In calculus today I was talking about how to find the average height of a function. Some kids just have a hard time understanding the concept. I always show them a few functions on certain intervals and I ask them what they think the average height would be. Just to initially test their intuition on the concept.
Some see it, and understand it; some don’t. All certainly have trouble articulating why they chose that value.
So I have two things that work for me, when explaining this. There’s some handwaving, but the focus is on the idea, and building intuition.
The first thing is we talk about how we would approximate the average temperature somewhere:
We take a bunch of temperature readings, and we add them together and divide by the number of readings.
How do you make it more accurate?
MORE READINGS!
How do you make it more accurate?
INFINITY OF READINGS!
What helps us deal with infinities and infinitessimals?
CALCULUS!
So that’s how we get started.
Then when I want them to understand the formula — 
So an EVIL mathematician has an almost 2 dimensional fish tank. Really thin. Sad for the fish. Which are almost 2 D. And the mathematician likes to lay a strip of plastic on top of the water, and constrain the fish in these weird shapes.
(In this case, the mathematician is constraining the fish in an 
You come along and want to GIVE THE FISH WHAT THEY WANT: a normal rectangular water to swim in.
So you yank the plastic strip away, and what happens to the water?
IT ALL LEVELS OUT!
What shape does it make?
A RECTANGLE!
Does the amount of water change?
NO!
What’s the height of the rectangle?
THE AVERAGE HEIGHT OF THE FUNCTION!
So by then, we have on the board:
And since the amount of water didn’t change, they know that the area of the red rectangle and the area of the blue rectangle are the same.
That makes sense to them.
I then threw this up and almost all of ’em got it!
So that’s my way of building their intuition when it comes to average height of a function. It’s not like it’s hard for them to apply the formula, but I think this little thing makes it more conceptually manageable. And if they forget the formula, they can just do the “fish tank problem.”
Continuous Everywhere but Differentiable Nowhere 
I love the water tank visual. Thanks for the post!
I also like the fish tank. I say something similar, but I just use the imagery of building a weird sand-castle and you get a hole on one side and castle on the other, then you can fill it back in. I think the semi-traumatic-PETA-enraging moment with the fish, though, may be an image that sticks with them longer.
Oh yeah, and the connection to “Mean” is cute, too.
I have them imagine jello under the curve, and when it gooshes down, how high it would be. (There are people who do jello sculpture, by the way.)
Jello sculpture? Wow!
I use something like this to teach averaging to my middle-school kids: Imagine [whatever you’re measuring] as juice in glasses, and then we dump all the glasses into a pitcher and pour them back out so everyone has the same amount to drink. This works very well with bar graphs, because it’s easy to imagine the bars as glasses filled to different levels. And, as with the fish tank, we can easily imagine the liquid seeking its own average level.
=)
When did your blog go orange? I think I must be reading it 95% of the time through a reader.
Anyhow, I love the swimming alphas – very clear, great probability that a kid will move from “not get it” to “get it” on that model.
I don’t like the three-D-ification of the tank – but that’s me.
Jonathan
It went orange only yesterday (or maybe the day before)… White was getting too hard on the eyes. I know, I know, the orange is really bright…