Vectors and Dancing – A Dangerous Mix

I came across this activity from Jim Noble at www.teachmaths-inthinking.co.uk the other day and tried it out with my precalculus class. We defined four different dance steps with vectors, and combined them into a routine.

We then tried out the routine while listening to Donna Summers’ “Hot Stuff”. As you can see, we could have spent a little more time practicing!

I liked this activity, because it got girls up and moving. We had a good discussion of the various ways to combine the steps to reach the same destination, and they mapped them out using the vector definitions. Jim Noble has made available some excellent worksheets that go along with this project.

If you’re trying to teach simple vector addition and scalar multiplication give this activity a try!

A Simple Way To Teach Polar Coordinates

The courtyard outside my building is laid in a circular pattern, which makes it a perfect setting to teach polar coordinates:   I took my Precalculus class outside and stood on the polar axis. I assigned an r and theta-value to each student. They then had to start at the pole and figure out where their given polar coordinates would position them. It was a great way to incorporate physical movement with math. They even understood what happens when is negative! To wrap things up, we worked out what the equations theta = pi/4 and r = 6 would look like, and we compared them to their rectangular counterparts. Then, back inside to apply our new knowledge to some problems!

The Wheels On The Bus Go Round and Round…

A quick post today about a lesson in precalculus that went a lot better than I expected (and isn’t it nice when that happens!). The objective was to understand linear velocity and angular velocity, and the differences between them.

I like to reduce as much as possible the number of formulas students memorize, so when we discuss this topic, I try to get them to use unit cancellation to arrive at the desired result. We talked a little about how to convert rpm to rad/sec, and mph to rpm. Then I sent them out in teams of four to the parking lot, armed with rulers, to figure out the rpms necessary for the wheels of cars traveling at 35 mph.

I didn’t tell them what to use the rulers for, so they had to figure out for themselves that they needed to calculate their car’s wheel circumference. When they returned to the classroom, they got right to work, and wrestled with the proper setup for their expressions. Eventually, every team got a good answer, and they didn’t use the same process to arrive at their result (which is great!). Here’s one team’s work:

Comparing the teams’ different results led to a nice discussion on how wheel size affects the rpms needed to roll at 35 mph, and why cars need differential gears. Merely getting my students out of the classroom and moving for about ten minutes really energized them, and made them interested in figuring out the answer to an admittedly simple problem. Whenever possible, I need to incorporate movement and outside activity into my lessons, even if it’s only for a few minutes.

Water, Water, Everywhere, and Parabolas to Boot

I’ve done this activity before, but never with GeoGebra, the dynamic geometry and function graphing program. So for those of you who would like to try it with that open-source (meaning, it’s completely free!) program, here are the steps I’ve found to work best.

First, you need a picture of water that is following a parabolic path. Fortunately, that happens at most drinking fountains:

Now we need to paste this into GeoGebra. You can drag it directly into the graphics pane of GeoGebra, or go to Edit/Insert Image From…File:

Now you can resize and drag the picture around any way you like. (Helpful tip: It really simplifies things if you make the water stream pass through the origin.) Hit the little Graphics toggle at the top left of the Graphics pane, and click on the grid button:

You’ll notice that you can’t see the axes or the grid behind the picture, so click on the picture to select it, then right-click on it to choose “Object Properties”. Under the Color tab, you can then change the opacity of the picture. Something around 50% usually works fine. (If you don’t see an Opacity slider, click on the pic1 label in the left pane of the Preferences window).

You can also change the color of the grid by going to Options/Advanced, select the “Preferences – Graphics” button (it has a little circle and triangle) and clicking on the Grid tab:

Now you’re ready to actually model the water’s path! Place the picture so that it passes as close as possible to a couple of grid intersections. Put three points on the water’s path using the point tool. In the Algebra pane on the left, you’ll notice that the points’ coordinates are automatically calculated.

Using your three sets of x and y-values, you can set up a system of three equations in the form of $y=a{{x}^{2}}+bx+c$. Students can use any method they wish to solve that system. Mine were comfortable enough with matrices and graphing calculators to use them to solve the system:

Enter the quadratic function with the calculated a, b, and c values into the Input bar at the bottom of GeoGebra. Then, see your function model the water’s path perfectly!