IB Biology SLSkill Set 3 — Maths for BiologyPaper 1 & 2Core skill~11 min read
Graphing Skills
Graph questions are some of the most predictable marks in the whole course. Half of them are for drawing things properly — axes, units, a sensible scale — and the other half are for reading a gradient or a trend off a curve. Both are learnable in an afternoon, and both are thrown away constantly.
📘 What you need to know
Independent variable on the x axis, dependent on the y axis, every time.
Label both axes with units, use a linear scale, and choose one that fills the space you are given.
A line of best fit can be straight or curved. It shows the trend, with points balanced either side.
Gradient = change in y ÷ change in x, which for a rate means change ÷ time.
For a curve, take a tangent to find the rate at one moment. A tangent at time zero gives the initial rate.
Interpolation reads between your points and is safe; extrapolation goes beyond them and is a guess.
A box-and-whisker plot shows the median, quartiles and range in one picture.
Choosing the right graph
Graph
Use it when
Biology example
Bar chart
The independent variable is categoric, so bars have gaps between them
Mean height of three plant species
Histogram
Continuous data grouped into classes, so bars touch
Distribution of leaf lengths in a population
Scatter graph
Looking for a relationship between two continuous variables
Light intensity against rate of photosynthesis
Line or curve graph
A continuous variable changing, usually over time
Volume of gas released during a reaction
Logarithmic graph
Values that span several orders of magnitude
Bacterial population growth
Pie chart
Parts of a single whole
Proportion of each base type in a DNA sample
Box-and-whisker plot
Comparing the spread of two or more data sets
Shell length at two shore sites
Drawing a graph that scores
Notice the line does not touch every point, and it should not. It shows the trend, and the scatter around it is the honest part.
🧩 The checklist before you draw anything
Which variable did you control? That one goes on the x axis.
Pick a linear scale that lets every point fit and fills most of the grid. Awkward scales like 3 squares per unit cost marks and cause plotting errors.
Label both axes with the quantity and the unit, written as “rate / cm3 s−1“.
Plot accurately, in pencil, with a small neat cross or dot.
Draw the line of best fit — straight or a smooth curve, whichever the data shows.
Only go through the origin if the data and the biology justify it. Zero substrate really does mean zero rate; zero light does not always mean zero growth.
Lines of best fit
They can be curves. “Best fit” does not mean “straight”.
They do not have to pass through every point, or even any point.
They should pass through the majority of the pattern, with points spread evenly above and below.
A curve should be drawn smoothly, in one movement, not as a series of joined straight sections.
Joining the dots is not a line of best fit — that is a different thing, and it earns no marks in a “draw a line of best fit” question.
Gradients and rates of change
The gradient of a graph is the rate of change: how fast the dependent variable moves as the independent one changes.
Gradient, or rate of change
change in the dependent variable ÷ change in the independent variable
For a straight line the gradient is the same everywhere, so you can take any two points that are far apart and read them off. The further apart you choose them, the more accurate your answer.
WORKED EXAMPLE
On a straight-line graph, oxygen volume is 12 cm3 at 10 s and 44 cm3 at 50 s. Find the rate of oxygen production.
Step 1: change in y44 − 12 = 32 cm3Step 2: change in x50 − 10 = 40 sStep 3: divide32 ÷ 40 = 0.80.8 cm3 s−1The unit comes straight out of the division: cm3 divided by s gives cm3 s−1. Never quote a gradient without one.
Tangents and the initial rate
Enzyme experiments almost never give a straight line. The reaction starts fast and slows down as substrate runs out, so the gradient is changing the whole way along. To get the rate at a single moment you draw a tangent: a straight line that just touches the curve at that point and matches its slope there.
The initial rate is the rate at the very start, where time = 0 — and it matters because that is the only moment when the conditions are exactly what you set up. Later on, substrate has been used and product has built up.
The curve flattens because substrate is running out. That is why the initial rate, taken before anything has been used up, is the fair value to compare between conditions.
🧩 Drawing a tangent accurately
Use a ruler and a pencil, and pick the point on the curve first.
Line the ruler up so the gap between it and the curve looks equal on both sides of the touching point.
Position the ruler so it does not hide the curve — you cannot judge the gap on a line you cannot see.
Draw the tangent long, so the triangle you read off is big and small measuring slips matter less.
Read the rise and the run from the tangent, not from the curve.
WORKED EXAMPLE
A tangent drawn at the start of the curve above shows 60 cm3 of product released in the first 24 seconds. Calculate the initial rate of reaction.
Step 1: gradient is rise over runrise = 60 cm3, run = 24 sStep 2: divide60 ÷ 24 = 2.5Initial rate = 2.5 cm3 s−1“Rise over run” is worth remembering: any change up or down, divided by any change across.
Other features worth naming
Changing gradient — take tangents at several points to describe how the rate changes as a reaction proceeds.
Intercept — where a line crosses an axis. A y intercept above zero often means something is happening even at zero of the independent variable.
Maximum — a peak, where the gradient goes from positive, through zero, to negative. An enzyme’s optimum temperature sits at a maximum.
Minimum — a trough, where the gradient goes from negative, through zero, to positive.
Plateau — the gradient reaches zero and stays there, meaning something has become limiting.
Interpolation and extrapolation
An enzyme graph extrapolated past its optimum would predict faster and faster rates. In reality the enzyme denatures and the rate collapses.
Interpolation uses the line of best fit to read a value that lies between your data points. You have evidence either side, so this is reasonably safe.
Extrapolation extends the line beyond the data to estimate values you never measured. It assumes the trend continues, and in biology it very often does not.
If you extrapolate in an answer, say that you have, and say what could make the prediction wrong.
Box-and-whisker plots
These pack the median, the quartiles and the full range into one small picture, which makes comparing two data sets almost instant.
Because the box ignores the extremes at both ends, a box plot is a fair way to compare two sites even when one has a single unusual reading.
WORKED EXAMPLE
Nine shell lengths in mm: 12, 15, 18, 19, 22, 25, 28, 31, 35. Find the median, the quartiles and the interquartile range.
Step 1: they are already in order, so find the middle value
Nine values, so the 5th is the middle: median = 22 mmStep 2: Q1 is the middle of the lower half (12, 15, 18, 19)(15 + 18) ÷ 2 = 16.5 mmStep 3: Q3 is the middle of the upper half (25, 28, 31, 35)(28 + 31) ÷ 2 = 29.5 mmStep 4: interquartile range29.5 − 16.5 = 13Median 22 mm, IQR 13 mmOrder the values first. Every mistake in this calculation starts with someone working from an unsorted list.
Sketch graphs and biology-specific diagrams
A sketch graph has no numbers on it. It shows the shape of a relationship — proportional, inversely proportional, levelling off — and it still needs labelled axes. Use one when the question asks what happens, not how much.
Three diagram types are specific to biology and get their own conventions:
Dichotomous keys — paired either/or statements that lead to a species name.
Food chains, food webs and pyramids of energy — arrows always point in the direction the energy travels.
Pedigree charts — family relationships and how a trait is inherited through generations.
💡 Exam tip
Axes, units, scale, points, line. Work through the five in order every time you draw.
Choose a scale that fills the grid. A graph squashed into a corner loses marks even if every point is right.
Take gradient readings from two points far apart, and always quote the unit.
Use a big triangle for a tangent — the bigger it is, the smaller your reading error.
Describe a curve in words as well as numbers: rises steeply, then levels off at around 60 cm3.
Say “interpolation” or “extrapolation” by name when you read a value off a graph.
Decide deliberately whether (0,0) belongs on your graph. If it does, it is usually your most reliable point.
⚠ Common mix-up
Variables on the wrong axes. What you changed goes on x.
Axis labels with no units, or units written beside every number instead of in the label.
Joining the dots and calling it a line of best fit.
Forcing the line through the origin when the data does not support it.
Taking a gradient from the curve instead of the tangent. The tangent is the straight line you drew.
Quoting a gradient with no unit, so it could mean anything.
Extrapolating an enzyme curve past the optimum and predicting rates that could never happen.
Working out quartiles from an unordered list.
Up next: Skill Set 4 — The Inquiry Process — turning all of these tools into a full investigation, from research question to conclusion and evaluation.
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