Algebra

Properties

• In the transition from arithmetic to algebra, attention shifts from arithmetic operations (addition, subtraction, multiplication, and division) to use of the properties of these operations.
• All of the facts of arithmetic and algebra follow from certain properties.

Variable

• Quantities are used to form expressions, equations, and inequalities.
• An expression refers to a quantity but does not make a statement about it. An equation (or an inequality) is a statement about the quantities it mentions.
• Using variables in place of numbers in equations (or inequalities) allows the statement of relationships among numbers that are unknown or unspecified.

Equivalence

• A single quantity may be represented by many different expressions.
• The facts about a quantity may be expressed by many different equations (or inequalities).

Solving Equations & Inequalities

• Solving an equation is the process of rewriting the equation to make what it says about its variable(s) as simple as possible.
• Properties of numbers and equality can be used to transform an equation (or inequality) into equivalent, simpler equations (or inequalities) in order to find solutions.
• Useful information about equations and inequalities (including solutions) can be found by analyzing graphs or tables.
• The numbers and types of solutions vary predictably, based on the type of equation.

Proportionality

• Two quantities are proportional if they have the same ratio in each instance where they are measured together.
• Two quantities are inversely proportional if they have the same product in each instance where they are measured together.

Function

• A function is a relationship between variables in which each value of the input variable is associated with a unique value of the output variable.
• Functions can be represented in a variety of ways, such as graphs, tables, equations, or words. Each representation is particularly useful in certain situations.
• Some important families of functions are developed through transformations of the simplest form of the function.
• New functions can be made from other functions by applying arithmetic operations or by applying one function to the output of another.

Modeling

• Many real-world mathematical problems can be represented algebraically. These representations can lead to algebraic solutions.
• A function that models a real-world situation can be used to make estimates or predictions about future occurrences.