Related Rates, Yet Another Redux

I posted in 2008 how I didn’t actually find related rates all that interesting/important in calculus. The problems that I could find were contrived, and I didn’t quite get the “bigger picture.” In 2011, I posted again about something I found from a conference that used Logger Pro, was pretty interesting, and helped me get at something less formulaic.

I still don’t know how I feel about related rates. I’m torn. Part of me wants to totally eliminate them from the curriculum (which means I can also possibly eliminate implicit differentiation, because right now I see one of the main purposes of implicit differentiation is to prime students for related rates). Part of me feels there is something conceptually deeper that I can get at with related rates, and I’m missing it.

I still don’t have a good approach, but this year, I am starting with the premise that students need to leave with one essential truth:

Often times, as we change one thing, it affects a number of other things. However, the way that the other things are affected can vary greatly.

Right now, to me, that’s the heart of related rates. (To be honest, it took some conversation with my co-teacher before we were able to stumble upon this essential understanding.)

In order to get at this, we are starting our related rates unit with these two worksheets. A nice bonus is that it gets students to think about the shape of a graph, which is what we’ll be embarking on next.

The TD;DR for the idea behind the worksheets: Students study a circle which has it’s radius increase by 1 cm each second, and see how that changes the area and circumference. Then students study a circle which has it’s area increase by 10 cm^2 each second, and see how that changes the radius and circumference. The big idea is that even though one thing is changing, that one thing affects a number of different things, and it changes them in different ways.

[.docx] [.docx]

(A special thanks to Bowman for making the rocket and camera problem dynamic on Geogebra.)

It’s not like this is a deep investigation or they come out knowing anything super special. But the main takeaway that I want them to get from it becomes pretty apparent. And what’s really powerful (for me, as a teacher trying to illustrate this essential understanding) is seeing the graphs of how the various thing change.

***

I had students finish the first packet one night. Before we started going over it, or talking about it, I started today’s class asking for a volunteer to blow up balloons. (We got a second volunteer to tie the balloons.) While he practiced breathing even breaths, I tied and taped an empty balloon to the whiteboard.

Then I asked our esteemed volunteer to use one breath to blow up the first balloon. Taped it up. Again, for two breaths. Taped. Et cetera until we got a total of six balloons taped.

Then I asked what things are measurable in the balloons.

Bam. List.

(We should have listed more. Color. What it’s made of. Thickness of rubber.]

Then I asked what we did to the balloon.

Added volume. A constant volume (ish) in each balloon.

Which of the other things changed as a result?

How did they change?

This five minute start to class reinforced the main idea (hopefully). We changed one thing. It changed a bunch of other things. But just because one thing changed in one particular way doesn’t mean that everything changed in that same way. For example, just because the volume increased at a constant rate doesn’t mean the radius changed at a constant rate.

***

This is about all I got for now. I’m going to teach the rest of the topic the way I always do. It’s not up to my personal standards, but I still am struggling to get it there. I suppose to do that, I’ll have to see a more nuanced bigger picture with related rates, or find something that approaches what’s happening more visually, dynamically, or conceptually.

PS. The more I mull it over, the more I think that geogebra has to be central to my approach next year… teaching students to make sliders to change one parameter, and having them develop something that dynamically illustrates how a number of other things change. And then analyzing how those things change graphically and algebraically.

(A simple example: Have a rectangle where the diagonal changes length… what gets affected? The sides, the angle between the diagonal and the sides of the rectangle, the area, the perimeter, etc. How do each of these things get affected as the diagonal changes?)

1. Cindy Hallman-Morris says:

I like the activity with the balloons – an excellent visual to get to your list of things that change. I wrap up related rates with a project. Students have to make their own Related Rates problem, solve the problem, and then for fun decorate a shoebox diorama for their problem. The best part of this project is the day in class when students switch problems with each other. So many excellent ideas are tossed around as students try to make sense of each others problems. They discover things are often not changing as they think. It is hard to give a whole day of class to this but the discussions are so worth the time. This year I had a student modeling the water coming into the airplane that landed on the Hudson River. He took some liberty with the model and used the airplane as a cylinder but the water level was certainly not changing as any of the cylinders in the text books! See a few of them here. http://www.buncombe.k12.nc.us/Page/17017

1. Hi Cindy,

Thanks! I’d love to have more instructions on what you actually do… what instructions you give, any handouts you provide, how much class time they get, how they present their work, how they get graded, etc. I honestly see this as a great thing for an AP class, to get them to flex their muscles and create their own problem (the height of knowing something is the ability to create a good problem from it, methinks), but I wonder about my class… And the algebra involved with solving some of the problems they might pick… (We go simple on the algebra.)

Might you be willing to do a guest blogpost to talk more in depth about your project, and share your thoughts about what makes it work so well? The pictures are stunning! If so, my email is samjshah at gmail.

Sam

2. TJ says:

I am not old enough to “know” (who is?), but i suspect that related rates problems are just warm-up/scaffolding for modeling and differential equations tasks later in the standard sequence of college courses.

at many places, mine included, differential equations is no longer required for all mathematics majors, which leaves the calculus curriculum without an ultimate goal.

It is a bit like learning to peel potatoes carefully, without ever being told about the wonders of cooking meals with potatoes… Or learning to conjugate french verbs without ever hearing about Paris.

But you seem to enjoy teaching modeling. Instead of losing related rates, why not restore the context and give students more stuff that ends up with differential equations? Your balloon task is a good opportunity. If they are going to learn some calculus, they should see _why_ it is good to learn calculus.

3. jg says:

For me, it’s yet another case where the chain rule is the most important functional tool in calculus. So many things can be framed in terms of it (both diff. and int. calc.), so I don’t see it as a separate topic, just an application of something they already know

1. That’s currently how I view it. It just is extra practice of the chain rule. Not really anything interesting.

4. Jen says:

Hi Sam- Have you ever used “Calculus in Motion” software by Audrey Weeks? If not, it is an amazing program– especially for related rates. It does precisely what you mention in your “P.S”. It has sketches of the scenarios in numerous related rate problems, has sliders to change parameters, dynamically shows movement and how other “things” change accordingly, shows what is happening graphically, etc. I highly recommend it. My students loved seeing their pencil-and-paper problems come to life and said it really helped them to understand the concepts much better. At any rate (no pun intended), here’s the website: http://calculusinmotion.com/
And thanks, Sam, for always sharing your good stuff with the world!

1. Hi Jen! Yes, I have calculus in motion! I think those visualizations help A LOT in understanding what the actual scenario is… I rarely use them / I forget about them. Thanks for reminding me. We tend to do easy problems (nonAP), but even being presented with more complicated problems and conjecturing how other things change if you change one thing — and then seeing if their conjecture is right? That’s some nice conceptual work right there. Even without the algebra!

5. Ever filled up a fountain drink? Bam. Related rates. You could do some testing with cups of various sizes. Have them do some CSI work on a few videos. Make comparisons between the soda and the water/tea spouts. I want to teach calculus specifically for that lesson.

Let it be said I always stand in awe of Geogebra prowess, I wish I had the class size/student access that would make it work for me.

1. So dV/dt is constant, but higher/lower based on which thing they use? I suppose I don’t quite get what you’re imagining… Is it how the height of the drink in the cupchanges based on the cup, and based on the fountain used? Or something else?

I’ve often thought about making something where water is pouring into various shaped vases (cylindrical, conic, wavy, etc.), at a constant rate… And making plots of the height of the water over time… and then asking the backwards question… You want the height of the water to be increasing constantly in these various shaped vases… Plot dV/dt to make this happen? (Just sketches, no calculations…)

I haven’t done this yet. But who knows. Next year?

1. I suppose the idea revolves around the difference speed at which you can fill the cup, how dh/dt compares with higher/lower values of dV/dt. You could also compare the difference in dh/dt between the cup and something like a water cooler or tea pitcher.

Not being a calculus teacher I’ve never fleshed the idea out, if anything it makes for a relatable scenario to introduce the idea. Inspired by all the problems that revolve around water troughs when it comes to the subject.

6. I like related rates because they’re an accessible application of calculus to beginners. Are the problems contrived? Maybe.

My favourite problem is the merry-go-round problem. Who is travelling faster, the person on the outside of the merry-go-round or the person on the inside? It’s something we can all relate to, and anyone who has ridden on a merry-go-round knows the answer. Calculus formalizes that.

1. I feel silly but I don’t see how this is a related rates problem… Is the equation you’re thinking s=r*theta, so taking the derivative gets you your velocity? Actually, okay, I now see this. But it also seems super easy to prove using precalculus and without calculus, assuming the merry go round is moving at a constant rate…

I wonder if we can extend the problem somehow so the merry go round is not moving at a constant rate… Maybe it starts not moving, and then it moves up to a constant rate, and then it slows down… Eh, this isn’t going anywhere. Maybe if we have a person walking from the outer part to the inner part as the merry go round starts taking off… and then stays there for a while… and then heads back to the outer part… So we have graphs of r(t) and theta(t), and we need to make a decent graph of ds/dt… I don’t know… I’m just thinking here…

Sam

7. andrew says:

http://0-www.jstor.org.library.bennington.edu/stable/2691482
might be of interest. Related rates problems were invented by *math teachers* in the 19th century to make calculus seem more “applied”. I’ve never seen any evidence that a related rates problem has been used in an actual applied problem, ever.

(Of course, one does some similar reasoning in setting up differential equations, which definitely are useful. But most kids run through the related rates wringer will never get to differential equations.)

8. andrew says:

And with that in mind, I am in total agreement with TJ’s comment above. If the argument is, teach related rates because they are a warmup for differential equations – well, teach differential equations instead. Simple ones are really not that hard, and they are much more applicable and universal than related rates.

That is, of course, if you have the choice of what to cover.

9. l hodge says:

You can do a pretty nice demo with just a rope. Have two students grab an end. Ask what will happen if one student starts walking. What can you say about their speeds? Everyone agrees the speeds will be the same.

Now have one student stand at the front of the room and the other at the back. Ask what will happen if the person at the front of the room moves along the wall at a steady rate and the other person moves forward towards the original position of the first at the front (i.e. their directions are perpendicular to each other). Then do it. It will be very clear that when the person moving along the wall moves at a constant rate, the person headed towards the front of the room is speeding up.

10. I guess my example (merry-go-round) is contrived and can be solved using pre-cal unless you want students to produce a general formula. What I like about related rates, or really, anything to do with calculus, is the “relative” part. I also really like the balloon idea.

11. when a watermelon grows, does it add a constant volume (because it can only steal so much water/nutrients from the soil) or does it grow more at a constant radius? that’s something that i have always been curious about that seems really related rates-y to me.

with all related rates problem the structure i like to do is ….

given constant dV/dt, what is dr/dt when the volume is
a. 10 cm^3
b. 100 cm^3
c. 1000 cm^3

based on that, what is happening to dr/dt?

or something like that. totally agree that the problems are contrived when you are only looking for a specific time like the textbook problems, but i think they are more interesting when you look at them over time (like you are doing with the rocket and with the balloons). i actually do find that interesting.

12. Thanks for this awesome post – inspired a great lessons with my Y10s (14/15 year olds) :)

1. COOL! Thank you for sharing. Super fun to read, and love the pictures!

13. This is my first exposure to your blog. I look forward to exploring your site because I am teaching a new calculus class next year. It will be the first time I have explored calculus since the three unpleasant experiences I was dealt with a decade ago as a student. I noted in this post: http://thegeometryteacher.wordpress.com/2013/01/28/beyond-geometry-help-me-please/

So, if you have good ol’ standbys that are great resources that I might not find here, please send them along. I need all the resources I can get. Thanks for putting your expertise on display for those of us who need the help to look at.