Class I -- Pre-Algebra Review I. Introduction to Class II. Order of Operations III. Positive, Negative, and Absolute Value IV. Fractions A. Addition and subtraction B. Multiplication and division V. Decimals and Percents A. Operations on decimals B. Converting between percents, decimals, and fractions VI. Word Problems I. Introduction to Class No single person is responsible for the invention of algebra. The basic building blocks of algebra as we know it today were invented in the Arab world from about 700 to 1400 A.D. The term algebra comes from an Arabic book "Hisab al-jabr almuqa-balah". "Al-jabr" literally means reunion. The author of the book is al-Khowarizmi, whose name is the basis for the word algorithm, meaning a set of instructions used to solve a problem. Arithmetic is a way of dealing with numbers of known values. Algebra gives us a way of manipulating and doing arithmetic on numbers whose values we don't know. The basic idea of algebra is the unknown, a symbol that stands for a number. We use can do everything to the unknown we could do to numbers: add to it, subtract from it, multiply it, divide it, square it, etc. This ability to maniuplate unknown or arbitrary numbers makes algebra a powerful method of solving problems, and of representing relationships in the real world. We'll begin the class with a review of the basics leading up to algebra. II. Order of Operations Consider the following arithmetic expression: ((2 . 3^2 + 1) - 3 (2 . 4 - 3)) + (2^3 - 4)^2 How would we evaluate this? To do so, we need to recall the order of operations: parentheses, exponents, multiplication/division, addition/subtraction. A useful mnemonic device for this is "Please excuse my dear Aunt Sally": PEMDAS. Let's apply this to the expression on the board. Have class go through evaluating the expression. Note: when you evaluate an expression like this, you should immitate the notation you're seeing here. Other notation is not correct, and will not be accepted. III. Positive, Negative, and Absolute Value The next topic I'll remind you about is positive and negative numbers, and absolute values Consider this expression: -5^2 - (-5)^2 - ((-2) (-3) + (-4))^4 Let's evaluate this. Remember the rule: a negative times a negative is a positive, a negative times a positive is a negative. Also remember that negative signs are like multiplications: unless the parentheses say otherwise, apply the exponent before you apply the negative sign. Walk class through evaluating expression. OK, let's add one twist: the absolute value sign Absolute value is very simple: it's just like a parenthesis, but once you've evaluated everything inside it, take the positive of that number Here's a practice example: -|3 (-4) + 5|^2 + |6 (2) - 5 (3)| Let class evaluate. IV. Fractions Next review topic: fractions A. Addition and Subtraction Let's begin with adding and subtracting fractions Consider: 5/6 - 3/8 To evaluate this, we need to first find a common demoninator Walk class through finding common denominator and solving B. Multiplication and Division Multiplying and dividing fractions is easier, because there's no need to find a common denominator For multiplication, just multiply top times top and bottom times bottom For division, flip the top and bottom of the second number, the multiply Example: 1 (1/8) / (3/2) Walk class through evaluating this Putting together addition / subtraction and multiplication / division |(1/4) / (1/3) - 5/6|^3 Walk class through evaluation. V. Decimals and Percents Next, decimals A. Operations on decimals Operations on decimals are easy, because they work just like operations on integers -- you just have to keep track of the decimal point. Example: 3.2 . (5 - 0.9) Walk class through evaluating this expression. The result is 13.12. B. Converting fractions, decimals, and percents How would be write 13.12 as a percent? Recall that the way to convert a decimal to a percent is just to move the decimal point two places to the right. Walk class through this. How would we write 123.456% as a decimal? Walk class through this. OK, now how do we convert to and from fractions? Let's start with converting from a decimal to a fraction, because that's very simple. The rule is, just move the decimal point to the right until you get all the non-zero digits to the left of it. For every place you move the decimal point, add a zero in the denominator. Example: write 123.456% percent as a fraction Walk class through this. To go the other way, the rule is just to do the division. Example: write 19/16 as a decimal. Walk class through evaluation. VI. Word Problems Of course the point of all this is to let us solve problems in the real world, not just manipulate expressions on paper To this end, let's go through some of the word problems on the placement test. Go over problems 5 and 6 from placement test.