How to do well in (my) Precalculus class

General Student Tips

• Reflect on (or measure!) how much time you’re spending on the class outside of class.

  • A successful student may average anywhere from 5-10 hours per week outside of lectures, either studying, working problems, or revising/organizing old material. There’s a huge amount of variation here, of course, but often the students that lose the most points are spending the least amount of time practicing outside of class.

• Take good notes.

  • You don’t have to capture everything in lectures, but there are often many small details in lecture that are hard to remember. Try to record general formulas and derivations, of course, but also listen for things like common mistakes, restrictions on how to use formulas (e.g. radians vs degrees), hints about topics that will be on quizzes/exams, or things the instructor points out are particularly important.

• Do all of the work available to you, and use all class materials.

  • If there are materials given to you for the class (such as extra problems or worksheets) and you’re not completing them, this is a lot of wasted potential. Curated materials from your exact class are by far by the best things to work and study to do well in your class.
  • Similarly, if there are videos, notes, discussions sections, office hours, discussion boards, group chats, or really any other aspects of the class that don’t happen in a lecture, take full advantage of them!

• Revise.

  • That is, don’t just work a problem and/or turn it in and just forget about it. Check your solutions ASAP, and if you get something wrong, have a systematic way of marking those problems and revisiting them.

• Go to office hours!

  • Many students don’t quite understand what office hours are, so let me frame it this way: it’s like having a private tutor working for you, for free, who also happens to be the exact person grading your assignments and writing your exams. Your professors, teaching assistants, or graduate student assistants are there to help you, and moreover they want to help you!

• Model your solutions off of posted solutions

  • If any written-up solutions for your class are posted, you are extremely lucky! They are giving you a fantastic model for exactly what they consider to be a good solution. Absolutely use this to your advantage, and structure your own solutions and write-ups to mimic their format.

• Don’t focus on what specific things you might use “in the real world”!

  • Not every class is meant to be job training!

    Do you have similar expectations from your philosophy class?

    Your art appreciation class?

    Your literature class?

    Mathematics is a technical skill, to be sure, but its broader use is to understand methods of solving problems.

  • We use mathematical equations because they’re much simpler than real life: there actually is a single correct solution in many cases, you can explain the algorithms and methods relatively easily, and the problems are accessible without years of extra scientific or industry-specific training.

  • Depending on your career, no one may ask you to solve for $x$ for the rest of your life. But skills like being able to learn and communicate a technical process, abstracting away difficulties, getting yourself “unstuck” in problems, and keeping an organized account of your work, all give you tools to deal with whatever difficult problems you do end up caring about.

  • In any case, many fields are becoming more data-driven and thus mathematical, and you may very well have to solve for $x$ at some point! Mathematics shows up in surprising places. I personally had to use logarithms and cosines/sines multiple times in one of my industry jobs, you just never know.

Writing Tips

• The process is more important than the solution.

  • Teach a man to fish! In other words, having a solution in hand to one specific problem with specific numbers isn’t so valuable. However, having a process, a recipe, an algorithm for solving that problem, is immensely useful, because that’s what can be used, modified, and adapted to unforeseen new problems.

• Rarely write just a solution!

  • Try to communicate to your reader a recipe, a logical series of steps they could retrace to arrive at the same solution from scratch. Teach your reader the algorithm!

  • Bonus: this also provides strong evidence to your instructor or grader that you are really and truly internalizing the material.

• Know your audience.

  • At this point in your academic career, you aren’t writing up solutions in order to demonstrate the correct answer or show someone that you did the work. Instead, you should transition to thinking about your solution as a piece of mathematical writing, which (importantly) will be read by other human beings.

  • In particular, you should frame your writeup as though you were explaining it to someone in the first week of your course, and include all of the information they’d need to work the problem themselves.

• Differentiate “scratch work” from what your present.

  • In previous classes, we’re very used to writing our work out on some paper, possibly erasing things and reworking until we solve the problem, and turning that piece of paper in in.

    Instead, it can help to do your intermediate work on scratch paper, setting that paper aside, and cleanly writing up a series of logical steps (usually equations) in a clean and organized way on the paper you’ll actually turn in.

• Write your solution as a “mathematical sentence”, i.e. a series of logical steps,

  • For example,

    \[\begin{align*} &&6x + 2 &= 5x + 1 \\ \implies&& 6x + 2 - 5x &= 1 \\ \implies&& x + 2 &= 1\\ \implies&& x &= -1. \end{align*}\]

  • Notice that each line is a statement that two things are equal.

  • Also notice that I’ve put a period at the end! This is the emphasize that you should be able to read your solution out loud. For example:

    • “Six $x$ plus two equals five $x$ plus 1,”
    • “Which implies that six $x$ plus 2 minus five $x$ equals one,”
    • “Which implies that $x$ plus two equals one,”
    • “Which implies that $x$ is equal to negative one.”

• Your solutions should tell a story.

  • What I mean here is that there should be some kind of “narrative arc” to what you’ve written, as opposed to a collection of mathematical symbols on a page. There should be a beginning, a middle, and an end.

    You should generally start with a known equality or equation, take a series of logically justified steps, and eventually arrive at a destination: the solution to the original question being asked!

• Write the general form of any formula you use before you use it.

  • Again, just imagine that your reader is a student in Week 1, and they haven’t seen the formula yet. Remind the reader of what it is, and importantly, what it means.
  • For example, you might use the fact that the area of a circle is given by $A(r) = \pi r^2$. Definitely write this general equation before you use it, but don’t stop there! Tell the reader that this is a formula for the area, and that $r$ denotes the radius of a circle. Indicate what every symbol means!

• Box and/or highlight the actual solution to the problem 100% of the time.

  • Every mathematical problem is asking a question. What’s the answer? At the end of doing all of your work, always go back to the original statement, and ask yourself, “What question was this problem originally asking?” Then write the answer to that question and distinguish it from the rest of the work: box it, highlight it, put a period on it, whatever works!

Mathematical Tips

• Draw a picture!!

  • Ask yourself for every problem, “What’s the picture?”

    It doesn’t have to be precise every time, sometimes just having a schematic diagram of the situation can be very helpful.

• If you use a picture to reason precisely, label it precisely 100% of the time.

  This often comes up when graphing functions, in which case you should always

  • Label the dependent and independent variables.

    We’re used to these being $x$ and $y$, but in most applications of mathematics there are other variables with other names like $t,r, \theta$, etc.

  • If you plot the graph of one or more functions, label each one by name: $f(x), g(t)$, etc.

  • Label with numbers enough tick marks to communicate the scale of the graph.

    For example, do the ticks each represent $1$ unit? $1000$ units? $0.00001$ units? Are the scales on the $x$ and $y$ axis the same or different?

  • Label several “interesting” features, such as coordinates of points or lines. These can include

    • $x$ or $y$ intercepts
    • Points computed in a table
    • Points where a piecewise function changes
    • Intersection points of two graphs
    • Horizontal or vertical asymptotes
    • Two or three points on the graph of a function.

• Don’t modify equations “in-place” in write-ups.

  • For example, in the above problem it might be tempting to look at

    \[6x+2 = 5x + 1\]

    and write

    \[\begin{align*} 6x+2 &&=&& 5x+1 \\ -5x && && -5x \\ \\ x+2 && = && 1. \end{align*}\]

    This is totally fine in scratch work, but tends to allow errors to creep in and generally makes the “bookkeeping” in your solution hard to track as you modify the same thing multiple times.

    Instead, an alternative is just not including that $-5x$ step in your writeup at all, or explaining off to the side how you got from one step to the next. For example,

    \[\begin{align*} 6x+2 &= 5x+1 \\ \implies x+2 &= 1 && \text{subtracting $5x$ from both sides} \end{align*}\]

  • Note that this also applies to dividing both sides of an equation by a number, exponentiating both sides, taking a log of both sides, applying $\sin$ or $\cos$, etc. It’s more clear to just write the result of the operation and explain in words what you did.

• Carefully distinguish between equalities and approximations with $=$ and $\approx$ respectively.

  • Saying $a=b$ else is a very strong assertion, and can introduce issues in many contexts where numerical precision is important. Calculators can only hold so many digits, and some real numbers have infinitely many!

  • It’s important to use the $\approx$ symbol to communicate to the reader when error from approximation has potentially been introduced.

    As a general rule, the second you plug something into your calculator, you’ve traded in an exact answer for an approximation and should be using $\approx$ instead of $=$ in your notation.

• Practice not using your calculator!

  • At least in my class, you will rarely need a calculator, and often plugging things into one actually makes things more difficult.

    It is perfectly fine (and often preferable) to leave something in a form like

    \[12 + 7 \log_4(5) \over 2\log_5(3^3) - \sqrt 7\]

    instead of approximating it with a calculator.

• Don’t plug anything in until you absolutely have to!

  • In other words, don’t use your calculator until the very end!

    This will save you a ton of writing – a single letter is much easier to write out than a multi-digit number – but also produces the most accurate and exact answer.

    (If you take science courses, this becomes a necessity: approximating at intermediate stages can compound errors and significantly throw off important calculations.)