Posts Tagged With: Algebra

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

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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Categories: College, Course Notes, Math | Tags: , , , | Leave a comment

Avoiding Mistakes in your Algebra Studies

Wrong algebra solution.

Really?

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!

Categories: Math | Tags: , , , , , | Leave a comment

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