Introducing Trigonometry

Introducing Trig - NZmaths

This week I will be introducing Trigonometry to Year 9 (UK Key Stage 3, age 13-14). I always like to begin trigonometry with students actually measuring lengths in triangles, I believe they get more of a feel for the meaning of the ratios of the sides of a triangle if they have actually measured the length of the sides and calculated the ratios themselves.

I decided what I need is some accurate drawings of triangles of various dimensions that they could work on. It took a few seconds (the third entry in the search results for introducing trigonometry) to discover not only the drawings I wanted but a perfect recording sheet! From NZmaths (New Zealand Maths) – a site I have mentioned before for its excellent resources comes Introducing Trig. As well as the resources, teachers’ notes are provided. (Scroll to the end, past the teachers’ notes  for the resources)

This slideshow requires JavaScript.

So after an introduction including a reminder of Pythagoras they met last year, we’ll just need to be clear on the terms opposite, adjacent and hypotenuse, how to complete the recording sheet and the measuring can begin! This is a very able group of students and I suspect several of them to be telling me for example by the end of the lesson that sin 60° is the same as cos 30°.

Checking a few more links in the results of the search I see the excellent Math Open Reference site which I have referred to on several occasions. I also see that I am in very good company in my desire to get the students measuring themselves and using the nzmath resource – see Dan Pearcy’s post.

Something else I like to do when discussing trigonometry is to discuss all the possible types of problems that can come up because whether they are disguised as buildings / trees / ladders or whatever there are still only a limited number of problem types, eg find the angle given the opposite and hypotenuse. The students can work out how many problems there are.


Desmos – tangents to curve

At the moment I am introducing calculus to my sixth-form classes and I wanted to use Desmos to illustrate drawing a tangent to a curve. A quick search found a page I copied to my own account, I then modified the page to show a simple quadratic (f(x) = x2).

Desmos tangent

Desmos – tangent to a curve

Something I have been meaning to do for a while is try the folder feature on Desmos, a rather neat way to create a tidy looking set of items.

Desmos page notes

Desmos page notes

Show / hide folder contents

Show / hide folder contents

Folders are easy to create:

Desmos Folders

Desmos - add folder item

Desmos – add folder item

Having created a folder, press enter and the next item will be placed in the folder. If you want to add items later, ‘nudge’ your item into the contents of the folder.

I recommend that students use Desmos to help them understand any questions they do; finding the equation of a tangent to a curve is a good example, Desmos can be used very quickly to illustrate a correct (or not!) solution.

For checking calculus examples students can use WolframAlpha, examples are included in the fourth slideshow here).

Update ….and along came @Desmos! (select either picture for graph page)
Desmos tangent tweet

Tangent to curve - Desmos improved version

Tangent to curve – fabulous version by Desmos!

Thank you @Desmos! (Think I’ll put in a few requests!)

Learn Desmos – Tangent Line Slider

This week…

…such a busy one – so I’ll be brief (but can’t break that new year resolution of January 2011, the only one I have ever kept, to write a post each week!)

So this week – I have been using my name cards – I’m learning names fast! My classes all know that I think WolframAlpha and Desmos are rather useful and we have had some great discussions on the learning behaviours I’ll be noting about them all on ClassCharts.

I must recommend the free samples from Bring On the Maths, the Core 3 activity – Logarithmic equations worked really well with my Year 13 class and next week I’ll use the C4 Binomial Expansion resource when we are talking about the validity of a given expansion. I have used the Trigonometric Ratios resource before – and will again; there are several other great resources in that list of samples – explore!

Now to hand over to a master (and his son), I love this TED talk from Hans and Ola Rosling. Can you do better than the chimps? Why we are all so statistically ignorant! (More on The Ignorance Project)


Diagnostic Questions

Logs & ExponentialsPlanning some lessons for the week I realised that I will be telling all my students about Craig Barton’s and Simon Woodhead’s wonderful Diagnostic Questions website so thought I would write this week’s post on that (again!). By some amazing coincidence as I started typing – into my inbox comes notification of a new post from Craig! He has been writing lots of lovely new questions and we can look forward to more (thank you so much Craig!).

Diagnostic Questions can make a great start (or any time!) to a sixth form lesson. I will be teaching Year 13 (UK age 17-18) this week about logs and exponentials. The Advanced Pure section has a growing and great selection of questions. Checking Logarithms and Exponentials I think those will do nicely as part of my lesson! As a registered user I was able to create a quiz consisting of all these questions. By creating a quiz one can order the questions exactly as required and also very easily create a PowerPoint slideshow for offline use, alternatively or as well – simply download the pdf version.

I find the easiest way to create quizzes is to use the Instant Quiz Facility which I have written about before; I thought it would be worth putting all the instructions together so created the following slideshow showing how I created the quiz on logs and exponentials. To create a new quiz I make sure that the Instant Quiz has no questions currently in it so have got into the habit of clearing it out once I have created a new quiz. The instructions for doing so are included here.

I created the following PowerPoint:
Logarithms & Exponentials diagnostic questions and to see the pdf version choose this file: Logs & Exponentials Diagnostic Questions. To view the quiz online then follow this link.