Introducing Trigonometry

I love introducing trigonometry - it's one of those big new concepts to get excited about teaching. It simply blows my mind that a few simple relationships between the dimensions of a right-angled triangle underpin the mathematics of a vast array of diverse fields; astronomy, acoustics, optics, seismology,  electronics, civil engineering and mechanical engineering to name just a few. I feel it's kind of important to kick it off with a good start!

I have seen trig introduced a number of different ways ranging from the mega investigative approach outline by Jo Morgan in her awesome post on resourceaholic.com through to... well, it been indigently given no real introduction at all! :(

My approach tends to vary depending on the time available and the type of class but one thing that now always ends up in the mix is a little GeoGebra. I have found that in particular the sketch below is fantastic for developing students conceptual understanding of trig ratios in even a relatively short space of time. This makes it a particularly great tool for introducing the topic when time is short, for revision purposes, or to summarize students findings after a longer paper based investigation. This sketch can be used in many different ways but I will outline a common approach I use that dips in and out of it over the course of a few lessons.

First off I get everyone in the class to draw an accurate right angle triangle that is any size they like but has another given angle, lets say \(36^o\) - a nice bit of practise in constructing a RHS triangle. 

Next up I introduce the terminology of 'adjacent' and 'opposite' sides and get them to measure and record these lengths, then finally I'll get them to  calculate the opposite/adjacent ratio. I'll take some answers and put them on the board... 

"That's weird, you all seem to have roughly the same answer...is that weird?"

This usually provokes an interesting discussion where terms like 'similar triangles' and 'gradient' often feature. At this point I'll show them my GeoGebra construction - I'll set theta to \(36^o\) first and the click the lower check box to reveal the tangent ratio. Playing around with the size of the triangle reiterates what they have just found as a group - the ratio is independent of the size of the triangle.

A nice game is then to find out who drew and measured most accurately; this gets students sharpening their pencils and honing there protractor and ruler skills as they attempt to draw a second triangle with a different angle - this can get quite competitive. If you find it's a tie you may need to resort to ramping up the the number of decimal places displayed in GeoGebra (I explain how here).

Its then back to GeoGebra to check - I might go for a third triangle or maybe we'll try to figure out a couple of ratios:

"What will the ratio be when \(\theta = 0\)?"

"How about when when \(\theta = 90\)?"

"\(\theta = 45\)?"

At this point I'll use GeoGebra to demonstrate that each angle has a unique ratio and trace up \(y=Tan(\theta)\)  by checking the green trace button and sliding theta (I won't mention what happens outside of the range 0-90 at this point). Note that if you want to plot the actual function click anywhere in the right-hand panel, then in the 'input bar' at the bottom type \(y=Tan(x^o)\) - GeoGebra always uses x as the horizontal axis variable and you must include the degree symbol to ensure the plot is in degrees not radians. Hint: Press Alt+o (as in the small-case letter) to get the degree symbol.

Why is GeoGebra having trouble calculating the ratio?

I will then explain that you can access the ratio for any angle by looking it up in a table or more conveniently using the 'tan' button on your calculator! From here I'll develop the idea of how we can use the tangent ratio to find the adjacent if we know the opposite and visa versa and we'll tackle some typical textbook type problems using the tangent ratio.

Over the next couple of lessons we'll come back to the GeoGebra sketch to explore the Sin and Cos functions in a similar way, again emphasising the questions: 

"What will the ratio be when \(\theta = 0\)?"

"How about when when \(\theta = 90\)?"

to get students really thing about what is happening to the ratios as theta changes.

"Will the sine and cosine ratios ever be the same? When and why?"

"Do we actually need all three ratios to solve right angled triangle problems?"

"Why do the sine and cosine functions have some similarities in their graphs but the tangent graph is totally different"

This sketch also makes a great starting point for introducing the trigonometric graphs via the unit circle which I will discuss in a subsequent post. I hope this sketch is useful to you whichever way you introduce trig - thanks for reading.

Equivalent Fractions

This week I delivered a training session on GeoGebra to my department, I showed them a variety of different sketches to illustrate how GeoGebra could be used to teach topics from each of the four main areas of the KS3/4 curriculum; algebra, geometry, data and number. We looked at linear graphs, transformations and cumulative frequency distributions but I think it was the fractions sketches that were the most popular.

A colleague who is full of creative teaching ideas and who has been teaching for many years told me that one sketch in particular was possibly the best single resource that he'd ever seen for teaching fractions - that made me smile so I thought I'd better share it here.

Experiment with manipulating the blue sliders to change the left hand fraction. "Will the fraction be bigger or smaller if I increase the numerator"..."Why do you say that?... What will happen to the picture?" demo "What if I increase the denominator ...",  "What if I reduce the denominator so that its less than the numerator?" 

I have used this demonstration several times with different ability groups and students always find these seemingly simple questions very challenging - they really get to the heart of understanding fractions.

Now onto the idea of equivalent fractions...Use the green sliders to perform an operation on the numerator and/or the denominator. "If I double the numerator whats going to happen"  demo, reset "If i double the denominator whats going to happen" demo, reset "What if I double both?"..."Surely not that's crazy we're doubling both the numerator and the denominator; all that effort and the fraction's going to remain the same size. Why is that?" You can use the verify slider to double check the fractions overlay exactly - especially useful when it's a close call.

Rack up the values on the blue fraction a bit and then switch the operation to division and you can explore look at the idea of simplifying fractions.

Great that all works nicely, time for some addition ... "What if I add two to both the numerator and the denominator"..."eh"..."Why doesn't that work?" Ah, if only we lived in Farey Land, our lives as maths teachers would be so much easier!

There are so many questions you can ask with this simple tool that really probe students understanding of equivalent fractions; you can use it at the board or get the students using it independently. My colleague and I were discussing after the session that we should use this for the AS level transition lessons we run in September.We find that even at A-level some students have serious problems when it comes to algebraic fractions as they don't have a basic conceptual understanding of what a fraction is and how they can/cannot be manipulated. I can't think of many other types of resource that you can use from KS2 right the way up to KS5 - I will post up some more of the sketches I use to teach fractions in the near future.

y = mx + c (part 3)

The link at the base of my first post should have landed you at the GeoGebra Tube applet page (y = mx + c). Geogebra applets are great for students to experiment with or for teachers to use who aren't that confident with GeoGebra. They allow access to a GeoGebra sketch without opening the full program so although they offer limited functionality compared with opening the sketch in the program, they are very simple to use and display.

To try out the ideas in this post you will need to open up my sketch in the full program (the Chrome App or GeoGebra Applet do not offer full functionality of the second sketch I will share). One way to to do this is to click on the share or copy link at the bottom of the applet page. You then have the option to download the sketch as a GeoGebra file:

I should add at this point that if you haven't already done so you will need to download GeoGebra to open this file; you can do this here.

Ok, so after recapping some of the questions I asked in the first lesson on y = mx + c this is how I would develop students knowledge through questioning whilst using the following sketch;

Click on the background then hold down shift and use the right/left keys to zoom in/out on the x-axis only (up/down controls the y axis zoom). Is the gradient of the line still the same? Zoom the axes back to how they were.

Can you give me the equation of a line parallel  to this one? - Use the input bar to enter an equation and check. I have included a "clear" button in the sketch to quickly delete any additional lines (this is especially useful if the sketch is accessed as an applet as you can't just click on the line and press delete).

Before the next question hit clear, hide the axes labels and change the original equation using the sliders.

Can you give me the coordinates of a point on the line? - Enter the coordinates in the input bar; multiple points can be entered and the 'Show Point Coordinates' button used to, well you guessed it ... show coordinate points but it also colours incorrect points red for clarity. When you want to change the equation use the "clear" button to declutter.

When we're ready to move onto perpendicular lines I'd click the line tool, select an integer point on the line and then invite a student to select a second point such that the line drawn will be perpendicular - I'd check the gradient using the slope tool ask; Is their a link between the gradients of the two lines? 

                

Time to introduce another sketch; 

I'd explain to the students that this time I have defined the red line such that it’s always perpendicular to the blue line. I'd demo this by changing m₁. This sketch uses the spreadsheet view of GeoGebra to record the gradients of both lines so that the relationship between them can be examined. To toggle record to spreadsheet 'on/off', click both of the grey/red buttons at the top of columns one and two.

I like to start from m₁ = 1, record up until m₁ = 8 then toggle record off. I'll then ask the students to look for a rule connecting m₁ and m_2. Next I'd toggle record off and move the gradient slider to -1 and then toggle on and record through to m₁ = -8; does your rule hold for negative gradients? A further more challenging question is when won't this rule work and why?

Once you have the rules for parallel and perpendicular lines established you can explore a range of problems problems;

Can you give me the gradient of a line perpendicular to:

$y = 4x + 7$

$y = -\frac{x}{2} -2$

$y = -\frac{3x}{5}$

To show the last two lines using the sliders you will have to first change the increment setting for them. Right click on the slider and click Object Proprieties. You then have the option to change the increment. I originally set it as 1 to avoid generating an excessive number of values when recording, if you make it too fine its difficult to control using a small slider, 0.1 works well for what we are doing here. Incidentally I usually explain what I am doing to students when building or editing a sketch on the fly - I want them to see that the models we are using are dynamic and it's not just a series of pre-programmed tricks I am showing them. Often I will have to tweak something in a sketch or build a new one to respond to a question from a pupil.

Can you give me the equation of a line that is perpendicular to: 

$y=\frac{2x}{5}+2$

How many answers are there to this question?

Ok then how about the equation of a line that is perpendicular to the line 

$y=\frac{x}{2}+4$ 

that passes through the point (0,-3) ... How many answers are there to this question?

What about one that passes through the point (1,2).

As students propose answers to these questions I will use GeoGebra to check them at the board. You will notice that I have the sketch set-up to show the equations in decimal form (it is much more of a faff to show them with neat fractions embedded). I like to present the equations in the questions as fractions as this is an interesting discussion point and they understand that they can be displayed in both forms.

As always and comments, further ideas or questions are most welcome. 

y = mx + c (part 2)

Since the sketch I shared in my first post focused on linear graphs, I thought I would continue with this theme and share some ideas of how I might choose to develop this topic over the next few posts. Once students have a handle on how m and c affect linear graphs, here's a fun game they can play. If they have access to a laptop, tablet or even a smart phone then using a graph plotter is a good way for them to check their answers as this way they can see what's gone wrong if they make a mistake.

Each player needs a set of the cards (A-F) in a pile (I get them to cut their own out first). Playing in pairs or small groups students should all turn over the same card at the same time and try to write down the equations of all of the lines on it. Once each team member has written down their predictions or after an agreed time has elapsed the group can use a graph plotter to check if their equations are correct. One point is scored for each correct c value, one for each correct m value and one for an equation in the form x = a.  There are six rounds of the game and then early finishers can try and design some interesting cards of their own. The cards can then be stuck in books and annotated to explain any misconceptions.

If each group has access to a tablet or laptop a convenient way for students to access GeoGebra is via the new Chrome App (at the sign-in panel thy can click continue without signing in) . For this exercise they should select the algebra view and all they need to do is type in their equations in the left-hand panel. Once they have completed a card their equations can be selected and deleted to clear the screen. 

At a push students can use the Chrome App on a smartphone but Desmos which is an online graphical calculator has a much friendlier interface for small screens. Desmos is actually pretty sophisticated; if you start typing an equation into Desmos such as 'y = mx +c ' it will give you the option to set m and c up as sliders as in GeoGebra. Getting students access to laptops or tablets isn't always easy in my school so if I just want students to plot something up I usually get them to use Desmos. I know pupils in many schools now have access to tablets in every lesson -  I'd love to experiment with this in maths - I can imagine it would revolutionize the way I teach.

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y = mx + c

I was first introduced to GeoGebra in 2011 during my teaching training at the University of Leeds. I remember being shown how to use sliders to manipulate the x-coefficient and constant term in a linear equation that was controlling a graph. I was struck by the power of this demonstration for teaching links between functions and graphs. The graphical response of manipulating the m and c values was immediately apparent and with careful questioning a deep understanding of the mathematics at play could be obtained by a student in a matter of minutes. For a student to explore these relationships using a pencil and paper would take hours of graph plotting and even with a graphical calculator at their disposal the relationships would take much longer to explore and become clear. The session was only short but the snapshot of GeoGebra I’d seen was impressive.

Unfortunately I didn’t endeavor to actually use GeoGebra during my training year; I was too busy planning three part lessons, filling in paper work and writing essays. Indeed over three years passed before I saw it in use again, not in a school classroom but back in the same seminar room at the University of Leeds. This time, whilst enrolled on the excellent TAM course run by the MEI, I was shown how GeoGebra can be employed to help students obtain a deeper understanding of calculus (this will be the subject of a blog post in the near future).

I think there are several barriers aside from being very busy that meant I did not embrace GeoGebra sooner;

1.       Whilst I was impressed by what I had seen of GeoGebra whist training I was only shown its use in a fairly limited capacity – I was unaware of its amazing potential to aid in understanding a vast array of concepts linking number, geometry, algebra, statistics and calculus.

2.       I did not come across any teachers using GeoGebra at the schools where I trained or have worked and so I had little inspiration or support to develop its use in my teaching.

3.       Although there are a lot of excellent tutorials on the web on how to use GeoGebra and a plethora of GeoGebra worksheets available via GeoGebraTube, few authors have attempted to explain how they actually integrate GeoGebra into their day-to-day practice as teachers and use it as they cover different topics across the maths curriculum.

At the beginning of this year I took some time to really get my teeth into developing and using GeoGebra resources. Now, whilst I wouldn’t call myself an expert; if I can visualise a way to represent a mathematical concept - I can usually make it happen using GeoGebra and the result is generally far more eloquent, informative and engaging than I could achieve with a whiteboard and a pen.  My motivation for writing this blog is to allow you to look over my shoulder in the classroom, to share some ways in which I use GeoGebra and hopefully help some readers to overcome the three barriers that I have discussed. I welcome any comments, ideas, corrections or questions that you have relating to any of my posts or resources.

The first applet I will share here is a very simple GeoGebra sketch based on the one I was first shown that I have eluded in this post. Thanks for reading. 

Some suggested questions you might ask are:

What effect does changing c have on the graph?

What about m?

What can you say about the line if m is negative?

What will happen to the line if m is set to zero?

Can the line ever be vertical and if so how would you write down its equation?

Can you think of a way to find out where the line crosses the y-axis from the equation?

What about where it crosses the x-axis?

If you know the x-value of a coordinate point on the line how could you find its y-value? e.g. The point (3,A) lies on the line y = 3x + 10; find A.

If you know the y-value of a coordinate point on the line how could you find its x-value? e.g. (B,5) lies on the line y =2x - 15, find B.

How could you check if a particular coordinate point lies on the line using just its equation? e.g. Does the point (2,7) lie on the line y = 6x - 5?

If you were given two points how could you find the gradient of the line going through them? e.g. (5,2) and (3,1).

Could you then find the lines' equation?