# Mullet Ratios, Rational Functions and Asymptotes

We never would have guessed that we would have a post with that title.

Yesterday we ventured into a Grade 12 Advanced Functions class to do a lesson on Mullet Ratios inspired by Mr. Vaudrey’s blog.   The original plan was to do the lesson as Mr. Vaudrey did but we wondered if we could make it connect better with the curriculum of the Grade 12 course.

Our preliminary “planning” thinking went like this:

We started experimenting with rates.  Like what would happen if my mullet grew by 2 cm’s every month? Imagining that our “party” was 12 cm’s to start and our business was 4 cm’s to start (see Mr. Vaudrey’s lesson for a better explanation of this), we made a table:

Of course we got tired of calulating the ratios for every month on the table so, as mathematicians do, we thought about another way to model it – an equation!!!  And if we think about what happens as time goes to infinity we can see that the limit is 1!!!!  We needed to try this out with some classes, so clad with mullet wigs and a good youtube version of Achy Breaky Heart, we went to work with some Grade 12 Advanced Functions classes.

Our “mullet ratios” lesson went like this:

We gave students clickers, had them rate mullets, gave them a definition of a mullet (party in front, business in the back), had them rate mullets, gave them the mullet ratio, had them rate mullets, and had them find their own mullet ratio.  With a well established understanding of mullets, we were now able to present our ‘question du jour’:

Many students felt they were finished after just a few minutes.  “The mullet ratio will just decrease,” they said.  “But how far will it decrease?” we asked them.  We had already talked about whether or not it was possible to have a 0 or negative mullet ratio.

Kids started talking about exactly the things we hoped they would talk about… some of them made equations and saw the asymtote, some of them just thought about how the party and business would start to look almost the same (so close to 1) but would never actually be exactly the same (hence never actually equal to one… they understand asymtotes!!!).  Some drew graphs.  One student asked for a graphing calculator so that he could plot the function he had come up with (yes!!!).

Here’s a sample of some of the great solutions we saw (we picked several to present their thinking to the class).

We had a bit of time left… so our next questions were as followed:

Students got right to work.  They used some of the things they had learned from their classmates presentations to help them with these problems… almost all realized that the first question would have an asymtote of 0, and that the second would get infinitely bigger.  A couple of groups even observed that the second part is a linear equation.

We ended the class with an exit card (ordering the mulletness of several well-known mullets) which demonstrated that the students really had grasped the concept of mullets.  Even more than that though, we were impressed with the language and understanding these students had expressed about increasing and decreasing functions as well as asymptotes.  The students we were working with had already finished a unit on rational functions, however the classroom teacher felt that this solidified the concepts for them.

We’re curious how it would go as the first activity in introducting rational functions. Let us know if you try it!!