Why use this resource?
Students are asked to estimate the gradient of three points on a sine graph, without being given a scale. They should find that they have to decide between using degrees or radians on the \(x\)-axis. This gives an opportunity to discuss what the gradient function of \(\sin x\) is, and why it will be different depending on the units of \(x\). This can help to explain why we use radians when doing calculus.
Preparation
Students should have a copy of the sine graph with points A, B and C marked on, to use for their estimations. You can download a printable page with the graphs on here.
If students are not familiar with the idea of sketching the gradient function of a graph, you may want to use one or two of the cards from Gradient match to introduce the idea with more straightforward graphs.
Possible approach
You may want students to estimate the gradients by themselves initially, as their differing approaches could lead to a rich discussion about which methods are more appropriate and perhaps more accurate. Some students may not make a conscious decision to use radians or degrees and so sharing students’ results should draw out the problem of not having a scale on the axes. Once it is understood there are two possibilities, students should be asked to sketch the gradient function in degrees and in radians, and compare the results.
Key questions
- How can we estimate the gradient of a curve?
- What scale have you used for the \(x\)-axis?
- Can you sketch the gradient of \(\sin x\) in radians? In degrees?
Possible support
Students might try and estimate the gradients by measuring distances on the graph. To help them see why this might be incorrect and suggest a different method, you could ask them to label any values on the axes that they know.
Possible extension
The derivative of \(\sin x\) when \(x\) is in degrees is \(a\cos x\). Students could try to work out what the value of \(a\) is. They may want to consider how you would transform a sine graph with a period of \(2\pi\) to one with a period of \(360^{\circ}\).
- What is the scale factor that is involved?
- How does that impact the gradient?
