Powerful imagery using Geogebra

I absolutely love Geogebra, I use it in nearly every lesson to the extent that I’m not sure how I would teach certain topics without it!

I’ve written before about the power of starting from a blank sheet (angles in parallel lines, trigonometry, circle theorems), but recently I have found and used some excellent visualisations that other users have created and kindly uploaded. I feel like this aspect of Geogebra has improved considerably over the last couple of years in particular the search. I often have an idea in my head for what I want to show students and within a few seconds I have found exactly what I need by searching.  Using dynamic geometry that you can narrate to (or not) is so much better than just playing a YouTube video.  I try to think of points where I can ask “what will happen if…” type questions.

Here are my latest finds that I have used in class recently. Click on them to take you directly to the Geogebra. I’m sure this collection will be added to as I find more.

 

 

 

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Hippocrates First Theorem

Another one from the fabulous Don Steward:

You could of course just go straight for the algebraic proof but it does require a level of confidence with surds.  So you might want to scaffold this task. Maybe start by putting some numbers in for the radius of the smaller semi-circle, maybe 2.  You could then do it again with 4 and ask students if they are convinced by that.  (Here are some examples to warn against the dangers of extrapolating what appears to be a pattern).  If you do take the numbers approach it’s good calculator practice.  Can you type the whole expression for the area of the curved shape into the calculator to get an exact answer?

And here is a little GeoGebra drawing to go with it.

 

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!

 

Generalising with Areas of Circles

So, at #MathsMeetGlyn last weekend Don Steward briefly put this picture up on the screen but then decided to talk about something else as time was pressing.

That’s a bit like a “Wet Paint” sign.  You have to touch it just to see if it really is wet…

So I had a little play with these. I couldn’t find anything about it on his blog but of course there is lots of other lovely stuff there.

A few different questions you could pose here:

• What is the ratio of red area to blue area?
• What is the red area as a fraction of the blue area?
• What is the red area as a fraction of the whole circle?
• What is the blue area as a fraction of the whole circle?

And then of course, can we generalise?  What about when there are n small circles.  What happens as n gets bigger? Why? Can we find a general expression for the area?

Have a go.  It actually turns out to be quite simple but depending on which question you start with, you can get into a lot of practice with ratio and dividing fractions.

Geometry Problems

Having now taught this lesson, I’ve edited a few things. 

This problem was posted on Twitter last week by @solvemymaths.  I must confess I was pretty slow to solve this! It’s a lovely problem – so easy when you know how.  I created this Geogebra file.  It didn’t help me solve it directly but it did answer the question of whether this is a fixed shape or not, which I was struggling to visualise from the diagram. It isn’t fixed and that gave me an idea for a lesson. Rather than just give my students the problem, we will draw it first to provide that hook, the hunger to answer the question, “Why does that happen?”. I also provides a bit of practice using a compass, a skill that is needed at GCSE and one which we don’t spend much time practising.

Here’s how I would now run the lesson (key maths words italicised):

1, On plain paper everyone draw a circle using a pair of compasses.  You chose the radius, anything between, say 1cm and 6cm. Make sure you write down your radius. Calculate the area of this circle.

2, Then, using a ruler, carefully draw a tangent, doesn’t matter where it is on the circle.  Make the line nice and long, using a 30 cm ruler if you have one.

3, Then, using compasses again, mark 5cm either side of where you tangent touches the circle. Label the points A and B.

4, Next draw a second concentric circle, going through A.  It should also cut B if you have drawn accurately. You should have a shape looking something like this:

or this 

5, Now, calculate the area of the big circle.  You’ll obviously need to measure the radius of the big circle first.

6, Now find the area of the orange ring. Compare your area with that of others around you.

Where you go next will depend on the class. I’m not going to give you a full solution here, that would spoil the fun! But the answer is 25π…