CourseNotes: MAC 1105 College Algebra

I am your friend! - Calculator

I am your friend!

When I enrolled in college, my placement scores allowed me to skip college algebra. I decided to take it anyway because it had been several years, and I was sure I was rusty. I’m so glad that I took it. I’m currently studying calculus and analytical geometry, and I gotta tell you. If you’re taking algebra, don’t slack. If you do you will seriously regret it when you get to calculus. You can breeze through pre-calculus and trigonometry without being an algebra ninja, but not with calculus. It’s just like my calculus professor said–the hardest part about calculus is algebra. You’ll repeatedly use algebraic techniques that you thought you would never see again. The slope formula? That thing will beat you silly the first week in calculus if you don’t learn it now. In calculus, you will literally use everything you even just glanced at in algebra. Furthermore, your calculus professor will be disappointed that you didn’t see more things in algebra.

I don’t have a lot of advice for students taking algebra, but following are some of my thoughts. I’m sure your professor will drill it into you, but I should probably say it too… good note taking and practice are the key.


Modelling problems, otherwise called “application problems” or “word problems” are often the most difficult part of mathematics for most students. It takes me more time to solve a modelling problem than it takes to solve other problems, in fact, I would even say that modelling problems are more difficult than other problems. However, where I differ from a lot of math students is the attitude I have toward them. I like them. I find them challenging. To me, they’re puzzles, and I love solving puzzles. There is a very real benefit to doing modelling problems and that is that it takes all that abstract mathematics and turns it into real, useful, tools.

Most of the modelling problems encountered in college algebra involve geometry, interest calculation, mixtures, uniform motion, rate of work done, or proportions. My suggestion is that you try to figure out a general procedure for each different type of problem. Some of these procedures involve the applications of formulas. For example, for simple interest calculations we use the formula I = prt (interest = principal * rate * time) and for uniform motion we use d = rt (distance = rate * time). For other types of modelling problems, you can learn specific tactics that make it easier to solve any problem of that type. For example, draw the problem if it involves geometry, and for mixtures; make a table. The takeaway here, is that general solutions/equations/procedures are extremely helpful when you’re ready to solve specific instances.

Absolute Value Equations and Inequalities

I’ve always had trouble with absolute value equations and inequalities, and so I think do a lot of other people. I try to remember the specific procedures, for example; |x| = a means x = a or x = -a, but come test time, I end up confused and unable to remember the procedures. The only thing that saves me is to think logically about a given case. It often helps me to draw a number line and then mark it up with interval notation and arrows to show the values that x can take.

When working with absolute value, it helps me to think of it as distance. For example |x – 3| is just the distance between x and 3. Distance is always positive–just like absolute value. If given the problem |x – 3| < 9, I would start by drawing a line with an arrow at each end (my number line). I know that the distance between x and 3 is less than 9, so I mark a spot for 3, a spot 9 units to the left of 3, and 9 units to the right of 3. I know that x must be between those outer two points in order for it to be within 9 units of 3. Then my answer is simply 3 – 9 < x < 3 + 9 or -6 < x < 12. What is |x|, you may ask? Well, it’s just the distance between x and zero.

Your Calculator

Your calculator is your best friend. Learn it well. Your calculator will save your life repeatedly when you’re in the midst of battling an exam… assuming you’ve learned how to use it. You must learn how to graph functions and use the trace and intersection features. This will allow you to double check your algebraic work as you go along. You also need to know how to enter an equation and then use the table and VAR features to try different values for x. These skills are guaranteed to save your ass and to save you time in the long run.

Also, be sure to read my post on avoiding mistakes in algebra.

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Analyzing and Improving your Math Test Scores

My Calculus Test Grade

My Calculus Test Grade

Yesterday I finally got my calculus test results for the test I took last week. My grade was 90% (plus 3 percentage points for extra credit work I did). That the class average was under 64% is irrelevant. As a perfectionist with a 4.0 GPA, my grade is unacceptable. The question is, how am I going to make sure it doesn’t happen again?

The first step is a critical analysis of where I went wrong. The second step is to figure out what was wrong with my learning and test preparation strategies, and the last step is to adjust my learning and test preparation strategies so it doesn’t happen again.

It has been my experience that errors I make on a math test fall into three different categories:

  1. Format errors
  2. Calculation errors
  3. Conceptual errors

The first, format errors, occur when you don’t obey the formatting rules outlined or assumed by the professor. For example, if you’re asked to show work and you don’t or if you’re supposed to call a nonexistent limit “DNE” instead of “it no work”, I would call those formatting errors. The second is calculation errors, and those are caused when you don’t pay attention to detail, or you fail to check your work. The third is the worst kind of error because it signals a conceptual misunderstanding of the material the test is testing you on.

  1. Format errors: Sloppiness, failure to follow guidelines
  2. Calculation errors: Failure to check work or pay attention to detail
  3. Conceptual errors: Failing to understand the material at the conceptual level

Before the test, our professor repeatedly told us that we have to show work in order to receive credit. Well, silly me, on a two-part problem which required the same kind of thinking on both parts, I decided it would be sufficient to show work on one part and just do the next part in my head. Ouch, -1 point. On another problem, I got a -2 for not being explicit enough with my answer. I answered “discontinuous at x = 3 because undefined”. The correct answer was “f(3) is not defined”. I think my professor was getting tired or something when she graded that one. On another one I used absolute value bars when I should have used brackets and a negative sign. I still got the right final answer–my method was just not the traditional one. All in all, 4 out of 10 of the negative points I got was due to avoidable sloppiness–format errors.

On two of the problems, I demonstrated a failure of understanding at the conceptual level. I lost 5 points for mistaking a tangent line to a curve at a specific point for the derivative function of the curve. It took me an hour to figure out where I went wrong–a definite failure at the conceptual level. The one other point loss came from an inability to remember (or to figure out) whether or not a function is differentiable at a removable discontinuity.

To fix errors of the first kind (i.e. format errors) I need to pay more attention to my professor’s guidelines, and I need to follow them to the letter. It wouldn’t hurt to show way more work than I think is necessary. It probably would be a good idea to give my answer in multiple ways just to make it obvious that I know what I’m doing. I didn’t make any calculation errors, but these are fairly easily fixed by checking work religiously and by at least two different methods. Errors of the third kind (i.e. conceptual errors) can only be fixed by analyzing my learning strategy and adjusting it to reinforce and check whether I’m actually understanding the material at the conceptual level. A good review done prior to the test should also help keep the conceptual stuff fresh come test time. So here’s how to fix each of these math test errors:

  1. Format errors: Pay attention to and follow your professor’s guidelines and generally accepted math syntax. Don’t forget your units.
  2. Calculation errors: Check your work as you’re going along then later come back and double-check it with a completely different method.
  3. Conceptual errors: Study, study, study. Constantly review to keep the conceptual stuff fresh in your mind. You need to know the “definitions” so you have a conceptual foundation to fall back on.

That’s it. We’ll see what my grade is on the next calculus test…

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Math Tests and Hominid Skulls

Last week was horrible. I had a pre-calculus exam that covered the previous month of study and a trigonometry mid-term on the same day. I love math, but I don’t think it’s the greatest idea to be taking two math courses at once.

I spent every free hour of the week drilling the trigonometry into my brain. Most of pre-calculus is just a review of concepts I’ve covered before, so I thought it would be better to focus on the trig. Besides the pre-calc test was just one of four and the trig test was one of two, so my score on the trig test would have twice as much of an impact on my overall score in that course compared to the pre-calc test and course.

So the day of the course comes, and I have to drive about 30 miles south of home to take the 4:00PM pre-calc test on my college’s southernmost campus. It turns out I should have studied more. The pre-calc test was a bitch. It would’ve been easy if I had reviewed for several days prior to taking it. Still, I persevered and took almost the whole two hours, and in the end I did get 100% on the test. After that, I hopped back in my truck and headed 45 miles north to my college’s nothernmost campus for the 7:00PM trig test. This one was easier, but only because I had spent the whole week reviewing. The first half had to be done without calculator, so we could prove a good conceptual grasp of the subject. That’s why I love trig–most people hate deriving formulas and proving identities, but I think it’s the best part of math, and trig is filled with it. I took the entire 2 hours, checking and double-checking my work. I haven’t received my score yet, but I feel very confident. I finally got home at 9:30 that night. I think I would have vomited if I saw another equation.

The next several days I took it easy. Bad idea! This week I’m still suffering from a lack of ambition. Is it mid-semester blues? Am I burned-out from all the math last week? Taking several days off probably didn’t actually help me. Instead of being revitalizing me, it made me lazy.


Today, news sites are filled with the story of a hominid skull found in Georgia (the European country). Apparently, the researchers believe this skull could mean a rewrite of hominid taxonomy. According to one article, the skull “could be evidence that early hominids are actually all members of a single species.” Why is this important to me? Well, just two weeks ago I did a research essay for my anthropology course where I proposed this very “rewrite” on the basis of previous evidence. Now I feel validated. When I had written the essay, I assumed that my position was probably wrong–that I simply did not have the background in anthropology to make sweeping proposals like that. However, given the little knowledge I did have of the subject it seemed to me that the evidence suggests that early hominids were all the same species. I’m sure my knowledge was too limited and my reasoning to shallow to convince anthropologists on the basis of my essay alone, but it feels good to know that I may actually have had some insight into a subject that is still new to me.

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Lazy Math Students

There it is!

There it is!

New Commandment: Any math student that leaves a test early and receives anything less than an A, shall be considered a failure and unworthy of the precision, the utter beauty, and the pure truth of mathematics.

Every math test that I’ve taken so far has been scheduled for a full two hours. Invariably though, half the students have left by the one hour mark. By the time I’ve triple-checked every equation, I’m one of the few that’s left. Always, I think, wow! this course has a lot of brilliant students! However, when I check the current average grades for the course, they’re around 65-80%. WHAT THE HELL?!?! Obviously, some of those students leaving early should keep right on walking and never look back.

What in bloody hell could possess students to leave a math test early AND do poorly on it at the same time? Do they not triple-check their work? Are there really students that are fine with C’s? Are there really people who pay good money to do things which will be on their permanent academic record and affect the rest of their academic life and subsequent career, AND they’re fine with mediocre?

Maybe those people should withdraw from college so that poor people that truly want an education can get the financial aid they need. I’m just putting it out there. If you don’t give it your everything, maybe you don’t deserve it.

Image credit: Math question image copyright (c) 2008 Jerry Paffendorf and made available under Attribution 2.0 Generic (CC BY 2.0)

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Joining the Math Club

Can you find the cat in the junk pile?

Can you find the cat?

Today I visited my college’s math club for the first time. We met in a room in the mathematics department–all six of us. Out of about 11,000 students enrolled for college credit, the math club consisted of 3 professors and 2 other students. It is my understanding, however, that there are more than 5 members–they just didn’t all show up today for one reason or another.

It wasn’t exactly what I had anticipated. To be fair, I’ve never been part of an academic club, so I probably shouldn’t’ve’d (should not have had) preconceptions. For some reason, I expected the math club meeting to be a more challenging and more participatory version of a math lecture. It wasn’t. We watched the last part of a documentary on the Riemann Hypothesis (apparently they had watched the first part in last week’s meeting) after which they bandied about esoteric words like “topology” and tried to find a well-hidden cat in a photo of a massive junk pile, because, well you know–mathematicians are blind people in a dark room looking for a black cat which isn’t there.

I did find the experience interesting. I think I will officially join the club and begin attending it as regularly as I can. It’s an opportunity to network with like-minded people, and it’ll expand my horizons when it comes to number theory.

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For the love of math, man!

2013-08-21_2214This evening I went to my trigonometry orientation. Man, was I in for a rude awakening. About a dozen students were gathered outside of the lecture hall, waiting for the professor to finish her pre-algebra orientation. It all started innocently enough. One of the students mentioned that this was her second go at the trig course. Wow, she must really love trigonometry, I thought. Then another student said the same thing, head hanging in shame. Wait a minute… they failed the course on their first try, I realized. Then the conversation drifted to the horrifying as the students professed their mutual hatred for all things cosine and for math in general. Such blasphemy! I wanted to cover my ears and go, “lah-la-lah-la” and pretend I never heard a thing.

I love math. If I could, I would have sex with it and then cuddle with it. When I start a math course, my heart starts fluttering. Not with nervousness, mind you, but with a feeling of pure lust. When I’m at a party, I can’t wait to get home and do the nasty with sines and cosines. I even profess my love for math on my clothing. One of my favorite t-shirts has a symbol of pi and i. Pi is saying “Get real” and i is saying “Be rational”.¬† I love that shirt. It makes me feel all warm and fuzzy inside when I wear it in public.

I haven’t always loved math. When I was in Amish grade school, I was at the top of my class. In other words, I was mediocre and sometimes left with B’s. My teachers simply refused to show me math in its real naked glory. As a result, I found math rather boring. But then, wait for it… I saw it… or felt it, rather. My inner mathesexuality.

It happened when I developed an affinity for layman-oriented books on number theory. When I learned that there is no easy way to determine if a number is prime (other than dividing it by every smaller natural number), I was skeptical. I mean, seriously… that makes no sense. The structure of the number line–the distribution of primes–it’s such a straightforward thing, how could there not be a simple formula for determining whether a number is prime? I played around with it, and the hours stretched into days as I tried to figure it out and develop the magical formula. I never did find it, but my fascination with the distribution of prime numbers remains erect… I mean unabated.

Then I learned about e. It had been sitting there on the number line all this time. The utter amazingness of e is so profound that I can’t even find the words for it. And then there it was… something even holier…

e^{i \pi} + 1 = 0

Somebody is making this shit up, right? I mean that’s just too damn convenient. I still haven’t the foggiest clue what it means, but its utter beauty gives me a intellectual erection the likes of which I’ve never experienced in all my life of lusting after equations. It was love at first sight. I’m not a religious person (anymore), but that simple little equation is the closest I’ve ever come to a religious epiphany.

The only math I’m not good at is combinatorics. When it comes to calculating probabilities, I can’t make heads or tails out of coin flips. Well, actually, that part is easy. It’s the permutations and combinations that get me. It’s like math on PMS–about all I can do is tiptoe around it. Double counting fallacy? I will count my apples as often as I damn well please, thank you very much! One of these days I will take a discrete math course. That’ll teach me!

Anyway, I think I hear trig puzzles calling me. Maybe I should return to my homework. I hope this has helped you understand how much I love math. I have just screamed it from the rooftops. One last thought… I like my math how I like my women–abstract and well-defined. Hmm, yeah, I can see that becoming a nerd meme…

Happy mathing!

P.S. That thing I said about liking my women abstract… I think I may have tangled that simile a bit. I actually have a real life fiancee. She is real (not abstract), well-defined, gorgeous, and I love her second only to math.

Image Credit: Get the shirt here

Categories: Math | Tags: , , , , | 5 Comments

Avoiding Mistakes in your Algebra Studies

Wrong algebra solution.


Algebra is an eminently useful endeavor and a fun (I know I’m a nerd) way to spend time. However, few pursuits offer the variety of potential mistakes that algebra does. I am far from mistake proof in my algebra doings, but I am much better than I used to be. Here are some tips I’ve gathered along the way to reduce stupid mistakes:

  • Algebra typically involves doing a lot of calculations. Therefore, the probability of making a mistake is higher. Train yourself to pay attention to every detail–both in the given problem and when solving it. I cannot tell you how often I’ve pulled my hair out because my solution didn’t check, only to eventually figure out that for some reason my brain insisted that 3 + 4 is 12. I mean I could tell you. Never once did I actually pull my hair out. I feel that I may be losing my point here…
  • If a problem seems unreasonably difficult, or if it gives some long-winded decimal expansion when you were expecting an integer, check to make sure you’re working on the given problem and not some transcription error (i.e. you copied the problem incorrectly when transferring it to your notebook). If the problem on the test has a minus sign, but the one you’ve been working on the last ten minutes has a plus sign, then you should feel silly.
  • For problems with a lot of computation (e.g. factoring large polynomials), it is important to have a clean and organized solution system. That way you can quickly and clearly see how each step progresses, and you can check your work at each step since checking the final solution might take too much time.
  • Draw a box or circle around your solution as soon as you write it down. That way, if you check your solution by writing calculations underneath your solution, you’re not as likely to accidentally write the wrong thing onto the test paper.
  • When checking your solution, always check against the one given on the test (not the one you wrote on scratch paper) just in case you transcribed the problem incorrectly when transferring it to your scratch paper.
  • Always check your solutions! You may think you know what you’re doing, but sometimes you don’t. Trust me!
  • Every time you make a mistake, get to the bottom of it. Were you careless? Were you not paying attention? Do you not understand the material? Vow never again to make that mistake.
  • …and of course, always check to see if your answer can be simplified or factored further.

Do you have any tips on avoiding mistakes that I haven’t covered? Please let us know!

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