Refresh your High School Math skills

In my first article “The most enlightening Calculus books”, I argued the importance of maintaining high standards for mathematics education and suggested deep and inspiring calculus books for those of you who are interested in pursuing the joy of learning mathematics. The feedback has been overwhelming and I wish to follow up with an article that addresses a couple of remarks that I’ve received by email.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general – and mathematics, technical and scientific education in particular – are key for the development of every country and ultimately for good of humankind.

However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them.

Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as pre-calculus, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

Simple Precalculus Questions:

1) Factor the following polynomials:

  1. [tex]\displaystyle x^{2}-6x+9[/tex]
  2. [tex]\displaystyle x^{2}+x-6[/tex]
  3. [tex]\displaystyle x^{3}-27[/tex]

2) Solve for x:

  1. [tex]\displaystyle 3x^{2}+5x-2=0[/tex]
  2. [tex]\displaystyle |x^{2}-x|=3[/tex]
  3. [tex]\displaystyle x^{4}-8ax^{2}+16a^{2}=0[/tex]
  4. [tex]\displaystyle \frac{x^2+x-6}{x+3}=0[/tex]
  5. [tex]\displaystyle 2\sqrt{x} = x – 15[/tex]

3) Find the values of x for which:

  1. [tex]\displaystyle x^{2}>9[/tex]
  2. [tex]\displaystyle |2x-3| \leq 5[/tex]
  3. [tex]\displaystyle |2x-1| > 9[/tex]
  4. [tex]\displaystyle |x-1| + |x-3| \geq 8[/tex]

4) Evaluate:

  1. [tex]\displaystyle \log_{2}{1}[/tex]
  2. [tex]\displaystyle \ln{e}[/tex]
  3. [tex]\displaystyle \log_{2}{1024}[/tex]
  4. [tex]\displaystyle \frac{4^{8}2^{4}}{2^{12}}[/tex]

5) Solve for x:

  1. [tex]\displaystyle 5^{x}=10[/tex]
  2. [tex]\displaystyle \log_{3}{7x} = 2[/tex]
  3. [tex]\displaystyle \log_{x}{9}=2[/tex]
  4. [tex]\displaystyle \ln(3x-2)=0[/tex]
  5. [tex]\displaystyle 3^x+x=4[/tex]

6) Solve for x, where [tex]\displaystyle 0\leq x \leq 2\pi[/tex]:

  1. [tex]\displaystyle 2\sin{x} = 1[/tex]
  2. [tex]\displaystyle \tan{2x} = \frac{\sqrt{3}}{3}[/tex]
  3. [tex]\displaystyle \sin{3x} = 1[/tex]
  4. [tex]\displaystyle \cos^{2}{x} – x = 2 -\sin^{2}{x}[/tex]

7) Write the equations of the following curves in the Cartesian plane:

  1. Parabola
  2. Hyperbola
  3. Circle
  4. Ellipse

8 ) Find the vertex, focus, and directrix of the parabolas given by the equations:

  1. [tex]\displaystyle x^{2}=16y[/tex]
  2. [tex]\displaystyle y^{2}+4y+12x=-16[/tex]

9) Find the center, vertices, foci, and eccentricity of the hyperbola given by the equation:

[tex]\displaystyle \frac{x^{2}}{4}-\frac{y^{2}}{36}=1[/tex]

10) Find the equation of a circle whose center is at [tex](2, -3)[/tex] and radius [tex]3[/tex].

11) Determine the center and radius of the circle with equation:

[tex]\displaystyle x^{2} -4x+ y^2-18y = -4[/tex].

How did it go? Did you experience many struggles and the feeling that “I used to know this stuff”? If so, then it is a good idea to go for a refresher before attempting calculus right away. The following are two books that you may find useful to respectively learn and refresh basic math in a well organized manner:

  • Precalculus by Michael Sullivan: a big book, which is quite extensive and thorough. If you want an all-in-one book that covers all you need to know about precalculus and more, in a clear but college oriented manner, than this is without doubt an excellent choice. It will likely make the step up to Calculus quite easy.
  • Schaum’s Outline of Precalculus: it has a less prosaic approach but it’s still very clear and easy to read. If you were pretty good at math in high school and you just forgot a few things because you haven’t touched these topics in a while, then pick this book up. It is adequate for already mathematically inclined people who are in a rush to brush up the skills they once had.

If you feel entirely clueless and would like a “for dummies” type of book, the following two titles seem to have a good table of contents and excellent reviews:

If you would like to use some free resources available online instead, here are some lessons:

If you know of any other resources that are available for free, or if you successfully used other books for these purposes, please feel free to use the comment section to add to the discussion.

Sponsor’s message: Math Better Explained is an insightful ebook and screencast series that will help you deeply understand fundamental mathematical concepts, and see math in a new light. Get it here.


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