Things you need to know before you can be successful in pre-algebra

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Most of the time, students come to a class having "forgotten all" over the summer. When this happens, it's worth a teacher's time to go back and review the basics and the things that most would consider to be "obvious." Therefore, the following is a list of the things that students should know in order to be successful and things to study and review before you get going.

Contents

The Basics

All maths deal with numbers. Numbers represents amounts. To make our lives easier, we chose to represent these amounts with symbols rather than dots. Can you imagine how many dots it would take to do trading in the stock market? Holy cow!!!

We use symbols that are grouped into tens because we have ten fingers. If we had six fingers, we would count in groups of 6. Our number system starts with zero (a number invented by the Mayans) and goes up to 9. Once we hit 9, we start over, and put a number in the tens place. We continue this cycle, putting numbers in the hunderds, thousands, and so on, until we get tired of writing them. Really, they keep going, we are just much too lazy to do it... That's why we invented calculators and computers!

Numbers

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. That's it. There are no other numbers besides those.

What about 10? Ten means we have started over with 0 in the ones and put a 1 in the tens spot to remember how many we've gotten to. So, the number ten has a 1 and a 0, meaning we put a 1 in the tens spot to help us remember that we've already got ten of them, and then we start over with a zero in the one's spot! So we're not using any numbers that we don't already have, just the same numbers over and over, using place values to keep track of it all!

The number 237 means that we've already counted to ten 23 times and then counted to 7.

Decimals

Decimals work the same way, but going the other way. Again, we use the same numbers and place values, but they get smaller as we go, not larger, as explained before.

Therefore, the number 0.01 is smaller than 0.1 and 0.001 is smaller than 0.01.

Fractions

Fractions are also known as portions of a whole. if you took a cake and cut it into 12 equal sized pieces to serve it, then each piece would represent \frac {1}{12} of the whole cake. Now, if you decided to eat the whole thing by yourself over the course of the next twelve days, you would eat an increasing amount of the cake, starting with the original \frac {1}{12} on the first day, followed by \frac {2}{12} on the second day, \frac {3}{12} on the third day, and so on until you had eaten \frac {12}{12}, or one whole cake.

Some important rules to remember are:

When adding or subtracting two fractions, you must have common denominators (those are the numbers on the bottom)...
\frac {2}{3} + \frac {3}{4} must become \frac {8}{12} + \frac {9}{12}
When adding or subtracting two fractions, only add/subtract the numerators (the numbers on the top), and not the denominators.
\frac {8}{12} + \frac {9}{12} = \frac {17}{12} or 1 \frac {5}{12}
When multiplying two fractions, remember to multiply the top times top, bottom times bottom.
\frac {2}{3} \cdot \frac {3}{4} = \frac {6}{12} or \frac {1}{2}

This last example may confuse you, but realize that \frac {3}{4} is the same as writing \frac {1}{4} + \frac {1}{4} + \frac {1}{4}. So if we take two of the 3 (or \frac {2}{3} of them), then we get \frac {2}{4} or \frac {1}{2}, just like we expected by multiplying them!

As it turns out, many students find it easier to visualize the amounts that decimals represent over the amounts that fractions represent. This is fine until you get into the higher maths. The higher maths (Geometry, Trigonometry, and Calculus) are much easier to understand if the student has a real good understanding of fractions.

Operations

Here's where we actually start doing something with our amounts. As stated before, numbers are nothing more than symbols that represent amounts. But if you want to change your amount, either by spending or earning, then we need other symbols to help us see what's going on!

Back in the olden days, and I mean real old, if you started out with 100 sheep and 50 cows, how would you ever determine how many animals you had if you couldn't add them?!

Image:Sheep02.gif

Addition

Some words that mean addition are: sum, total, add, increase, combine, and more. We use the symbol + to tell us that we are increasing one amount by another. For instance, if you have 100 sheep and 50 cows, you have \small 100 + 50 = 150 animals! Addition never gets any harder than this, but it does get uglier.

For instance...

Subtraction

Some words that mean subtract are: minus, decrease, take away, separate, remove, or less. We use the symbol - to tell us that we are decresing one amount by another. When we are introduced to negative numbers, we learn that if we subtract 50 from 100 (as in \small 100 - 50 = 50), then it is the same thing as saying \small 100 + (-50). Therefore, subtraction is the same thing as adding negatives!

One of the best ways to help you see what's happening when we add or subtract with negatives is to use a Number Line. You can see a piece of the number line below.

It is very important to remember what you are doing when you are adding and subtracting with integers. Some rules you can remember are:

Whe adding two positives, like \small (+) + (+), you will always get a bigger positive (farther right on the number line).
\small 13 + 27 = 40
When adding two negatives, like \small (-) + (-), you will always get a bigger negative (farther left on the numberline).
\small (-13) + (-27) = -40
When adding two numbers with opposite signs, like \small (+) + (-) or \small (-) + (+), then subtract the two numbers and take the sign of the larger number.
\small (-13) + 27 = 14, and
\small (13) + (-27) = -14

Note that the 27 and the -27 determine the sign of the answer.

Multiplication

Multiplication is nothing more than repeated addition. For instance, if you add 2 + 2 + 2 + 2 + 2, you will get the same result taking 2 x 5 (the answer is 10, by the way). It is very important for students to remember their multiplication tables, since they get used so very often in every math class they take from 6th grade up through college. The invention of the calculator simplified things quite a bit, but the calculator is no substitute for the complexity and power of the human brain!!! (That last sentence sounds so much more profound if you read it with an deep booming voice...)

Below is a copy of the multiplication table. It can be used as a reference, if you need it...

× 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225

Notice that the numbers along the diagonal (the ones in bold) are the square numbers. This means that they represent a number times itself, like \small 5 \cdot 5 = 5^2 = 25. We call them square numbers because of the superscript 2 above the number, or in other words, the "number squared."

Also notice that the top part of the multiplication table (above the square number diagonal) is a mirror image of the bottom half. For the astute student, one would realize that this simplifies things quite a bit, in that you now only have to worry about half the multiplication table, especially if you remember that multiplication is commutative (see below in Properties of Arithmetic).

Multiplication with negatives can be very simple if you know your times tables. There is only one rule you need to remember when multiplying numbers with negative signs.

The Even & Odd Rule - Every two negatives make a positive. The word negative means opposite, and the opposite of a negative is a positive. For examples,
\small (-3) \cdot (5) = -15
\small (3) \cdot (-5) = -15
\small (-3) \cdot (-5) = 15
\small (-2) \cdot (-3) \cdot (-5) = -30
\small (2) \cdot (-3) \cdot (-5) = 30
\small (2) \cdot (3) \cdot (-5) = -30

The multiplication is the same as it were without the negatives, except that when there is an even number of negatives, the answer is positive, and when there is an odd number of negatives, the answer is negative! I hope you see the pattern.

Division

Division follows the same rules as multiplication does, but it is like repeated subtraction. If you have ten pieces of candy that you want to share with yourself and four friends (yeah, that's 5 people total), then you give the first one 2, the second one 2, the third one 2, and so on. Each person gets two candies! This is the same as \small 10 \div 5 = 2 or \small 10 - 2 five times...

Division can also be done in the form of a fraction. A way to say the same thing would be:

10 \div 5 = \frac {10}{5} = 2

So, the fraction \frac {2}{3} says to take two things and split them into 3 equal parts. Obviously each part will be less than a whole, so this fraction is the same as a portion, as mentioned above in Fractions.

Properties of Arithmetic

There are some things that are equivalent (equal, or the same). The following rules, or properties, will help you remember stuff in the long run. It's important that you understand them and are able to use them.

Commutative Property

The commutative property says that adding and multiplying two numbers can be done in any order.

a + b = b + a for addition, and
a \cdot b = b \cdot a for multiplication.

For example,

6 + 4 = 4 + 6 = 10, or
5 \cdot 2 = 2 \cdot 5 = 10

Associative Property

The associative property says that when you add or multiply more than two numbers, it can be done in any order.

a + (b + c) = (a + b) + c for addition, and
a \cdot (b \cdot c) = (a \cdot b) \cdot c for multiplication.

For example,

2 + (3 + 4) = 2 + 7 = 9 = 5 + 4 = (2 + 3) + 4, or
2 \cdot (3 \cdot 4) = 2 \cdot 12 = 24 = 6 \cdot 4 = (2 \cdot 3) \cdot 4

Distributive Property

The distributive property combines multiplication and addition and allows you to do the arithmetic in any order.

a(b + c) = ab + ac

For example,

2(3 + 4) = 2\cdot3 + 2\cdot4 = 6 + 8 = 14 = 2(7) = 2(3 + 4)

Primes, Composites, & Factors

Prime numbers are those that are only divisible by 1 and themselves. The first twenty-five prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Composite numbers are the numbers that are not prime (except for the number 1... it's neither prime nor composite). For instance, the number 6 is a composite number because the numbers 2 and 3 multiply to make 6. Any number that has factors other than 1 and itself is a composite number.

If a number is divisible by another number, for example, 6 is divisible by 3, then the smaller number is known as a factor, or divisor. These factors are very important, especially prime factors. Prime factors help us do a number of things. For instance, prime factors help us to simplify fractions and radicals (you'll learn about those in class).

You can find all the prime factors of a number by making a factor tree. Here is an example of a factor tree, showing the prime factors of 66.

Or for 96...

An example of using prime factors to simplify a fraction would be this:

\frac {66}{96} = \frac {2 \cdot 3 \cdot 11}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = \frac {2 \cdot 3}{2 \cdot 3} \cdot \frac {11}{2 \cdot 2 \cdot 2 \cdot 2} = \frac {11}{16}.

We'll be using prime factors a lot in class. So, be sure you understand how to find the prime factors of any number.

Make sure you remember all these simple things, and you'll do just fine in Pre-Algebra!

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