Intermediate Algebra

Well, what do you already know, and what do they cover in INtermediate Algebra? I tutor it but we’ve got a couple levels at the the developmental level (hey, we’ve gotten past calling it remedial :-)).
For the basic class, knowing how to add fractions with different denominators, being solid in positivce and negative integers, and at least knowing the basics for exponents would put most folks ‘way ahead.

For the next level up, y = mx + b … knowing how to plot lines with point-slope or two points, knowing what perpendicular slopes are like, knowing that a y=5 line is horizontal (with slope 0) and x= 3 is vertical (with undefined slope)…

And for the third level… really know those exponents and the distributive property - take a look at your book…. try the first problems… what are they about? Does the course start with the inevitable “commutative property” review?

What kinds of issues give you grief and woe?

First of all, I’d get rid of your lead pencils and particularly the erasers. Get a whole bunch (a nice ream, 500 sheets) of nice clean white looseleaf paper ant Staples, and several easy-to-use pens, such as rolling writers. Now keep saying to yourself “I’m here to do MATH, I’m not here to learn erasing.” This will immediately double or triple your math productivity.
I am not kidding at all. Most students are so trained in erasing that it takes up the majority of their time and effort, and sorry, you don’t get grades for nice erasing.

Second, you can use you Barron’s basic algebra book, although I would prefer Schaum’s review books (nice simple plan).

I use books like these with the attitude that there is always something new to learn, that review is a way to gain mastery and consolidate. So I don’t skip around picking and choosing, I start in at page 1 and work through. Stuff you really know goes fast, and stuff you need to learn will take more time. You find your own level pretty quickly.
Find a space that is really comfortable and well-lit and schedule yourself a time. We all know that “when I get around to it” will be in the next life. If you feel less stressed with nature shows or news on the TV, fine. If you lie on a sofa with a clipboard in one hand and a Coca-cola in the other, fine, as long as the math goes onto the paper. The idea is to make this doable.
It’s better to do the work with pen and paper than in a workbook. If you make a mistake, run a line through it and do it again. If you make a big mistake, crumple up the paper and throw it across the room — good for venting spleen and saves that wasted erasing time. And this way if you misunderstand something and have to redo the whole section, fine, redo it.
Read the explanation, which in these books are in short bites, do the problems for that section, and then go to the answers and correct yourself immediately. DON’T erase or rewrite wrong answers — your goal is to be an adult and learn from your mistakes, not to prepare a fake perfect paper for Miss Jones. Just X out the errors — then go back and re-work the problem. If it still won’t go, save (make a list) it to discuss with a tutor. Aim at getting 90% correct on review sections, maybe 80% on newer work. Try to find a good tutor to see once a week for those tricky bits; if you have your list prepared you can cover a heck of a lot of ground in an hour.

If you work like this honestly an hour per evening, you will be way, way ahead of the game when you start your class — and you will have developed an effective study method that you can carry through on your class assignments. You will also have built up your confidence by reviewing what you *can* do and getting it more automatic as a foundation.

Now keep saying to yourself “I’m here to do MATH, I’m not here to learn erasing.”

BRAVO !!!! Great advice and well said. You’ve made my morning and maybe the entire day. :)

John

Well, for two semesters in a row, I have been learning the linear equations with different variables with the whole numbers and numbers to a power. And, linear equations where you have fractions you have to clear and make into whole numbers.And when you have linear equations with decimals. These two topics really and truly took me two semesters of study to learn.

Then we learned scientific notation and the powers of ten and problems using that. Now we are learning of powers and exponents and roots; where you add them, subtract them, multiply them, divide them…and write them in exponential notation and whatnot. I have been working this topic very hard because I find it to be very abstract.

I think that the last thing we cover is foil and factoring. We also have done some very elementary geometry. My final is 100% comprehensive and as someone who has taken the class for the second time, I am studying for the final now.

I have the text and a nice fellow who is to tutor me for intermediate level algebra and I have almost an Elementary Algebra book written out in longhand at this point! I am still somwhat “old school” because I fold a piece of paper in half and write the equation on one side written out correctly, and then I read over the equation until I can do it by heart without looking at anything. I respect the viewpoint of doing math in pen, I think that is good discipline. But, sometimes I copy equations wrong and I am just used to pencil. I do wish, after reading of these posts you folks were kind enough to do, to one day be able to use pen in math…but I still have to learn to read equations better first.

I know that for this Intermediate Algebra you learn the Quadratic formula and I have that in my calculator already. I assume that the Elementary Algebra is the same as pre algebra as to how pre algebra is taught now a days in the seventh grade. the Intermediate level Algebra is the one that I assume prepares you for lower division University level course work.

So what if you make a mistake? You think I don’t make mistakes?? I make a ton of them, same as everyone else. But I *move on* from my mistakes, instead of spending an hour erasing. Repeat, use pen, and teach yourself to X out and throw away. Stop focusing on doing things wrong and start looking at what you do right.

As far as all those types of equations, here is something very very important:
BIG PICTURE
An equation is a *balance* (Insert picture here of a balance like the image of statue of Justice, a stick and two pans hanging off the ends, balanced in the middle, which is the equals sign.)
If you add something to one side, you have to add an *equal* amount to the other side to keep it in balance. If you take something off one side, take the *equal* amount off the other side to keep it in balance. If you triple one side, triple the other side to keep it in balance. If you take half of one side, take half of the other to keep it in balance. And so on.

RULE
You can do any MATHEMATICAL OPERATION that you want to an equation, as long as
(A) You do **exactly the same thing** to *both* sides.
(B) You do the operation to the *entire* side of the equation (you don’t double part, you double *everything* in the equation balance pan.)

Simple rule, and very very powerful. It works in pre-algebra, algebra 1, algebra 2, pre-calculus, calculus, and every other field you can name. Tht’s the *good* thing about real mathematics — it *is* simple, big-picture, and powerful.

MATHEMATICAL OPERATION: Add, subtract, multiply, divide, and other operations developed from those such as square, square root, exponent, etc. Really simple. That’s the point, simple is easy! Don’t make it harder on yourself than you have to.

*NOT* a mathematical operation: picking up numbers and shoving them around — that’s what you do on a felt board teaching your kindergarten class.

Now, you have probably been looking for easy ways out — just the answer to this problem, just the formula for this shape, just what you need for this course. Trouble is, what looks like a shortcut now usually leads to a dead-end (been there in a new town?) Most shortcuts and quick tricks are actually very very long and slow. Ummm — how many years did you say you’d been fighting this algebra thing already? If the tricks were so quick, wouldn’t they have gotten you out of it before now? Honest, they don’t work. Period. People have been trying them for centuries, and nope, still don’t work.
Trying something new is always difficult; we prefer sticking to our old ruts. But when the old rut is a rut of repeating the course over and over and failing it, take a deep breath and dive into the new.

BTW, I like the idea of this pen thing. I have heard of the same thing for written language. My student with dysgraphia writes much more neatly with a felt tip or on the white board. I think the thing is he can’t hold it in a death grip or lean too hard on it. It’s sort of OT so I won’t go on, but I think there is merit in just Xing out and going on.

—des

I am sorry, but I should have gone into more detail, I guess. I have a mathematics learning disability (dyscalculia)and then I have a learning disability not otherwise specified (map reading, certain things pertaining to hand eye co-ordination, and the sciences are harder for me than most). I have a sixth grade level of the abstract thought needed for math. Since August I have been doing Elementary Algebra, non stop, even worked it duiring the school breaks…took me until four weeks ago to learn linear equations with variables, and decimals and fractions, and whole numbers. Now, I am not looking for an easy way of of diddly squat. I do not attend a University that gives math waivers at all. I do not even attend a University that has a Student Disabilities office worth all too very much (I had to beg for the simple accomodation rule about being given a tad bit more time for tests, and was called every name in the book for exercising my rights on that end, and my University gets mad wicked federal funding). Now due to family obligations I am stuck where I am at for right now. But…

I was just being curious about what to go over with a tutor prior to entering Intermediate Level Algebra. The. End. I figure there is no harm in going over a math course prior to taking it, and was just looking for some advice on that end. Nothing more. I do use my T.I. 83 calculator and do not care. Now, as someone who is working towards an English degree, I want to learn enough to take enough math to earn my degree and to take the GRE’s. The two University level math courses that I am dying to take in the fall and spring contain all there is to know for the mathematics portion of the GRE. In this regard, I am indeed trying to take the east way out becuase when I qualify to take Universit ylevel math, I only need these two math courses for my degree and then I get to spend the whole summer of 2005 studying for the GRE and doing independant research in John Donne.

What I most wish for, and I guess it is pushing it to ask is… I would love to see a list of topics of what most teachers wish for their students to enter Intermediate Algebra already knowing and what most teachers wish for their students to have learned after completing Intermediate level Algebra. I figure if I know of this, I could be a tad bit better prepared than most when I do Intermediate Algebra during the summer.

The pen versus pencil debate is cool. I really just have it ingrained in my skull to do math in pencil and I have the death grip thing going on, so I use 9m.m lead and can break that! I actually have broken ink pens in my day too. These things I do not mean to do. Maybe how well you read your equations has something to do with ink versus pencil? I copy out equations wrong all the time without meaning to and can get very confused if I just mark through something becuase I cannot remember why or if I meant to later on in the day when I study, that is why I use pencil. However, if you are a younger person trying to learn your math, then I would imagine ink is very good discipline…and I think discipline is 99% of math becuase if you do not have the discipline to sit down and learn your math and do your best in the classroom setting and with your tutor…if you do not have the discipline to try your best to do that, you do not have all too much.

One of the “language bridge” things I do with exponents is to walk through a review that multiplication is a lazy-mathematician-shortcut of adding the same thing more than one time (hence the word “times”). Then reflect on just how much faster something gets bigger when you multiply it than when you add. (Generally I *don’t* toss the fraction concept in here… but I do explain that when it comes ‘round on the guitar.)
IF it’s a visual learner I walk through that “multiplication” with rows and columns.
Mathematicians being power hungry as well as lazy, they couldn’t stop there. They had to try the same trick wtih multiplication — *multiplying* someting by itself again and again. And that makes things bigger even faster — it’s like going into another dimension. And for my visual folks, I will show lines, squares, and cubes. I stress taht teh exponent number probably isn’t going to be in the problem at all — that it’s the directions for what to do, more like the “x” that says multiply. So 4^3 isn’t going to have any 3’s in the problem, just its abstract concept as manifest by the three fours in 4 x 4 x 4.
Then we get to negative exponents — I emphasize that that “negative” has absolutely nothing to do with the *number* being less than zero — that it’s more like the line that you make when you make a fraction. (For my verbal-logical folks, I will mention that if a positive exponent multiplies, then a negative exponent divides, since the idea of negative means “opposite” — but not everybody is helped by that).
Radicals are a major bunbite for us non-visual thinkers — instead of being in a sequence they’re “around” the number, so we tend to just ignore it or leave it on when we shouldn’t. I try to express it as a special kind of division (again w/ a visual of a square and explanation of area vs. one side). That one really does sometimes just take a whole lot of practice thinking and writing that the square root of 64 is 8… and the square root of “whatever” squared (be it a number or a dance troupe or a complicated mathematical expression) is “whatever,” by definition, and “i squared” is always just negative one.

Hey, we’re compulsive algebra addicts… p’raps we’re the ones shoujld eb apologizing! I know you’ve got LD’s but sometimes approaching the stuff from the verbal angle can help it make a little bit of sense so there’s one less thing to memorize. And if you have to do the negative exponent thing… (besides, some other algebra addict might just google to these posts and have an algebra-nirvana moment!)

I know our college has a little booklet at the bookstore that has a list of the topics in each of the courses, with sample problems of each — but then, there are faculty & staff & admins who rather actively try to figure out if such things would be useful, and that doesn’t sound like your situation.
Each syllabus is a little different — so if you’ve got the text book and the class syllabus, that’s what I’d be focusing on.
Ya better keep us posted, too :-)

merlin —

You say you have a Grade 6 abstract thinking level. Right there is an important key to what is bugging you *and* to what you can do to help.

Math is based on abstract thinking, and you know that. But math is also used to *teach* and *develop* abstract thinking, especially after elementary arithmetic.

There is a very annoying attitude that just because you’re good at something academic, especially math, it comes easily to you without doing any work. You know, when you see the champion figure skater or the winning swimmer or whatever, you don’t walk up and say “Wow! You’re lucky that that comes to you so easily and effortlessly!” No, you know that these people spend four hours a day training and devote their lives to it. Same goes for academic talent and skills. Those of us who are good at math have sat up many, many nights wrestling with a new concept, filling up garbage cans with crumpled paper and staring at the same *&&&^^ problem with burning eyes and aching head. We do know about it. OK, so the ones that give me headaches now may be in abstract group theory, but the general system is the same; I did go through the headaches in algebra 2 and geometry back in the dark ages when I was young.

This is what I was trying to get at earlier when I was advising avoiding so-called “easy” and “quick” tricks, which in the long run are neither easy nor quick.
I was talking about the *big picture* in equations, and Sue went on about the *big picture* with exponents.

You keep asking for a nice list of things to memorize. Trouble is, what math professors want is *not* to have things memorized and repeated; what they want is to see you *working* with the big-picture *concepts*.

I have had this discussion with far more than one math student and with a number of Chinese students learning English; you simply cannot do the whole thing by memorizing lists (imagine trying to give vocabulary lists for the whole English language, which has the largest lexicon in the world!) and as long as that is the question you are asking, you are going to be frustrated and upset with the answer.

The Chinese students needed to learn to use context and use root forms and ask questions to learn about new words and drop their comfort zone of having everything told to them. The math students need to learn to look for big-picture concepts and to drop their comfort zone of having a recipe for everything.

Dropping your comfort zone and going into uncharted waters of thought is not easy. It is necessary.

There’s an easy way and a hard way to do this.

The hard way is to jump off the end of the dock into deep water and sink or swim, usually sink. This is what has been happening with you walking straight into algebra classes with weak abstraction skills. You can get thrown off the dock lots of times without learning to swim, and you can take lots of classes and still be repeating algebra. This method is very rarely successful.

The easy way — and I know this because I have done it myself learning new material — is to take very small steps and do the shallow water before the deep.
You want to learn the concepts a little at a time, you want plenty of practice relating those concepts to things you already know, and you want a very good grounding in the “why” before you get tangled up in the details of the “how”. You want to *learn* improved abstraction at the same time as you are learning to fill in blanks. The answers to the equations will change every day, but the *logic* and *methodology* will always be the same.

This is why I suggested taking an hour a day, actually *reading* the *explanations* in the books, and working right from Page 1 upwards. It is amazing, when you go back over a subject like this, just how much you learn. I still learn things from the primary-school books I use to teach little kids …
In particular, don’t skip over things either because you think you already know them, or because they look too hard. Re-doing something you think you know is a wonderful way to get a sudden flash of insight into how and why this whole thing fits together. And the hard parts are the new material, the stuff you are there to learn; doing the hard stuff once now will make it look much easier when it comes up in class later.

Once you drop the comfort zone of memorizing and reciting everything, look for patterns and consistencies in your work.

Try for 70% to 80% correct on topics that are absolutely new to you and 80% to 90% correct on review. As long as you are at that level, you can go on to the next unit or chapter. Later, you can come back and review the things you are learning now and aim at the higher level.

It’s really, really hard to do that if you’re afraid (or just pretty darned certain, based on lots of experience that’s taught you) that you really can’t understand it. A good tutor (who will be honest about what is worth working through, and which things you really can just memorize) can help. At a certain point you do what you can to get the necessary credits.
However, if it’s *all* unrelated memorized stuff … well, one of the disadvantages to not understanding the math is that then you don’t understand just how much less you’d have to memorize if you worked with just a couple of the concepts. THe possible combinations of equations increases exponentially with each new concept. If you understand the principles of one concept, though, sometimes teh “new” concept really isn’t new, it’s just “here’s how that concept works in a different situation.” (I had to learn social skills that way when memorizing stopped working :-))

I thank you all for responding to my post. I am sorry but I have been doing a mad wicked amount of studying for finals. I appreciate every last one of your posts and I thank you all.

We’re with ya :-) Good or bad, let us know how it went.