Is algebra unnecessarily quadratic intensive?

Question

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?

Answer 1

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).

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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.

Answer 2

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.

Answer 3

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...

Answer 4

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.

Answer 5

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