## To CAST, or not to CAST

Today I was working with some Y12 students on solving equations involving trigonometry, the type where they have to find all the solutions within a given domain. *Incidentally, always refer to.it as a given domain, rather than a given range. I know that it is a “range” of values, but the **range** of that sine curve you’re using is -1 to 1, not 0 to 2pi. If you refer to this, as I’ve seen any resources including textbooks do, you’re setting the students up for trouble when they come to look at domains and ranges in more detail.*

So we were looking at finding values in a given domain, discussing the graphs and how they could help us etc and then once they’d got their head around that we discussed the CAST graph.

When I was an A level student I preferred the CAST graph, I found it easier to use than the full trigonometric graphs and found it speeded up my working. For this reason I encouraged it’s use when I started teaching, but now I have a different view on the matter.

Each year I discuss with students the different approaches they can take to tackle different problems, giving them the widest possible armoury to tackle problems in exams, and always discuss with them which way they prefer. Over the years my students have consistently told me that they prefer using the trig graphs, rather than the CAST graphs.

In the first few years I couldn’t understand why, students explained that they “just understood it better”, but that still didn’t quite get why. I was reflecting on it today, as we were discussing it, and I think I have much more of an understanding now. I think that the CAST graph is a procedural short cut, but to fully understand it you need to fully understand the trig graphs and other angle properties, and the relationship that exists between the graphs and the ratios and the angles you are looking at. I think when students fully understand the trig graphs they can use them easily, and the process is quick. To then learn the CAST graph, an additional piece of knowledge and set of rules, to speed up that process fractionally seems silly.

As I’ve grown more aware of this over the years my teaching of the subject has shifted, I still show them the CAST graph, but I spend more time concentrating on the trig graphs to ensure a fuller understanding. Sometimes I wonder if there is even a point to doing the CAST graph at all.

*Do you teach both methods? Which method do you prefer? What about your students? Why do you/they prefer that one? I’d love to hear your thoughts either in the comments or via twitter.*

It’s a puzzler for sure as I also learned the CAST diagram and then the relationships between sin and cos as opposed to intimate knowledge of the graphs.

When it came to teaching it, the trig graphs are the way to go. Much like any shortcut, I have seen it misapplied in error and unlike a poorly applied graph it is not as useful to explain WHY it’s wrong as it is to explain the relationship between the trig functions and their graphs. It’s easy to see whether your answers make sense using a graph as opposed to working through a method.

I’ve also taught the same topic to students who haven’t been taught through GCSE higher and A Level but who are having to do Level 3 Maths for access and foundation degree courses.

Once they get the shapes of the graph they find it quite easy to visualise the solutions rather than applying an additional method. I just see it as having to teach another layer or method that doesn’t necessarily add any additional understanding. Sketching the graphs is as quick and easy as the CAST with practice plus it links to good calculator, sketching and layout of work skills as any in the curriculum.

Great post that as its a topic I’ve queried a lot with myself over the past few years and remains one of the few instances where I teach counter to the method I was taught myself.

Thanks for the comment Chris. Interesting insights. One massively common theme arising is that people usually teach the method they were taught, and I think we need to be wary and question that, as I have recently on this and so it seems have you.

I am a bit torn on this. I have taught both and think in the past I would largely prefer my students to use the graphs, but accept that the CAST diagram would be used by many.

However, my thinking is changing on this. In dismissing the CAST diagram as a procedural shortcut that precludes all understanding is to miss the potential richness of the unit circle. I don’t believe there is anything in the unit circle that requires the graph to understand it.

The signs of the trig ratios in each quadrant do not, of course, require the graphs to understand. Once the horizontal and vertical vectors from the origin to a given point on the circle are understood to represent sin(x) and cos(x), the signs are apparent – and tan(x) can easily be understood as the quotient of these and it’s sign derived quickly. I think for many the ‘CAST’ letters serve to obscure this deeper understanding.

Other trigonometric facts are much more readily available from the unit circle, also – the identity sin^2(x) + cos^2(x) = 1 immediately springs to mind as it is simple Pythagoras on any triangle in the unit circle.

So I guess I am saying don’t throw the baby out with the bath water. The unit circle is a rich and perfectly valid representation of how the trigonometric ratios vary with angles and can be understood on its own terms. The CAST diagram, though often used by students as a ‘black box’ without understanding, is directly derived from this – perhaps we need to think on about engaging students more with the unit circle.

I certainly agree there is a richness in the unit circle that enhances the understanding immensely. The CAST graph, however, tends to be taught often with no reference to the circle. Often with just the axes and two straight diagonals through the origin.

I teach both. It took a year but I once finally persuaded a very capable student (now studying at Cambridge) that the unit circle was they way forward. I still think it is the best way to show all the trig inter relationships and enlargement. I had never heard of SOHCAHTOA until I started teaching and had always used the unit circle. In my head I still use the unit circle to help me plot the graphs and if I do that I assume pupils should be able to adopt it. However it is not as useful for solving problems with Sin (ax)

This year I have found Y12 accepting it for finding second solutions

They are all tools that we should be using to develop a deeper understanding and I would question anyone who didn’t at least show pupils the unit circle

Aye, I agree. I perhaps should have mentioned the unit circle. I use.it to introduce the graph. The CAST diagram isn’t the same though. It tends to be the axis and two straight diagonals, no circle involved.

I think its best to give them all 3 and let them pick. If you are a good teacher you should always have multiple ways of doing things like you have.

However my argument towards the cast diagram would be efficiency. I regard speed in a method as the reason why you should pick one over another, as long as you can be correct of course.

I agree that a multitude of methods is the best approach. I think the speed element was why I prefered CAST as a student.

Thanks for this, was really helpful for me. Just finished teaching trig – I start with the graphs (I never learnt the CAST method at school!) as it definitely helps them with knowledge of graphs elsewhere too, but then explain CAST through using the unit circle. Having learnt the unit circle & CAST off a colleague I think it is a bit quicker, and it’s also helped my understanding too – my students, having learnt both deeply, are very divergent as to which one they prefer! The best bit is they’ve also started to see that both methods are very useful in different circumstances. Thanks again!

Thanks for your insights. I think I should have probably discussed the unit circle in the post in hindsight.