Key Concepts

• Phrases such as sumincreased 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