Home > Student Resources > Pre-Algebra – Quick Reference Sheet

Pre-Algebra – Quick Reference Sheet

Properties of Algebra

The Cartesian Coordinate System

	Addition	Multiplication
Identity	The additive identity is 0 $a+0=a$	The multiplicative identity is 1 $a \cdot 1 =a$
Inverse	The additive inverse of a is $-a$ $a+(-a)=0$	If $a \neq 0$ , the multiplicative inverse of a is $\frac{1}{a}$ $a(\frac{1}{a})=1$
Associative	$(a+b)+c=a+(b+c)$	$(ab)c=a(bc)$
Commutative	$a+b=b+a$	$a \cdot b = b \cdot a$

Distributive

$a(b+c)=ab+ac$ and $a(b-c)=ab-ac$

Property of Inequalities
When multiplying or dividing each side of an inequality by a positive number an equivalent inequality is produced.
$2x<6 \rightarrow \frac{2x}{2} < \frac{6}{-2} \rightarrow > -3$
When multiplying or dividing each side of an inequality by a negative number the equivalent inequality is produced by reversing the direction of the inequality.
$2x<6 \rightarrow \frac{2x}{-2}<\frac{6}{-2} \rightarrow x >-3$
Order of Operations (PENDAS)

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

The Cartesian plane contains two axis, the x-axis and y-axis. The coordinates of point are written (x,y). You can tell what quadrant a point lies in by the signs of the coordinate.
Quadrant 1: (+,+)
Quadrant 2:(-,+)
Quadrant 3:(-,-)
Quadrant 4:(+,-)

Probability

Definitions
Probability: the likelihood that an event will occur
Equation
$P(success)=\frac{# Possible\quad successes}{# Possible \quad outcomes}$

Multiplication Rule: When there are multiple independent events happening together the individual probabilities are multiplied together. Such events are drawing two cards from a deck or flipping a coin.
Example:A coin flipped and a 6-sided die is rolled at the same time. What is the probability of getting heads and 3?
$P(rolls\quad 3) = \frac{1}{6}$ and $P(heads)=\frac{1}{2}$
$P(heads\quad and\quad rolls 3)=\frac{1}{6} \times \frac{1}{2}=\frac{1}{12}$

Addition Rule: When there are two mutually exclusive events (outcomes cannot happen at the same time), add the two individual probabilities together.
Example: A bag of marbles contains 1 green, 3 red, and 2 blue marbles. One marble is drawn. What is the probability that it is either green or blue?
$P(green)=\frac{1}{6}$ and $P(blue)=\frac{2}{6}$
$P(green\quad and\quad blue) = \frac{1}{6}+\frac{2}{6}=\frac{3}{6}=\frac{1}{2}$

Ratios and Percentages

Definitions
Ratio: when two qualities are divided as a means of comparing them
Percentage: a fraction (or ration) out of 100
Converting From Fraction to a Percentage
To convert a fraction to a percentage first write the fraction as a decimal then multiply the decimal by 100 and add the percent symbol (%).
Example:

$\frac{3}{5}=0.6 \times 100 = 60%$

This space was intentionally left blank for spacing purposes.

Next Table

Linear Equations	Prime Factorization
Definitions Standard form: $\textbf{ax+by=c}$ Where a, b, and c are constants, and a and b are not both zero Slope intercept form: $\textbf{y=mx+b}$ Where m is the slope of the line, and b is the y-intercept Point slope form: $\mathbf{y-y_{1}=m(x-x_{1})}$ Given a point $(x_1,y_1)$ and a slope m	Definitions Prime numbers: are positive integers that are not divisible by any integer besides one and themselves Compositve numbers: are integers that are the product of prime factorization. Example Prime Factorization of 1092
Functions	Mean,Median, and Mode
Definitions Function: the mathematical rule that takes an input value and converts it to a single output Input: also called the independent variable is denoted by x Output: also called the dependent variable is denoted by y The Vertical Line Test You can tell if an equation is a function by performing the vertical line test. Look at the graph of an equation. Imagine moving a vertical line back and forth across the graph. (You can use your pencil or edge of a ruler for this.). If there is any point at which the vertical line passes through the graph of the equation more than once, then it is not a function.	Definitions Mean: the average value Median: the value that falls in the middle when the values are arranged from lowest to highest value Mode: the value that appears most frequently Example Find the mean, median, and mode of the following data:{6, 3, 1, 6, 4} First arrange the numbers from least to greatest. {1, 3, 4, 6, 6} The middle value is 4, so our median is 4. The mode is 6, because it appears twice, which is more than any other value. Mean = $\frac{1+3+4+6+6}{5}=\frac{20}{5}=4$
Domain and Range	Pythagorean Theorem
Definitions Domain: is all of the possible inputs (x-values) for a function. For each x-value in the domain, there must be a valid output. Sometimes there are x-valeus that cannot be used, because plugging them into the function results is an undefined point. If all real numbers can be used as an input, the domain is “all real numbers” or $(-\infty,\infty)$ Range: is all of the possible y-values for a function. For each value y in the range, there must be a valid input x in the domain such that f(X)=y	A triangle is a triangle with one $90^{\circ}$ angle. The longest side is called the hypotenuse, and the two shorter sides are called legs of the triangle. For a right triangle with leg lengths a and b, and hypotenuse c. $a^2+b^2=c^2$
	The Distance Formula
This space was intentionally left blank for spacing purposes.	distance $= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Previous Table