First I have to express an obscene amount of gratitude to Bowman Dickson for illuminating what will be challenging for students learning related rates, and sharing how he deals with it. I basically just took his post and reorganized it into a lesson that will work for me. This post will probably make more sense if you read his first.
Second, I have been thinking this morning about what this lesson has to do with the recent discussion at dy/dan. There's probably a way to turn these into a problem we could pose without words through the cunning use of video production skills I don't have. It's really fun to think about.
Here's what I'm giving the kids. Here are relevant documents: handout for the kids, smart notebook file, ggb's that I made.
Update: Mimi Yang, a.k.a. thebomb.com, revised the cone tank ggb to reflect a constantly changing volume. That file is in there too.
SUPA Calc Lesson 4-6 : Fold this paper in half to hide the bottom half. Please don’t look at the example problem while we are doing the investigation. It will just get in the way of your learning.
1. Blow up a balloon!
2. Go here: http://bit.ly/calcballoon
3. Figure out everything you can about rates with the balloons. Record your observations below… (there is no one right way to do this. Make it make sense to you.)
4. Which one is more like inflating a real balloon and why? Write about it.
5. What is going on with the other one? Write about it.
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
(Here, we'll set up a solution with a diagram, givens, equation, etc, in a very structured way.)
6. Go here: http://bit.ly/calclad
7. The model depicts a 10-foot ladder leaning against a wall. If the bottom of the ladder slides along the floor at a constant rate, what happens at the top of the ladder? Why? How did you figure it out? Write about it below.
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
8. Go here: http://bit.ly/calccone
9. A conical tank is filling with water. Use the slider to change the height of the water in the tank. How are the height and radius related? How are the height, radius, and volume related? Write about it.
10. Imagine you are standing in a municipal pumping station, watching this tank being filled with water. What do you think is more likely: (a) the height of the water is changing at a constant rate, (b) the radius of the water is changing at a constant rate, or (c) the volume of the water is changing at a constant rate? Why?
A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.