Or, Why Rates And Proportions Are Important.

It’s Monday morning.  I am having a medical procedure done in which medicine is pumped into my arm through an IV.  My nurse is calm, efficient, and experienced.  She is also training someone new.  No problem: Everyone is new at one time or another.

After the experienced nurse inserts the IV, the two nurses leave me to retrieve the medicine.  This is the gist of the the conversation I hear through the curtain:

The amount of medicine is 100 ml and it needs to be infused over half an hour.  The machine needs the infusion rate to be entered in ml per hour.  What rate should you enter on the machine?

100?

The medicine is 100 ml and it needs to be delivered in half an hour.  What do you enter on the machine?

50?

Think.  We have to infuse the entire 100 ml in only half an hour.  How many ml/hr do we need to enter on the machine?

Oh, 200!

Oh.

My.

You can be sure I was paying close attention to this conversation.  And even though the trainee eventually got to the right answer, I was still relieved to see the experienced nurse check the machine.

One thing I noticed:  The strategy employed by the experienced nurse was essentially to repeat the question, with very little modification, and to calmly ask the trainee to “think.”

My approach would have been different.  When the trainee first answered “100,” I would have said something like, let’s think about that.  If we enter 100 ml/hr into the machine, and we have 100 ml of medicine, how long will it take the medicine to be completely infused?  I presume the trainee would have said it would take an hour.  (At least I hope so.)  At that point, I would have said, now if we want to infuse the medicine in half that time, do we need a faster rate or a slower rate?  What rate do we need to enter?  

I think this approach would have made a better connection for the trainee and allow her to see how her answer needed to change.  It would also better model the thinking needed to solve the problem.

What would be an even better way to have handled this situation?

Welcome!  Here is my contribution to Math Twitter Blog-o-Sphere Mission #1:  What is one of your favorite open-ended/rich problems?  How do you use it in your classroom?

I was looking for a way to open a discussion of functions with one of my tutoring students last summer and found a lesson on the Internet called The Shapes of Algebra, which (at least in one version) included a great Ice Breaker activity.  (See references below.)  This activity asks students to consider five functions – represented in five different ways (verbal, tabular, graphical, function notation, and as an application problem) – and determine which representations belong together.  One example is shown below.

What I Did

After printing the file, cutting apart the different representations and mixing them up, I gave my student the pile of cards with only the briefest of explanations:  Which ones go together?  It was wonderful to watch how engaged he was as he first figured out what he was looking at, and then went to work puzzling out which cards described common situations.  His initial groupings contained several errors, but he corrected most of them on his own as he worked through the set of cards.  To finish the activity, he taped each set of representations to a piece of paper labeled with the name of the function, creating a personal reference for the rest of our lessons.

In addition to creating a introductory experience with different types of functions, this activity generated lots of great conversations about interpreting graphs, identifying slopes and rates of change, as well as the meaning of function notation.  Each of the five functions are completely different (linear, quadratic, reciprocal, absolute value, and exponential) and nicely set the stage for the rest of my planned function lessons.

How I Would Use It

In a classroom, I would definitely use this lesson as a group activity.  I would create “kits” of the cards so that (ideally) each group would have a complete set for a different function along with lots of cards for other functions so they had opportunities to compare and contrast different representations.  I think this activity would spark a great discussion and I would follow it with The Shapes of Algebra explorations using the Function Explorations document I created.  The document both structures the activity and provides a nice way to collect students’ observations about the way the functions change.  Desmos is a great tool for students to use for graphing the functions, but a graphing calculator (or even pencil-and-paper!) will also work.

What would you do differently?

References

There are several links for The Shapes of Algebra handout (including one for an online version), but I can’t seem to find the link where I found the ice breaker activity.  You can use my link, or Google (including the quotation marks) “Mary owes her mother $7” and you should find the file.