Your U-1, P-2 Trigonometry Takeaways

​Do you consider the Understanding-first right-angled trigonometry example to be valid?

Do you use a similar approach?

Are you inspired to adopt this approach ... or to modify the version you currently use? 

​Can you see how to apply these principles to other units of work so that you foster less passive learning and more active learning in your students? 
​

Some past comments ... 

Will Martinez
I like the idea of the brick wall strategy - Definitely something I use but always good to have an explicit reminder. I would make a few changes to the worksheet to encourage students to think and not make things too procedural. Adding in a question where students should use Pythaoras' Theorem, including angles/side lengths that are decimal, surds or algebraic, including an equilateral triangle and not including the size of the angle.
Richard​
Adapt to your heart's content, Will. Glad to hear you are resonating.

Maja Rogic
I thought this was the only way one can teach trigonometry :)
Richard​
It is surprisingly uncommon, Maja. Did you work through it closely? Or did you assume from the start this is what you do so skipped it? Do you use an intyerleave worksheet as per the example?
Maja Rogic
Yes, I went through the whole thing and saved it. I’m terrible in memorizing things, that’s why I find it important to increase understanding of my students. And trigonometry, being a crucial part of further math studies, is the topic I pay special attention to. Hope it makes sense, I use interleave worksheets since that’s also how I’ve been taught. I had a great math teacher :)
Richard​
You were one of the lucky ones!!

Waris Huang
I like reading about using interleaved practice rather than blocked practice. I used to do a lot of blocked practice myself as a student and have seen students in auto-pilot mode with blocked practice and then make mistakes when shown with interleaved questions. It's a small area I didn't really notice until my director pointed it out and now I get a wholistic understanding on. I will definitely implement the strategy to my teaching.
Richard​
Great to hear Waris! Enjoy.

Michelle Millar
I've used these resources a few times and have found the students have a better understanding of Right-Angled Trig than by using more traditional methods - separating the ratios and teaching them individually at first.
Richard​
Great to hear, Michelle. Do you apply these principles to other topics? If you haven't, are you now tempted?

Your comment here ... 

More past comments ... 

Deb Morton
I would have said that 2a and 2b are more procedural than helping students to think for themselves and understand/make connections/links. In 2a, students must first learn how to identify then label the side lengths - a procedure, then how to decide which 2 of the 3 side lengths to choose - another procedure, memorise/learn what SOHCAHTOA stands for and finally the procedure of linking the 2 side lengths to the correct ratio. Do they understand why we use the words Sine, Cosine and Tangent? Do they understand why we need 3 different ratios? Do they understand what the word itself does? Do they understand why they need to connect the 2 sides to one of these words? In 2b, learning how to write the trig sentence is also a procedure - what is the order of the components of the sentence? - where do we get the information for each of these compoents from on the triangle? Do they understand why we need the sentance? Do they understand the function of each of the components? As an equation - this looks very different to any other equation they have previously seen before simply due to the fact there is a word in there - is it an operation? a number? a something else? At what point do we work through how to actually calculate the side length once we have the equation? What if they still can't solve or rearrange equations easily? I do love the geogebra file and definitely agree that is U1. Does changing the fraction/ratio into a percentage help long term as they will probably never do that again in the topic and it's the easiest part of the whole process to understand - once it's gone... do they get the ratio/fraction concept well enough?
Richard​
You make an interesting point re 2a and 2b. I guess they are a blend. I'm definitely advocating infusing a lot of understanding in that phase, as the questions you pose suggest. Re 'At what point do we work through how to actually calculate the side length once we have the equation?' This is a natural part of the same process. Re 'What if they still can't solve or rearrange equations easily?' ... then we support as we would in any other situation. (I never found either of these two to be issues) Re 'Does changing the fraction/ratio into a percentage help long term as they will probably never do that again in the topic, and it's the easiest part of the whole process to understand - once it's gone... do they get the ratio/fraction concept well enough?' ... I simply found that using a % helped at the early conceptual phase. As for 'does it work' ... in my experience this approach is far superior to the traditional, compartmentalised approach. Hope that helps, Deb.

Heath Barlin
I have realised that a number of my questions are too compartmentalised and need to have more mixed examples. After all, our assessments are mixed, too.
Richard​
YES! I think you are saying you use too much blocked practice and want to interleave the questions more. That is the correct terminology (not my terminology, btw). Compartmentalising is more about how we teach (teach sine, then cosine, then tangent ... teach fractions separately, then decimals separately, then percentages separately. And so on.
Heath Barlin
Yes, I did mean blocked practise.

Angus Vos
This is a very valid teaching strategy and one that I will definitely be trying to use more and more in my lessons. Students are able to share what they know and have freedom too even if they get the questions incorrect, felt listened to and hopefully gain understanding.
Richard​
Exactly. Although U-1, P-2 is more of an over-arching approach than a strategy!

Indar Deo
Agree - It does make sense to teach all the 3 trig ratios together. U-1 approach is very clear and concise. The mixed questions and the throwing in the angle questions without warning is really going to ask the kids to put on their thinking caps on as the need to explore on their own. Definitely i am going to use this strategy from now to teach this topic. Just a question. Would it be better to teach the side names first before anything else.
Richard​
'Would it be better to teach the side names first before anything else?' ... yes, I would. Maybe ask the kids to invent some first and then segue to the actual names. Glad you can see the merit in U-1, P-2. Re the trig example - it requires students to engage their brains AND p[events them from going into auto-pilot, robot mode, blindly following the procedure on the board.

Jasmyn Du Plessis
I agree with teaching all the ratios and the concept of the 'sentence'. I usually start by having students write the ratios for each rule for the same triangle. making them realise that the H is always the denominator as it is the largest side. How to realise their answer doesn't make sense. For example, if they are calculating the H then the answer has to be longer than the other two sides etc. Also take the same triangle and change the location of the angle. Then label the side again and write the ratios again. what is the same? What is different?
Richard​
Great!

Thomas Allen
I think some of the ways in which you justify the activities as U-1 could be equally used to conclude that they are P-1 activities. For example, you say that your worksheet is 'highly U-1, P-2' because you mix up the trig ratios, whereas doing them one-by-one would be 'highly P-1'. I feel like I could just as easily argue that it's P-1, U-2 because you are compartmentalizing trig ratios (i.e. all the questions are trig) so the 'procedure' of doing the trig calculation is just expanded to include the step of 'choose the right trig ratio based on the side you have and want'. In my opinion, mixing in different trig ratios doesn't magically make it U-1, P-2, it is just a good thing to do to prepare students for the actual contexts where they will use trig in the future. The whole planned series of lessons is U-1, P-2, as you mentioned in one of the other activities. Any activity where you give the students a bunch of questions will inevitably result in procedures being used and student understanding making their lives easier, but it does seem strange to rate a worksheet based on how 'understanding-first' it is when the worksheet is not teaching anything. I used to use a similar kind of structure to teaching trig the first few times I did it. This year I've tried starting with the unit circle and defining the sine, cosine, and tangent as functions and moving onto using them as trig ratios after. I'm still not sure on my favourite way to teach it, but I did spaghetti trig for the first time this year which turned out quite well with my middle ability year 8s.
Richard​
Thomas, thanks for this. I always love it when people challenge the message. I do think you are unintentionally splitting hairs, although that might not be the best term to explain what I mean. First up, P-1 vs U-1 is a continuum, not a dichotomy. Re 'you say that your worksheet is 'highly U-1, P-2' because you mix up the trig ratios, whereas doing them one-by-one would be 'highly P-1' ... the reason I refer to this activity as U-1 (ie on the P-1 vs U-1 continuum but towards the U-1 end of the continuum) is because when I have observed students work through it they are definitely not in 'machine mode' as they often are when just working through sine-length questions, or any 'ten questions of the same type'. The simple act of requiring students to decide whether a question is sine, cosine or tangent is enough to put sufficient cognitive load on them so as to require them to think their way through the work more. Sure, if you or I worked through such a worksheet we would be in 'machine mode' but to kids seeing this for the first time, not so, and therefore that worksheet, by definition, falls closer to U-1 than P-1. This is the perspective I'm coming from when explaining U1-P1. It doesn't make your view wrong ... it is obvious you are seeking out Understanding-first anyway. Hope that makes sense.
Thomas Allen
I understand what you're saying and I completely agree that the way you're suggesting doing that worksheet is better than if you hadn't split up the trig ratios. It just seems to me that trying to classify each activity in a series of lessons as U-1 seems a little forced in the sense that some of the activities help students to understand the concept and some activities help students to practice procedural work using their understanding. My point was that the worksheet is a tool to help the students utilise their understanding and practice the procedures. Since it doesn't teach anything, it seems strange to call it an 'understanding first' activity. The 'first' surely applies to the activities you did prior to that worksheet and the worksheet is now part of the 'procedures second' that you are helping the students to practice. I am splitting hairs a little because it felt like the message from that section was 'look at how everything is at least mostly U-1, which is good'. However, I felt like the main point could have instead been: 'Notice how we first build understanding before we look at any procedural practice over the whole scheme of teaching trig'. In one of the activity descriptions you pointed out yourself that the activity was procedural but since you tried to help students understand the content in a previous lesson it counted as U-1. Maybe it's just how I read it but that was what occurred to me. I think it is just a semantic argument when everyone is in agreement about the fundamental ideas going on here. I'm just thinking about how someone coming from a P-1, U-2 perspective might look at those activities and descriptions. Anyway, that's my ramblings. I hope that made sense! 😀
Richard​
Haha all good Thomas.

David Slade
So, first of all, I loved the geogebra file and I loved the questions. Coming from a primary background, the discussion and questioning reminded me of the anchor tasks in the Mastery Approach we use. Discussion is fantastic, especially when it appears that there are no wrong answers. I particularly enjoy this sort of introduction when looking at data. I remember, years ago, watching Dan Meyer take the labels and information off of graphs and seeing the impact it had on my data lessons. Apart from the geogebra file, which is key to it, this U-1, P-2 trig example has some similarities with my approach. I always vary the questioning (what's the point otherwise?) and I like to give students the opportunity to move on in a low key way. I'm not great with Geogebra, so look forward to a future course becoming proficient with it!
Richard​
That's great David. Glad to hear you resonate so well. If/when you do the GeoGebra PD, get ready for a head explosion. It happens.