# Is algebra unnecessarily quadratic intensive?

I don't know about all algebra courses, but the following is the basic outline of my algebra years:

• Alg. I - Lots of linear graphs and solving systems of equations.

• Alg. II - Lots of quadratics, some harder systems of equations, and light touch on some topics like exponential functions, the meaning of 'functions', complex numbers, etc.

Now, I can understand why we teach what we teach in Alg. I, it is easy/good start and marks a foundation to manipulate equations, something you can hardly do with out. Not to mention, linear graphs are pretty much the simplest graphs, yet you can get fundamental knowledge from them (slope, familiarity with graphs).

On the other hand, I'm not so sure I understand the Alg. II curriculum. From my experience, Alg. II grinds quadratics into you so hard that at times I was outright done with parabolas and factoring. Why is it that they do this? Sure, quadratics are the next step beyond linear functions, but I most certainly do not feel the amount of time we spend over the subject was worth my sanity.

Why not do less of the quadratics and more of other topics? For example, we could inquire more about higher degree polynomials. In my area (USA), we are taught trigonometry first, then Alg. II, so we could solve cubic polynomials using trig functions, which has a derivation that isn't excruciatingly hard. We could talk about so many other topics, like inequalities and such...

So: Why not do less of the quadratics and more of other topics?

• There is a reason we don't teach cubics and other, higher degree polynomials. See the formula at: math.vanderbilt.edu/~schectex/courses/cubic Otherwise, I agree that an inordinate amount of time is spent on degree 2.
Sep 26 '16 at 3:15
• @Adam I think we could at least talk more on general second degree equations like hyperbolas and circles! Sep 26 '16 at 14:02
• What about splines? It is getting so much used, in so many areas, that it might be a time st start look at fitting it into high school math? Cubic splines ghives a specific reason to look at third degree polynomials. Dec 1 '17 at 14:41
• In my Algebra II course (also in the US), we spent a lot of time on quadratics because our curriculum included the study of conic sections, both from a geometric perspective ("ellipse is the set of points where the sum of distances from the foci is a fixed value") and the algebraic (x-h)^2/a^2 + (y-k)^2/b^2 = 1, which I found rather useless until having to take my third semester of calculus as an undergrad, where it not only generalized to the quadric surfaces, but also led the way to other ideas like the normal form of a plane, etc., which only deepened my understanding of the topics. Dec 4 '17 at 21:51

Here are some brief [and admittedly inchoate] thoughts, as I'm not quite ready to invest in writing a formal essay about your very important question with respect to what an Algebra 2 course should contain. (I am also purposefully avoiding a discussion over whether an Algebra 2 course should be required for students to complete secondary school, or even whether it should be a college/university requirement.) That said/unsaid, my thinking right now is roughly as follows:

I believe that the study of linear equations is important, and that linear algebra is an area that can be pursued with theoretical interest and/or for its practical applications. So, I think that an Algebra 1 and Geometry experience that really builds an understanding of lines and systems of linear functions is important. For Algebra 2, I like the idea of segueing from lines to piecewise linear functions to a specific example of a piecewise linear function: the absolute value function. I think that a lot of analytical thinking and reasoning can be built into a rigorous approach to absolute value functions, equations, and inequalities, and that this can lead nicely to quadratics, which have analogies with absolute values (concavity; x-intercepts; vertex; y-intercept; defined by three parameters/constants) and can pave the way towards the more general study of polynomials. (I also think that insufficient attention to absolute values is unfair for students who will pursue Calculus courses that use delta-epsilon proofs, or go on to major in mathematics and take courses on, for example, Real Analysis.)

Through solving quadratics, one can get some wonderful experience with rigorous mathematical thinking (including the broaching of complex numbers). I wrote of a problem set that I used last year in my Algebra 2 class in my answer to MESE 12977, which should give a rough impression of the sort of questions that I think an Algebra 2 student can be prepared to answer. In my own experience, it is possible to pursue quadratic functions, equations, and inequalities in a way that sets students up to launch into polynomials, or complex numbers, or rational functions, or limits ("end behavior"), or a variety of other topics. However, I feel that there is a complete overfocus on factoring quadratics (and, for what it is worth, would prefer to see a greater number theoretical focus on factoring whole numbers at other points in a school curriculum).

As an addendum (just for fun) here is a blitz treatment of quadratics that I think can be informative: Consider the two primary ways of writing quadratic functions.

• standard form: $f(x) = ax^2 + bx + c$

• vertex form: $f(x) = a(x-h)^2 + k$

We can move from one representation to the other by expanding the latter and matching coefficients.

But, in my classes, we note that $x=h$ corresponds to the line of symmetry for a parabola, on which the $c$ from standard form has no effect; thus, in particular, we set $c=0$ to find roots at $x=0$ and $x=-b/a$, so that a line of symmetry between them occurs at $h = -b/2a$. Since $k$ is the $y$-coordinate for the vertex that has $x$-coordinate $h$, we note $k = f(h)$. This is enough information to move from standard form to vertex form.

To move in the other direction, we rearrange to find $b = -2ah$ and observe that $c = f(0)$ is simply the $y$-intercept of $f$. Bearing this in mind, we derive the quadratic formula rapidly from vertex form:

$$a(x-h)^2 + k = 0 \implies x = h\pm\sqrt{\frac{-k}{a}}$$

which yields the quadratic formula in terms of $a$, $b$, and $c$ by using the above substitutions of $h = -b/2a$ and $k = f(h)$, although the latter is a bit tedious to compute. All of this is to say, I find the idea of bazillions of quadratic factoring exercises to be a suboptimal use of time, since one can really get into some deep mathematical thinking by experimenting with graphs (see also the end of my answer to MESE 12286) and then diving into algebra in ways such as the derivation above.

It is unfortunate that some courses may overemphasize quadratics to the detriment of other topics. Quadratics is a very important topic, but focusing exclusively on them could get needlessly tedious. Personally I have never seen an Algebra I course that did not cover at least the basics of quadratics, but I suppose some curricula might put quadratics off until Algebra II.

In the U.S., most states have adopted the Common Core Standards. There are about 50 standards in the Common Core that might be included in a traditional Algebra I/II sequence, of which 6 involve quadratics. If all 6 standards are put off until Algebra II, that could make for up to one fourth of the course involving quadratics. This could be a bit much. If some of those standards are treated in Algebra I, it should make for a balanced Algebra II curriculum.

• I like this idea very much. When I took Algebra, they put off quadratics until Alg. II and it was too much for pretty much everyone. I think, that as a result, the students performed the worst on quadratics and were unable to perform as well as they could've on other areas. Sep 27 '16 at 19:55

Quadratic formula is key part of most chemical equilibrium and rate problems (big part of college frosh chem). Many kinematics problems (intro college physics) also involve it (acceleration is a second derivative of position). It is also needed to solve the characteristic equation of the second order diffyQ with constant coefficients (most important ODE for applications, comes up in EE, quantum mechanics, controls/systems, etc.) Also helpful wrt some trig derivations.

Now actually answering if it is "too much" attention involves knowing what percentage, x, of the algebra 2 course is spent on it, and if x is "too much". But I sense you aren't strong on applications, so sharing the above.

Of course classes will differ. I remember it not being covered that heavily when I took it, but I was in an advanced class and most people just got it quickly. Lot of other material to spend time on since we did alg2/trig in a year and you have to hit exponents, logs, etc. polynomial division, etc. also. Perhaps weaker classes spend more time on the quadratic and less time on other "college algebra" topics.

• It's really nice, looking on this a few years later and having seen where this has all been used. Nov 29 '17 at 23:39
• Hope some of my comment was actually validated by your experience. Nov 30 '17 at 2:15
• (for Ben's answer, can't comment there) I had inequalities and absolute value in a semester long pre-calc course called "functions" (after algebra 2, before calc). It was interesting and sound theory background. But not as vital as understanding exponent rules and the like. Also a pre-calc analytic geometry course covered foci, vertices, translation and even rotation (remember that last one being a lot of work): after "functions" and before calc. I would be hesitant to lard the normal algebra 2 course down with inequalities, etc. They cover enough and have enough issues as it is. Nov 30 '17 at 2:18
• @guest Interesting. I find the naming conventions around some of these courses downright puzzling! "Algebra 2" makes it sound as if there are some fixed number N of Algebra courses, and "Pre-Calculus" (or "Functions") gives little to indicate what it'll actually cover. (Although each is better than schools that opt for Math 1, Math 2, etc.) Nov 30 '17 at 19:18
• (post 1 of 2) My exp,: 80s public school but fancy suburb. "GT" (high track). 7th grade: pre-algebra (equations in X). 8th: Algebra ( x and y line). A year ahead normal track, supports AP calc in 12th. 9th: geometry (proofs and construction, light on solid geo). 10th: Algebra 2/trig (normally 2 semesters for alg 2 and 1 for trig, but GT combined them). Pretty much all the basics in algebra 2 (exponents, logs, polynomial division, etc.) from early 20th century Hart College Algebra (plus intro to vectors). Trig was plane only but other than that classical. Nov 30 '17 at 21:23
1. In a certain sense quadratic equations are the simplest that have TWO solutions. Students might believe equations have only one solution.
2. The very important No-Zero Divisors Theorem is first seen when solving quadratic equations and so...
3. Quadratic Equations provide a motivation for factoring.
4. In a certain sense the simplest function with a local max/min is a quadratic function.
5. They are required later. Students SHOULD be able to factor in their sleep. Learning later takes time away.
6. In a certain sense they are the simplest non-affine functions for students to graph.
7. They exhibit axial symettry about their extremum.
8. Their discriminant introduces students to the problem of a square root of a negative number.
9. Word problems involving the area of a rectangle, usually lead to quadratic equations.
10. They test students on B|E|MD|AS
• I've never heard of the "very important No-Zero Divisors Theorem". What theorem do you mean? Nov 30 '17 at 12:01
• Point 4 about max/min is an important one. Something stronger can be said. Any sufficiently smooth function looks like a quadratic function near a max or a min. This silly sounding observation underlies the study of what physicists call "small oscillations", that near a stable equilibrium a mechanical system can be approximated by one with a quadratic potential energy. Nov 30 '17 at 12:05
• @Dan Fox: I've never heard (for algebra 2 level material) of the "No-Zero Divisors Theorem", but I would guess this is the theorem "$uv=0$ implies $u=0$ or $v=0$", with the name borrowed from the abstract algebra term (when ring theory basics are introduced). Nov 30 '17 at 13:53
• @JpMcCarthy In k12, I've generally seen this called the "zero product property." If you are thinking about the material from the perspective of abstract algebra, then you may ask whether a ring $R$ can have a $k$-product property for nonzero elements, i.e., can there be a $k \neq 0$ for which, for all $a, b \in R$, if $ab = k$, then $a=k$ or $b=k$. With a bit of work, this can be shown impossible for rings with at least four elements. (Note that $R= \{0, 1, 2\}$ under addition and multiplication mod $3$ has a $2$-product property.) Nov 30 '17 at 19:25
• @BenjaminDickman Thanks for that. It was never referred to by name in particular while I was at school (no in the US) but that is what I would call it now. Nov 30 '17 at 21:34

IMHO, the problem is not overemphasis on quadratics, but rather that many students completely miss the point of algebra. They think algebra means "Solve for $x$". Really, they should be thinking about algebra as the generalization of arithmetic, that algebra allows us to reason with a class or classes of numbers rather than known constants.

The main problem with a lot of algebra classes isn't the overemphasis on quadratics but the underemphasis on algebraic generality.

Here is a sample of how weak algebraic reasoning is at a community college, i.e. a place where all students have graduated high school.

As you can see, overemphasis on quadratics may have harmed them, but far more conceptual "damage" was done by missing the most basic properties of algebra. These students have not yet learned the generality of grade 1 arithmetic.

THEY HAVEN'T GENERALIZED GRADE >>ONE<< ARITHMETIC!!

But they were good enough at following fixed math procedures to graduate high school, though. I bet they applied the quadratic formula with fidelity...