Sunday, 6 December 2015

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.





Saturday, 28 November 2015

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.
.



Sunday, 15 November 2015

Introducing the Normal Distribution (Part 1)


I can’t really say I enjoyed stats lessons in school; I found S2 particularly confusing and difficult, kind of like going bare-handed fishing whilst blindfolded. It wasn’t just me, when attempting to solve a problem nobody in my class, least of all my teacher seemed to understand what we were doing or why we were doing it . It felt like were memorising a book of recipes and then to answer a question we had try to work out which particular dish it was asking for before cooking it. We didn't accept any variations on dishes - if it wasn't on the menu you couldn't have it! Stats is not an easy subject to teach and I hate to point my finger at our teacher but they certainly had a very procedural teaching style and most things were presented in a pretty abstract context. I think we were seriously malnourished in terms of conceptual understanding.

Somehow I managed to muddle through but I never really came round to enjoying stats until I started applying it during my PhD, it was then that things started to really click into place. Now as a teacher of S2 I am determined to avoid focusing too heavily on teaching procedural knowledge in abstract contexts so I have set myself the following goals;

1. When introducing a new statistical technique or distribution do so by framing it in a practical context.

2. Wherever possible collect some real data to use in examples.

3. Whilst I don't think it's necessary for students to follow from 1st principles the derivation of every formula they use, I want to give them at least a feel for how/why formulas work and where they come from.

To put these ideas into context I'd like to share some highlights from a few lessons that I have recently taught on the normal distribution. In this first post I will focus on introducing the normal distribution then in a second post I'll discuss using the normal distribution as an approximation to the binomial distribution.

I have a particular keen pianist in my S2 group and I thought I'd draw on him for inspiration here and take a look at hand-spans.  Hand-span is one of my go to bits of continuous data to collect from  a class. Unlike height, hand-span takes seconds to measure but students still find it interesting to see who has the biggest! With a small sample size (around 20) and a mix of male and female students I wasn't really sure what to expect but I figured whatever happened it would lead to some interesting discussion points.




As students settled down I defined what I meant by hand span and asked students to measure each others to the nearest millimetre and asked them to just shout them out; as they did so I punched the numbers into the spreadsheet column (X) of this sketch.





So what's this chart you see appearing? A histogram? Well kind of but not as your used to seeing one at GSCE - you can see that the areas don't add up to the total frequency, what do you think I've done?

Well, I normalised the bar heights to give a total area of one so that it's comparable to a continuous probability density function - my students were familiar with this concept from an earlier lesson on pdf's. If you want to have a play round with the data set I collected from my class then I have uploaded it to GeoGebra tube for you here.



As you can see from my data it worked out kind of OK; fairly normal. We had a discussion about if this was what they'd have expected from the data.  For comparison I had some secondary data up my sleeve with a larger sample size to take a look at; it came in pretty handy (sorry I couldn't resist). This is from the website smallpianokeyboards.org which questions the one size fits all approach to key-size.


Some nice points here are that female distribution is pretty normal, the male distribution not so much. Get your students to scrutinise the table below (or even just the frequency curve) for reasons why this may be so.

An obvious difference is sample size. The other point to make here is that if the sample size was the same or you normalised the frequencies, if you combined the data into one distribution then this maybe bi-modal or possibly a bit flat topped! My group were mixed gender but the distribution looked OK if a little shaky round the edges so we proceeded. I did however explain that there are formal tests which can be used to determine whether the normal distribution was an appropriate model to use.


Going back to GeoGebra if you check the show $f(x|μ,σ)$ check-box then the normal distribution curve and its equation will pop up.



Now this equation is either hideous or beautiful depending on your point of view but you may wish to discuss some of the features of the equation or at least brace yourself for at least one student asking what $pi$ and $e$ are doing there. I was pleasantly relieved when someone asked because I would have questioned whether I had done a decent enough job in conveying the wonder of these constants in core lessons if they hadn't.

Anyhow without an understanding of multivariate calculus integration we can't really formally derive the normal function but we can convey an intuitive sense of why $e$ and $pi$ should appear. First lets take a look at $pi$.

Starting with a dart board if you aimed at the centre you would expect to get more darts clustered around the centre and less as you move out. This scenario can be formalised to act as the starting point for the derivation of the normal distribution function. Anyway, skirting round any formality, if you imagine you plotted dart density as a function of horizontal and vertical position (x and y) in a 3D plot you'd get something like this*.


For this to be a pdf you need the area under it to be equal to one but in this 3D plot that is the volume. Well we can't integrate a multivariate function yet but your students will recall;

$Volume \; of \; a\; Cone = \frac{1}{3} \pi r²h$

$Volume \; of \;a\; Sphere = \frac{4}{3} \pi r^3$

It shouldn't therefore come as a surprise that due to the shape of the plot that pi comes into play here and you can see this on the contour plots. The plot can basically be sliced into disks and integration sums the volume of each of these disks.






You can plot these live in WolframAlpha - just type in $e^{-(x^2+y^2)}$.

Now your students may be wondering why you just plotted $e^{-(x^2+y^2)}$ to represent the dart density. Well you could choose any other positive number (a) aside from e for the base and get a similar bell shaped distribution but as the derivation of the normal function continues you'd find another scaling factor ($ln(a)$) cropping up. If we chose $e$, the magical propety $\int{e^x }.dx=e ^x {\;}(+c)$  means that things run a bit more smoothly and no awkward scaling factor is required.

Getting back on track, students need to know that the normal distribution function is dependent on two parameters; $μ$ and $σ$. Play around with these using the sliders to give students a feel for how they change the shape of the distribution. In this case you want to set them to the values of your data buy typing in the pre-calculated values of $μ$ and $σ$ and see how well the normal curve models your distribution.  Ask your students for some comments: A key thing to notice is that if you cut the bits of the histogram off that stick out above the curve - you should be able to sandwich them between the curve and the histogram; The total area under each should be one.

Maybe you could now modify the sliders to give a distribution that is distinctly different from the data you've collected and ask the question;



Can you describe in everyday terms  how a class whose hand-spans could be modelled by this distribution differs from our class?  

Moving on;

Suppose we use our class data as a sample to reflect the hand-spans of all year 13 students in the school. What is the probability that a randomly chosen person in year 13 will have a hand-span that is less than 190cm.

Let $X$ denote the random variable hand-span and assume $X$ is normally distributed $X∼N(μ,σ^2)$. In other words;

$f(x)=\frac{1}{\sigma\sqrt{2 \pi }}e^{\frac{-(x-\mu)}{2\sigma²}}$

We are assuming from our sample $μ=213.88$ and $σ=18.72$

 $f(x)=\frac{1}{\ 213.88\sqrt{2 \pi }}e^{\frac{-(x-\ 213.88)}{2\times\ 213.88²}}$

Great, now we have a continuous probability density function so thinking back to the work we have done on pdf's how can we evaluate $P(X \leq 190)$?....Anyone particularly good at integration?

Nah, unfortunately it's not possible to analytically integrate this function (or at least to my knowledge)  but thankfully there is another way round.

 At this stage I introduced my students to the cumulative normal distribution tables found in the formula booklet.

 If you take a look at these  some kind sole has found the integral for you numerically and tabulated the results. Unfortunately they have only done it for one very specific distribution;

$Z∼N(0,1)$

At point we discussed standardising the distribution by firstly subtracting μ off X resulting in a mean of zero and then dividing by σ the give a variance of one. I think the first step is pretty obvious but the second bit more conceptually challenging. I used this sketch that I have discussed in more detail in a previous post on coded data - I have added an option to change the transformation so that it is comparable to a normal standardisation.  The key points to emphasise here are;

1 - If a scale factor is applied to a distribution then standard deviation will increase proportionally by the same scale factor.



2 - If the original data set has a mean of zero or is first transformed so that is has a mean of zero then applying a scale factor will not affect its mean.



Thus first subtracting μ and then dividing by σ (or the root of the variance) will transform the distribution to $Z∼N(0,1)$.



You can show this in the 'Normal Distribution' sketch by checking Standardise Data' and then holding shift and using the cursor arrows to zoom to see the standardised distribution. Finally check 'Show $\phi(z) $' to show the standardised curve and equation.


At this point we calculated a couple of fairly arbitrary probabilities using the standardised tables just to get a bit of practise using tables. We then went back to our distribution and amongst others tackled the question I posed earlier;

"What is the the probability that a randomly chosen person in year 13 will have a hand-span that is less than 190cm"

One last thing that I should mention is the GeoGebra Probability Calculator. You may well be getting your students to use graphical calculators to calculate probabilities rather than looking them up in tables. GeoGebra also has this functionality conveniently packaged in its Probability Calculator that is very easy to use and comes in handy for checking answers. You can access this by pressing ctrl+shift+p or by going here:



I really like the clear visual displays which come in handy when talking about complementary probabilities and the like. For some strange reason sometimes when I open the calculator screen it is cut-off at the bottom, just click on the bottom of it and drag it down if this happens to you.

I think I've covered more than enough ground for one lesson so I'll leave it here for now and talk about approximating a binomial distribution  by using a normal distribution in a second post on this topic. Any questions or comments are as always very welcome.

*Thanks to vonjd, mathstackexchange contributor for the idea to use WolframAlpha to show these plots.

Friday, 9 October 2015

Comparing Fractions

OK, so your students have mastered the concept of equivalent fractions and it's time time to start comparing fractions - I have made two sketches to help. 

I find that you really need to keep revisiting the fundamentals when advancing a topic; if students forget the basic ideas then everything suddenly becomes very abstract indeed.   I have designed these sketches so that the ideas of how to find equivalent fractions and equivalent fractions being...well... equivalent, are constantly recapped and reinforced.


Chose two fractions to compare by adjusting the sliders.



Click show equivalent fractions.



Move the central sliders to experiment with finding equivalent fractions.



Settle on a common denominator.



Verify the fractions are equivalent by using the top grey slider.



Use the right hand grey slider to make a comparison.



What's the difference?


If you don't like pizza and prefer to eat bars then check the 'model type' box.




I like to mix it up but bars are a fantastic tool for modelling proportions that are easier for students to draw and compare than pizzas! Check out William Emeny's amazing post on bar modelling. I have started to re-design all of my fraction models to offer a bar model but I haven't since tested this one extensively so let me know if you experience any issues.


Next up is comparing multiple fractions and ordering them. Try this:

Show some fractions - how can we go about ordering them?



Maybe some pictures would help (click the 'bars' check-box) - but you wouldn't want to draw them all out by hand every time!




Would finding fractions with a common denominator help? How can we visualise what we are doing here?




Maybe this is too complicated to start with and you just want to compare two or three fractions. Use the left hand check boxes to hide some of the fractions.





You can set up your own fractions - by adjusting the left hand sliders. I always encourage this in my GeoGebra sketches so that you can be dynamic and respond to pupils questions and enable students to visualise any question they like. However in this case, just because it takes a while to dial in all the fractions I have built in five examples that you can access by adjusting the vertical slider on the right-hand side.



I have made the bars drag-able by clicking on the point at the bottom-left corner of each bar. This enables you to put them next to each other to make a closer comparison if you wish.


I have found these sketches to be extremely powerful and would highly recommend you give them a go. If you notices any glitches or have any suggestions for improving them the please just leave a comment. Thanks.



Sunday, 20 September 2015

Understanding the Effects of Coding on Mean and Standard Deviation

Calculating summary stats from coded data has the potential to be to one of the driest topics in statistics - we're talking Weetabix without milk. I referred to my text book for some inspiration;

"Coding is useful as it simplifies the arithmetic of calculating the mean"

Well that's sold it! Please, we live in the 21st century, we have calculators. 

The main purpose of teaching coding data in S1 should be to develop a deeper understanding of measures of central tendency and spread. 

Is the standard deviation and mean of a data set sensitive to the units of the data?
If I have two sets of data that have been recorded using different datums how can I expect this to affect the summary stats?

These are the sort of questions that I want my students to be able of answer - they're not really about churning through calculations; they're about understanding. In addition I want this lesson to lay the groundwork for seeing how and why a data set can be transformed and modeled using a standard normal distribution which is an important later lesson.

I made this sketch to demonstrate what is happening when a data set is transformed/coded.



The $x_i$ data points can be moved around. Just to recap the concepts of mean and standard deviation try moving a point or two and asking the questions How will increasing the value of $x_4$ affect the mean? What about the standard deviation?

You can the start to ask questions about what will happen if the parameter's b or a are changed so that the data set is coded and then use the sketch to examine any predictions by checking the $y_i$, $\bar{y}$ and $σ_y$ boxes. Although I haven't included it on the sketch, don't forget to bring in how variance scales with respect to the standard deviation.

Some nice additional questions to ask are; Will scaling the data always affect the mean? Can you give me an example of a set of data points where changing the parameter $a$ will not affect the mean? Does the data set have to be symmetrical for this to be the case?

I'm not saying this sketch is going to set your coding lesson on fire but some students find coding quite confusing and I have found this visual certainly helps out. Give it a go!







Wednesday, 16 September 2015

Introducing e

$π$, φ, $\sqrt2$, $e$, $i$

You can count the number of mathematical constants that you are likely to have come across in secondary school on one hand - introducing a new one is pretty  rare and special opportunity. For me, Euler's number is the most challenging of these constants to get to grips with. I wanted to make sure that when I introduced e for the first time to my A2 group that I would be able to give them a feel for why this number is so special and not just give them them skills to deal with exam questions in which it features. Here are some ideas on the topic that I would like to share;

A FUNCTION THAT IS ITS OWN DERIVATIVE

As a warm up get pupils doing a bit of differentiating, if they don't yet know about $e$ we're probably just talking polynomials. Tricky question:

Is there a function that is its own derivative?
Could a polynomial function ever differentiate back to itself? 

Most likely you will get a reason based on the power law for why polynomials can't but you may wish to use this sketch to illustrate the point graphically:



An interesting attempt from one of my students based on this sketch was; where $x>0$, $\lim\limits_{ a \to \infty } x^a=f'(x^a)$ ... a nice reminder of how misleading graphs can be if you neglect the scale of the axes and just look at one part of the graph!

Another valiant attempt having not studied derivatives of trigonometric functions was $y=Sinx$. 


What's the gradient of the tangent where $Sinx$  is at a maximum? What is the value of $Sinx$ at this point?... Nice try 

Well they were struggling so I gave them a hint:

$y=a^x$ .... Desmos

You may need to explain how to plot a derivative in Desmos - after defining f(x) you can type in d/dx followed by f(x) or go here:




then type $f(x)$ so you have $\frac{d}{dx}f(x)$.

Rather than Desmos you might want to use this GeoGebra sketch at the board;


This sketch enables you recap differentiation; to take a step back and examine what we are really looking for here.

Set the slider to 4 to plot $y=4^x$ and move the red point (not the slider) along the curve.

Whats happening to the gradient as x increases? 
Hit show tangent and move the red point again.
Now if we plot these values we'd have the function $f'(x)$. Hit trace tangent and again move the red-point.

Now if I decrease the base number (a), what will happen to the the function $y=x^a$?... What will happen to the gradient function?... Un-check 'Trace Gradient' move the slider to 2, then recheck 'Trace Gradient' and move the red dot to explore.




So you as you probably found if we set a to a certain value then f(x)=f'(x)?

2.7 .... are you sure? ... Un-check 'Show Tangent' and 'Trace Tangent' and click 'Show Gradient Function'. Now set a to 2.7. This looks pretty good, lets just look a bit closer. Zoom in (one easy way to do this is by holding shift and using the cursor keys; left/right for the x-axis and up/down for the y)



See if the students can get any closer - now that this value of a where f(x)=f'(x) is so important that we give it it's own letter just like $π$; this time it's not a Greek letter (the first known algorithm for calculating $π$ came from the ancient Greek mathematician Archimedes) it's simply the letter e coined by the Swizz mathematician Euler.  Type in the letter e into the input box for the parameter 'a' (2.72 will appear) then zoom in to validate.(Note you can just type in e for Euler's number in GeoGebra as long as there is no other object defined as e - if you ever get a "redefinition error" press 'alt e' instead). No matter how closely we zoom in these lines will always overlay precisely - there will never be any difference.

Ah but is e actually 2.72, someone may have once told you pi was 3.14, but they were kind of selling you a bit short! Click options; rounding; 15 Decimal Places. Ah that's a bit better but still not the true number; sadly even if I was to write a number on every atom of the universe I still wouldn't be anywhere close to writing out e in full; what type of number do you think it might be?... yes just like $π$ its irrational and also transcendental

To change the degree of rounding, on the menu bar click - options; rounding.

OK so that's nice, so you've convinced your students that the function $y=e^x$ has a cool property but where else does $e$ crop up and what is it useful for? Let's explore just one area; growth.


RECAPPING THE CONCEPT OF EXPONENTIAL GROWTH

If students are going to understand the significance of e to growth they first need a solid understanding of exponential growth, lets recap...

I'd start by posing a couple of simple problems -

e.g. You stick £1000 in a savings account offering 5% compound interest per annum. How much will you have in the account after a year?... 3 years?... 20 years?

You can use the following sketch to recap these ideas.



We're talking about discrete growth here, your bank has chosen to pay you interest once a year. What would happen if your bank was to pay you the interest in two, six month installments so that they put half of the annual interest into you account after six months, would this change anything? 

Let's take a closer look....

THE CONCEPT OF A CONTINUOUS GROWTH FUNCTION




The idea for this sketch came from reading Khalid Azad's fantastic Intuitive Guide to Exponential Functions and e - I would highly recommend this if you need to refresh your knowledge before teaching this topic. The idea is ask...

If you have £1 in an account and are promised a return of 100% after a year (some return I know!) then how much would you have after a year?...£2... obviously. On this graph the initial investment is represented in blue and the return in green. 

Now what about if half the interest half is paid after six months and left in the account until the end of the year, how much will you have at the end of the year?

Move the slider to 2.

What does the red bar represent?...It's the interest earn't on the green interest.

How would we calculate this? 


At this point you may need to recap the the basic idea of compound interest;

$$Total = Investment × (100\%+ Interest Rate(\%))^n$$

and discuss that interest rate here as a decimal will be 1 (as we are dealing with 100% return here) divided by n  (the number of time periods we are splitting the year into). Hence you have the formula

$y=\left(1+\frac{1}{n}\right)^n$ 

What would happen if we split the year into three chunks?

Move the slider to 3.

Can you see that little cyan bar?...What does that represent?

Just to make it clear how the colouring works; green is interest earn't on the blue, red is interest earn't on the green and cyan is interest earn't on the red.

What do you think will happen if we keep increasing the number of time chunks? ... Infinite money?...

Move the slider a bit more.

Are you sure?

Ramp it up.

What is that number that the total interest is heading towards? 

It was quite tricky to work out a a way to create this sketch in GeoGebra; in the end I ended up relying on using the spreadsheet function to generate a matrix to control the colored lines (GeoGebra nerds please let me know if yo have a more eloquent solution!). The downside to this method is that the sketch can get quite hungry so I've only shown the first six 'interests' as distinct colors and the rest are shown but are amalgamated into black. This only matters if you try to zoom in which you can do but just be aware of this limitation of the plot if students ask.






The other thing is you can't go for more than 99 increments - the model will sadly break.

Anyhow at this point I would plot up the formula:


$y=\left(1+\frac{1}{x}\right)^x$







to show $e = \lim\limits_{ n \to \infty } \left(1+\frac{1}{n}\right)^n$. 

So e is the maximum amount  you can get by compounding 100% growth on one unit (e.g. a pound), over a time period time period i.e by continually compounding the growth.

So what if you compounded over two time periods?...  $e^2$ ...Three time periods.... Half a time period? 

Hang on a minute we seem to have something here...$y=e^x$ will give us the amount of growth (+the original amount) if we continuously compound. 

You can link this back to a basic compound growth formula;  

$Total = Investment × (100\%+ Interest Rate(\%))^n$

if students don't see why it's $e^x$ rather than $xe$. 

You can now head back to the previous sketch and plot $y=e^x$ to see that this function perfectly models continuously compounded growth (just type $y=e^x$ into the input bar).


Coming back to original question;

What would happen if your bank was to pay you the interest in 6 months installments?

Well yes of course you get more interest and it would be even better if they paid out the amount accrued from continuously compounding interest. They certainly take all of this mathematics into consideration in their models but they usually quote an Annual Equivalent Rate (AER) which tells you how much interest you will get calculated once per year even if they agree to deposit it more frequently.


COMPARING MODELLING FUNCTIONS

So now we have got to the point where we have a function $y=e^t$ that can be used to model exponential growth but it's a bit limited in that so far it can only model continuously compounded growth with an initial amount of 1 and a growth rate of 100%. How could we make this more adaptable to model different scenarios? 

I considered a couple of ways to build rate into the equation but in the end took the textbook route of looking at equivalence between the $y=ab^t$ and $y=ae^{kt}$. To do this they need to be introduced to the natural logarithm and then it can be done in a few steps:

Let's modify our percentage multiplier ($b$) by taking its natural logarithm to give a new parameter ($k$).

$k=\ln{b}$

or in other words

$b=e^k$

Substituting this into $y=ab^t$ yields;

$y=ae^{k^{t}} = ae^{kt}$

Now we have something very similar to $y=e^{t}$ except it contains the extra parameters $a$ (our initial investment) and $k$ which is an alternative growth constant that can be calculated from $b$ using $k=\ln{b}$.  $y=ab^t$ and $y=ae^{kt}$ are identical functions where $k=\ln{b}$ and can both be used to model exponential growth - you can confirm they are the same by plotting them both and watching them overlay. In fact you could use any base to model exponential growth but e is by far the most common and the one you need to be used to dealing with.

In summary we have discovered

A  $y=e^x$ is its own derivative
B  The number e is reached as

$\lim\limits_{ n \to \infty } \left(1+\frac{1}{n}\right)^n$

C  $e$ is the maximum possible result you get if you continuously compound 100% growth  for one time period.
D $y=e^x$ is the function of continuously compounded growth
 $y=ae^{kt}$can be used to model any form of exponential growth. 
and we've only scraped the surface!

I couldn't resist displaying Euler's Identity and the Gaussian Integral to round of this introduction.

$e^{i\pi} +1 = 0$


$ \int^{\infty}_{-\infty} e^{-x^{2}}.dx = \sqrt{\pi} $

Marvelous