IB Biology HLSkill Set 3 — Maths for BiologyPaper 1, 2 & IAPractical skill~13 min read
Graphing Skills
A graph is an argument with the arithmetic already done. Draw it well and the pattern in your results does the explaining for you. Draw it badly — squashed into a corner, axes unlabelled, points joined dot-to-dot — and you have thrown away marks that had nothing to do with biology.
📚 What you need to know
The independent variable goes on the x-axis; the dependent variable goes on the y-axis.
Label both axes with units, use linear scales, and make the graph fill the space you are given.
You should be able to draw bar charts, histograms, scatter graphs, line and curve graphs, logarithmic graphs, pie charts and box-and-whisker plots.
A line of best fit can be straight or curved. It shows the trend, passes through most of the points, and has the points balanced on either side.
The gradient of a straight line is the rate of change. For a curve, draw a tangent to get the rate at one point.
The initial rate is the gradient of the tangent at time = 0.
Interpolation reads values between your data points; extrapolation extends the line beyond them.
Maxima are peaks, minima are troughs, and intercepts are where the line crosses an axis.
Sketch graphs versus plotted graphs
A sketch graph has no numbers on it. It is there to show the shape of a relationship — whether two variables are proportional, inversely proportional, or level off. If a question says “sketch”, you are being asked for the trend, not for accurate points.
A plotted graph is the opposite: real data, accurate scales, points marked exactly where they belong. Everything below is about plotted graphs.
Choosing the right graph
Pick the graph that suits your data, not the one you find easiest to draw.
Graph
Use it when
Biological example
Bar chart
The x-axis holds separate categories or groups
Mean height of plants in four different soils
Histogram
Continuous data has been grouped into ranges
Number of leaves in each length band
Line or curve graph
Both variables are continuous and one follows the other
Rate of respiration against temperature
Scatter graph
You are looking for a relationship between two measured variables
Leaf width against leaf length
Pie chart
You are showing parts of one whole
Proportion of each species in a community
Box-and-whisker plot
You want to show median, quartiles and spread together
Comparing the range of body masses in two populations
Logarithmic graph
The values cover a huge range
Bacterial population growth over time
Getting the plot right
🧩 The checklist examiners mark against
Independent variable on the x-axis, dependent on the y-axis. The independent one is what you changed; the dependent one is what changed as a result.
Choose a linear scale that lets every data point fit on the grid, then fill the space. A graph crammed into the bottom-left quarter loses marks.
Label both axes and add the units. “Time” is not enough — write “time / s”.
Plot the points accurately, in pencil, so you can fix a slip.
Draw a line of best fit, straight or curved, that shows the trend.
Decide about the origin. Include (0,0) only if the data and the biology allow it — if they do, it is often your most reliable point.
Four of these points sit above the line and four below. That balance is what “line of best fit” means — it is not a line joining the dots.
Lines of best fit
People hear “line of best fit” and immediately reach for a ruler. A line of best fit does not have to be straight.
It can be a straight line or a curve, whichever matches the trend in the data.
It shows the general trend. It does not have to pass through every point.
It should pass through the majority of the points, with the rest spread evenly above and below.
If it is a curve, draw it smoothly in one movement. A wobbly, hairy curve is marked down.
Joining the points dot-to-dot is the single most common graph mistake. Every measurement carries uncertainty, so a zig-zag line is claiming your data is perfect. The best fit line says something more honest: this is the trend, and the points scatter around it.
Gradients, tangents and the initial rate
For a straight-line graph, the gradient is the same everywhere, so the rate of change is easy: pick any two points and divide.
Gradient
gradient = change in y ÷ change in x
Enzyme experiments almost never give you a straight line, though. They give a curve that starts steep and then flattens, because the reaction is slowing down as substrate runs out. A curve has a different gradient at every point, so to get the rate at one moment you draw a tangent: a straight line that just touches the curve at that single point and matches its slope there.
The initial rate is the rate right at the start, where time = 0. It matters because that is the only moment when nothing has been used up yet, so it is the fairest rate to compare between experiments.
The tangent leaves the curve almost immediately, and that is fine — it only has to match the slope at the one point it touches.
🧩 Drawing a tangent that actually works
Extend the curve in your head if you need to, so you can see where it is heading at the point you want.
Use a ruler and a pencil. The line must be dead straight.
Line the ruler up on the point so the gap between ruler and curve looks equal on both sides of where it touches.
Try to keep the curve visible rather than hidden under the ruler — it is much easier to judge.
Draw a big triangle on the tangent and read off the change in y and the change in x. Bigger triangle, smaller rounding error.
Divide: rise over run, and put the units on.
WE 1
Calculating an initial rate
Using the graph above, calculate the initial rate of reaction. (3 marks)
Step 1: draw the tangent at time = 0
The tangent touches the curve where the reaction starts.
Step 2: read a triangle off the tangent
The tangent reaches 60 cm3 after 20 s.
Step 3: gradient = rise over run
60 ÷ 20 = 3.0
3.0 cm3 s−1read the triangle off the tangent, never off the curve — the curve is already slowing down
Gradients change along a curve
Because a curve of best fit has a different gradient everywhere, you can take several gradients and watch how the rate changes as the reaction goes on.
WE 2
Comparing rates at two times
On the same curve, 70 cm3 of product had been released at 40 s and 91 cm3 at 80 s. Calculate the average rate over that interval and explain why it differs from the initial rate. (4 marks)
Step 1: find both changes
change in volume = 91 − 70 = 21 cm3
change in time = 80 − 40 = 40 s
Step 2: divide
21 ÷ 40 = 0.525
0.53 cm3 s−1Step 3: explain the difference
This is much slower than the initial rate of 3.0 cm3 s−1 because the substrate has been used up, so fewer collisions with enzyme active sites happen and the curve flattens.
the explanation is worth as much as the number here — always link the flattening curve back to substrate running out
Intercepts, maxima and minima
Feature
What it is
Why it matters
Intercept
Where the line or curve of best fit crosses an axis
The y-intercept tells you the value when the independent variable is zero
Maximum
A peak. The gradient passes from positive, through zero, to negative
The optimum — the temperature or pH at which an enzyme works fastest
Minimum
A trough. The gradient passes from negative, through zero, to positive
The lowest point of a cycle, such as a population crash
Describe the shape, then explain it. “The rate rises to a maximum at 40 °C and then falls sharply” is a description. Adding “because above the optimum the enzyme denatures” turns it into an explanation. Most graph questions want both.
Uncertainty bars on a graph
An uncertainty bar shows the absolute uncertainty of a plotted value. It is drawn the same way as an error bar: usually vertically for y-values, but horizontally for x-values when the uncertainty is in what you set.
The length of the bar shows how much uncertainty there is in that measurement. Short bars mean confident values.
Bars let a reader judge whether the differences between your points are real or could just be measurement wobble.
Your line of best fit should ideally pass through, or very near, most of the bars.
Interpolation and extrapolation
Once you have a line of best fit, you can use it to read values you never actually measured. There are two ways to do it, and they are not equally safe.
Interpolated values sit between real measurements, so they are well supported. Extrapolated ones are a prediction, and organisms have limits that a straight line knows nothing about.
Extrapolate a graph of enzyme rate against temperature and the line cheerfully predicts a wonderful rate at 90 °C. The enzyme, meanwhile, has denatured. If a question asks you to evaluate an extrapolated value, that is the point to make.
Graphs that are specific to Biology
Beyond the standard graphs, a few diagrams turn up only in this subject. You need to be able to read and draw them.
Diagram
What it represents
Dichotomous key
A branching series of paired questions used to identify an organism
Food chain and food web
The feeding relationships in a community, and the direction energy flows
Pyramid of energy
The energy available at each trophic level, drawn to scale
Pedigree chart
How a genetic condition is inherited through the generations of one family
WE 3
Choosing and justifying a graph
A student measured the dry mass of seedlings grown at five different light intensities, with five repeats at each intensity. Suggest how the results should be presented, and justify your choice. (3 marks)
Point 1: identify the variables
Light intensity is the independent variable, so it goes on the x-axis. Dry mass is the dependent variable and goes on the y-axis.
Point 2: choose the graph
Both variables are continuous, so plot the mean dry mass at each intensity as a line or curve graph, not a bar chart.
Point 3: show the spread
Add error bars showing standard deviation, because there were five repeats at each intensity.
Line graph of mean dry mass against light intensity, with error barswhenever a question mentions repeats, the mark scheme usually wants means plus a measure of spread
💡 Exam tips
Write the unit into the axis label with a slash: “time / s”, “mass / g”. It saves you writing it beside every number.
Use a scale that goes up in 1s, 2s, 5s or 10s. Scales going up in 3s or 7s make plotting slow and mistakes likely.
Take your gradient triangle as large as the graph allows.
For an initial rate, the tangent must be drawn at t = 0, not at the first plotted point.
Describe a trend using the numbers: quote values from the axes rather than saying “it goes up a lot”.
If you extrapolate, say in one line why the prediction may not hold.
⚠ Common mistakes
Joining the points dot-to-dot. Draw a smooth line or curve of best fit instead.
Swapping the axes. What you changed goes on the x-axis, every time.
Squashing the graph into one corner. Choose a scale that uses the whole grid.
Leaving the units off the axis labels. That is a mark gone before you have plotted anything.
Taking the gradient off the curve instead of the tangent. The tangent is the straight line you drew.
Forcing the line through the origin. Only include (0,0) if the biology actually says the value is zero there.
That completes Skill Set 3 — Maths for Biology. The four notes work as one toolkit: handle the numbers, give them units, admit their uncertainty, and then show them on a graph that argues your case. These skills are marked in Paper 1, Paper 2 and every piece of internal assessment you write.
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