Category Archives: geogebra

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.

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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.

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

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

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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:

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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:

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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.

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And then, of course change the angle again.

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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.

Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

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In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

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So, what was this wonderful method?  Well, it might be nothing new to you dear reader, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

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3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

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5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

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I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: Geogebra uses the US terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

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And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

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This is an example.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…

 

 

 

 

The angle defines the ratio of side lengths in a right angle triangle

This post shows how to use Geogebra to demonstrate this fundamental truth in geometry and hopefully demystify Trigonometry to a certain extent.

As with all things Geogebra, I always try to start with a blank sheet (see other posts on this here and here).  This time, I’m not using the Geogebra app itself but just launching it from within a Chrome browser window which works pretty well.

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Once it is launched, I right click in the middle to remove the axes, but I am going to leave the grid on.

Then I create the triangle by constructing a line, a perpendicular line…

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…and a third point which I then join to create a triangle using the polygon tool.

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Next, measure the base angle of the triangle remembering the convention that angles are measured in an anti-clockwise direction.

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The next bit is a tad fiddly. You need to right click on the line segment to change the label to “value”.  Then do the same for the other two sides of the triangle so that you now have one angle and all three side labelled.

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So far, this has taken about 2 minutes to create from a blank screen.  You could do it in advance of the lesson, but I think it is worth doing it in front of the class, maybe having practiced it a couple of times.  Using “something I created earlier” is less powerful – it looks like some sort of trick, somehow.

Now you have everything set up you can start moving the points as shown here.

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I start by moving point B, thus keeping the angle fixed.  I would ensure students have calculators in front of them and ask them to calculate opposite divided by adjacent. Then move the triangle to get different values for side lengths. Then do the calculation again. The answer is the same, of course.  I might ask them how they could get that directly from the angle (tan angle).  Depending on where the discussion goes with that, I might then move on to look at sin and cos.

Finally, I always like to talk about how things were done in the old days, being careful to point out that I’m not that old and that I didn’t actually use these…crc_trig_tables.jpg

I explain that the sin button on your calculator is basically just looking up the values in the sin column of a table like this – not actually true, I know, but it helps understand what’s going on so that’s OK for me!

 

 

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.

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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.

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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?

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

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

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

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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.

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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.

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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.

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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.feb-01-2017-18-50-17

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.

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

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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.

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

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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°.

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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.

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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.

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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.

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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:

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:

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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:

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And here is the same idea for Corresponding Angles:

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

Fundamentals of Area

Area is a topic where students are likely to have remembered a process (you multiply the lengths to get the area) but may not have a secure understanding of what area actually is.  It is worth spending time discussing what we mean by area.  I would prefer my students memorise “Area is the amount of space inside a 2D shape” rather that “Area is length times width”.

Counting the Squares is a good place to start even with students who feel like they know how to calculate area.   The concept of 1cm² as a quantity, an amount of something that bigger shapes have more of is powerful.  Estimation can be useful.  Physical pieces of 1cm² card and a series of rectangles, some with grids, some without is worth doing.

Reinforcing units is important.  Reminding students of the commutative law and rewriting 2cm x 3cm as 2 x 3 x cm x cm can help explain why we write cm² and link it back to algebra.  You could even play with 3cm x 2m.  Does make sense to say that this area would be 6 cmm?

A key step in the process is moving from:

Screen Shot 2016-07-14 at 09.57.31.png…without losing sight of the underlying concept that the rectangle is made up of 12 little squares, each one 1cm².

Moving on to rectilinear shapes, I like this question as a way of challenging thinking.

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The first misconception here would be multiplying 3 x 4 to get 12.  But of course the correct answer is “we don’t have enough information”, an answer which shows depth of understanding and one which I would like my students to be presented with more often.