What you need to know
- Unit 3Plot data on a suitable scale, describe the relationship, and find a gradient with a large triangle where the line is straight.
Use the Maths Clinic for extra algebra practice. Here the graph is biological data you were given. A line of best fit is only for a relationship that is actually linear. These figures are illustrative.
By the end, you should be able to…
- Assign axes, units and a scale that uses the grid.
- Describe the relationship without claiming more than the line shows.
- Calculate a gradient from a large triangle, including the unit.
Build your understanding
Axes, scale and the points
Put the independent variable on the x-axis and the dependent variable on the y-axis. Label both with units. Choose a scale that uses at least half the grid and has equal intervals. A scale that jumps from 10 to 50 to 60 compresses the pattern. The origin can be omitted if starting at zero would squash the points, but the scale must still be even and labelled.
Plot the points you were given, including one you have queried, unless the question tells you to omit an identified anomaly. A line of best fit is appropriate when the relationship is linear. If the points rise and then level off, do not force a single straight line through all of them. Joining points dot-to-dot is a sketch of the path, not automatically a best fit.
Say the relationship you can see
Comment on the shape: as the independent variable increases, the dependent variable increases, decreases, stays the same, or increases and then levels off. Quote a pair of values if the question asks for evidence. “It changed” is too vague.
Stop at the relationship unless you are asked for a cause. One graph does not prove a mechanism, and it is not a statistical test. If the question asks you to determine a constant from a straight section, that constant is the gradient, not a second description of the same sentence.
A large triangle, then the unit
On a straight section, draw a triangle that covers a large fraction of the line. Gradient = change in y ÷ change in x. Read both changes from the axis scales, not by counting squares if the squares have different scales.
Illustrative line: at 1 minute the volume is 4 cm³, and at 6 minutes it is 19 cm³. Change in volume = 15 cm³. Change in time = 5 minutes. Rate = 15 ÷ 5 = 3 cm³ min⁻¹. Using 19 ÷ 6 instead would treat the line as if it passed through the origin and started at zero minutes. Include the compound unit. A curve needs a tangent at the stated time; the gradient of that tangent is the rate at that instant, not the mean rate for the whole curve.
NOT: a straight line through the origin is not automatically the right line.
- Label both axes with the quantity and the unit
- A broken scale must be shown if you use one
- Quote the gradient with its unit
- An anomaly is checked, not rubbed out without a reason
Two traps this diagram blocks
- Do not calculate a gradient from a tiny triangle on a steep line.
- Do not give a gradient without units.
From the diagram alone
Question
Explain how to find the gradient of a straight line on a graph of rate against concentration. [3]
Show a model answer
Draw a large triangle on the line. Divide the change in rate by the change in concentration, using the scale on the axes rather than millimetres on the page. State the unit of that gradient.
Graph skills: SLAPUK and choosing a scalePractise SLAPUK and scales on interactive grid paper.
Make your reasoning visible
Read the scale before the squares
A “steep” line on a cramped scale can be a small change. Quote axis values when you describe or calculate.
Large triangle, labelled changes
Mark the two coordinates you used. A triangle a few millimetres wide makes the reading of the gradient the largest uncertainty.
Key explanations
- Axes need the quantity and the unit.
- A point that does not fit should be repeated or explained, not silently omitted.
Avoid this misconception
The claim: A line of best fit should pass through every point and through the origin.
The correction: It should follow a linear pattern, with points scattered on both sides. It passes through the origin only if the evidence and a zero start justify that.
Check your understanding
Try each question before opening the suggested answer.
Practice question
Retrieval and explanation · suggested answer
An illustrative straight line passes through (2 min, 5 cm³) and (8 min, 23 cm³). Calculate the rate.
Show suggested answer
Change in volume = 18 cm³. Change in time = 6 min. Rate = 18 ÷ 6 = 3 cm³ min⁻¹. Do not divide 23 by 8.
Practice question
Retrieval and explanation · suggested answer
The points rise from 10 to 40 arbitrary light units and then stay almost level to 80 units. Why is one straight line of best fit inappropriate?
Show suggested answer
The relationship is not linear across the whole range. A single straight line would hide the plateau. Describe the rise and the levelling off, or fit a line only to a section that is straight.
Practice question
Retrieval and explanation · suggested answer
What must be written on both axes besides the numbers?
Show suggested answer
The quantity and the unit, for example “Light intensity / arbitrary units” and “Rate of bubble production / min⁻¹”.
Take the idea further
Apply the same specification knowledge to an unfamiliar problem. Any extra assumptions are supplied in the question.
Stretch question
An unfamiliar curve of product against time is steep at the start and then flattens. You are asked for the initial rate. Explain how to obtain it, and why the concentration change divided by the full time is a different quantity. No coordinates are supplied.
Show a hint
A tangent uses a straight line that touches the curve at the start.
Explore the reasoning
Draw a tangent at time zero, or at the earliest point, and calculate its gradient from a large triangle, with units. That estimates the initial rate. The overall change divided by the full time is the mean rate for the whole period, including the slow later stage, so it is smaller than the initial rate if the curve is flattening. The assumption is that the tangent can be placed unambiguously on the figure you are given.
Recall this lesson
Lesson recall questions
Question 1 of 5: A graph of absorbance against time is a straight line through the origin for the first 60 s of a beetroot investigation. Outline, in order, how to find the initial rate using a large triangle.
Unit 3 processing results
Practice question
Retrieval and explanation · suggested answer
A graph of absorbance against time is a straight line through the origin for the first 60 s of a beetroot investigation. Outline, in order, how to find the initial rate using a large triangle.
Lesson: R2: Scales, lines and gradients
Build the sequence from memory first. Then reveal the model order.
Tap the steps in the order you recall. Tap a placed step to send it back.
- 01Next step
- 02Waiting
- 03Waiting
Tap to place
Beyond the classroomA line can look more certain than the measurementsData · official statistics · Enrichment, not required recall
National statistics guidance treats uncertainty as part of presenting results. A school graph does not need that publication method. It does need a scale that does not exaggerate a wobble, and a line that follows only the relationship you can justify.
Back to the spec: Unit 3 processing: plot an appropriate scale and comment on the relationship the graph actually shows.
Office for National Statistics: uncertainty and how we measure itQuestion
Why might stretching a small change to fill a graph, without labelling the scale clearly, mislead a reader?
Explore a possible response
The slope looks dramatic even when the change is small. Clear axis labels and an even scale let the reader judge the size of the change. The picture is not a substitute for the scale.