Key Concepts

• Phrases such as sum, increased by, difference, decreased by, of, and, etc. can be translated into mathematical operations and notation to help solve a problem.
• To add or subtract fractions, make sure they each have the same denominator first.
• When dividing fractions, use the phrase keep-change-flip to remind you to multiply the first fraction by the reciprocal of the second one.
• Just like the digits of a whole number increase to the left by powers of 10, the digits after the decimal on the right decrease by fractions of powers of ten.
• When using inequality symbols, the smaller side of the symbol faces the smaller number and the larger side faces the larger number.
• The product or quotient of two negative numbers or two positive numbers is always positive; the product or quotient of two differently signed numbers is always negative.

Key Expressions, Equations, and Inequalities

• $a=b$ is read as “$a$ is equal to $b$.”
• $a\neq b$ is read as “$a$ is not equal to $b$.”
• $a\gt b$ can be read “$a$ is greater than $b$” or “$b$ is less than $a$.”
• A fraction is written $\dfrac{a}{b},$ where $a$ and $b$ are integers and $b \neq 0$. In a fraction, $a$ is called the numerator and $b$ is called the denominator.
• The distributive property, $a(b+c)=ab+ac$, represents the distribution of multiplication over addition or subtraction.
• The property of one states that any number, except zero, divided by itself is one. That is $\dfrac{a}{a}=1$, where $a \neq 0$.
• if $a, \text{ } b, \text{ and } c$ are numbers such that $b \neq 0, \text { } c \neq 0,$ then $\dfrac{a}{b} = \dfrac{a\cdot c}{b\cdot c}$.
• To multiply fractions, $\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}$

Glossary

absolute value

a number’s distance from zero on the number line, which is always positive

constant

a number whose value always stays the same

distributive property

a number multiplying an expression inside of parentheses distributes to each term in the contained expression

equation

two expressions connected by an equal sign

equivalent fractions

fractions that have the same value

exponent

a number in a superscript position that tells how many times to multiply the base by itself

expression

a number, a variable, or a combination of numbers and variables and operation symbols

improper fraction

a fraction $\dfrac{a}{b}, \text{ } b \neq 0$ is proper if $a \lt b$, and is improper if $a \geq b.$

inequality

two expressions connected by an inequality sign

integers

counting numbers like 1, 2, 3, … including their opposites (negatives) and zero

like terms

terms where the variables match exactly (exponents included)

operators

symbols that represent arithmetic operations such as addition, subtraction, multiplication, and division

order of operations

the universally accepted order to perform operations when more than one is present in an expression, often represented by the acronym PEMDAS

real numbers

fractions, negative numbers, decimals, integers, square roots, and zero

reciprocal

two fractions are reciprocals if their product is $1$

simplified fraction

also called a reduced fraction, or a fraction in lowest terms, a fraction having no common factors in the numerator and denominator

term

a single number, variable, or a product or quotient of numbers and/or variables

variable

a letter that represents a number or quantity whose value may change