What basic algebra skills and techniques are most important for calculus students to know?

Question

In my experience, algebra is one of the biggest stumbling blocks to calculus students. For instance, sign errors are common, and exponent laws (and log laws!) cause a lot of headaches.

Many courses have algebra pre-tests to let students know if they are "prepared" or not; however, these pre-tests often cover material that has nothing to do with the course itself.

Is there any published information on what algebra skills and techniques are most important for student success in calculus?

(I saw this question, but it covered a different aspect of the issue).

Answer 1

Sam Shah has this post on his blog: http://samjshah.com/2012/06/01/algebra-bootcamp-in-calculus/

I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.)

I teach my calculus course in an order that I think will help students learn. I have four units:

• Unit 1 includes history, graphing functions, slopes of tangent lines by approximation, algebraically finding the derivative using the limit (which we do not carefully define yet), seeing the similarities between velocity, rate of change, and slope, average versus instantaneous velocity, derivative from a graph, (estimated) derivative from a table of values.

• Unit 2 includes derivative properties needed for polynomials, graphing, limits, product and quotient rules, and trig derivatives.

• Unit 3 includes chain rule, derivatives of exponential functions, implicit differentiation, derivatives of inverse functions (ln x and inverse trig functions), and related rates.

• Unit 4 includes integration (finding area under the curve), anti-derivatives, fundamental theorem of calculus, and substitution method. If there is time we include volumes of rotation (which I think is a perfect ending for the course).

Algebra Skills needed for Unit 1

Algebra

• Determine the equation of a line given two points, or a point and a slope, or a graph of a line,

• Find the average rate of change over an interval given a function or its graph,

• Clearly express what is happening to an object given a position versus time graph,

• Evaluate f(x+h) for any given function f(x),

• Rationalize the numerator (to find the derivative of the square root function) ,

• Simplify complex fractions (to find the derivative of the 1/x function).

Algebra with Calculus Concepts

• Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph,

• Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change,

• Sketch a velocity versus time graph given a position versus time graph,

• Construct the formal definition of the derivative by modifying the definition of slope,

• Apply the formal definition of the derivative to simple polynomials and to simple square root functions.

Algebra Skills needed for Unit 2

Algebra

• Multiply out the expression (x+h)n (necessary to understand the proof for the derivative of y=xn),

• Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,

• Sketch the basic shape of a rational function,

• Identify an equation for a rational function given a sketch of the function,

• Explain clearly what a hole and an asymptote are,

• Construct the equation of a piecewise function given its graph,

• Sketch the graph of a piecewise function given its equation,

• Work with inequalities,

• Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,

• Evaluate sin x, cos x, and tan x at all multiples of π/6 and π/4, without a calculator,

• Understand trigonometry identities, including and sin(x+h)=sin x cos h + sin h cos x,

• Graph y = sin x and y = cos x.

Algebra with Calculus Concepts

• Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,

• Follow complicated logic (in the definition of limit).

Algebra Skills needed for Unit 3

Algebra

• Understand composition of functions,

• Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,

• Understand properties of exponents,

• Be able to graph exponential and logarithmic functions.

Algebra with Calculus Concepts

• Think in terms of composition of functions to determine outer and inner functions, in order to use the chain rule.

Algebra Skills needed for Unit 4

Algebra

• Work with summations.

Answer 2

Most problems I see (in classes after the first calculus courses) are basic misunderstandings in what is allowed to do with equations (and even worse with inequalities). There are also misconceptions on the meaning of strings of equations/inequalities, i.e., $p^2 < p^2 + u \le p^2 + p < p^2 + 2 p + 1 = (p + 1)^2$ (here $1 \le u \le p$). There are problems with the concept of a variable (i.e., it doesn't vary, it is a fixed, perhaps unknown value or some value we don't want to nail down for the sake of generality; note that this are two different notions).

Answer 3

I think its more about the complexities of equations, such as rational equations with a quadratic under a radical in the denominator where you need to complete the square to simplify.... even when you know all the basic rules things start to get a little bit more complicated when everything is thrown in together and you're trying to do something new like integrate at the same time. I thought I had a good foundation in the basic concepts but pretty quickly realized I didn't know it as well as I thought

Answer 4

I think the main thing is a comfort with working multi-step equations. (Doing same thing to both sides, showing your work, having sufficient care/practice to make a very low level of mistakes.) At the simplest level, think of the steps in solving for x in first order, one unknown problems. But this same trait, in other problems. I would say that this can still be drilled, emphasized "as you go" in teaching calculus. For example at a help center. And if you are teaching weak students, SHOW EVERY STEP. Get them to show every step. No "throwing something from one side to the other and flipping sign" but "subtract same from each side, showing it pedantically step by step".

Obviously there are some specific items of knowledge (e.g. exponent rules) that are important also. It's basically "most of it" from the precursor classes, so it's almost as helpful to specify what parts of the curriculum are NOT important. And this is not to say they shouldn't be taught. Just for a kid, trying to progress, with some gaps, they are low bang for the buck.

On the list of things that are lower priority (for weak kids, because life is prioritization, because they ain't ever getting to Rudin, but we'd like them to get their nursing degrees):

1. Most of geometry (side angle side, proofs, etc.). Only mensuration area formulas are needed.

2. Conic sections, except for the very basics of a centered circle and parabola and hyperbola on y-x plot. But not foci and directrix and ellipses and vertical/sideways displacements.

3. Most of the trig identities (just sinsq plus cossq). [If double angle or the like needed later, review it then.]

4. Determinants. [Review it when needed in a calc class for weak students or even just eschew entirely...it's usually "enrichment" anyhow, even within calculus or diffy screws.]

5. Rotation and displacement of coordinates (analytic geometry).

6. harder (often, but not always asterisked, sections of Alg 2): synthetic division, partial fractions, induction proofs, series, etc. For series or partial fractions, if that is part of the calculus course, make sure to cover the required algebra WHEN they do the calculus topic (and show all the steps). But don't expect precursor competence. Look at it more as an opportunity to brush up that topic they didn't handle/remember in 11th grade.

If you want a build up (versus tear down) approach, the list from Kahn Academy is decent:

https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ap-ab-about/a/ap-calc-prerequisites

Please note, I'm assuming a context of community college or other remedial work. Not precalc at Thomas Jefferson. They should have a tough course. But then they don't match Brian Rushton's posited set of students struggling with standard calc 101 because of poor algebra.

Also note: don't underestimate the importance of algebra. It's at the core of calc and diffyQs in terms of the actual working the problems. And is required in most chem, physics, engineering classes. Just to work problems (often not needing calculus even!)

Not also that this means when you DO teach a calc class for weaker students, you should be MUCH more receptive to the "here's an example, here's a method, practice small variations of the algorith, test recognizable problems versus the examples". After all, even if it is "mechanical", it gives them a golden chance to practice multistep algebra, itself.