Tutorial
A Conceptual Approach To Teaching Right-Angled Trigonometry
Showcasing the principles of an Understanding-first, Procedures-second mindset
Before We Begin …
When we have taught a particular topic ‘a hundred times’ and then come across a tutorial based on that topic we are tempted to dismiss it - “What could possibly be new for me in this tutorial?” My answer is maybe nothing. However, I’m confident you will find something here worth implementing, so stay tuned. However, you’ll need 20-30 minutes of reasonable headspace. If you don’t have that right now, why not save the URL and set a calendar reminder to come back to it.
Two Common Problems We Face When Teaching Right-Angled Trigonometry
The way I see it, there are two common problems most of us face when we teach trigonometry:
- The unit takes longer than it should, and
- From the very beginning of the unit, the level of student understanding is much lower than it should be. Sure, some students get it, but many end up confused and switch off. Sadly, trigonometry is one of those ‘nail in the coffin’ units that has many students saying “I’m going to drop out of mathematics as soon as I can”.
Procedures
You may have some fantastic way of approaching trigonometry - you take your class sailing and work out the trig ratios of the various tasks … or you build a few scale-model sheds and use trig to calculate the various pitches of the rooves … or you knock on Space-X’s door and convince Elon to allow your students to calculate, by hand, some of the Falcon Heavy trajectories (!)
OK, I got carried away a bit there. But, regardless of how you approach right-angled trigonometry, at some point in the unit, you’ll need to teach students the procedures for calculating side lengths and unknown angles in right-angled triangles, correct?
And this is the point in the unit where students start disappearing down the proverbial gurgler.
OK, I got carried away a bit there. But, regardless of how you approach right-angled trigonometry, at some point in the unit, you’ll need to teach students the procedures for calculating side lengths and unknown angles in right-angled triangles, correct?
And this is the point in the unit where students start disappearing down the proverbial gurgler.
Block Practice - A Barrier to Learning
The procedures absolutely need to be taught, and my guess is that most of us do a pretty good job of teaching them. But it isn’t the procedures that are the problem. The problems arise when we compartmentalize the learning into Block Practice.
Most of us, I suspect, use Block Practice when we teach trig - and, in fact, when we teach most units. And this, it turns out, is a real problem.
Before you implement the conceptual approach to trigonometry below it will help if you understand the way Block Practice (too much of it) negatively impacts student learning. Also, you need to understand why the process of Interleaving is an excellent alternative to Block Practice and to see how Interleaving is specifically applied to this conceptually-based, right-angle trigonometry unit.
However, this is the subject of a separate, short article called ‘Why We Need To Use More Interleaving And Less Block Practice When Teaching Mathematics’. Feel free to have a read of that now.
Most of us, I suspect, use Block Practice when we teach trig - and, in fact, when we teach most units. And this, it turns out, is a real problem.
Before you implement the conceptual approach to trigonometry below it will help if you understand the way Block Practice (too much of it) negatively impacts student learning. Also, you need to understand why the process of Interleaving is an excellent alternative to Block Practice and to see how Interleaving is specifically applied to this conceptually-based, right-angle trigonometry unit.
However, this is the subject of a separate, short article called ‘Why We Need To Use More Interleaving And Less Block Practice When Teaching Mathematics’. Feel free to have a read of that now.
Assuming you have returned from reading the recomended article and now that you have been schooled on the advantages of Interleaving and on the pitfalls of Block Practice, let’s explore a way to present right-angle trigonometry that:
- Will have your students understanding the work for (almost) ALL lesson time
- Will have your students much more engaged in their work
- Will save you between 2 and 3 lessons out of a 10 to 12 lesson trig unit.
The Tutorial
A Conceptual Approach To Teaching Right-Angled Trigonometry
This tutorial covers four aspects of the conceptually-based trigonometry unit. Note that each of these is integral to the success of the unit:
Importantly, the tutorial is premised on the principles of an Understanding-first, Procedures-second mindset, designed to have students understanding what they are working on for the majority of lesson time, as outlined in this article.
- The Conceptual Introduction. Using a pre-made GeoGebra file, students gain an understanding of what trigonometry is; an understanding of the concepts that right-angle trigonometry is based on.
- The Trigonometric Sentence. Students are shown how to ascertain whether a trig question is a sine, cosine or tangent situation. A PowerPoint resource is provided to support this process.
- Interleaved Trigonometry Practice. A worksheet is provided that requires students to work conceptually - i.e. to think through questions and make decisions (about the trig nature of each question) rather than to blindly follow one procedure. This results in students having to calculate side lengths and angles using all three trig ratios through interleaved practice.
- Completing The Unit: The transition onto worded problems, bearings and ‘real-life’ mixed questions occurs as per usual. However, this transition typically occurs 2-3 lessons sooner than conventionally, and with happier students!
Importantly, the tutorial is premised on the principles of an Understanding-first, Procedures-second mindset, designed to have students understanding what they are working on for the majority of lesson time, as outlined in this article.
Step 1: Watch the video below to see the conceptual introduction demonstrated.
This introduction to the unit should take about 10-20 minutes and is best delivered interactively.
Step 2: Teach The Writing Of The Trigonometric Sentence.
- How to determine whether a question is a sine, a cosine or a tangent situation
- How to write the initial trig sentence, e.g. sin 36 = x/12.
The provided Powerpoint file is designed to help. You may choose to use the PowerPoint interactively with your students or it might simply give you some ideas on how to better convey this understanding to your students in other ways.
Step 3: The Trigonometry (Interleaved) Worksheet Series.
This 4-page Interleaved series (with hand-written solutions) is a major key to this approach. However, Steps 1 and 2 need to be successfully completed before students work through the provided Interleaved Worksheet Series. This is because students first need to have a solid understanding of the trig concept as well as the ability to ascertain whether each question is a sine, cosine or tangent situation.
This is explained in the video below ...
This is explained in the video below ...
Step 3 Continued
Demonstrate several solutions to questions from page 1 of the Interleaved Worksheet Series first, then have your students work through all questions at their own pace.
No Block Practice?
In the article ‘Why We Need To Use More Interleaving And Less Block Practice When Teaching Mathematics’ (referenced above) you may recall I included the following quote: ‘Some blocked practice is useful, especially when students encounter a new concept or skill’.
However, in this conceptual trig unit no Block Practice is included because I firmly believe it is unnecessary and would likely be detrimental to the outcome. The only reason you may sense the need for some Block Practice would be if aspects 1 and 2 were not successfully grasped by the students - this is why it is critical that aspects 1 and 2 are covered well before students tackle the interleaved worksheet series.
However, in this conceptual trig unit no Block Practice is included because I firmly believe it is unnecessary and would likely be detrimental to the outcome. The only reason you may sense the need for some Block Practice would be if aspects 1 and 2 were not successfully grasped by the students - this is why it is critical that aspects 1 and 2 are covered well before students tackle the interleaved worksheet series.
Step 4 Completing the unit.
Worded problems and bearings. Once students are able to readily determine whether trigonometry examples are sine, cosine or tangent situations, apply the appropriate trig sentence and correctly determine both side lengths and angles, they are then ready for your worded, ‘real-world’ problems and bearings. There is no need to provide questions for you here - simply use the questions you would normally use.
Differentiation
To differentiate the learning, and once students complete the Interleaved worksheet series, you may choose to start students at an appropriate point in your text or question series depending on the competence level of each student.
Student Engagement
You should find that, via this conceptual approach, students are much more engaged by the time they transition to the mixed, worded questions.
It is worth reiterating that a successful implementation (with an 'average' class) should result in students being ready for these mixed/worded questions by lessons 3 or 4, which is 2 to 3 lessons earlier than typically occurs with a traditional, compartmentalized, Block Practice approach (based on 40-ish minute lessons).
It is worth reiterating that a successful implementation (with an 'average' class) should result in students being ready for these mixed/worded questions by lessons 3 or 4, which is 2 to 3 lessons earlier than typically occurs with a traditional, compartmentalized, Block Practice approach (based on 40-ish minute lessons).
Why Not A Practical, Inquiry-Based Approach?
Note: Trigonometry can be presented with a much stronger focus on practical inquiry than the approach given here. However, the point of this approach is to demonstrate an interleaved, conceptually-based approach within a relatively traditional classroom context, without a heavy requirement for technology or out-of-class excursions. In other words, anyone can successfully use this approach, and it does not require students who are initially highly engaged in order to work. However, you are encouraged to combine this approach with your more practical approach should you use one.
All the files here
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