7.3 Mathematical and data-presentation skills
- Syllabus
- 0625–2026–2027
- Topic
- 7.3
- Level
- —
Learning objectives
7.3.1Numerical and algebraic calculations• Use arithmetic, decimals, fractions, percentages, ratios, reciprocals, comparisons and standard form; round final answers appropriately; estimate; use indices, consistent units, formulas, direct and inverse proportion, mathematical models and Δ for changes in variables.7.3.2Geometry, measurement and unit conversion• Use geometry, circumference, areas, volumes, scale diagrams, Pythagoras’ theorem, angles, mathematical instruments, compass directions and metric unit conversions in physical contexts.7.3.3Graphs, charts and statistics• Draw and interpret graphs and charts; interpolate and extrapolate; determine gradients and intercepts; select scales and axes; use y = mx + c, recognise direct proportionality and calculate means.7.3.4Record readings and tabulated data• Read instruments to half a smallest division where appropriate; record precision, units, repeated readings and significant figures correctly; use quantity/unit table headings and express ratios as x : y.7.3.5Plot and analyse physical data• Plot variables on labelled axes with sensible scales; mark points accurately; draw and interpret best-fit lines or curves, anomalies, intercepts and gradients using suitably large triangles.7.3.6Extended trigonometry and curve gradients• Use sine, cosine, tangent and inverse trigonometric functions, and determine the gradient of a curve by drawing a tangent.
Carry out numerical and algebraic physics calculations
Physics calculations require arithmetic, decimals, fractions, percentages, ratios, reciprocals, standard form, estimation and algebra, all attached to physical quantities and consistent units.
| Step | Control |
|---|---|
| 1 | identify known and unknown quantities with symbols and units |
| 2 | convert to one consistent unit system |
| 3 | select or construct the physical equation |
| 4 | rearrange algebraically before substituting |
| 5 | calculate without rounding intermediate values |
| 6 | round the final result appropriately and check by estimation |
For direct proportion, doubling one quantity doubles the other; for inverse proportion, doubling one halves the other. A mathematical model must match the stated physical conditions.
The symbol Δ means a change: Δx = final x − initial x. It is not automatically the final value itself.
Write standard form as a × 10ⁿ with 1 ≤ |a| < 10. Preserve positive whole-number indices correctly in algebraic expressions.
Round only the final result unless an intermediate approximation is explicitly required. Early rounding can shift the final answer outside justified precision.
Use geometry, measurement and metric unit conversion
Physical geometry uses lengths, angles, areas, volumes and directions to model real objects and vector relationships.
| Shape or relation | Required result |
|---|---|
| circle | circumference = 2πr; area = πr² |
| rectangle/triangle | area = length × width; area = ½ × base × height |
| rectangular block/cylinder | volume = lwh; volume = πr²h |
| right-angled triangle | a² + b² = c² |
For a scale diagram, state the scale, draw each length and direction accurately with ruler and protractor, construct the resultant, then convert its measured length back to the physical value.
Metric conversion factors must be raised to match the dimension: 1 cm = 10⁻² m, so 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³.
Use N, S, E and W and specify clockwise or anticlockwise angles from a clear reference direction.
Do not apply a linear conversion factor unchanged to area or volume, and do not read a scale diagram before stating or identifying its scale.
Draw and interpret graphs, charts and statistics
Choose a graph or chart that represents the variables clearly, with each axis labelled by quantity and unit and with a scale suited to the data range.
| Feature | Meaning or method |
|---|---|
| gradient | Δy/Δx with units from the axes |
| y-intercept | y when x = 0, read or found by justified extension |
| y = mx + c | straight line with gradient m and intercept c |
| direct proportion | straight line through the origin |
| interpolation | estimate within the data range |
| extrapolation | estimate outside the range, with greater uncertainty |
| mean | sum of values divided by number of values |
A graph reveals direction, shape and proportionality. Distinguish a straight relationship with non-zero intercept from direct proportionality.
Joining every point does not necessarily represent the relationship. Use an appropriate best-fit line or curve and retain the evidence of scatter.
Record scale readings and tabulated data correctly
Where appropriate, read an analogue instrument to the nearest half of its smallest scale division, including interpolation between marks.
| Requirement | Correct practice |
|---|---|
| precision | decimal places reflect the instrument's detectable difference |
| units | include the unit in the quantity/unit heading, such as time/s |
| table body | record numbers only, without repeating units in cells |
| repeats | record every reading where repeats are appropriate |
| measured significant figures | match the instrument used |
| calculated significant figures | match the least precise raw quantity used |
| ratio | express in the form x : y |
A micrometer with 50 divisions across a 0.50 mm thimble movement has resolution 0.50/50 = 0.01 mm; record readings to that precision after checking zero error.
A calculated value should not claim more significant figures than the raw measurements support, and units belong in headings rather than after every table entry.
Plot and analyse physical data accurately
Transfer the table's quantity/unit headings to the axes, place the independent variable on the horizontal axis unless instructed otherwise and select simple 1, 2 or 5 × 10ⁿ scale steps.
| Feature | Standard |
|---|---|
| graph use | data occupy more than half the grid in both directions where possible |
| points | small clear crosses, plus signs or encircled dots; plotted within half a small square |
| best fit | one thin smooth line or curve with scatter balanced on both sides |
| anomaly | identify and omit from the fit only when clearly anomalous |
| intercept | read to about half a small square after justified extension |
| straight-line gradient | use a marked triangle spanning at least half the best-fit line |
Calculate gradient = change in vertical quantity/change in horizontal quantity, using points on the best-fit line rather than necessarily raw data points. Give two or three significant figures and derive its unit from the axes.
A best-fit line need not pass through any individual point. Do not force it through the origin unless direct proportionality is supported.
Use trigonometry and find the gradient of a curve
For the Extended route, use sine, cosine, tangent and their inverse functions to connect sides and angles in right-angled triangles.
| Relation | Use |
|---|---|
| sin θ = opposite/hypotenuse | opposite side or hypotenuse with angle |
| cos θ = adjacent/hypotenuse | adjacent side or hypotenuse with angle |
| tan θ = opposite/adjacent | two perpendicular components |
| θ = sin⁻¹(...), cos⁻¹(...) or tan⁻¹(...) | find an angle from a side ratio |
To find the gradient of a curve at a chosen point, draw a tangent that just touches the curve there and follows its local direction. Choose two well-separated points on the tangent, then calculate Δy/Δx.
Use a large gradient triangle to reduce the percentage effect of reading uncertainty, and include the gradient unit formed from vertical-axis unit divided by horizontal-axis unit.
A tangent is not a chord joining two points on the curve. Calculator angle mode must match the required degrees, and inverse trigonometric functions are not reciprocals.