Year Six of teaching this pickle of a topic in Calculus. I keep changing it up hoping I'll happen upon the magic elixir that will allow students to eagerly gobble up the problems and spit out the correct answers with joy and understanding oozing out of their pores.
* Finding problems that aren't the Wisconsin of all Cheese Balls.
* Reading and interpreting the problem.
* Being able to translate known information and subtly given information into Math Speak.
* Finding the right equation that links all the variables together.
* Navigating the Dangerous Path of Implicit Differentiation.
* Plugging back in at JUST the right time.
* Finishing things up with a tidy bow.
I have a good feeling about this year! And yes, I say and mean that every year, so I am allowing my bubble to remain floating blissfully in the air yet again.
First of all, cheesy problems: ladders sliding down walls, circles mysteriously expanding and contracting, lots of liquids being poured into and leaked out of cylinders/cones/prisms, cars passing in the night at some perpendicular intersection.
I started off this year's spiel with just a blathering of why this topic is important:
BP oil spill, how fast is it spreading? How will they know how much resources to devote to the problem?
The rain recently in Austin, TX, people's houses were ruined, water was rising at various rates, what could that mean for drainage capabilities and emergency personnel?
Fires in Bastrop, containment, speed of spreading vs speed of dousing ....
The amusement park problem on some previous AP exam about rates of people in park and number of employees needed ...
But then alas, we had to start on the cheeseball problems. But I prefaced it with, "no one really cares about a balloon being expanded or contracted, but as with all math modeling, we may have to start with a simplified version of reality just so we can gain understanding and maybe add difficulty and more accuracy later. Think about some real life things that could be modeled by a malleable sphere."
That was about as far as I got on that THEN. Wait for it .....
Also, I spent a class period JUST setting up problems with the students, and their homework was to differentiate and solve. I found Bowman's set up with the table for variables very nice and tidy and useful. I'm a convert. In the past, I just set the students on their merry ways to attempting each problem on their own. This year I was uber bossy and said I wanted things set up in a specific way - enter the drawing and table and something new I added: a highlighted box or equation that I labeled SUBTLE information. There are some things that are not specifically mentioned in the problem, but you can figure needed information out like a detective. I wanted the students to be on the lookout for such things and to know that they exist.
I think this is the NEW component I'm adding to the homework next time. They will get the goofy circle/square/car problems, but with EACH problem or maybe just one, they have to do extra. This is where my crowd surfing comes in. I want them to think of a NEW specific real life situation that (say) could be modeled by this simplified version or expanded upon, and discuss what rates are needed and why anyone would care.
I'm looking forward to seeing their responses.