# Appendix B: Maths requirements and examples

In order to be able to develop their skills, knowledge and understanding in Environmental Science, students need to have been taught, and to have acquired competence in, the appropriate areas of mathematics as indicated in the table of coverage below.

Overall at least 10% of the marks in assessments for Environmental Science will require the use of mathematical skills. These skills will be applied in the context of environmental science and will be at least the standard of higher tier GCSE mathematics.

The following tables illustrate where these mathematical skills may be developed during teaching or could be assessed.

This list of examples is not exhaustive. These skills could be developed or assessed in other areas of specification content. Other areas where these skills could be developed have been exemplified throughout this specification.

Table 1. Arithmetic and numerical computation
Mathematical skill number Mathematical skills Exemplification of mathematical skill in the context of A-level Environmental Science (examples are not limited to those given below)
MS 0.1 Recognise and make use of appropriate units in calculations Students should demonstrate their ability to:
• convert between units such as length and volume, eg calculating surface area: volume ratios in energy conservation
• select appropriate units and values for a calculation, eg estimating water and organic matter content of soils.
MS 0.2 Recognise and use expressions in decimal and standard form Students should demonstrate their ability to:
• use an appropriate number of decimal places in calculations, eg calculating mean population density from multiple sample sites in a habitat
• carry out calculations using numbers in standard and ordinary form, eg when comparing production of different energy resources
• convert between numbers in standard and ordinary form, eg when using masses in biogeochemical cycles.
MS 0.3 Use ratios, fractions and percentages Students should demonstrate their ability to:
• calculate percentage yields, eg in pollution control
• calculate surface area to volume ratios and relate this to heat loss
• calculate and compare percentage loss, eg of rain forests over a given time period or of declining populations of endangered species.
MS 0.4 Estimate results Students should demonstrate their ability to:
• estimate results to sense check that the calculated values are appropriate, such as when calculating residence times in different water reservoirs.
MS 0.5 Use calculators to find and use power, exponential and logarithmic functions Students should demonstrate their ability to:
• interpret population growth curves
• compare noise values quoted in decibel units.
Table 2. Handling data
Mathematical skill number Mathematical skills Exemplification of mathematical skill in the context of A-level Environmental Science (examples are not limited to those given below)
MS 1.1 Use an appropriate number of significant figures Students should demonstrate their ability to:
• report calculations to an appropriate number of significant figures given raw data quoted to varying numbers of significant figures, eg in calculating indices of biodiversity
• understand that calculated results can only be reported to the limits of the least accurate measurement, eg in estimating lifetimes of mineral reserves.
MS 1.2 Find arithmetic means Students should demonstrate their ability to:
• find the mean of a range of data, eg mean power output of a wind farm.
MS 1.3 Construct and interpret frequency tables and diagrams, bar charts and histograms Students should demonstrate their ability to:
• represent a range of data in a table with clear headings, units and consistent decimal places, eg to compare the energy density, production cost, carbon intensity and mean load factor for a range of energy resources
• interpret data from a variety of tables, eg data relating to aquifer flow rates
• plot a range of data in an appropriate format, eg atmospheric carbon dioxide levels, atmospheric temperature and solar output over time represented on a graph
• interpret data from a variety of graphs, eg change in electricity cost from renewable energy sources, industrial output and level of financial incentives/tax over a number of years.
MS 1.4 Understand simple probability Students should demonstrate their ability to:
• use the term probability appropriately when investigating causal relationships such as the link between human health problems and urban pollutants.
MS 1.5 Understand the principles of sampling as applied to scientific data Students should demonstrate their ability to:
• analyse random data collected by an appropriate means, eg use Simpson’s index of diversity to compare the biodiversity of habitats exposed to different pollution types or management regimes
• analyse systematic data along a transect to monitor impacts of pollution with increasing distance from a copper smelter.
MS1.6 Understand the terms mean, median and mode Students should demonstrate their ability to:
• calculate or compare the mean, median and mode of a set of data, eg of yields of fish farmed under different conditions or fish from commercial catches.
MS 1.7

Use a scatter diagram to identify a correlation between two variables

Students should demonstrate their ability to:
• interpret a scatter graph, eg to compare human development index with environmental footprint of different countries.
MS 1.8 Make order of magnitude calculations

Students should demonstrate their ability to:

• compare storage volumes of natural water reservoirs and transfer rates
• calculate national energy use from population and individual use data.
MS 1.9

Select and use a statistical test

Students should demonstrate their ability to select and use:

• the chi-squared t test (goodness of fit and association)
• the Student’s t-test
• Spearman’s rank
• Mann-Whitney U test
MS 1.10

Understand measures of dispersion, including standard deviation and range

Students should demonstrate their ability to:

• calculate the standard deviation, eg of crop yield for a given nutrient input
• understand why standard deviation is a useful measure of dispersion for a given set of data, eg for comparison with other data sets with different means such as populations of endangered species under different management regimes.
MS 1.11 Identify uncertainties in measurements and use simple techniques to determine uncertainty when data are combined

Students should demonstrate their ability to:

• calculate percentage error where there are uncertainties in measurement, eg estimating total population using sub-samples in a preliminary study.
Table 3. Algebra
Mathematical skill number Mathematical skills Exemplification of mathematical skill in the context of A-level Environmental Science (examples are not limited to those given below)
MS 2.1 Understand and use the symbols : = < << >> > and ~. No exemplification required.
MS 2.2 Change the subject of an equation

Students should demonstrate their ability to:

• use and manipulate equations, eg nutrient transfer rates, energy conversion efficiencies and fish maximum sustainable yields.
MS 2.3 Substitute numerical values into algebraic equations using appropriate units for physical quantities

Students should demonstrate their ability to:

• use a given equation, eg Simpson’s index of diversity:
 D = N (N – 1) Σ n (n –1)

to assess the impact of a new habitat management regime.

MS 2.4 Solve algebraic equations

Students should demonstrate their ability to:

• solve equations in an environmental context, eg calculations using the universal soil loss equation to assess the effectiveness of soil conservation programmes.
MS 2.5

Use logarithms in relation to quantities that range over several orders of magnitude

Students should demonstrate their ability to:

• use a logarithmic scale in the context, eg of noise pollution levels.
Table 4. Graphs
Mathematical skill number Mathematical skills Exemplification of mathematical skill in the context of A-level Environmental Science (examples are not limited to those given below)
MS 3.1 Understand data presented in a variety of graphical forms

Students should demonstrate their ability to:

• interpret data in a range of graphical forms, including line graphs, which may involve logarithmic scales, bar charts, stacked bar charts, histograms, kite diagrams, pie graphs, scatter graphs, 3-dimensional graphs, flow diagrams, Sankey diagrams and circular (radar) diagrams to enable a wide variety of data to be analysed.
MS 3.2 Translate information between graphical numerical and algebraic forms

Students should demonstrate their ability to:

• understand that data may be presented in a number of formats and be able to use these data, eg dissolved oxygen levels, soil erosion rates.
MS 3.3 Plot two variables from experimental or other data

Students should demonstrate their ability to:

• select an appropriate format for presenting data, bar charts, histograms, graphs and scattergraphs, eg organic matter and oxygen depletion, nutrient inputs and yield increase.
MS 3.4

Understand that y = m x + c represents a linear relationship

Students should demonstrate their ability to:

• predict/sketch the shape of a graph with a linear relationship, whether with a positive or negative correlation, eg the relationship between insolation and solar panel output
MS 3.5 Determine the intercept of a graph

Students should demonstrate their ability to:

• read an intercept point from a graph, eg the temperature at which oxygen levels fall too low to support particular aquatic species.
MS 3.6 Calculate rate of change from a graph showing a linear relationship

Students should demonstrate their ability to:

• calculate a rate from a graph, eg rate of infiltration through rocks with different permeabilities.
MS 3.7

Draw and use the slope of a tangent to a curve as a measure of rate of change

Students should demonstrate their ability to:

• use this method to measure the gradient of a point on a curve, eg rate of heat loss through double glazing with varying gaps.
Table 5. Geometry and trigonometry
Mathematical skill number Mathematical skills Exemplification of mathematical skill in the context of A-level Environmental Science (examples are not limited to those given below)
MS 4.1

Calculate the circumferences, surface areas and volumes of regular and irregular shapes

Students should demonstrate their ability to:

• calculate the circumference and area of nature reserves to assess the impact of the edge effect on wildlife conservation programmes
• calculate the surface area and volume of cylinders or spheres, eg to estimate rates of heat loss in energy conservation programmes.