numbers once more after the conclusion is drafted. The words ‘on the supplied data’ are not a sign of weakness. They make the range of the answer clear. A decision grounded in five tested settings is not a claim about all possible settings.
Keep the PSLE foundation of evidence-limited conclusions in view: PSLE science: keep the claim inside the evidence. At SEC G3 level, extend that foundation with formal ratios, graphs, uncertainty, constraints and complete working.
Mathematics decision check 01: The missing axis unit
A graph has horizontal marks 0.2, 0.3 and 0.4, with no variable or unit. Its vertical axis shows the numbers 2 to 8.
The reader cannot tell whether the horizontal values represent amperes, seconds or coil turns. A numerical scale without a defined quantity has no scientific meaning. Label current in amperes and the outcome in washers; do not invent a secondary axis.
Try the correction yourself: Provide the two complete axis labels for the fictional trial and state which variable the teacher deliberately changes. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 02: The graph starts at 0.30 A
A classmate draws a trend line through only the last four rows and calls it a graph of the whole investigation.
The 0.20 A observation was omitted without explanation. A restricted plot may be legitimate if the range is explicitly stated, but it cannot silently stand for the entire dataset.
Try the correction yourself: Explain which evidence is excluded and how this changes any statement about the original five-setting decision. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 03: Three readings, one total
At 0.40 A the results are 5, 6 and 5. The student enters 16 as the mean.
Sixteen is the total across the three repeats, not their arithmetic mean. Divide 16 by 3 to obtain 5.33 recurring and interpret this as an average count. The total and mean answer different questions.
Try the correction yourself: Show the calculation with the denominator and a sensible final rounding. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 04: An average cannot hang
The learner rejects a mean of 6.67 because a coil cannot hold two-thirds of a washer.
An individual observation is a whole-number count; a mean of several counts need not be an integer. Fractional means summarise repeated outcomes without suggesting a fractional object was actually used.
Try the correction yourself: Write one sentence distinguishing the observed count from the computed mean. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 05: Median before sorting
For counts 7, 7 and 6, a learner selects the first number in the notebook as the median.
The median is the middle member after ordering the data; it is not automatically the first or last recorded value. Sort to 6, 7, 7 and identify 7 as the median.
Try the correction yourself: State the correct median and why its value does not determine whether every repeat passed. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 06: The range as a percentage
The range of 8, 8 and 7 is recorded as 1% because the maximum minus minimum is one.
Range inherits the units of the variable; here it is one washer, not one percent. A relative spread could be calculated separately, but it must specify a denominator and the reason for using it.
Try the correction yourself: Correct the unit and explain why the row’s range is only a descriptive spread. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 07: All means the minimum
The threshold is seven washers in every repeat, but the student accepts 0.50 A because its mean exceeds 6.5.
A mean threshold and a minimum threshold are not equivalent. One result of six washers fails the stated every-repeat rule. Test the row minimum against seven before accepting the setting.
Try the correction yourself: Write the inequality min(counts) ≥ 7 and apply it to 0.50 A. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 08: The smallest among tests
The class chooses 0.60 A and writes that no lower current anywhere could work.
Only five current settings were tested. The smallest observed qualifying setting is 0.60 A. Untested values between 0.50 A and 0.60 A cannot be classified from the table alone.
Try the correction yourself: Qualify the conclusion by naming the tested set and the missing interval. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 09: A ratio with zero current
Someone extends the data table to zero current and divides the mean washer count by zero to check proportion.
Division by zero is undefined. A direct-proportion model includes the origin as a theoretical feature, but its ratio test is applied only where the independent variable is nonzero. Do not invent a measurement at zero.
Try the correction yourself: Explain separately how an origin check and the nonzero ratio calculation are used. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 10: Constant differences, changing ratios
A fabricated table rises by three washers when current rises by 0.10 A each time. The learner calls that direct proportion automatically.
Constant change over equal horizontal steps is evidence of a linear relationship. Direct proportion additionally requires a zero intercept. A line can rise by a constant amount and still begin above the origin.
Try the correction yourself: Construct y = 30I + 2 as an example of linear but not direct-proportional behaviour. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 11: Changing gradient sign
The points rise from left to right, but the learner reports a negative gradient because the subtraction order was mixed.
For gradient (y2 − y1)/(x2 − x1), the numerator and denominator must use the same ordering. A rising straight-line segment has a positive gradient when x increases.
Try the correction yourself: Calculate a positive secant gradient between the 0.20 A and 0.40 A mean observations. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 12: A model calculated from one point
One data pair is used to declare y = 12I for all five settings.
A single nonzero point can suggest a constant k under an assumed direct-proportion model, but cannot test the assumption. Use every row and compare ratios or residuals before accepting any range-limited approximation.
Try the correction yourself: Test y = 12I against the 0.60 A row and record observed minus predicted. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 13: A missing experiment control
The higher-current tests also use a different coil core.
The dataset then mixes current and core changes. The arithmetic may still be correct for the recorded rows, but the scientific claim about current alone is not justified. Identify the changed control in the interpretation.
Try the correction yourself: Separate a mathematical association from a causal conclusion. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 14: A rounded mean at a boundary
A mean is 6.666…, and the rule is at least 6.67. The student rounds early to 6.67 and declares success.
A rounded display value can cross a close threshold. Compare the unrounded mean 20/3 with 6.67; it is slightly below that exact decimal. Preserve full precision until the decision is made.
Try the correction yourself: Show the inequality using a fraction or calculator precision. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 15: The threshold changes halfway
The worksheet begins with ‘at least seven each time’ and ends with ‘on average at least seven’.
The decision rule has changed, not the evidence. This is a reading and Mathematics error, and it can reverse which setting is chosen. Rewrite the problem statement in symbols before calculating.
Try the correction yourself: Identify which function of the repeats—minimum or mean—belongs in each rule. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 16: Relative increase confused
The washer mean changes from 7/3 to 23/3. A student says this is a 329% increase.
The new mean is about 329% of the old mean; the increase itself is about 229%. Subtract one from the final-to-initial factor before multiplying by 100 to obtain an increase.
Try the correction yourself: Write the difference between ‘as a percentage of’ and ‘percentage increase’. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 17: A trendline forced through origin
The class forces every best-fit line through (0,0) because magnetic effects begin when current flows.
The observed washer count is a proxy with a threshold, handling effects and an untested zero-current row. Its graph need not obey the same idealised relationship as a simple theoretical field model. Let evidence and the problem’s instructions determine the fit.
Try the correction yourself: Explain why a physically plausible origin alone does not validate a zero-intercept data model. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 18: Scale disguises a difference
A vertical axis spans zero to ten thousand washers, so all five results appear on the bottom line.
A mathematically valid but unhelpfully wide scale hides differences that the question asks the learner to compare. Choose a readable scale covering the actual counts while preserving honest labels and spacing.
Try the correction yourself: Select a suitable vertical range and describe how to show points at fractional means. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 19: A bar chart drawn like a continuous law
The student joins all individual washer counts from different repeats into one zig-zag line against current.
The graph mixes repeats and can suggest a continuous physical trajectory when the points are separate observations. Plot separate repeat series or the stated summary, with clear labels and matching current values.
Try the correction yourself: Choose a representation appropriate to comparing mean performance across current settings. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 20: The wrong graph goes into the answer
The task asks for a plot of mean count, but the student plots only the highest of the three repeats.
Selecting maximum rather than mean changes the outcome definition and can conceal unreliable lower trials. Calculate the mean for every row before plotting the requested values.
Try the correction yourself: State one reason the maximum can be a poor substitute for all-repeat reliability. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 21: A gradient with no units
The learner obtains a gradient of about 12.5 and leaves it as a pure number.
If the vertical axis is washers and the horizontal axis amperes, the gradient is expressed in washers per ampere. Units preserve the meaning of the model and prevent confusion with actual magnetic field measurements.
Try the correction yourself: Supply the rate’s units and identify what the rate measures. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 22: A data point deleted
The 0.50 A repeat of six is erased because the other two repeats were seven.
An inconvenient observation is evidence unless a documented measurement fault justifies exclusion. Deleting the low result turns a failing every-repeat row into a misleading pass. Keep it and investigate its cause.
Try the correction yourself: Explain how deletion would alter the row minimum and hence the decision. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 23: An outlier declared by dislike
One result differs by only one washer, so the learner calls it an outlier without an agreed rule.
With three discrete observations, a one-count difference does not automatically establish an outlier. Describe the variability, repeat under the same procedure and disclose any defensible exclusion rule.
Try the correction yourself: Write a neutral note that reports the discrepancy without prejudging its cause. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 24: Current as a ratio denominator
The mean count divided by current varies from roughly 11.7 to 13.3, and the student says the ratios are identical.
Approximate values can be near without being mathematically equal. The proposed direct-proportion model is not exact for these figures. State whether an approximate model might still be useful for a limited predictive purpose.
Try the correction yourself: Compare two ratios numerically and use the words ‘approximately’ and ‘tested range’ correctly. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 25: Unjustified future setting
A line is extended to 1.50 A, and the group recommends that setting without any more tests.
The recommendation extrapolates beyond 0.60 A and ignores the teacher’s operating constraints. Mathematical extrapolation may be calculated in a textbook exercise, but it is not permission or measured evidence of performance.
Try the correction yourself: Separate what a formula predicts from what the fictional experiment actually established. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 26: An optimum without constraints
The group selects the greatest current because the mean washer count is highest.
If the question asks for smallest current that reliably holds seven washers, the objective is not maximum count. Optimization depends on the stated criterion, and unknown safety considerations cannot be assumed away.
Try the correction yourself: Write the objective and constraint in two separate lines before making the selection. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 27: Comparison across different washers
Trial A uses small washers, Trial B large ones, but the final table treats their counts as comparable.
A count of objects can lose meaning when object size or mass changes. Repeated counts require the same unit specification to support fair comparisons. The numerical operation can be valid while the empirical comparison is poor.
Try the correction yourself: Propose a controlled washer specification and state a limitation if it was not held fixed. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 28: Cumulative total mistaken
The student adds the number of washers held across all five currents and claims the result describes one coil setting.
The sum describes the entire notebook under five different conditions. It is not a characteristic of any single setting, and it depends on how many trials were run. For a setting decision, compare each row separately.
Try the correction yourself: Explain why aggregate totals across different experimental conditions are usually not the desired statistic. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 29: Exact value hidden by presentation
The data sheet shows 6.67 as though it were an exact measured mean, with no indication that rounding occurred.
The exact mean of 7, 7 and 6 is 20/3. A displayed 6.67 is a convenient approximation. Preserve the distinction when testing thresholds close to that value.
Try the correction yourself: Show exact and rounded forms side by side and identify which should govern a close comparison. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 30: One independent variable becomes two
The group decides to increase turns and current together and then draw one simple current-count scatter graph.
A scatter graph can show association with current, but turns were also changing. The graph does not isolate how current affects the result. Maintain a control table or test the two factors in separate comparisons.
Try the correction yourself: Write one revised two-stage plan: a current comparison with fixed turns, then a turns comparison with fixed current. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 31: Graph direction mistaken for proof
A rising line prompts the sentence ‘current causes an exact rise in magnetic field, confirmed by our count data’.
The count data concern held washers, not a direct magnetic-field measurement. Controlled variation offers a useful empirical comparison but does not turn a proxy into an exact field formula. State the observed quantity and the inference separately.
Try the correction yourself: Rewrite the sentence using ‘in these trials’, the tested range and the actual dependent variable. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 32: An axis starts above zero
A display chart starts its washer-count axis at five to emphasise the differences between 0.40 A and 0.60 A.
Truncated axes can be legitimate when clearly marked, but the visual impression can exaggerate relative differences, especially with bars. Interpret values using the labelled scale, not bar height alone.
Try the correction yourself: Describe a fair presentation method and what the reader must be told about the baseline. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 33: An unlabelled feasible set
A solution checks 0.60 A but never shows why the preceding tested setting fails.
A correct optimization answer needs a comparison. State that 0.50 A has a minimum count of six and fails the all-repeat threshold; then show that 0.60 A has a minimum of seven.
Try the correction yourself: Write a two-row feasibility check that makes the decision transparent. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Mathematics decision check 34: An arithmetic fix that changes evidence
A student replaces 7,7,6 by 7,7,7 to make the mean a neat whole number.
Changing a recorded value is not mathematical simplification. The correct mean is 20/3. If a value was a proven transcription error, correct it with an audit note; otherwise keep the observation.
Try the correction yourself: Explain why preserving raw data matters even in a fictional textbook exercise. In a marked response, show the relevant calculation or decision inequality, name its units where they exist and finish with the smallest conclusion that the supplied data justify. Do not spend marks on a more ambitious claim that the numbers cannot support.
Five worked examination-style questions
Worked question 1. Calculate the mean and range at 0.30 A. The readings are 4, 4 and 3. Their sum is 11, so the mean is 11/3 = 3.67 washers to three significant figures. The range is 4 − 3 = 1 washer. Both values describe the same row, but only the minimum of three controls a rule requiring every trial to reach a threshold.
Worked question 2. Compare the current factor and the average-count factor between 0.20 A and 0.60 A. The current factor is 0.60/0.20 = 3. The mean-count factor is (23/3)/(7/3) = 23/7, about 3.29. They are not equal, so the measured counts are not exactly directly proportional to current across these two rows. No claim about all possible currents follows.
Worked question 3. Find a gradient between 0.30 A and 0.50 A using the means. The mean difference is 20/3 − 11/3 = 3 washers. The current difference is 0.20 A. The secant gradient is 3/0.20 = 15 washers per ampere. This is a rate for the two selected mean data points, not a guaranteed global gradient.
Worked question 4. A school rule allows only settings not exceeding 0.50 A. It also requires at least seven washers in every repeat. Which row qualifies? None. Although 0.50 A has two sevens, its third reading is six. The 0.60 A row meets the holding condition but violates the current limit. A correct answer can be ‘no feasible tested setting’ when the constraints conflict.
Worked question 5. A teacher changes the rule to a mean count of at least 6.5 washers and permits the five tested settings. The smallest qualifying current is 0.50 A, because its mean is 20/3, about 6.667, while the mean at 0.40 A is 16/3, about 5.333. The changed result illustrates why the original wording cannot be replaced by a convenient interpretation.
Revision route and conclusion
Use one session to reproduce the table of means by hand, one to draw and label its graph, one to solve the two different threshold rules, and one to diagnose ten of the model errors above. Mark each mistake by type: computation, units, graphing, interpretation or constraint. Repair only the first weak link before repeating the hardest task.
A good Mathematics answer respects what the numbers mean. It does not treat a washer count as a field measurement, it does not confuse an increasing trend with exact direct proportion, and it does not declare an option best without reading the selection rule. Those habits transfer to finance tables, science readings, schedules and the real-world problem section of K310.
The next volume applies the same evidence discipline to English situational writing. Students will have to address a particular audience and purpose, select only the observations that support the message and avoid publishing an elegant but misleading claim about the demonstration.
