Application: Geometric Integrals

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SWBAT calculate integrals using area formulas

Big Idea

In this lesson, geometry students find out that they know enough to do integral calculus.

Physics Background

15 minutes

To begin, each student gets a copy of Caren's Problem. We'll read the first page on Prerequisite Physics Knowledge together.

Since this lesson is steeped in the context of 1-Dimensional Motion and Physics, I'll need to do some work to make sure students are comfortable with the concepts.

First I explain what is meant by 1-D motion. I draw a number line all the way across the whiteboard, clearly indicating the positive side and negative side. Then I explain that we will be considering only motion back and forth along that axis. So (as I walk out away from the board) even if I am walking diagonally, we are only concerned with how my position on the number line changes, not how far I am from the wall.

Next I distinguish speed from velocity. I demonstrate as I run in the positive direction that my velocity, say 100 mph (because it sounds cool) is positive. Then I turn around and run in the other direction at the same speed and explain that now my velocity is -100 mph. Repeating the same scenario, I explain how my speed does not change. It is 100 mph regardless of the direction. 

Next I develop the concept of acceleration. I'll start by running in the positive direction, supposedly going 50 mph. Then I'll indicate that I'll be running faster. Therefore my velocity will be increasing from 50 mph to, say, 75 mph. That's a positive acceleration.

Then I act like I'm running 50 mph in the negative direction and ask, "What will happen to my velocity if I start to run faster?" Eventually, we discuss that my velocity will go from -50 mph to -75, for example, which means that my velocity is actually decreasing. That's a negative acceleration.

Then I generalize by asking students to imagine the direction of the force needed to make the change in velocity in order to know if the velocity will increase or decrease. If I am going forward and want to speed up, I'll need a force going in the same direction as I am (positive). If I am going in the negative direction and want to speed up, I'll need a force that is also pointing in the negative direction, so velocity will decrease. Likewise if I'm going in the negative direction and decide to slow down, I'll need a braking force that is acting in the positive velocity will be increasing.

So on this type of thing goes with me acting out various scenarios based on what I can tell students are not understanding.

Guided Practice

15 minutes

After the physical demonstrations, it's time to see if students grasp the concepts. Working on the second page of Caren's Problem, I'll pose the question, #1 for example, then give students a minute or two to write their own answer and discuss with a partner. Then I'll show and discuss the answer. We continue in the fashion until we complete #6. Then I give students 5 minutes to craft a response to #7, letting them know beforehand that someone else, possibly the whole class will be seeing and/or hearing what they wrote. After that, I have students switch papers to read each other's writing and I also call a few non-volunteers randomly to come to the document camera and show/read what they wrote.

Finding Integrals

20 minutes

At last it's on to the final page of Caren's Problem, where students actually get to use some Geometry. This section is meant to be student-centered, self-directed so I allow the students to go through it as I go around coaching, clarifying, and giving feedback.

When I can see that most students are nearly completing #8, I'll stop the class to make sure they have a solid understanding of integrals (as solid as is needed for this lesson). I'll want to play up the idea of unit analysis as that is one of the major math practices in this section. Once I feel confident that students are understanding what the areas under the curve mean, I will back off and let them finish problems 9a - 9d. There is some problem solving they need to do, especially on 9d, so I give that process time to happen, resisting the temptation to jump in and give answers.

When there are about 20 minutes remaining in class, I'll announce that students are free to move around the classroom to consult with classmates. I explain that everyone will need to be prepared to explain 9a through 9d in case they are called up to explain. I tell them that they'll have 10 minutes to get prepared and then I start the clock.

At the end of the 10 minutes, I'll select 4 students at random to come to the document camera and explain 9a through 9d.