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Do students learn implicit equations (such as $x^2+y^2-r^2 = 0$) and parametric equations (e.g., $x=a t^2,\;y= 2 a t$) in a first course in algebra, which in the US would be early high school, maybe 9th grade? Or not until pre-calculus or calculus, late high school or early college? Perhaps parametric equations are not part of standard curricula?

I'm trying to gauge how much it is reasonable for me to assume when writing for high-school students.

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  • $\begingroup$ It's not really obvious to me what it would mean to "learn implicit equations." Knowing that $x^2+y^2-r^2=0$ is the equation of a circle is more a matter of understanding the Pythagorean theorem along with understanding the notion of Cartesian geometry. The implicit function theorem is a topic that my community college teaches in first-semester calculus, but I don't think it's necessary to know the implicit function theorem in order to interpret this equation. $\endgroup$
    – user507
    Commented Nov 2, 2020 at 21:34
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    $\begingroup$ The idea that a curve would naturally be given by $y=f(x)$ is more like something that students would pick up in a first-year college calculus course that included some analytic geometry. For high school students, I don't think that's an idea that has ever been planted in their brains, so they don't need to make any transition. An implicit equation should seem at least as natural. I think issue is more like how much exposure they've had to the Cartesian approach. I think the answer is that they get their first exposure to this in a precalculus course, which might be in 11th or 12th grade. $\endgroup$
    – user507
    Commented Nov 2, 2020 at 21:54
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    $\begingroup$ Are you asking when they learn or when they are taught? $\endgroup$
    – shoover
    Commented Nov 2, 2020 at 21:56
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    $\begingroup$ In my experience: parametric equations are not taught until calculus. $\endgroup$ Commented Nov 3, 2020 at 11:39
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    $\begingroup$ @Gerald Edgar: parametric equations are not taught until calculus --- They are covered in some U.S. high school precalculus courses, but this is quite rare. Nearly all U.S. high school students will not see parametric equations until BC calculus, the higher level of AP calculus that only about $3.8\%$ of U.S. high school graduates currently take each year ($140,000$ divided by $3.7$ million). $\endgroup$ Commented Nov 3, 2020 at 18:06

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I am in a US high school. The implicit equations look like "conics", and are part of the junior (3rd year) class typically called Trigonometry with Algebra. Parametrics are part of the course labeled pre-calculus, which comprises either juniors who will take calculus as seniors, or as seniors in preparation of college calculus.

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The community college where I teach puts an introduction (lines, circles, ellipses, parabolas) in precalculus, and then covers it again in vector calculus. So, this would be first year and again in second year of undergrad.

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Neither topic is covered as a Common Core standard as such.

Somewhere in middle school, students learn that a circle cannot be a graph of a function because it fails what we call the Vertical Line Test. (I could look up the exact standard and grade level if you're curious about it.) In Geometry (tenth or eleventh grade typically, but it can vary by district), students learn to calculate the standard and general equations for a circle based on its radius and the coordinate of its center and working backwards from there, but we don't describe it as an implicit function. There are no common standards for pre-calculus at the moment, but people generally cover conics to some level of detail. Implicit differentiation is covered in the AP Calculus standards (both AB and BC), but IME we don't miss the lack of prior knowledge.

When I was in high school 35 years ago, we did cover parametric equations in what was essentially pre-calculus, and I believe differentiating parametric equations came along with studying the Chain Rule in Calc AB. I can easily imagine that some college courses would still cover it, but it is no longer in the AP Calculus standards.

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    $\begingroup$ Thanks, Matthew. So much of computer graphics depends on parametric representations, 'tis a shame it is not more central in the curricula. $\endgroup$ Commented Nov 5, 2020 at 20:10
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I think college prep kids (your general population at a "Sister" school) will have seen this stuff. But it will not be as ingrained as polynomial differentiation or the quadratic equation. So you should somewhat understand the bottom half of your classes, having a slight hiccup, needing to actually use these concepts.

Circle equation:

That's classic "algebra 2", as part of the "conics". They will learn the equation and graph it on Cartesian coordinates. Not the 9th grade first algebra class, but a standard part of the 11th grade. (10th for kids who are accelerated a year and heading to AP Calc).

So, yes absolutely standard high school topic. Maybe not as strongly learned as polynomial graphs. There is a sort of veneration and cleaving to "functions" as opposed to "relations" as if God loved the former more. And they are simpler, with the whole no more than one y per x. And functions are probably more important in technical subjects like chemistry or physics. So the circle (or ellipse) equation can be a bit of a sidelight. But I'd argue even here, it's got some importance as we think about centripetal forces or orbits or the like. In any case, yeah "they had it", even if it (and arguably a lot of stuff you teach) is sort of off the classic, functional, cannon-ball-hitting-things, algebra to calculus track.

In precalculus (nominally 12th grade, 11th for "GT crowd"), the circle equation probably gets reinforced slightly. Still remember learning the function concept and the relation concept (relations get no love here, look at the arctangent debates). But the circle is a classic example of a relation, so it was getting double duty as an example (in Cartesian coordinates). I.e. getting some reinforcement.

Parametric equations:

I had this in pre-calculus (second semester, "analytic geometry"). Nominally 12th grade class, but all the "GT crowd" kids had it in 11th, setting up for AP. Decent public high school, but not top of the heap. 1980s. And if you look in books, you will still see it. I remember graphing cloverleafs or the like, on r-theta graph paper. (Loved that paper...love all cool graph papers, like semilog or maneuvering boards or triangular phase diagrams. Actually...now that I think about it, if your kids are weak, I would definitely hand out some of that graph paper and have them do stuff by hand, at least a little bit...it can be powerful learning to hand graph as opposed to only relying on the computer magic black box.) So r theta was definitely covered.

I could see how weaker kids might not have it until college, and even in the 80s, there was this trend of calculus textbooks to say "with analytic geometry". So, there's a possibility, they haven't seen r-theta yet, as freshman in one of your classes. But if they had college calculus, they should have seen it. All in all, it's still not as central a concept as x-y equations and cartesian coordinates. (Still, important, shouldn't be excised, but not AS important.)

P.s. You know you could just look at some typical textbooks or public high school curricula, to research questions like this. Not a snipe, or to stop from asking the forum also. But...multiple methods...

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    $\begingroup$ Who are you? I regularly see answers, coming from you, but you are always presented as a new contributor and indeed your reputation does not seem to increase. $\endgroup$
    – Dominique
    Commented Jul 10 at 13:45
  • $\begingroup$ Thanks. I should clarify that I am not using this in the classroom (I teach at a college in the US), but instead for writing books accessible to high-school students. $\endgroup$ Commented Jul 10 at 21:04

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