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Saturday, December 3, 2011

Still Relating Those Rates

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., 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:

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:

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:

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.


  1. >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.

    No way will they hold off looking!

    Can you put the later stuff on a second piece of paper and hand it out when you want them to see it?

  2. Yeah yeah yeah. :) I was trying to be Eco and not use twice as muc paper.

  3. So when your kids "Go here:", are they on smartphones? Do they each have a laptop? Is there one computer in a corner and they're taking turns? Do they each have an ipad? How do you make this work?

    I have a bunch of computers in a separate room where, once we're there, it's hard to get students organized to do anything *not* on a computer. Switching back and forth between on-line and on-paper in the same class period seems really challenging.

  4. We have a cart of 30 laptops we can wheel right into my room. I have not had the same problems managing work with laptop + paper.

  5. I love these! Thanks for sharing. :) But, I'd like to try to see if I can modify the cone to control change of volume directly via the slider. Can you please send me a link to the actual GeoGebra file if you still have it? Thanks!

  6. It's in the folder of documents I shared in the post. I would like a slider that changes the volume, too. :) Let me know if you figure it out.

  7. Try this: I changed k to represent how much of the volume has already filled up.

    It's fun to see how when k=0.25, k=0.5, k=0.75, what the cone looks like. Thanks for your file!

  8. (Although visually it looks not that intuitive. I'll have to go back and double-check my math and make sure everything is kosher. If I find a bug, I'll let you know!)

  9. I added it to the shared folder with the other stuff from this lesson. Hope that's ok. Thank you! You're

  10. That's OK. Thanks for the shoutout! :) I'm glad you liked it.

  11. PS. Making a GGB file modeled off of yours really saved me a LOT of time. I think it's neat that we essentially collaborated on the technology aspect even though we're physically separated. Go internets!


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