The warm-up prompt for this lesson asks students to predict what happens to the area of a parallelogram when one base is shifted with respect to the other without changing the altitude. Students saw this question before, on the unit pre-test. I ask students to follow our Team Warm-up routine, which involves sharing their responses with the other members of their cooperative learning team. I choose students at random to write the team's chosen response on the board.
I briefly review the team answers on the front board. I summarize the predictions of the class, and praise teams who provide thoughtful reasons. I am careful to highlight any arguments based on area formulas, such as "both areas are base times height". While sound reasoning, such an argument takes area formulas as a given. In this unit, we are going to look at where those formulas come from. (MP3)
There may be an opportunity to address the misconception that "area is base times height", which I find that many students have committed to memory, use indisciminantly, and are reluctant to let go of. I will chip away with a series of questions:
Finally I will say, "Today, we will see that the area of a parallelogram is--in fact--equal to the area of a rectangle with the same base and height, and we will be able to explain why that is so."
Before moving on I display the agenda and learning targets for the lesson and briefly review them with the class. I plan to make a relatively formal announcement to accompany the targets:
Today we are going to learn how a mathematician named Cavalieri might have answered the question about the parallelogram. His way of analyzing and comparing the areas of two shapes is very powerful, and we will use it again when we study volume. We will also see how the strategy of shifting the bases of a parallelogram without changing its height -- called a shear -- is used in a proof.
Cavalieri was one of a handful of mathematicians -- Archimedes among them -- whose thinking anticipated calculus. Most mathematicians were very uncomfortable with the concept of infinity or with the idea that an infinite number of very, very small quantities could be added together to get a finite something -- a plain old number. That is, until Leibnitz and Newton--who were contemporaries of Cavalieri--showed how useful calculus could be for solving problems and convinced the world to drink the cool-aid.
With the help of an animated slide, I use the question about the area of a parallelogram whose bases have been shifted relative to one another to explain Cavalieri's principle.
To illustrate the principle, I ask students to consider one face of a deck of cards. The area is the sum of 52 very small areas, the areas of the thin faces of the cards. Does that area change when the deck is shifted so that the face becomes a parallelogram? (MP3)
I also like to bring in a Profile Gauge, a tool used by tile-setters, because it illustrates the concept nicely and kids think it is neat.
I then ask the class to consider how Cavalieri's principle could be applied to a real-world problem--comparing two designs for the sails of a racing yacht. Displaying the slide, I ask students to answer the question silently to themselves, then cold call on students to invite them to share their opinion and reasoning. The slides are animated, allowing me to summarize the arguments put forward by students or to guide the class through the process of understanding the problem (MP1). The main sails of the two boats are identical, so the question comes down to comparing the triangular foresails, both of which have the same base and altitude (MP4, MP7).
The concept of a shears -- a parallelogram whose bases are shifted relative to one another while the altitude and area remain constant -- is used in a number of proofs of the Pythagorean Theorem. I pass over Euclid's own proof with its famous "bridal chair" in favor of a more straightforward proof attributed to Howard Gardner.
Using an animated slide, I lead the class through the proof. At each step, I ask students to explain why the total area of the highlighted region or regions remains unchanged.
Then I distribute the Portfolio Problems for the unit. Problem #1 asks students to explain the reasoning behind the proof -- specifically, why the total area at each step is unchanged -- either in their own words or by referring to a postulate, theorem, or principle that applies.
Students will turn in this problem as part of their Learning Portfolio for the unit. I ensure that the class completes the first step of the proof in problem #1, so that they will understand what is required. Using remaining class time, I help students get started.
The lesson close follows our Individual Size-Up Routine. The prompt asks students to explain or show how Cavalieri's Principle can be used to argue that the area of a triangle is unchanged when the apex is shifted parallel to the base. Students write their answers in their Learning Journals. We will not summarize Cavalieri's Principle in course notes for several more lessons.
For homework, I assign problems #7-9 of Homework Set 1 for this unit. Problems #7-8 ask students to use Cavalieri's Principle to show that the areas of two shapes are equal. Problem #9 provides more practice using the Pythagorean Theorem to find the missing side of a triangle.
In addition, students should begin Portfolio Problem #1.