Maths Skills 1 (OCR A-Level Biology A): Revision Notes

Maths Skills 1

Mathematical skills are essential for analyzing biological data and understanding quantitative relationships in biology. This note covers the key mathematical techniques you'll need for A-Level Biology, from handling numbers to creating and interpreting graphs.

Working with numbers

Standard form and powers

Biologists study organisms and structures across an enormous range of sizes—from blue whales weighing $190\,000\,\text{kg}$ to bacteria with masses of $1 \times 10^{-15}\,\text{kg}$. To handle these extreme values, we use standard form (also called scientific notation).

In standard form, numbers are written with one digit before the decimal point, multiplied by a power of ten. For example:

• $100\,000\,000\,000 = 1 \times 10^{11}$
• $0.000\,001 = 1 \times 10^{-6}$

Multiplying numbers in standard form: Add the powers together.

$2.0 \times 10^{4} \times 3.3 \times 10^{5} = 6.6 \times 10^{9}$

Dividing numbers in standard form: Subtract the powers.

$\frac{4.6 \times 10^{6}}{2.0 \times 10^{3}} = 2.3 \times 10^{3}$

Fractions and reciprocals

A fraction consists of a numerator (top number) and a denominator (bottom number). Always simplify fractions: $\frac{2}{8} = \frac{1}{4}$.

To calculate a fraction of a number: multiply by the numerator, then divide by the denominator.

$\frac{1}{2} \text{ of } 8 = \frac{1}{2} \times 8 = \frac{8 \times 1}{2} = 4$

A reciprocal is calculated by dividing $1$ by a number. For example, the reciprocal of $25$ is $0.04$. Sometimes reciprocals are multiplied by $100$, $1000$, or $10\,000$ to create whole numbers that are easier to work with and plot on graphs.

Decimals and rounding

In scientific writing, use a full stop (not a comma) for decimal points.

To multiply or divide decimals:

• Multiply: Move the decimal point right for each factor of $10$
• Divide: Move the decimal point left for each factor of $10$

When deciding how many decimal places to use, apply common sense. Don't make results more precise than your original measurements. If you measure time to the nearest second, don't report calculated results to three decimal places.

Significant figures

Significant figures (sig figs) indicate the precision of a measurement or calculation. All non-zero digits are significant, but zeros have special rules:

• Zeros at the beginning are not significant: $0.032$ has two sig figs
• Zeros at the end of whole numbers are significant: $7090$ has four sig figs
• Zeros between sig figs are significant: $601.4$ has four sig figs
• Zeros after a decimal point are significant: $0.780$ has three sig figs

Calculations—best practices

Always write out all steps in your calculations, even when using calculators or spreadsheets. This approach:

• Helps you think through what you're calculating
• Creates a record for checking errors
• Allows examiners to award partial credit if your final answer is incorrect

Units of measurement

Biological measurements use the SI system (Système Internationale). The table below shows the units you'll most commonly encounter:

When writing units, never use the solidus (/) to mean 'per'. Instead, use negative indices:

Measurements from microscopy

Magnification is the ratio between the size of an image and the actual size of the object.

Key formulae:

$\text{magnification} = \frac{\text{size of image}}{\text{actual size}}$

$\text{actual size} = \frac{\text{size of image}}{\text{magnification}}$

$\text{size of image} = \text{actual size} \times \text{magnification}$

When images include scale bars:

1. Measure the scale bar length in millimetres
2. Measure the object length in millimetres and divide by the scale bar length
3. Multiply by the length represented by the scale bar

Descriptive statistics

Creating tables

Tables organize data clearly. Follow these rules:

• Plan your table before starting—decide how many columns and rows you need
• Use at least half the available space
• Draw outlines in pencil with a ruler
• Write clear, brief column headings with appropriate SI units
• The first column should contain the independent variable
• Subsequent columns should contain dependent variables
• Arrange the independent variable in ascending order
• Keep entries brief—single words, numbers, ticks, or crosses
• Use the solidus (/) to separate quantities from units in headings (e.g., Concentration/g$100\,\text{cm}^{-3}$)

Calculating averages

The term average describes the center of a data range. There are three types:

Mean ($\bar{x}$)—the arithmetic average, calculated by:

$\bar{x} = \frac{\sum x}{n}$

where $\sum$ means 'sum of', $x$ is any data value, and $n$ is the total number of values.

Median—the middle value when data is arranged in order. For an odd number of values, take the central value. For an even number, calculate the mean of the two central values.

The median is most useful when data is skewed (not normally distributed).

Mode—the most frequently occurring value. Useful for whole numbers (e.g., counting organisms) and identifying the modal class in grouped data. When two values occur most frequently, the distribution is called bimodal.

Spread of data

The range shows the spread of results—either as the difference between maximum and minimum values, or by stating both values (e.g., $75\,\text{mm}$ to $113\,\text{mm}$, or $38\,\text{mm}$).

Standard deviation (SD) measures how widely data is dispersed around the mean. It's calculated from the variance:

$\text{variance} = \frac{\sum(x - \bar{x})^{2}}{n-1}$

$\text{standard deviation (SD)} = \sqrt{\frac{\sum(x - \bar{x})^{2}}{n-1}}$

Error bars on graphs can show either the total range or the standard deviation, helping visualize data variability.

Percentages and percentage change

Percentages allow valid comparisons between different totals:

$\% = \frac{\text{number with feature}}{\text{total number}} \times 100$

Percentage change compares a change to the original value:

$\% \text{ change} = \frac{\text{difference between final and original}}{\text{original number}} \times 100$

Percentage changes can be positive (increase) or negative (decrease).

Calculating areas and volumes

Prisms are solid objects with identical ends, flat sides, and constant cross-sectional area. You should know how to calculate surface areas and volumes for cuboids and cylinders. For spheres, formulae will be provided if needed.

Circles:

$\text{circumference} = 2\pi r$

$\text{surface area} = \pi r^{2}$

Surface area to volume ratios

Surface area to volume (SA:V) ratios are important for understanding diffusion, gas exchange, and transport in organisms. To calculate:

1. Calculate the surface area
2. Calculate the volume
3. Express as a ratio (e.g., $0.6:1$)

Graphs and charts

Line graphs

Line graphs show relationships between continuous variables. The independent variable goes on the $x$-axis; the dependent variable on the $y$-axis.

Drawing guidelines:

• Use at least half the grid in both directions
• Draw in pencil
• Scale axes using multiples of $1, 2, 5$, or $10$ per $20\,\text{mm}$ square—never use multiples of $3$
• Label axes clearly with quantity and units (e.g., Time/s, Concentration/g$\text{dm}^{-3}$)
• Plot points clearly using encircled dots ($\odot$) or crosses ($\times$)
• Decide if the origin should be included
• Draw straight lines, smooth curves, or point-to-point lines as appropriate

Analyzing graphs

Graph analysis involves:

1. Describing the overall trend or pattern
2. Providing specific details with data from the graph
3. Processing data (calculating rates, percentage changes)

For linear relationships, the equation is:

$y = mx + c$

where $m$ is the gradient and $c$ is the $y$-intercept. If $c = 0$, the relationship shows direct proportion.

Bar charts

Bar charts compare discrete categories. The blocks:

• Should be equal width and not touching
• Can be arranged in any order (descending size helps comparisons)
• Don't need shading or color-coding
• Should have equidistant spacing on the $x$-axis

Label the $y$-axis with appropriate units and scale it properly.

Histograms

Histograms display continuous numerical data organized into classes (frequency diagrams).

Creating classes:

• Number of classes = $5 \times \log_{10}(\text{total readings})$
• Range of each class = total range $\div$ (number of classes $- 1$)
• Ensure classes don't overlap (e.g., $3.0$–$3.9$, then $4.0$–$4.9$)

Drawing rules:

• Use at least half the grid
• Blocks should touch (continuous data)
• Block area is proportional to class size
• $x$-axis shows the independent variable (continuous)
• $y$-axis shows frequency or number
• Label blocks clearly (e.g., first block: '$3.0$–$3.9$' includes $3.0$ but not $4.0$)

The modal class is the class with the highest frequency.

Scattergraphs

Scattergraphs reveal correlations between two variables.

Positive correlation: As one variable increases, the other increases

Negative correlation: As one variable increases, the other decreases

No correlation: No clear relationship between variables

Uncertainty in measuring

Uncertainty is half the smallest graduation on measuring equipment. For example, if a syringe has $1.0\,\text{cm}^{3}$ graduations:

• Uncertainty = $\pm 0.5\,\text{cm}^{3}$
• A reading of $6.3\,\text{cm}^{3}$ is recorded as $6.3 \pm 0.5\,\text{cm}^{3}$

Percentage error calculation:

$\% \text{ error} = \frac{\text{uncertainty}}{\text{measurement}} \times 100$

Remember!