Math Mondays: Functions can be challenging

Functions can be quite an adventure to teach.   Students have been working with x and y, and suddenly they have this weird symbol thing where the y was. I, like so many others, use the machine analogy. You give the function an input i.e. put something into the machine, the machine processes the input and an output comes out. We play with the function machine a lot -  I design machines, and then students design their own. (A shoebox makes a great "machine" for functions.)

The machine analogy gives students something to "hold on to" in terms of understanding functions.  The analogy also carries over well when we start inverse functions where now the inverse is working in reverse. If in the original function we put in 1 and got out 3, when we press the reverse button (inverse function) and put in a 3 we should get out a 1.
But when I get to transformation of functions (reflections, dilation, etc.), it gets a little harder to carry on the metaphor.  

I initially used the idea of tweaking the machine, and that works a little, but in the end we just work through notes and examples. And that isn't too fun, until... I had a student ask me why we even need to know these transformations, and I found I didn't have any idea. Basically, transformations were just the next thing in the curriculum. 

I accept that some math at this level is just going to be theoretical particularly if you are not applying it to a science or if you are not a mathematician, but I was curious. I asked a bunch of my friends who work in S.T.E.M. fields, and I couldn't find anyone who uses transformations of functions, a couple of them even needed a reminder as to what transformations were. 

Next, I did an Internet search.  I didn't find much, but one of the interesting applications was the use of function transformations in digital animation software. In animation the figure is moved a little bit or even sometimes stretched.  Now, I was excited about transformations of functions.  I'm sure you can guess what we did next. 

First, I showed students this brief animation video from animator, Glen Keane. Then, after copious notes on different types of transformations and a card sort activity,  we had some fun with transformations of functions.  I gave students flip book pages with graphs pre-printed on them. Students drew a function on the first page. It could be as simple as f(x) = x. 

Then on the second page they transformed the function, for example by graphing f(x)= x-1. They continued in this way tilting, reflecting and stretching their function.  We bound the pages together and had fantastic little flipbooks of transformations. Some of my students even used a tablet to create a stop motion film with their pages  (I'll upload the finished videos at a later date.)

The unit was a win in that students: 

  • Developed a strong understanding of functions and transformations, 
  • Saw a real life application of a math concept, and 
  • Were actively engaged in learning. 
Next challenge: if I'm not designing a telescope, why do I need to know the equation if a hyperbola? :(  I'll be looking for suggestions from the resources below or from your comments.

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.



  1. Check out this video for an application of hyperbolas (not a telescope). You will still get students that ask, "If I don't go sailing and don't care where the ship may be located, why do I need hyperbolas?" At least it is a different approach.


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