Trigonometry, another way

Trigonometry falls firmly into the camp of one of those areas which I don’t feel I have cracked yet.  If “cracked” means finding a bomb-proof way to introduce it to any class I encounter, then maybe I never will!  Either way, I am always interested in different approaches to teach this topic which many students seem to struggle with at first.

This approach builds on the presentation that Mike Ollerton gave at the recent Mixed Attainment Maths conference in Sheffield (keep an eye on the site for details of the next conference in November!)

I didn’t actually attend Mike’s presentation – I was too busy giving my own – but Mike has kindly shared his ideas and I have been thinking about how I might use Geogebra as a tool to aid learning.

As with most of my use of Geogebra, I am using it as an exposition tool to structure whole-class questioning and discussion around. In an ideal world, I might get students to do this themselves, but that is not practical in my classroom. I feel that starting Geogebra from a blank sheet can be nearly as powerful as them doing it themselves and is likely to be a much more efficient use of lesson time.

The basic principle used is that of rotating a fixed line segment, a “spinner” if you like, around a point.  We are aiming to explore the co-ordinates of the point at the end of the line as the angle increases from 0º to 90º (and beyond) in a table.

I must say that from this point onwards this post is not Mike’s recommended approach (which is here) – but my interpretation of it using Geogebra.

So, first step is to form the spinner by by plotting points at (0,0) and (1,0), zooming in and connecting the points with a line segment. There is something in observing what happens to the scale on the axes as we zoom in.

Next, we create the angle by using the Angle with Given Size tool. As the prompt says when you hover, the tool: select leg point, then vertex.  Rather than fix the angle, I want to make a slider so I can easily change it. I set the angle “a” making sure to leave the degree symbol in place (otherwise you get radians).  The slider then needs tweaking by double-clicking to set the max, min and increment.

Next, I need to make the line segment a bit bolder by right clicking on it…

..and change the properties of the point so it shows the coordinates to 2 decimal places, also using right click.

We now have a tool that can tell us the co-ordinates.  Before using this, however, I think I would want pupils to do some work on paper, using Mike’s handout to get a feel for the numbers and get their own results.  To fill in this table:

I feel that it’s useful that they have the opportunity to correct any measurement inaccuracies before the next step and this is where the computer helps.

As per the worksheet, a series of questions can be posed before using the calculator’s Sin and Cos functions to complete the following:

From there we can easily start exploring what happens when the side length is not 1 and use the ideas of scaling and similar triangles.

And then, of course change the angle again.

Once this is all set up, it’s easy to display / hide the coordinates and maybe so some miniwhiteboard work to assess how well the class has grasped the use of the Sin and Cos functions. And then keep the coordinates, but hide the angle to demonstrate inverse Sin and Cos.

With some practise and familiarity with Geogebra you are spending less than 3 minutes on the computer.  If you like the idea of “Geogebra from a blank sheet”, click on the Geogebra category link at the top to see posts on using the same idea for other topics.

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Exterior Angles of a Polygon and other shapes

I find some students like to memorise this, or at least the first two or three rows and then remember that we keep adding 180 for extra size.

Reason being – an extra triangle.

I feel that using exterior angles is more satisfying.  I find myself walking around an imaginary hexagon in my classroom with arm out stretched.

Here is a very nice Geogebra visualisation of this:

Exterior Angle Collections: REVAMPED! Modify at https://t.co/zXdRkaQEym @geogebra @panlepan @solvemymaths @giohio @gogeometry @NCTM #MTBoS pic.twitter.com/1dgxkNwhgh

— Tim Brzezinski (@dynamic_math) July 16, 2017

And here’s an activity which reinforces this idea.  Use a pencil as an arrow.  Does it always rotate through 360?  Does it always turn the same direction?  Use algebra, maybe?

 

 

Circle Theorems with Geogebra – Part 2

This is the second post in this series.  Part 1 is here.

 

1, Tangents

What is a tangent?  If you look at a line and a circle, it can either not touch the circle at all, cut it in two places, or if we are really careful, just touch it in exactly one place.

The special thing about a tangent is that there is only one line that can form a tangent at any point on the circle. If you move the line then the point of contact also moves.  Geogebra works by drawing both tangents from a point to the circle.  By putting points at the intersect you can see how they move.

As you connect the points on the circumference to the centre, stop to ask what we know about the lines that have been drawn here.  Then what do we notice about the two tangent lines and radius as we move them?

To prove the point we can measure the angle between the radius and the tangent:

And measure some lines too, whilst moving things around a bit:

We can deduce that DE = FD, by drawing in one more line and proving that we have congruent triangles AED and AFD.

2, Cyclic Quadrilaterals

What is a quadrilateral? How many lines, how many vertices? So what is a cyclic quadrilateral? Once you’ve established this, draw one.

Then go round measuring the angles. Before you move one of the points, ask students to predict what will change and what will stay the same.

 

 

 

 

Circle Theorems with Geogebra – Part 1

There are various applets available online and indeed good Geogebra constructions that seek to demonstrate the various Circle Theorems.

However, I think it is more powerful to construct them from scratch in front of the class using Geogebra, starting from a blank sheet. It requires a degree of familiarity with Geogebra and reliable IT infrastructure (something which my school sadly lacks) but I think it is a more instructive way of walking through them than just moving some points on an existing animation.

So, this blog post, is an attempt to show my progression through the Circle theorems.

1, Set up.

First, get the Geogebra page set up, remove the grid and the axes as they are distracting. Also change the menu options so your screen isn’t littered with labels.

2, Draw a circle!

Point a point for the centre and then draw a nice big circle and point a couple of points on the circumference, talking through this as you are doing it using the terms in bold.

3, Angles subtended by the same arc are equal

You need three points, but best not to use the one which was used to define the circle as this will change the size of the circle. Two of the points define an arc.  Then draw two line segments to subtend an angle from that arc.

Then measure the angle between the lines and show how it changes:

Maybe add another point to create a second angle subtended by the same arc. Ask students to predict what the angle will be before measuring it.

You might want to highlight the arc itself using the arc tool.

4, The angle subtended by a semi-circle is 90° OR The triangle in a semi-circle is right-angled

Next investigate what happens as you adjust the arc and look at the special case that occurs when the angle is 90°.

5, The Angle subtended at the centre is double the angle subtended at the arc

Now add two more line segments to create an angle at the centre (deleting the line you drew previously).  Measure that angle and ask students what they notice as you change the arc.

You might want to have another look at the special case of when the arc is a semi-circle and reason why the angle at the circumference is 90°. And indeed show that this works for reflex angles too.

An alternative way of discovering this theorem is to construct this special case and then reason that because you have an isosceles triangle formed by two radii, you can show the the other angle in the isosceles triangle will be 180 – 2x the base angle. Therefore the other angle at the centre will be 2x the base angle.

We’re not there yet, check back for Part 2 where we will go outside the circle.  Oooohh….

Angle as a measure of turn

Angles in parallel lines is a topic that doesn’t usually feel too tricky to teach, but I often feel that I am just telling them “this is how it is” without giving a good explanation.

Inspired by a Twitter discussion on this Brilliant task, I’ve had a rethink:

From @brilliantorg pic.twitter.com/UpoQQhanxm

— Jo Morgan (@mathsjem) February 2, 2017

I use Geogebra a lot. It’s a powerful tool which somehow seems more powerful when you start with a blank page rather than something that has already been created. And angles in parallel lines is quick and easy enough to create on Geogebra:

Whilst doing this, I would want to move things around a bit to show that the two lines can move but stay parallel whereas the third line can move by moving the points.

This is the point where I previously might have just started measuring angles and showing what stays the same and changes as I moved the lines.

But it has struck me that this is a good opportunity to reinforce the idea of degrees as a measure of turn.  I often use a simple “Guess the angle” game like this.

I ask students to estimate the angle and show me on mini-whiteboards. Something as simple as this can cause great excitement when someone gets it exactly correct!  But it also reinforces the idea of degrees as a measure of turn from one line to another.

So, back to parallel lines, I am trying to show that the reason that alternate angles and corresponding angles are equal is because after turning one way an then back the other I end up pointing in the same direction.  And the reason co-interior angles sum to 180° is because I end up pointing in the opposite direction.

So, here is one that I did make earlier.  I have set it up so you can see the actual lines turning with the degrees increasing as they do so. I need to convince my learners that you can move from the first arrow to the second arrow by moving down the transverse line without changing direction.  Because of that, I will need to turn through the same angle to land back onto the parallel line:

And here is the same idea for Corresponding Angles:

I’d be really interested to hear any thoughts on this as a pedagogy – please leave comments below.

Geometric proof

A quick post to share a useful site that I learned about today.  Thanks to Heather Scott  who shared this during this evening’s #mathscpdchat on Twitter.

I’ve always struggled teaching students how to explain their reasoning when solving angle problems to a sufficient level to gain full marks on a GCSE question.  Here is a nice site that scaffolds this for students by presenting a series of mathematically concise statements that students need to chain together into a coherent and complete reasoning.

I would get students to write these in their books so they get a good sense of what is required in a full and coherent proof.

There are lots of good examples on this site.

It was obviously built a while ago, and sadly it doesn’t cover circle theorems, but I have e-mailed the author, so I’ll post any updates here!

 

The paper napkin trigonometry trick with a smattering of Pythagorean triples.

Take a piece of paper and do the following:

1. Make it into a square (interesting discussion on best way to do this).
2. Fold in half then unfold so you have created crease along a vertical line of symmetry
3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??

Proving Pythagoras

Although I’m sure I’ve taught Pythagoras lots of times, I have never really looked at the proofs before either for my own subject knowledge or with students. This may be because I was always happy when students had the understanding of how to apply the theorem and were able to find the missing side and so I left it at that.

Looking at proofs is a good way to deepen understanding of a topic, but generally shouldn’t be attempted the first time the topic is introduced, one of the points made in this comprehensive review of literature on how students approach proof in mathematics written by Danny Brown.

There are something like 140 different proofs of Pythagoras, cut-the-knot.org lists 118 geometric proofs here.

I decided to work through three:

Proof 1

On squared paper, students draw two adjoining squares of side length a and b as follows:

Next they draw diagonal lines.  The first thing that needs proving is that these two lines are perpendicular which can be done by finding the gradient of each of them.

We are now starting to get closer to a square of side c. A bit of cutting and rearranging and hopefully they establish that the area that they started with, a²+b² can be re-arranged to form c².

Here is a lovely Geogebra showing how these squares could tessellate for form Pythagorean Tiles.

Proof 2

This one is worth drawing although the scissors won’t help much here.  This is a Geogebra drawing of it (click on it to adjust the lengths):

A few of my students went down a blind alley with this one assuming that a is double b.  That is why it is useful to have the dynamic drawing to show that this is not the case. The crux to this one is seeing that the red square in the middle has side length (a-b) and then multiplying out (a-b)² to get the area of that square.

Proof 3

The third one I chose is fairly simple if you can remember the formula for the area of a trapezium! And really, once you’ve played this video over 10 times, nobody will ever forget that!

 

 

 

A simple circle problem

My high attaining Year 9 class didn’t quite get this on their own yesterday but they enjoyed the challenge and were able to follow the explanation.

The crux of the problem is getting a right angle triangle with sides 1, (1+x) and (2-x).  It is the (2-x) side which is hardest to spot.  There were groans when I finally showed them.

Then it requires some algebra – namely expanding (1+x)² and (2-x)² which Year 9 hadn’t had much practice in, so it was good to show them why (1+x)²≠1²+x²

I gave them the problem printed out, here they are 2 to a page.

Here is link to the Geogebra file that I created this on.

I had a few Twitter responses to this including @ProfSmudge who kindly set us an extension question:

It’s an example of an Apollonian gasket, apparently (thanks to @mathforge for pointing that out!).  That gets properly hard, involving Cosine Rule.  Certainly not something I’d give to my students, but I’ve got a few teachers working on it!

 

Tools for demonstrating constructions

There are various websites, applets and tools available to show constructions on an Interactive Whiteboard, but I like this one as the compass actually behaves and sort of looks like the ones students use.  Not sure how much longer Flash will still work, so if there are any others similar, that you use, please share in the comments.