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.