Chapter 7
In algebra, letters are used to represent unknown numbers.These letters are called variables, and we usually use the rules of mathematics to solve for, or identify what the unknown number could be.
The lowercase letter "x" is often used in algebra which can cause confusion with the traditional multiplication symbol "×". They look a lot alike, don't they? To avoid confusion, we will use other methods to imply multiplication. The most common are:
A simple dot: 4 • x
Parentheses: 4(x)
No sign: 4x
All three of these examples represent the same expression: 4 is being multiplied by some unknown number.
SKILL CHECK: get into the habit of avoiding confusion with the multiplication symbol by practicing using one of the above methods to write out multiplication problems.
When evaluating mathematical expressions, be sure to follow the order of operations:
1) G: Groupings. These include parentheses, brackets, and braces.
2) E: Exponents. This includes roots and radicals.
3) M and D: Multiplication or Division. These both have the same "weight" so when you get to this step do whichever operation comes first when working from left to right.
4) A and S: Addition or Subtraction. These both have the same "weight" so when you get to this step do whichever operation comes first when working from left to right.
Get Everyone More Drinks And Snacks
According to an experiment run by Dr. Peter Price of the Classroom Professor website, about 75% of students get the wrong answer by not using the order of operations. In his experiment Dr. Price posted the following brainteaser:
7 - 1 × 0 + 3 ÷ 3 = ?
Out of the 6,000 respondents, only 26% arrived at the right answer. Try out this problem for yourself and see what you get.
. . . . . . . . .
Did you get 8? If so, congrats, you're using the order of operations! If you got a different answer check out Order of Operations 101, an article from the Calculator Site linked below for a complete explanation about GEMDAS (also known as PEMDAS).
You can also visit Khan Academy if you'd like more review and practice with the order of operations.
SKILL CHECK: practice simpliflying expressions by using the order of operations.
PROPERTIES TO KNOW
| PROPERTY |
DESCRIPTION |
EXAMPLE |
| Commutative Property of Addition |
When two numbers are added, the sum is the same regardless of the order in which the numbers are added. |
2 + 3 = 5
3 + 2 = 5
|
| Associative Property of Addition |
When three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped. |
6 + (4 + 3) = 13
(6 + 4) + 3 = 13
|
| Commutative Property of Multiplication |
When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied. |
3 • 5 = 15
5 • 3 = 15
|
| Associative Property of Multiplication |
When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped. |
6 • (4 • 3) = 72
(6 • 4) • 3 = 72
|
| Distributive Property |
The sum of two numbers times a third number is equal to the sum of each addend times the third number. |
5(7 + 2) = 45
5 • 7 + 5 • 2 = 45
|
| Identity Property of Addition |
Adding 0 to any number does not change the original value. |
76 + 0 = 76 |
| Identity Property of Multiplication |
Multiplying any number by 1 does not change the original value. |
19 • 1 = 19 |
| Inverse Property of Addition |
The sum of any number and its additive inverse is equal to 0. |
5 + -5 = 0 |
| Inverse Property of Multiplication |
The product of any number and its reciprocal is equal to 1. |
4(1/4) = 1 |
| Zero Property of Multiplication |
Any number multiplied by 0 is equal to 0. |
123(0) = 0 |
SKILL CHECK: practice using these rules, as well as the order of operations, to solve equations.
TYPES OF NUMBERS
- Natural Numbers (N), (also called positive integers, counting numbers, or natural numbers); They are the numbers {1, 2, 3, 4, 5, …}
- Whole Numbers (W). This is the set of natural numbers, plus zero, i.e., {0, 1, 2, 3, 4, 5, …}.
- Integers (Z). This is the set of all whole numbers plus all the negatives (or opposites) of the natural numbers, i.e., {… , ⁻2, ⁻1, 0, 1, 2, …}
- Rational numbers (Q). This is all the fractions where the top and bottom numbers are integers; e.g., 1/2, 3/4, 7/2, ⁻4/3, 4/1 [Note: The denominator cannot be 0, but the numerator can be].
- Real numbers (R), (also called measuring numbers or measurement numbers). This includes all numbers that can be written as a decimal. This includes fractions written in decimal form e.g., 0.5, 0.75 2.35, ⁻0.073, 0.3333, or 2.142857. It also includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line
EQUATION-SOLVING TIPS THAT MAY SAVE YOUR LIFE:
Be familiar with the rules.
Know the order of operations, basic mathematical properties, and how to use your calculator properly.
Read carefully.
Know what you know and know what you don't--especially when solving a word problem. Clarify for yourself what it is you have and what you're being asked to find.
Write clearly.
Don't skimp on writing things down. Write down the problem simply, in your own words. Set aside space to list out the variables in your problem, what they stand for, and what values are known to be associated with them. Color code to help you keep track.
PRACTICE.
Practice, practice, practice. Practice so much that the word practice no longer feels like a real word. Repeat the same problem multiple times.
Show all your steps.
Avoid skipping steps and definitely avoid not writing things down. This goes along with writing clearly: write each step on a new line. Don't sacrifice clarity for the sake of saving paper.
Double check.
As you solve ensure you're following the correct steps and using the correct values. Your instinct might be to rush in order to get the pain over with as fast as possible but please resist this temptation.
Video links
Order of Operations 101
