Trigonometric Graphs and The Unit Circle

After being introduced to the first 90 degrees of the trigonometric functions my students usually have to wait two or three years before I will let on that there is more. If you enjoyed my recent post 'Introducing Trigonometry' then you may be relived to know that you don't have to wait that long to see how I do it!

The first thing I do is rewind a few years and review the concept of trig ratios using this sketch; asking the same sort of questions as I did in my last post. Now to avoid visual clutter I try to keep my GeoGebra sketches as stripped back as possible but you may have noticed that in this sketch, on one of the triangle vertex's I left the coordinate point visible - there was a reason for this. One of the difficulties that students have with understanding the unit circle approach is that one minute they're used to applying the trig functions to right-angled triangles of all sizes and orientations and the next thing they're looking at a special-case of a triangle constrained in a unit circle. They sometimes struggle to understand how the results we are deriving could in fact be derived from any size of circle. Obviously it's easy to prove the results can be generalised algebraically but to help ease them into the circle here's what I do. I click on the vertex I just mentioned and stick a trace on it by right-clicking and going down to trace on.

I also make sure I  have a trace on the ratios in the right-hand panel. As you slide theta you should see a quarter-circle being traced out - I'll explain that the way I set this sketch up was by making the hypotenuse constant and thus forcing the opposite and adjacent side lengths to change as theta changes. If you think about it only two side lengths need to change to allow theta to change. If you now change the triangle's size a little and repeat then your students should notice that the graph has not changed - this is simply to re-emphasise the fact that trig ratios are independent of triangle size.

"Now I'm going to propose something pretty off the wall - what do you think will happen if we allow this point to continue on its journey around a full circle? Any ideas how the graphs may look...well lets investigate"

Since the trig functions are independent of the size of triangle I will suggest that we set the size of the hypotenuse to one unit for convenience - at this stage we're ready for the journey around the unit circle.

The unit circle is obviously nothing new and I expect that most teachers already use it for teaching trigonometric graphs. You may not however have a dynamic visual to use with your students and I hope this sketch makes it nice and clear.

If you are new to the unit circle then the basic idea is this. The fact that the hypotenuse is one unit means that the adjacent side length is simply \(1\times Cos\theta\) and the opposite length is \(1\times Sin\theta\). Thus the triangle's top vertex's horizontal displacement is equal to \(Cos\theta\) from the origin and the vertical displacement to \(Sin\theta\). Obviously I get my students to explain all of this to me...

"Why is the red line labeled with \(Cos\theta\) and the blue line \(Sin\theta\)"

"What will the length of the  blue line be when \(\theta=90°\) ?"

"What will happen to the length of the blue line once theta passes 90°?"

Meanwhile I will will check the \(Sin\theta\) box to trace out the function as I slide theta and test their predictions. I will then generally get students to make some longer range predictions on mini-whiteboards. They can then test their predictions using the sketch up to 720°, forgive me for not allowing you to go on  forever but you can at least wind it back down to -180°. As I mentioned in my last post if you want plot rather than just trace out these functions you need to type in \(y=Sin(x^o)\) into the input bar to make sure your plots are in degrees - press Alt+ o (as in the small-case letter) to get the degree symbol.

When you have spun through the sine and cosine functions you can turn to the tangent function. The question I like to ask is:

"Where does \(Tan\theta\) show up on the diagram"

I can't blame them really - its seems as if it's just too irresistible but someone will inevitable suggest without thinking it through that its the hypotenuse. Not the right answer but with further consideration their misconception leads them to the discovery of the identity:

\(Sin^2\theta+Cos^2\theta \equiv1\)

Getting back to the idea that \(Tan\theta\) is the ratio \(\frac{opposite}{adjacent})\ brings us to a second identity all within the space of about a minute - I guess trig identities are like buses!

\(Tan\theta  \equiv \frac{Sin\theta}{Cos\theta}\)

I love the fact that you could spend a life time doing standard textbook trig problems and never notice these relationships but when you're shown the unit circle they are just staring you in the face!

Now we know what ratio to plot we can trace out the tangent graph and discuss why it's such an awkward customer compared to the smooth talking sine and cosine curves.

Try taking it for a spin!

P.S. An obvious topic that links with trigonometric graphs is graph transformations. You might like to check out my posts on 'Single Graph Transformations' and 'Successive Transformations' to help.

After being introduced to the first 90 degrees of the trigonometric functions my students usually have to wait two or three years before I will let on that there is more. If you enjoyed my recent post 'Introducing Trigonometry' then you may be relived to know that you don't have to wait that long to see how I do it!

The first thing I do is rewind a few years and review the concept of trig ratios using this sketch; asking the same sort of questions as I did in my last post. Now to avoid visual clutter I try to keep my GeoGebra sketches as stripped back as possible but you may have noticed that in this sketch, on one of the triangle vertex's I left the coordinate point visible - there was a reason for this. One of the difficulties that students have with understanding the unit circle approach is that one minute they're used to applying the trig functions to right-angled triangles of all sizes and orientations and the next thing they're looking at a special-case of a triangle constrained in a unit circle. They sometimes struggle to understand how the results we are deriving could in fact be derived from any size of circle. Obviously it's easy to prove the results can be generalised algebraically but to help ease them into the circle here's what I do. I click on the vertex I just mentioned and stick a trace on it by right-clicking and going down to trace on.

I also make sure I  have a trace on the ratios in the right-hand panel. As you slide theta you should see a quarter-circle being traced out - I'll explain that the way I set this sketch up was by making the hypotenuse constant and thus forcing the opposite and adjacent side lengths to change as theta changes. If you think about it only two side lengths need to change to allow theta to change. If you now change the triangle's size a little and repeat then your students should notice that the graph has not changed - this is simply to re-emphasise the fact that trig ratios are independent of triangle size.

"Now I'm going to propose something pretty off the wall - what do you think will happen if we allow this point to continue on its journey around a full circle? Any ideas how the graphs may look...well lets investigate"

Since the trig functions are independent of the size of triangle I will suggest that we set the size of the hypotenuse to one unit for convenience - at this stage we're ready for the journey around the unit circle.

The unit circle is obviously nothing new and I expect that most teachers already use it for teaching trigonometric graphs. You may not however have a dynamic visual to use with your students and I hope this sketch makes it nice and clear.

If you are new to the unit circle then the basic idea is this. The fact that the hypotenuse is one unit means that the adjacent side length is simply \(1\times Cos\theta\) and the opposite length is (\1\times Sin\theta\). Thus the triangle's top vertex's horizontal displacement is equal to \(Cos\theta\) from the origin and the vertical displacement to \(Sin\theta\). Obviously I get my students to explain all of this to me...

"Why is the red line labeled with \(Cos\theta\) and the blue line \(Sin\theta\)"

"What will the length of the  blue line be when \(\theta=90°\) ?"

"What will happen to the length of the blue line once theta passes 90°?"

Meanwhile I will will check the \(Sin\theta\) box to trace out the function as I slide theta and test their predictions. I will then generally get students to make some longer range predictions on mini-whiteboards. They can then test their predictions using the sketch up to 720°, forgive me for not allowing you to go on  forever but you can at least wind it back down to -180°. As I mentioned in my last post if you want plot rather than just trace out these functions you need to type in \(y=Sin(x^o\) into the input bar to make sure your plots are in degrees - press Alt+ o (as in the small-case letter) to get the degree symbol.

When you have spun through the sine and cosine functions you can turn to the tangent function. The question I like to ask is:

"Where does \(Tan\theta\) show up on the diagram"

I can't blame them really - its seems as if it's just too irresistible but someone will inevitable suggest without thinking it through that its the hypotenuse. Not the right answer but with further consideration their misconception leads them to the discovery of the identity:

\(Sin^2\theta+Cos^2\theta \equiv1\)

Getting back to the idea that \(Tan\theta\) is the ratio \(\frac{opposite}{adjacent}\) brings us to a second identity all within the space of about a minute - I guess trig identities are like buses!

\(Tan\theta  \equiv \frac{Sin\theta}{Cos\theta}\)

I love the fact that you could spend a life time doing standard textbook trig problems and never notice these relationships but when you're shown the unit circle they are just staring you in the face!

Now we know what ratio to plot we can trace out the tangent graph and discuss why it's such an awkward customer compared to the smooth talking sine and cosine curves.

Try taking it for a spin!

P.S. An obvious topic that links with trigonometric graphs is graph transformations. You might like to check out my posts on 'Single Graph Transformations' and 'Successive Transformations' to help.

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.

Introducing Exponential Functions with Bacteria

Here's an idea for introducing exponential functions through the medium of digital bacteria.

Watch this video

In the spirit of Mr Meyer's; Three Act Tasks  ask "what's the first question that comes to mind after watching this video?"

A few of many potential questions;

Is this true?
How can this possibly be true?
How many bacterium would there be after 1 hour, 2 hours... 3 hours?
How many bacterium would there be after a week, a month, a year?
Can we write an equation linking number of bacterium to time?
Do bacteria really multiply like this?
If they do then why don't bacteria cover the entire surface of the earth?

Well we can certainly use GeoGebra to help tackle a few of these.

This sketch will enable you and your students to study some different hypothetical models of reproduction;

Starting with the green bacteria, play the animation; a few questions/tasks;

Sketch a graph of the number of bacterium against time (on mini-whiteboards or paper).
Can you write down an equation linking the number of bacterium to time?
How many bacterium do you predict there would be after 28 days?

If students want to re-examine some specific points in time, you (or they) can just drag the time slider back a bit. Once students have had a go at sketching you can then check the green box in the Graphics 2 window to plot up the number of bacteria. Then if you select the Graphics 2 window by clicking in it you can type a proposed equation into the input bar to validate it.

Uncheck the green boxes and repeat the same sort of thing for the hypothetical blue, pink and red bacteria. One point to note is that the petri dish can start to get pretty crowded and the red bacteria start to overlay each other. If all of the bacteria are shown at once, the other coloured bacteria are set-up to sit on top of the red bacteria so that they stand out - you can zoom in to take a closer look at just how dense the red bacteria are as "t" increases.

If students haven't come across exponential functions before they will probably need some help with the last equation. A nice hint to give is;

Ok so we need a more powerful expression but you need no more than the symbols you have already used, how can you rearrange them:

$+$      $x$      $2$

...$2^x$ I here you say, is that going to be different to $x^2$...why?

You might chose to get them to plot these in Desmos or using the GeoGebra Chrome app to make a detailed comparison.

I like the way that both the Petri dish and the graphs really demonstrate how powerful exponential functions are. Once students have had a go at sketching each of the graphs individually, try getting them to put them all on one sketch. You can use the slider in the Graphics 2 panel to change the aspect ratio so that students can see just how insignificant the other functions become relative to the exponential function as "t" increases.

By this point they should be all set to answer some of the other questions they came up with at the start of the lesson.

Hope you enjoy!

Match my Exponential Graph

Inspired by Michael Fenton's fantastic looking "Match my Line" and "Match my Parabola" resources I have created a similar activity to explore exponential functions using Desmos's new activity builder. I haven't actually road tested this yet but when I heard about the release of Desmos's activity builder I couldn't resist giving it a go and thought this would be a great way for students to explore basic exponential functions.

Match My Exponential Graph

Students have to come up with an equation in the form $y=a^x$ that goes through the coordinate points given in each challenge, there are also a couple of open questions thrown in. They should login at student.desmos.com using the class code that you as the teacher will be assigned to distribute. As the teacher you can then see students graphs and answers and share them for discussion (e.g. the misconception in fifth screen here).

Whilst solving each challenge students can experiment by plotting different graphs, this experimentation is the key to learning through this type of activity; students can quickly identify their own misconceptions and experiment to correct them.

My only concern with these 'Match my ... Tasks' is that students may use sliders to solve the problems very quickly without giving them much thought. Whilst sliders are a powerful tool for visualizing the effect of changing parameters I would like to have the the ability to disable them in certain challenges - I think the cycle of thinking about an equation, manually typing it in, checking its graph and then rethinking the answer is important to allow thinking time for students to develop new concepts. This thinking time may be avoided if the problems are solved quickly using sliders. Hopefully Desmos will make this possible in the future but until then I would encourage students not to use sliders for this particular activity or only a last resort.

I will definitely be trying this out at the first opportunity next year. This activity was really quick and easy to make - why not try creating one yourself ? Let me know how you get on.

Single Graph Transformations

I felt like I was jumping the gun a bit with my last post by talking about successive transformations so I thought I aught to take a few steps back. Single step transformations are something I've taught several times to GCSE classes and as part of C2. Some things I've learnt work well are;

1. FOCUS ON GROUPING

As a first task before even looking at any graphs I get pupils to organise the following transformations into sets of first two and then three groups as they see fit:

Getting students to group the transformations gets them carefully examining the equations for similarities and differences and encourages them to look for some structure in something unfamiliar. Later it provides a framework with which to help them understand the different properties of transformations and helps with recalling them. Here is a handout.

For more able groups or as an extension you can add some extras (obviously these need to be considered later anyway but I've found that to much to soon can muddy the picture);

An interesting misconception that often comes out here is that the equations containing  $y=f(x-a)$ and $y=f(x)-a$ should be grouped with $y=-f(x)$ and $y=f(-x)$. They expect because of the presence of the negative symbols these should all belong to one group although mathematically this is hard to justify; at this stage of the lesson I wouldn't correct them but towards the end of the lesson I would draw on this misconception and discuss how $y=-f(x)$ can be considered a special case of $y=af(x)$ with a coefficient of -1.

2. DEVELOP UNDERSTANDING

One approach to demonstrate the effect of the different transformations is to create some tables of values for different functions and plot them (handout here*). This method provides some valuable practice of substitution (which you may need to teach/review first) and is a good way to examine the effects of transformations on specific coordinates. Transformations parallel to the y-axis are intuitive and easily explained and understood but those parallel to the x-axis require more careful thought. Through this activity students can look at the relationships between the rows in the tables and think about why transformations parallel to the x-axis turn out the way they do. This is a challenging concept though so I also like approach it from another angle.

I really like Dan Meyer and Buzzmaths's collaboration; graphing stories. These videos are fantastic for developing an understanding of graphing and they can be extended to the topic of transformations. I like to use 'Height of Waist off Ground'. Play the first part of the video a couple of times and get students to plot the graph (do not use the half speed section). Then play the solution and get them to check it.

"Imagine you are able to look into the future by four seconds. You're watching the scene but seeing whats happening four seconds in the future, so at t=0 what will you see?" Replay the video and ask pupils to plot what they are seeing. "which way has the graph shifted?... If we define waist height above ground as $f(t)$ then what graph have you drawn?.... $f(t+4)$."

"Now imagine again your watching the scene but this time, time in the scene is moving twice as fast so it's as if everything your seeing is moving at double speed, draw what you would see"..."Describe the transformation in words"..."what about in terms of $f(t)$"..."$f(2t)$."

The nice thing about this activity is that without doing any 'real maths' it provides an intuitive understanding of x-direction transformations that students can recall.

3. EXPLORE, INVESTIGATE FURTHER AND STRUCTURE FINDINGS

Calculating tables of values and drawing by hand is pretty tedious but in an ideal world it would be nice to do this for lots of values of 'a' and several functions. This is where graphing software is invaluable.

This simple sketch enables students to put in any function they like and explore what happens as 'a' is changed. This works well in pairs. One student types in a function, and asks the other a 'what will happen if ...?' type question and then checks it using the sketch. Encourage students to be really pedantic when it comes to scrutinizing their partners description. 

Finally I ask students to revisit their diagrams from the first task and consider if they sorted them in the most useful way. To help with this I get them to add the following descriptions into their groups where each description can only be used once but some groups will contain more than one description. Once they have completed this task they can add annotations to their diagrams to explain how to sketch and describe each transformation.

The following diagrams essentially summarize everything they need to know and the groupings make the various transformations easier to understand and remember.

One final thing that I have to mention on this topic is how great Desmos is for transformations.

Desmos is very straight forward to use, it makes graphs look fantastic without any fiddling around. Something like this takes literally 30 seconds to make - find out how here. I really like the table of values feature; you can edit the type of transformation by clicking in the column header. So simple!

* This handout was adapted from TES contributor Kevin Bensley.