AS and A Level mathematical requirements
- Syllabus
- 9700–2028–2029
- Topic
- —
- Level
- A2
Learning objectives
—M.1—Prefixes, units, symbols and standard formUse biological units and prefixes including G, M, k, m, micro and n; select appropriate units; use decimal and standard form; interpret symbols such as less than, greater than, less than or equal to, greater than or equal to, proportional to, solidus in units and sum notation.—M.2—Calculator use, estimations and significant figuresUse a calculator for arithmetic, squares, square roots, reciprocals, logarithms and means; estimate calculation results; record experimental data with appropriate precision; keep justified significant figures in calculated quantities.—M.3—Magnification, actual size and geometryCalculate magnifications and actual sizes; calculate areas of triangles, rectangles and circles; calculate perimeters and circumferences; calculate surface areas and volumes of cuboids and cylinders.—M.4—Averages, ratios, percentages and experimental errorCalculate mean, median, mode and range; recognise and use ratios; calculate percentages and percentage changes; express errors in experimental work as percentage errors.—M.5—Graphs, data transformation and rates of changeTranslate information between graphical, numerical and algebraic forms; construct and interpret line graphs, pie charts, bar charts and histograms; choose suitable graph types; orient and scale axes correctly; choose straight joins or best-fit lines; calculate rates of change from gradients and tangents.
Units and prefixes preserve the scale of a biological quantity
A number is meaningful only with its unit and scale. Use the most appropriate unit, then convert all quantities to compatible units before comparing, multiplying or dividing them.
Prefix | Symbol | Multiplier
Giga | G | 10⁹
Mega | M | 10⁶
Kilo | k | 10³
Milli | m | 10⁻³
Micro | µ | 10⁻⁶
Nano | n | 10⁻⁹
To convert, replace the prefix by its power of ten. For example, 2.5 µm = 2.5 × 10⁻⁶ m = 2500 nm.
Use decimal or standard form as the scale demands. Read <, >, ⩽ and ⩾ as comparison limits; ∝ means directly proportional; Σ means sum. In a table heading or graph axis, write a quantity followed by a solidus and unit, for example length / µm, so entries contain numbers only.
Capitalisation matters: M is mega and m is milli. A bare value such as 2.5 cannot be compared with 3000 until both quantities have stated, compatible units.
Estimate, calculate, then report justified precision
Calculation quality has three checks: estimate the expected size, calculate without premature rounding, then report a precision justified by the measurements—not by the number of digits on the calculator display.
- Estimate the sign and order of magnitude; this detects misplaced powers of ten or calculator entry errors.
- Use the required operations: +, −, ×, ÷, x², √x, 1/x, lg and mean. Keep guard digits during working.
- Round the final calculated quantity to the same number of significant figures as, or one more than, the smallest number of significant figures in the data used.
- Record direct experimental readings with appropriate and consistent precision for the measuring instrument.
For 12.4 ÷ 3.2, an estimate of 12 ÷ 3 ≈ 4 makes 38.75 implausible. The calculator gives 3.875. The inputs have 3 and 2 significant figures, so the syllabus permits a final value with 2 or 3 significant figures: 3.9 or 3.88, with the required unit.
Decimal places and significant figures are not interchangeable. In 0.00450, the leading zeros are placeholders and the value has three significant figures. Do not round intermediate steps so aggressively that the final result drifts.
Match units and dimensions before calculating size
First decide whether the unknown is a length, area, surface area or volume. Convert measurements to compatible units, choose the formula with the correct dimension, and attach the resulting unit: unit, unit² or unit³.
Magnification = image size ÷ actual size; actual size = image size ÷ magnification.
Triangle area = ½bh; rectangle area = lw; circle area = πr².
Rectangle perimeter = 2(l + w); circle circumference = 2πr.
Cuboid surface area = 2(lw + lh + wh); cuboid volume = lwh.
Cylinder surface area = 2πr² + 2πrh; cylinder volume = πr²h.
A cell of actual length 20 µm is shown as 40 mm. Convert 40 mm to 40 000 µm, then magnification = 40 000 ÷ 20 = ×2000. If every linear dimension doubles, surface area becomes 2² = 4 times larger and volume becomes 2³ = 8 times larger.
Do not mix radius and diameter, and do not attach a linear unit to area or volume. Magnification has no physical unit because image and actual size are divided in the same unit.
The biological question determines the summary and denominator
Choose a summary that answers the biological question. For any ratio or percentage, state what the numerator represents and which reference quantity belongs in the denominator.
Question | Calculation
Typical value using every observation | mean = Σx ÷ n
Middle of ordered observations | median
Most frequent value | mode
Spread from extremes | range = maximum − minimum
Relative amounts | ratio a:b, simplified or scaled consistently
Part of a whole | percentage = part ÷ whole × 100
Change relative to the starting value | percentage change = (final − initial) ÷ initial × 100
Measurement uncertainty relative to the measured value | percentage error = absolute error ÷ measured value × 100
A mass increases from 10 g to 12 g: absolute change = 2 g and percentage change = 2 ÷ 10 × 100 = 20%. If a 10.0 cm reading has an absolute error of ±0.1 cm, percentage error = 0.1 ÷ 10.0 × 100 = 1%. These percentages answer different questions.
Do not divide percentage change by the final value. A mean can also conceal skew or an anomalous value; use median or mode only when the data and question justify them, not as interchangeable labels.
Data type determines the graph and rate method
A graph is a transformation of data, not decoration. Choose the representation from the variable and question, place the independent variable on x and dependent variable on y, and preserve units and numerical meaning.
Purpose/data | Representation
Separate categories | bar chart with separated bars
Parts of one whole | pie chart
Frequency distribution of continuous measurements | histogram with touching class intervals
Response across an ordered continuous IV | line graph or scatter plot with an appropriate straight or curved best-fit line
Label each axis as quantity / unit, choose a simple scale that uses the plotting area, plot accurately, and decide from context whether points represent a sequence to join with straight ruled lines or a trend needing best fit.
Rate of change = Δy ÷ Δx, with units from y per unit x. For a straight line, use two widely separated points on the line. For an average rate on a curve, use the relevant interval. For the rate at one instant, draw a tangent at that point and calculate the tangent gradient from a large triangle.
Do not join independent scatter points dot-to-dot or use a bar chart for a continuous IV merely because there are only a few values. A visually steeper line is not necessarily a larger rate: compare gradients only after checking both axis scales and units.