Category Archives: geogebra

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.

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:

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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.

Screen Shot 2016-06-07 at 21.30.34.pngWe 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):

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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

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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!

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Visualising volume of cuboid

The idea of these tasks is to get students to think more deeply about volume and to help visualise how different volumes “fit” together.

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Questions 1-5 can be done by effectively counting cubes. Multilink cubes would be good to help with the visualisation at this point.

Question 6 is the one to get the discussion going.  Is the answer 16 (if you picture them as solid cubes, rounding down) or 21.12 if you consider it as a flexible volume (e.g. a liquid)?

If you want to adapt this resource you can create your cuboids using this Geogebra resource.