Math 254: Multi-Variable Calculus Labs

Multi-Variable Calculus includes ideas which are very geometric as well as algebraic. I approached the course by providing geometric ideas first, followed by algebraic calculations. Throughout the course, I emphasized that students should know what type of "beasts" they were working with. This helped them distinguish between surfaces and curves, points in space and vectors, scalars and angles, etc.

In order to help students learn the importance of geometry, algebra, "beasts", and calculus content, I wrote a series of labs for my summer 2006 course. Students worked on the labs in groups, on white boards. At the end of the class, students and groups presented some of their work in a class wrap-up. This wrap-up allowed students to talk about mathematics. It also helped the students see the bigger ideas of the course and discuss the reasons behind their procedures or habits that they blindly used.

Overall, it worked very well to emphasize the geometry first, and it seemed that integrals were less confusing since they had had so much practice drawing surfaces throughout the course. The emphasis on drawing surfaces and the geometry also gave reasons for doing certain calculations that were otherwise "procedure". I believe the labs needed more class time (we had 50 minutes). In many of them, it was impractical to do the wrap-up the next day (i.e. lab 3 -- the position of the surface changes, and so they would have to redo the lab.), but many of the geometric ideas were understood enough by the students that a wrap-up the next day was fruitful.

The Labs

Lab 0: OSU Review. Throughout the course, we use circles, ellipses, parabolas, and "lines" in both polar and rectangular coordinates. This lab is a good review of material from MTH 111 and MTH 112 needed for the course. It creates "OSU" with a rectangular frame.

Lab 1: Surface Lab. In this lab, student groups work together to draw one or more surface in three dimensions. Groups had to work through drawing the surface, and decide if their picture matched their conceptual idea. The lab uses traces, which breaks the domain into different regions, and asks the students to specify whether the function would have positive, negative, zero, or no values in those regions. This lab is a good way for students to realize there is not "one" way to graph all surfaces, but that they must choose the correct method. Most groups used additional traces, although this did was not helpful for some groups.

Lab 2: Dot Product Lab. Done on Monday, July 3rd, so this lab could also be done individually as well. By the end of the lab, students found the length of the diagonal of a hexagon using vectors. They thought finding the diagonal length in a cubeoctahedron was difficult, until they held the three-dimensional models which I provided. The geometry made this problemm easy!

Lab 3: Gradient and Surface Lab. In the previous class, I introduced the idea of derivatives as rates of change. For functions of more than one variable, the rate of change can obviously be with respect to more than one direction. With that as their only background, the students entered the lab knowing nothing about gradient vectors! They were given an "x-y plane", which was a large piece of graph paper, a "z-axis", which was a dowel stuck in some clay, and a plaster of paris "surface". They then used straws to measure lengths and slopes on the surface in various directions.

At the start of the lab, I asked students to make a list of things they would want to know if they were standing on the side of a hill. Their list consisted of the following questions:

"Which direction is the steepest?"
"How steep is it in that direction?"
"Where would I walk to stay on a level curve?"
"Which way would a ball roll?"
"Which way in three dimensions would I travel if I walked up the surface in the steepest (or any) direction?"
"How do I know if I'm at the bottom of the hill?"
They were amazed that this gradient vector they had found could answer all these questions! During the wrap-up, the students were amazed that every group calculated a different vector for a different surface, yet it always pointed in the direction of steepest increase. They left the lab knowing a gradient vector's geometric description: It points in the steepest direction, and its magnitude is the rate of increase, and came to class the next day wanting to know all about this "magic". Some groups explored the idea that the gradient vector is coordinate independent by turning their surface on their "x-y plane".

This lab was a good review of many concepts. It helped students know why we use a right-handed coordinate system. It required students to practice identifying "Beasts" -- whether objects and numbers were rates of change, slopes, distances, directions, vectors, scalars, points in the domain, or range values. The lab also provided a good discussion of scale with some of the groups.

Lab 4: Tangent Plane Lab. Groups found the tangent plane to a surface at a point. The wrap-up emphasized this linearization of the hill. During the wrap-up, students also discussed things such as directional derivatives and the gradient at the top of the hill, which were good geometric questions. This lab reviewed concepts of gradient, dot product, vector addition, and the geometry of vectors, which was good review for the upcoming exam. This lab should be re-written.

Lab 5: Region of Integration Lab. Given 1 of three difficult integrals in rectangular coordinates, groups have to find a way to evaluate the integral. Students eventually realize they must re-interpret the region of integration in another coordinate system, and re-write the integral. From this lab, students realize why some integrals are more challenging than others. During the wrap-up, students discussed the correctness of "just switching the limits of integration" verses doing all the necessary steps to re-interpret the region of integration. This provided a good discusion of convincing arguments: The class was convinced by a picture that a student had done the right integral, but not by another student's technique which involved algebraic manipulations only.

Lab 6: Equivalent Triple Integrals Lab. Equivalent triple integrals. Groups are given one of 3 triple integrals, and the students have to write down the other 5 equivalent forms of their integral using different variations of dV in rectangular coordinates. It took much time for the groups to draw the region being integrated. Once they had done this, they could make cross-sections and the integrals for the cross-sections. Finally, they used an outer integral to add up all of the cross-section integrals.

This was the final lab for the class, and also was a review for the upcoming final. One student said this was exactly like what we'd done the first week in Lab 1, in that they used traces to draw surfaces.