As a math lover, I love that I share my birthday with τ (tau) Day. What is τ you ask? It’s 2π of course (τ = 2 x 3.14 = 6.28). Why give 2π a name and a day? Well, watch these 2 videos, one from Vi Hart and the other from Numberphile, to see what the fuss is all about.
Conic sections are the curves created when a cone is intersected by a plane. There are 4 types of conic sections: circle, ellipse, parabola, and hyperbola. Sometimes, you may see it stated that there are 3 types of conic sections, because the circle can be considered a special kind of ellipse (we’ll discuss that more another time). The picture below shows how the curves are formed from the intersecting planes. As you can see, cutting the cone so that you miss the upper cone, but slice through the bottom gives a parabola. Cutting the cone so that you do not intersect the top/bottom of either cone gives a ellipse (a circle if the cut is parallel to the top/bottom). And, cutting the cone so that you slice through both the top and bottom cones gives a hyperbola.
As with many concepts, doing a hands-on activity can help demonstrate conic sections, so why not make your own? You can easily make your own cone with modeling clay and slice some conic sections. To make your cone:
- Cut a half-circle from sturdy paper or card stock and a second half-circle from parchment paper or wax paper.
- Curl cardstock half-circle into a cone and secure with tape and then place parchment paper inside (do not tape parchment paper).
- Pack cone to the top with air-dry modeling clay. Use small pieces and press down to the tip of the cone. You can pull the parchment paper out a little to peek as you go, to make sure there aren’t any big gaps or air bubbles.
- Flip over onto a paper plate or sheet of parchment paper. Remove cardstock cone, then peal away the parchment paper.
Once you have your cone you can use dental floss to make your sections. Cut straight across near the top to make a circle.
Below your circle, make a slanting cut to make your ellipse.
Below your ellipse, make a cut that slants enough to pass through the bottom of the cone to make your parabola.
You could do another cut that goes straight down, but without the top cone, it is hard to see it as a hyperbola. So with my students, I chose to just show them a picture of the hyperbola. Once the pieces dry, you can color your curves. Below, is the cone that my 6-year old daughter made. She chose to color her circle blue (she also colored the bottom blue, since that is also a circle). She colored her ellipse yellow and her parabola orange.
Ready to see how good you are at drawing circles and finding the center of circles and angles, or making 90o angles? Here are two online games to try.
The Circle Drawing Experiment allows you to draw circles and then gives you a score based on how well the relationship between perimeter and area of your circle matches a perfect circle. And, it includes cute cat pictures! Hint: larger circles give better scores.
The Eyeballing Game asks you to use your “eyeballing” skills to make parallelograms, find midpoints of lines, bisect angles, and more.
Just be careful! These games can be addictive!
In a previous post, “Magnificent Pi”, I mentioned that pi is an irrational number, but it is also a transcendental number. A transcendental number is a number that is not algebraic; it is not a root of a non-zero polynomial equation with rational coefficients. What does that mean?
Numberphile has a video that explains what it means for a number to be algebraic or transcendental.