While none of the G.PE standards in the Common Core are considered to be widely applicable prerequisite standards, I love the opportunities they present for SMP 5,7,8. Before I tell my my students much of anything about conics, I send them to Desmos.com to explore circles, ellipses, hyperbolas, and inverse parabolas. They figure out how to use equations to produce their art.
It comes as a shock to many of my students in the fall when they find out I expect them to figure things out, but they quickly see the value in that. It’s not like math should be a scavenger hunt or they should have to prove the rational root theorem, but they are expected to grow their ability to think deeply and independently, asking for help electronically and in person when they get stuck. A simple rubric turns their creative projects into some rather serious analysis. They see the relevance, so they are ready to dive in from there.
Next we spend a few days studying the circle “formula” (Pythagorean Theorem), and how it makes sense graphically. Ellipses connect well because they have infinitely many (sort of) radii, the largest of which defines half the major axis and the smallest defines half the minor axis. To introduce hyperbolas, I briefly review circles and ellipses; and then I tell my story.
I explain that I have been married for a few decades and that my marriage looks like an ellipse. A circle is perfectly round. An ellipse is likewise connected but has balanced differences. My husband is a runner, and I am a walker. Actually, we are both Walker’s but… He has far more woman’s intuition than I have. He’s quiet and I have a hard time keeping quiet. If I talked to God as much as I talk to myself I would have an amazing spiritual life. Anyway, we get along very well:
I tell them we have a few tricks that keep us together:
- Divorce is not an option
- We work to keep it positive. It’s okay to disagree about things (critiquing reasoning), but we never attack each other. We keep selfishness in check.
- We don’t get mad at the same time: we take turns.
- We give the benefit of the doubt and ask, “What did you mean by that?” We forgive.
I explain that while circles and ellipses are connected with a plus (positive), hyperbolas are connected with a minus (negative). Negatives pit spouses against each other and their relationship begins to look like this:
Living like that must be a horrible existence, but I have never known anyone I couldn’t enjoy having relationship with. So I have trouble relating to how bad that must feel. As an education progressive, I often experience rejection; but instead of getting mad, I look at each rejection as an opportunity to grow more determined to love. If I can’t find something to love about another person, that’s my problem, not theirs!
I love integrating emotion into conics like this because, by anchor theory, students remember the +/- differences as they explore constructions, and it is a beautiful way to explain how to have lasting relationships. I always hope the relationship part sticks with them for the rest of their lives.
Lest anyone think I am relying on tricks, let it be known I wholeheartedly support Nixthetricks.com. While some teachers may “explain then tell them what to do,” my practice is to help them discover it, then they explain it, then apply it with explanations. When studying hyperbolas, I do not give my students a set of steps to graph them in a box. Rather we derive equations for the asymptotes from the equation of a hyperbola by applying limits that affect the constant term. The constant term is on the hyperbola, but is not part of the asymptotes. As x -> infinity, the constant becomes insignificant. I explain that I learned that in advanced calculus. They say, “wow,” and that’s reason #43 that they are capable of advanced mathematics even though they are taking my remedial class. The idea of calling a number (that could be large) “insignificant” blows their minds because I have been harping all year about having a rationale for “cancelling.” As the idea of limits affecting the constant soaks in, though, they begin to feel like they have superhuman analytical skills. Calculus isn’t difficult; it’s weaknesses in algebra. Algebra isn’t difficult; it’s weaknesses in fractions.
By the end of the intro to hyperbolas, my students have learned many things including:
- They are related graphically and algebraically to ellipses
- They are shaped by limits
- They remind us that good relationships depend upon positivity
Ha! Maybe hyperbolas should be considered “widely applicable prerequisite skills.”