My Favorite Calculus: Crazy Integrals

While reading on my computer today I received a notification from Dropbox that a shared file called "Crazy Integrals" had been updated.  This was pretty exciting since I created this activity four years ago and even though I am no longer teaching Calculus AB, the teachers are still using this activity.  This also reminded me that I could share this activity for the Explore MTBOS Week 2 blogpost challenge.

Before we launched into our integration chapter I did some pre-work to help students gain a strong understanding of both the notation and the concept of integration.  For Crazy Integrals I started by directing their attention the the diagram below.

 We talked about how finding a definite integral is like finding the area between a curve and the x-axis.  There are some finer details that need to be covered, and we brought them in both in the course of the activity and throughout the chapter.  The first two problems that we talked through are below.  Students noted on their diagram that the top piecewise curve is f(x) and the bottom piece-wise curve is g(x).

Students then discussed the best way to break up each area using vertical lines.  This helped them to rewrite each definite integral as two separate definite integrals.  Then they found each area.  We talked about how in Calculus an area below the x-axis is counted as negative.  Students worked in groups of 3 or 4 to complete the rest of the problems.  

Below are the other diagrams used for the activity.

There were all sorts of great questions that students brought up throughout the activity, most of which called on them to use precise definitions and language to answer.  For example, students wanted to know what to do with the smile and eyes/nose from the clown picture.  We had to use our working definition of definite integral to determine that they were extraneous to the problem.  Students also wanted to know how to find the largest/smallest x-values for the clown's hair.  We had to interpret the associated integrals (below) to know that those parts of the hair were not to be included.  

For the fish diagram students wondered why I only asked them to find the definite integral of the top curve.  Great question, and one that they could probably answer in their own groups.  

Overall super engaging activity with lasting pay-offs in understanding for the remainder of Calculus.  Students didn't want to leave class without finishing this assignment.

Note:  After the first couple of problems students stopped rewriting the definite integral as a sum of definite integrals, and instead got really into showing their work for finding the separate areas.  I figured this was fine as long as they demonstrated the understanding of the skill in the first part of the activity.

Visual Introduction to the Definition of the Derivative

One of the new teachers I will be working with this year will be teaching Calculus, and this has got me thinking about some of the more successful strategies that I used the last time that I taught this course. A bunch of ideas come to mind, but one that sticks out from the others is the visual intro to definition of derivative that I stumbled upon the second year that I taught Calculus.  The first time I taught the course students struggled with this concept.  Using dynamic geometry software made all the difference for helping students learn this concept the next time I taught it.

To demonstrate, say you want to use the defintion of derviative to find the value of the derivative of f(x)=x^2 at x=-2.  Use the diagram below:

I've added a couple of objects to my diagram to help out with this explanation.  First, we note that we are really looking to find the slope of the tangent line of the function f(x)=x^2 at x=-2.  I don't tell students that my software can simply graph this tangent line and give me the slope, nor do I tell them that there are formulas to find the derivative of a function that we can use to find the slope of the tangent line.  That all comes in time.  For now my fixed point, a moving point attached to f(x), and a secant line between these points is all we need.

The next step is a reminder of what a tangent line is.  I let students discuss with a partner and then tell the class what we are looking at.  If the diagram above with a secant line is provided, I can ask students what to do to the moving point to make my dashed line become a tangent line.  I start dragging the moving point towards the fixed points and ask students to tell me when to stop.  If they tell me to stop too soon, I ask them if we can move the blue point closer to the fixed point.  We are visually developing the idea of using a limit to find the slope of a tangent line by physically moving one point as close to another point as possible.  Eventually we put the moving point on the fixed point.

Students notice right away that the secant line is no longer present.  Since you can't turn a secant line into a tangent line using a fixed point and moving point, we will do the best that we can.  And the best that we can do is to move the moving point as close to the fixed point as possible, and use these coordinates to estimate the slope of the tangent line.  I ask students to write this symbolically.  We introduce the following notation:

Fixed Point: (a, f(a))
Moving Point: (x, f(x))

Then the slope of the secant line is

We practice finding the value of the derivative by dragging the moving point as close to the fixed point as possible, and then finding the slope of the secant line.  When we introduce the definition of the derivative of a function at a given value, we use similar language.  Students can tell you the following about the slope of the tangent line with little help:

The slope of the tangent line can be found by taking the limit of the slope of the secant line as the moving point approaches the fixed point.  When we have had this conversation, it is not such a big leap to 

With the visual introduction, students can interpret the parts of this formula and feel empowered when the formula yields the same slope as our estimations did. 

My school district has open enrollment for AP and honors courses, and because of this the Calculus AB classes are very heterogeneous.  There is a nice split between students that have taken all honors math courses, students that are choosing AP Calculus AB as their first honors math course, and students that have never taken a honors/AP course in any subject.  Because of this I chose to keep the functions simple when we were finding derivatives using the definition.  This allowed for a greater focus on the conceptual understanding of derivative, and I believe that the visual introduction using dynamic geometry software and the extra conversations around the topic helped students ace this topic on the following test.  

You can use this premade presentation to introduce the definition of derivative in your own class.  This tool allows you to enter a function as well as coordinates for your fixed point and moving point.  A smaller version of the tool is below, and you can try this out by dragging the moving point.  

You can also use this presentation when talking about when a function is not differentiable, because it is easy to see that the slope of the tangent line will not approach the same value on either side of the cusp.