To get straight into the meat of the post, I believe that one fundamental aspect has been missing from my teaching of exponential functions in the past.
To help students understand exponential functions, constant comparisons must be made to linear functions.
In the past, I’ve started discussions on exponential functions with a shocking statement (taken from Albert Barlett’s youtube lecture) – “The greatest shortcoming of the human race is our inability to understand the exponential function.” I’ve then tried to show, by getting students to do problems in which they struggle to predict the rapid, exponential growth, that this statement is on-the-whole correct. I still believe that this approach is valuable. However, when I reflect on how well students’ really understand the exponential function, there’s been something important missing from the learning equation. i.e. the connections and differences between exponential and linear growth.
The key difference between linear growth and exponential growth is that the rate of increase of linear growth is always the same, but the rate of increase of exponential growth always increases. It sounds obvious, but I believe that in making this direct comparison, students’ are more likely to understand why exponential growth is so rapid.
I saw the WTF problem below on Dan Meyer’s blog (The basic principle behind “What the F” problems is that they display a concept in a surprising way), and so started the lesson with what the students’ already know about growth.
Starter Problem – Here are some points in the plane:
(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11)
(20, 33), (7,7), (-5, -17), (10, 13)
Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?
Discussion leading from Starter –
Student: No matter which two points you choose, the gradient will always be two.
Me: Imagine explaining this to a younger brother or sister. What does it actually mean to say that “the gradient will always be 2?”
I wanted them to really understand that linear growth is constant. For every one you go across, you always go two up; the rate of increase is always two – no matter where you are on this line.
After that I showed the slide below. I obviously could have had them develop these questions themselves but I wanted this to be a high impact lesson; I had a clear goal in mind. I don’t think it took anything away from the learning in this instance.
In order to meet these objectives, I provided the following two tasks (10 minutes on each task in groups of 3):
1) Disease Epidemic– modified from an Nrich problem.
2) Rice Reward – the standard exponential growth (myth?) of rice on a chessboard.
You’ll notice that, in modifying these tasks, I’ve highlighted the point of comparing this new type of growth with linear growth. The main student comment which came out of these tasks was that the difference between linear and exponential is that you add the same thing every time with linear (in which I asked them to be more specific as to exactly what was being added to), and with exponential you multiply every time. This was a solid start but more to follow to really dig the concept deeper. Here comes the connection to Simple and Compound Interest – something we studied the previous year so wasn’t too difficult to re-jog their memories by re-introducing the material in manageable chunks.
Qualifier: I made sure with the bottom slide on compound interest that I connected to what the students had said previously about exponential growth by asking what was being multiplied every time (i.e. multiply by 1.02 in each step).
Next big question: “So why do the graphs both look linear? Why do they look the same?”
Great student response: Because the rate of increase at the start is pretty similar to the linear growth. The line will get steeper later on when the rate of increase gets bigger.
To me, that is exactly why people struggle to estimate exponential growth. Because it’s hard to imagine later on how much the rate of increase will have increased.
I asked students to sketch what both graphs would look like later on – to which everyone came up with the same sketch.
Then I asked for a prediction on the compound interest bank balance after 500 years before showing the Futurama clip alongside the 500 year graph.
Ended with this homework task:
So there’s the lesson. I’m sure there’s places that could be improved so any comments are appreciated. I realise that in this post, I haven’t used any new tasks but I believe that the difference in approach and the emphasis on a comparison to linear functions is absolutely the way forward to deepen students’ understanding of exponential growth.