Section Notes

Check here before you start each section in the software to see if I have any comments, notes, warnings, secret messages, etc. about the section to mention. These will be linked from the main Homework Assignment list for each section that I have notes for (so you can easily get here from there). Notes that I don't think of right away will be posted in the Announcements section on Blackboard as well.

Not all of these have notes yet; check back before you work each lesson to see if I have any notes for that lesson.

Lesson 1.1 The Real Numbers This lesson is a review of several of the lessons from Basic Math, relative to the real numbers. If you are having a lot of trouble with this, remember that the Learning Lab will provide additional help with the math concepts and with your pencil and paper homework. Also, Academic Systems has helpful information, including Interactive Math Study Tips, posted on their Interactive Mathematics Resources web page.

Lesson 1.2 Factoring and Fractions May people have trouble with fractions. Be sure you spend enough time here so you really feel comfortable working with fractions.

Lesson 1.3 Arithmetic of Numbers Please pay VERY close attention to the "order of operations" here. It is vital that you understand this very well before moving on. Also, watch out for problems with negative signs. These cause a lot of people problems.

Lesson 2.1 Algebraic Expressions This is the first lesson in which you will use the Expression Editor to enter answers to questions. If you did not take Basic Math with the Academic Systems software, it is important to be sure that they know how to use the Expression Editor. This looks like a calculator to the right of an answer box and is used to enter algebraic expressions involving fractions, exponents and other special symbols. The Expression Editor functions much like the Equation Editor in Microsoft Word. A tutorial on using the Expression Editor is posted on Academic Systems' Interactive Mathematics Resources web page .

• Any kind of "inverse" property means that you do the opposite. So, the "inverse property of addition" means you do the opposite of addition, otherwise known as subtraction. You use this to "move" things from one side of an equation to another, for example: 2x + 3 = 5 2x + 3 - 3 = 5 - 3 2x = 2 We used the "additive inverse" of 3 to eliminate it on the left. Or: 2x = 2 (1/2)(2x) = (1/2)(2) x = 1 We used the "multiplicative inverse" of 2 (i.e., we divided by 2, since dividing by 2 and multiplying by 1/2 are the same) to eliminate the 2 on the left.
  
  
• Any kind of "commutative property" means it doesn't matter which order you list things. We use this property to rearrange problems to get like terms together, usually: 2x + 5 + 7x + 6 = 2x + 7x + 5 + 6 Or, for multiplication: 3*x*2*y = 3*2*x*y
  
  
• Any kind of "associative property" means it doesn't matter which one we do first. In other words, we can rearrange the parenthesis using this rule (again, mostly to add together like terms): 5 + (2 + x) = (5 + 2) + x = 7 + x Or, for multiplication: 2(3x) = (2*3)x = 6x
  
  
• The "distributive property" is different from all the others because it requires both addition and multiplication (all the other rules involve only addition or multiplication. We use it to multiply through a parenthesis with addition inside: 2 (x + 3) = 2*x + 2*3 = 2x + 6 Notice that you have to "distribute" the 2 to both of the terms inside. If there were 3 or more terms, you would have to distribute to each one of them: 4 (2x + 3y + 6) = 4*2x + 4*3y + 4*6 = 8x + 12y + 24 Now, I have two important warnings about the distributive property:
  
  
  1. It only works if you have both addition (or subtraction) and multiplication. So, for example: 2 (x - 4) = 2*x - 2*4 = 2x - 8 uses the distributive property. However: 2 (4x) = (2*4)x = 8x does not use the distributive property. (Which property does it use?) Notice that I did not multiply the 2 times the 4 and times the x; you can't "distribute this through" because it only involves multiplication.
     
     
  2. The distributive property does not work across an exponent. So, for example: 2 (7 - 4)^2 does not equal (2*7 - 2*4)^2 (this is wrong) You would have to follow your order of operations and simplify inside the parenthesis first: 2 (7 - 4)^2 = 2 (3)^2 = 2*9 = 18

Lesson 2.2 Solving Linear Equations Again, watch your signs (especially remembering to distribute negative signs through parenthesis).

Lesson 2.3 Problem Solving Many students find this lesson particularly difficult because it involves "word problems". The applications include translations, number problems, age problems and geometry problems. It is VERY important that you don't skimp on the set-up for these problems. In particular, be sure that you remember to write down what all of your variables stand for and what it is you are trying to find. Don't be in too big a hurry to write down an equation to solve (though you will need to do that eventually, of course). Be sure to use algebra to solve these problems; that means you must construct an equation to solve (not just write down a bunch of numbers with a little arithmetic thrown in). Also, be sure to pay careful attention to the percent problems on the extra handout (I promise you they will be on the test).

Here is an example of a common mistake (and the correct way) that I often see with percentages:

Thelma ordered some books over the internet. Her total bill was for $54.49, including shipping and handling. If she knows that the site charges a 6% fee to cover shipping and handling, how much did her books cost BEFORE the shipping and handling fee?

Since you want to know the cost of her books before shipping/handling (S/H), let:

x = cost of books without S/H

Now, you know that:

Total cost of order = (cost of books) + (shipping and handling)

Well, shipping and handling is 6% of the cost of her books:

shipping and handling = 6% of cost of books = 0.06 * x

(A reminder: * means "times", so 6% times the cost of books. Don't forget that you have to change 6% to a decimal 0.06 to use it in an equation.) Thus:

Total cost of order = (cost of books) + (shipping and handling)
= x + 0.06 x

But, we know the total cost is $54.49, so:

54.49 = x + 0.06 x

To solve this:

54.49 = x + 0.06 x = 1 x + 0.06 x
54.49 = 1.06 x
x = 54.49 / 1.06 = 51.41
(rounding off to 2 decimals)

Thus, her books alone cost $51.41.

A warning: Some people try to work this problem by the following method:

Since she paid 6% for shipping and handling, that means shipping and handling cost 6% of 54.49, so:

shipping and handling = 0.06 * 54.49 = 3.27 (rounded off)

Now, just subtract that from her total:

54.49 - 3.27 = 51.22

Now, at first, this sounds good. Unfortunately, it is wrong. Why?

The reason is that she is supposed to pay 6% of the ORIGINAL price for shipping and handling (if her books cost $10, she should pay 6% of 10, or 0.06 * 10 = $0.60, for shipping and handling). However, in the work above, we multiplied the 6% times the FINAL cost (since we didn't know the original cost), which is not the same number.

You can check to see that this doesn't work: If her books really cost $51.22, then she should pay:

6% of 51.22 = 0.06 * 51.22 = $3.07 for shipping and handling

So, her total cost would be: 51.22 + 3.07 = 54.29, which is wrong. Think through this carefully until you really understand it. This is an important point.

Lesson 3.1 Introduction to Graphing

Lesson 4.1 Graphing Equations This is an important topic you don't want to rush through. It takes some students a little while to really make the connection between the visual graphs and the symbolic equations. Work some extra problems (and pay a visit to the Learning Labs) if you are feeling a little unsure with this.

Lesson 4.2 The Equation of a Line

Lesson 5.1 Solving Linear Systems Notice, there are different methods to solve these problems. The graphical method is great when you want to really see what is going on (for example, it helps you see how many solutions you have). However, if you want an accurate solution, the graphical method is really the best method; an algebraic method is much better at this. (Especially since often in "real life" the answers aren't always nice "whole numbers". If your answer is something like x = 3.0215, y = -2.3516, you aren't likely to be very successful finding the answer using graphs.)

Lesson 5.2 Problem Solving A variety of applications are discussed including number problems, interest problems, coin problems, and mixture problems. Again, please be sure you state what your variables stand for. (Also, notice that this section stresses solving these problems with 2 variables.)

Lesson 6.1 Exponents Please pay very careful attention to the rules for working with exponents and how those work in combination with parentheses.

Lesson 6.2 Polynomial Operations

Lesson 6.3 Polynomial Operations

Lesson 7.1 Factoring Polynomials I

Lesson 7.2 Factoring Polynomials II

Lesson 7.3 Factoring by Patterns

Lesson 10.1 Quadratic Equations I You will not have to memorize the Quadratic Formula for this class (but you will for later classes, so you might want to do so anyway).

Lesson 8.2 Rational Expressions II