Mathematical Requirements - A-Levels Biology

5d. Mathematical requirements

In order to develop their skills, knowledge and understanding in A Level Biology, learners need to have been taught, and to have acquired competence in, the appropriate areas of mathematics relevant to the subject as indicated in the M0-M4 table of coverage below.

The assessment of quantitative skills will include at least 10% Level 2 (or above) mathematical skills for biology. These skills will be applied in the context of the relevant biology.

All mathematical content will be assessed within the lifetime of the specification. Skills shown in bold type in the M0-M4 coverage table will only be tested in the full A Level course, not the standalone AS Level course.

The list of examples given in the M0-M4 coverage table is not exhaustive and is not limited to Level 2 examples. These skills could be developed in other areas of the specification content from those indicated.

Mathematical skills for biology - M0-M4 coverage table

Ref.	Mathematical skill to be assessed	Exemplification of the mathematical skill in the context of A Level Biology (assessment is not limited to the examples below)	Areas of the specification which exemplify the mathematical skill (assessment is not limited to the examples below)
M0 - Arithmetic and numerical computation
M0.1	Recognise and make use of appropriate units in calculations	Learners may be tested on their ability to: • convert between units e.g. $\mathrm{mm^3}$ to $\mathrm{cm^3}$ as part of volumetric calculations • work out the unit for a rate e.g. breathing rate	2.1.1(e), 2.1.2(s), 2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.1(a), 3.1.1(e), 3.1.2(a), 3.1.2(h), 3.1.3(a), 3.1.3(c), 4.1.1(b), 4.1.1(l), 5.1.2(c), 5.1.5(i), 5.1.5(k), 5.2.1(a), 5.2.1(c), 5.2.1(g), 5.2.2(i), 5.2.2(k), 5.2.2(l), 6.2.1(h), 6.3.1(b), 6.3.2(a)
M0.2	Recognise and use expressions in decimal and standard form	Learners may be tested on their ability to: • use an appropriate number of decimal places in calculations, e.g. for a mean • carry out calculations using numbers in standard and ordinary form, e.g. use of magnification • understand standard form when applied to areas such as size of organelles • convert between numbers in standard and ordinary form • understand that significant figures need retaining when making conversions between standard and ordinary form, e.g. $0.0050\,\mathrm{mol\,dm^{-3}}$ is equivalent to $5.0\times10^{-3}\,\mathrm{mol\,dm^{-3}}$	2.1.1(e), 2.1.1(f), 2.1.1(g), 2.1.2(s), 2.1.4(d), 2.1.4(f), 2.1.5(b), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.1(e), 3.1.3(c), 4.1.1(b), 4.1.1(l), 4.2.1(b), 5.1.5(e), 5.1.5(i), 5.1.5(k), 5.2.1(c), 5.2.1(g), 5.2.2(i), 5.2.2(k), 5.2.2(l), 6.1.2(f), 6.2.1(i), 6.3.1(b), 6.3.2(a)
M0.3	Use ratios, fractions and percentages	Learners may be tested on their ability to: • calculate percentage yields • calculate surface area to volume ratio • use scales for measuring • represent phenotypic ratios (monohybrid and dihybrid crosses).	2.1.1(e), 2.1.1(f), 2.1.4(d), 2.1.4(f), 2.1.5(d), 2.1.5(e), 3.1.1(a), 3.1.2(a), 3.1.3(a), 4.1.1(b), 4.1.1(l), 4.2.2(h), 5.1.2(c), 5.1.5(k), 5.2.1(a), 5.2.1(g), 6.1.2(b), 6.1.2(c), 6.2.1(h), 6.2.1(i), 6.3.1(b), 6.3.2(a)
M0.4	Estimate results	Learners may be tested on their ability to: • estimate results to sense check that the calculated values are appropriate.	3.1.1(a), 3.1.1(e), 3.1.2(a), 3.1.3(a), 5.2.1(a), 6.3.1(b), 6.3.2(a)
M0.5	Use calculators to find and use power, exponential and logarithmic functions	Learners may be tested on their ability to: • estimate the number of bacteria grown over a certain length of time.	6.2.1(h), 6.3.2(a)
M1 - Handling data
M1.1	Use an appropriate number of significant figures	Learners may be tested on their ability to: • report calculations to an appropriate number of significant figures given raw data quoted to varying numbers of significant figures • understand that calculated results can only be reported to the limits of the least accurate measurement.	2.1.1(e), 2.1.2(s), 2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.1(a), 3.1.2(a), 3.1.2(h), 3.1.3(a), 3.1.3(c), 4.1.1(b), 4.1.1(l), 4.2.1(c), 4.2.1(d), 4.2.1(e), 5.1.2(c), 5.1.5(e), 5.1.5(i), 5.1.5(k), 5.2.1(c), 5.2.1(g), 5.2.2(i), 5.2.2(k), 5.2.2(l), 6.2.1(h), 6.3.1(b)
M1.2	Find arithmetic means	Learners may be tested on their ability to: • find the mean of a range of data, e.g. the mean number of stomata in the leaves of a plant.	2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.3(c), 4.1.1(b), 4.1.1(l), 4.2.2(f), 5.1.5(e), 5.1.5(i), 5.1.5(k), 5.2.2(l), 6.2.1(i)
M1.3	Construct and interpret frequency tables and diagrams, bar charts and histograms	Learners may be tested on their ability to: • represent a range of data in a table with clear headings, units and consistent decimal places • interpret data from a variety of tables, e.g. data relating to organ function • plot a range of data in an appropriate format, e.g. enzyme activity over time represented on a graph • interpret data for a variety of graphs, e.g. explain electrocardiogram traces.	2.1.2(s), 2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.1(e), 3.1.2(h), 3.1.3(c), 4.1.1(b), 4.1.1(g), 4.1.1(l), 4.2.1(b), 4.2.1(f), 4.2.2(f), 5.1.2(c), 5.1.3(c), 5.1.5(a), 5.1.5(e), 5.1.5(i), 5.1.5(k), 5.2.1(c), 5.2.1(g), 5.2.2(i), 5.2.2(k), 5.2.2(l), 6.2.1(h), 6.2.1(i), 6.3.1(e), 6.3.2(a)
M1.4	Understand simple probability	Learners may be tested on their ability to: • use the terms probability and chance appropriately • understand the probability associated with genetic inheritance.	4.2.1(b), 5.1.5(e), 6.1.2(b), 6.1.2(c), 6.2.1(i), 6.3.1(e)
M1.5	Understand the principles of sampling as applied to scientific data	Learners may be tested on their ability to: • analyse random data collected by an appropriate means, e.g. use Simpson's index of diversity to calculate the biodiversity of a habitat.	4.1.1(b), 4.1.1(l), 4.2.1(b), 4.2.1(c), 4.2.1(d), 4.2.1(e), 6.3.1(e)
M1.6	Understand the terms mean, median and mode	Learners may be tested on their ability to: • calculate or compare the mean, median and mode of a set of data, e.g. height/mass/size of a group of organisms.	2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.3(c), 4.2.1(b), 4.2.2(f), 5.1.5(a), 5.1.5(e), 5.1.5(i), 5.1.5(k), 5.2.1(l), 6.2.1(i), 6.3.1(b)
M1.7	Use a scatter diagram to identify a correlation between two variables	Learners may be tested on their ability to: • interpret a scattergram, e.g. the effect of lifestyle factors on health.	4.1.1(b), 4.1.1(l), 4.2.1(b), 4.2.1(f), 4.2.2(f), 6.3.1(e)
M1.8	Make order of magnitude calculations	Learners may be tested on their ability to: • use and manipulate the magnification formula $\text{magnification}=\frac{\text{size of image}}{\text{size of real object}}$	2.1.1(e)
M1.9	Select and use a statistical test	Learners may be tested on their ability to select and use: • the chi squared test ( $\chi^2$ ) to test the significance of the difference between observed and expected results • the Student's $t$ -test • the Spearman's rank correlation coefficient.	4.2.1(b), 5.1.5(e), 6.1.2(c), 6.3.1(e)
M1.10	Understand measures of dispersion, including standard deviation and range	Learners may be tested on their ability to: • calculate the standard deviation • understand why standard deviation might be a more useful measure of dispersion for a given set of data e.g. where there is an outlying result.	2.1.5(e), 4.2.1(b), 4.2.2(f), 5.1.5(e), 5.1.5(k), 5.2.2(l), 6.2.1(i), 6.3.1(e)
M1.11	Identify uncertainties in measurements and use simple techniques to determine uncertainty when data are combined	Learners may be tested on their ability to: • calculate percentage error where there are uncertainties in measurement.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.3(c), 5.2.1(g)
M2 - Algebra
M2.1	Understand and use the symbols: $=$ , $<$ , $\ll$ , $>$ , $\gg$ , $\propto$ , $\sim$	No exemplification required.	2.1.5(d), 2.1.5(e), 3.1.1(a), 3.1.2(a), 3.1.3(a), 5.1.2(c), 6.1.2(c)
M2.2	Change the subject of an equation	Learners may be tested on their ability to: • use and manipulate equations, e.g. magnification.	2.1.1(e), 2.1.2(s), 5.2.1(c), 6.1.2(f)
M2.3	Substitute numerical values into algebraic equations using appropriate units for physical quantities	Learners may be tested on their ability to: • use a given equation e.g. Simpson's index of diversity $D=1-\sum\left(\frac{n}{N}\right)^2$	2.1.1(e), 2.1.2(s), 4.2.1(c), 4.2.1(d), 4.2.1(e), 5.2.1(c), 5.2.2(k), 6.1.2(f)
M2.4	Solve algebraic equations	Learners may be tested on their ability to: • solve equations in a biological context, e.g. cardiac output = stroke volume x heart rate.	2.1.1(e), 2.1.2(s), 3.1.2(h), 4.2.1(c), 4.2.1(d), 4.2.1(e), 5.2.1(c), 5.2.2(i), 5.2.2(l)
M2.5	Use logarithms in relation to quantities that range over several orders of magnitude	Learners may be tested on their ability to: • use a logarithmic scale in the context of microbiology, e.g. growth rate of a microorganism such as yeast.	6.2.1(h), 6.3.2(a)
M3 - Graphs
M3.1	Translate information between graphical, numerical and algebraic forms	Learners may be tested on their ability to: • understand that data may be presented in a number of formats and be able to use these data, e.g. dissociation curves.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.2(j), 3.1.3(c), 4.1.1(b), 4.1.1(l), 4.2.1(f), 5.1.2(c), 5.1.3(c), 5.1.5(e), 5.1.5(k), 5.2.1(g), 5.2.2(i), 6.2.1(h), 6.3.1(e), 6.3.2(a)
M3.2	Plot two variables from experimental or other data	Learners may be tested on their ability to: • select an appropriate format for presenting data, bar charts, histograms, graphs and scattergrams.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 2.1.5(e), 3.1.3(c), 4.1.1(b), 4.1.1(l), 4.2.1(b), 5.1.5(e), 5.2.1(g), 5.2.2(i), 5.2.2(l), 6.2.1(h), 6.2.1(i), 6.3.1(e), 6.3.2(a)
M3.3	Understand that $y=mx+c$ represents a linear relationship	Learners may be tested on their ability to: • predict/sketch the shape of a graph with a linear relationship, e.g. the effect of substrate concentration on the rate of an enzyme-controlled reaction with excess enzyme.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 3.1.3(c), 5.2.2(l)
M3.4	Determine the intercept of a graph	Learners may be tested on their ability to: • read off an intercept point from a graph, e.g. compensation point in plants.	5.2.1(a), 5.2.1(g), 6.2.1(h)
M3.5	Calculate rate of change from a graph showing a linear relationship	Learners may be tested on their ability to: • calculate a rate from a graph, e.g. rate of transpiration.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 3.1.3(c), 5.2.1(g), 5.2.2(l), 6.2.1(h)
M3.6	Draw and use the slope of a tangent to a curve as a measure of rate of change	Learners may be tested on their ability to: • use this method to measure the gradient of a point on a curve, e.g. amount of product formed plotted against time when the concentration of enzyme is fixed.	2.1.4(d), 2.1.4(f), 2.1.5(c), 2.1.5(d), 3.1.3(c), 5.2.1(g), 5.2.2(l), 6.2.1(h)
M4 - Geometry and trigonometry
M4.1	Calculate the circumferences, surface areas and volumes of regular shapes	Learners may be tested on their ability to: • calculate the circumference and area of a circle • calculate the surface area and volume of rectangular prisms, of cylindrical prisms and of spheres • e.g. calculate the surface area or volume of a cell.	2.1.5(d), 2.1.5(e), 3.1.1(a), 3.1.2(a), 3.1.3(a), 3.1.3(c), 5.2.1(g), 6.2.1(i)

Formulae used in A Level Biology

To address biology questions using mathematical skills, learners will need to be able to use and, in some cases, recall formulae and equations. Some of these will seem like pure mathematics, but will be deployed in biological contexts, while others are clearly biological equations, albeit manipulated using standard mathematical, algebraic techniques.

Formulae used in A Level Biology

	Biological	Mathematical
Recall	Magnification Rates $R_f$ $Q_{10}$ $SA : V$ Genetic biodiversity Cardiac output $RQ$ Log growth Efficiency	All of GCSE (9-1) Maths recall including (but not limited to): • circumference and area of circle • surface area and volume of cuboid • mean • percentage (to include %change, %yield and %error)
Provided	Hardy-Weinberg Simpson's index of diversity	Surface area and volume of cylinder and sphere chi squared $t$ -test paired $t$ -test unpaired Spearman's rank Standard deviation

GCSE (9-1) Mathematical formulae to recall

At AS and A Level Biology we assume knowledge of higher tier GCSE (9-1) Maths content. This includes (but is not limited to) the following list of formulae which learners will need to be able to recall.

Students should be familiar with the convention of using r for radius, h for height, b for base and l for length.

GCSE (9-1) Mathematical formulae to recall

Formula	Equation
Circumference of circle	$\text{circumference}=2\pi r$
Area of circle	$\text{area of circle}=\pi r^2$
Surface area of cuboid	$\text{surface area of cuboid}=2(bh+bl+hl)$
Volume of cuboid	$\text{volume of cuboid}=hbl$
Mean	$\bar{x}=\frac{\sum x}{n}$
Percentage change	$\text{percentage change}=\frac{\text{new quantity}-\text{original quantity}}{\text{original quantity}}\times100$
Percentage yield	$\%\text{ yield}=\frac{\text{actual amount}}{\text{theoretical amount}}\times100$
Percentage error (uncertainty)	$\%\text{ error (uncertainty)}=\frac{2\times\text{absolute uncertainty}}{\text{quantity measured}}\times100\%$

Biological formulae to recall

The following are the biological formulae learners will need to recall:

Biological formulae to recall

Formula	Equation
Magnification	$\text{magnification}=\frac{\text{size of image}}{\text{size of real object}}$
$R_f$	$R_f=\frac{\text{distance moved by the solute}}{\text{distance moved by the solvent}}$
Rates (e.g. enzymatic reactions, breathing (ventilation), transpiration, photosynthesis, respiration, reaction times, diffusion)	$\text{rate}=\frac{\text{change in quantity}}{\text{time taken}}$
Surface Area to Volume ratio	$\text{ratio}=\frac{\text{surface area}}{\text{volume}}$
Genetic biodiversity	$\text{proportion of polymorphic gene loci}=\frac{\text{number of polymorphic gene loci}}{\text{total number of loci}}$
Cardiac output as a function of heart rate and stroke volume	$\text{cardiac output}=\text{heart rate}\times\text{stroke volume}$
Respiratory quotient	$RQ=\frac{CO_2\ \text{produced}}{O_2\ \text{consumed}}$
Microorganism population growth	$N=N_0\times2^n$
Efficiency of biomass transfers	$\text{efficiency}=\frac{\text{biomass transferred}}{\text{biomass intake}}\times100$
Temperature coefficient ( $Q_{10}$ )	$Q_{10}=\frac{R_2}{R_1}$

Mathematical formulae that will need to be used but not recalled

The following mathematical formulae will be provided in the assessments where needed:

Mathematical formulae provided where needed

Formula	Equation
Surface area of a cylinder	$\text{surface area of cylinder}=2\pi r(r+l)$
Volume of a cylinder	$\text{volume of cylinder}=\pi r^2l$
Surface area of a sphere	$\text{surface area of sphere}=4\pi r^2$
Volume of a sphere	$\text{volume of sphere}=\frac{4}{3}\pi r^3$
Chi squared	$\chi^2=\sum\frac{(f_o-f_e)^2}{f_e}$
Spearman's Rank Correlation Coefficient	$r_s=1-\frac{6\sum d^2}{n(n^2-1)}$
Standard Deviation	$s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}$
Student's t-test - Unpaired	$t=\frac{\|\bar{x}_A-\bar{x}_B\|}{\sqrt{\frac{s_A^2}{n_A}+\frac{s_B^2}{n_B}}}$
Student's t-test - Paired	$t=\frac{\bar{d}\sqrt{n}}{s_d}$

Biological formulae that will need to be used but not recalled

Critical values tables, or appropriate excerpts from these tables, will be provided in the assessment where needed. Learners will need to be able to work out which degrees of freedom or r row, and which confidence column(s), is/are relevant to their analysis.

The following biological formulae will be provided in the assessments where needed:

Biological formulae provided where needed

Formula	Equation
The Hardy-Weinberg Equations	$p^2+2pq+q^2=1$ $p+q=1$
Simpson's Index	$D=1-\sum\left(\frac{n}{N}\right)^2$