MakeOverMonday Week 7: Postage Stamps

It is Week 7 of #MakeOverMonday problems!  Dan Meyer has issued the challenge to revise this postage problem and to be honest ... I'm struggling with this one as much or more than I struggled with the one last week.

In August, I know we will be introducing several parent functions to our Algebra 2 students.  Our Algebra 2 curriculum is built around the study of functions, consistently analyzing how the values for a, h, and k affect the graph, how the equations are used in context, and learning to manipulate and interpret the equations.

I'm thinking that both last week and this week's problems can be used as illustrations for basic functions.  (Although I'm still puzzled by last week's problem a bit.  I read someone's comment that tire marks should be a function of speed, and I agree.  It doesn't make as much sense to say speed depends on tire marks.  It makes much more sense to say tire marks depend on speed ... which would make the equation some sort of quadratic, instead of a square root function!)

OK ... so this week's is definitely a step function.  I know that my students will not have studied step functions in their Algebra 1 classes.  So I am curious how they will graph the data.   Here is the information presented from some textbook:

I would remove the term, "step graph," from the problem.  Instead, I want to put the data out there and talk about how it might be graphed.

Before we get started we might explore the history of the stamp briefly, and I would be curious if any of my students collect them.  (By the way, the American Philatelic Society has math activities on their website!)  I might also quiz students ... wonder if they know the current cost of a stamp ... I had to look it up!  Fortunately, Wikipedia has a record of stamp costs all the way back to 1863.

So ... after a brief discussion of the stamp, I would present data (I would add our current costs) and ask students to work in groups to graph the data.  I'll ask them to explain if the data is continuous or discrete.  I'll also ask them to identify the domain and range.

Each group will get an opportunity to share their graphs and explain why they graphed the data the way they did.  I am hoping that there will be some differences in the graphs so that we can discuss their reasoning.  I'd love to use "My Favorite No" strategy to discuss the positives and identify errors!

Depending on student work and our conversation, I'll end our session wrapping up the work involved in this function, explain how the greatest integer function works, and combine this work with an introduction to piecewise-defined functions.  I'll direct students to Cool Math's explanation of piecewise functions before giving students additional practice.


What would you do with this problem???