# Pre-Calculus discovers the elliptical orbit of each planet

Recently, our pre-calculus students were learning about conics. Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. The types of conic sections are circles, ellipses, hyperbolas, and parabolas. Each conic section also has a degenerate form; these take the form of points and lines.

During the lesson on conics, the students created a planet project, which focused on ellipses. According to math teacher Ms. Groszek, the students started with a discovery activity on eccentricity and how that relates to an ellipse. In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally, two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. Generally, the eccentricity helps to determine the curvature of the shape. If the curvature decreases, the eccentricity increases.

Students were assigned a planet and were then given information about their planet. Students then had to use that information to come up with the equation of the elliptical orbit of the planet, and once they had that, they had to figure out different things about that planet, including interesting facts, if it was a gas giant, a terrestrial planet, what kind of planet it was, etc. They also learned about Kepler's Law of Planetary Motion.

Kepler’s three laws of planetary motion can be stated as follows: (1) All planets move about the sun in elliptical orbits, having the sun as one of the foci; (2) A radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time; (3) The squares of the sidereal periods (of revolution) of the planets are directly proportional to the cubes of their mean distances from the sun.

The planets, and their corresponding information, were displayed outside the RES Library and are spaced appropriately for how far away they are from the sun.

**Why planets?**

“Planets actually have an elliptical orbit, so it was a natural connection. And some of the students are taking astronomy, so it was something that some of them were familiar with,” said Ms. Groszek.

**What were the challenges faced by the students?**

“The numbers are not nice,” Mrs. Groszek said. “They are normally used to working with really nice numbers from textbook problems and this is real-world data, so it doesn’t come out nice. They were getting odd decimals and rounding, so that was just a challenge of making sure that rounding was done correctly, that they were doing everything correctly since they did not get that automatic check of a whole number answer.”

**What has the feedback been with this project?**

“They liked it. They thought it was something really different. They did think it was challenging, especially if you had a planet that was really close to the sun, like Mercury and Venus, the numbers were very small, so they were dealing with 0.24, so they were getting frustrated saying ‘this is so small, everything is so close together’ but then people who had Neptune were getting frustrated because their planet is so far away that when they graphed their orbit, they had to add an extra page, because they miscalculated, not realizing how big it was. But they enjoyed having something different and real-world applications, rather than just problems 2-20 [out of a textbook],” Ms. Groszek said.