Students interpret equations like 1/x = 3 as two equations “1/x = 3 ” and “x ≠ 0 ” joined by “and.” Students find the solution set for this new system of equations.

Students learn to think of some of the letters in a formula as constants in order to define a relationship between two or more quantities, where one is in terms of another, for example holding V in V = IR as constant, and finding R in terms of I.

Students recognize and identify solutions to two-variable equations. They represent the solution set graphically. They create two variable equations to represent a situation. They understand that the graph of the line ax + by = c is a visual representation of the solution set to the equation ax + by = c.

Students recognize and identify solutions to two-variable inequalities. They represent the solution set graphically. They create two variable inequalities to represent a situation.
Students understand that a half-plane bounded by the line ax + by = c is a visual representation of the solution set to a linear inequality such as ax + by < c. They interpret the inequality symbol correctly to determine which portion of the coordinate plane is shaded to represent the solution.

Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically.

Students create systems of equations that have the same solution set as a given system.
Students understand that adding a multiple of one equation to another creates a new system of two linear equations with the same solution set as the original system. This property provides a justification for a method to solve a system of two linear equations algebraically.

Students use systems of equations or inequalities to solve contextual problems and interpret solutions within a particular context.

Students investigate a problem that can be solved by reasoning quantitatively and by creating equations in one variable.
They compare the numerical approach to the algebraic approach.

Students learn the meaning and notation of recursive sequences in a modeling setting.
Following the modeling cycle, students investigate the double and add 5 game in a simple case in order to understand the statement of the main problem.

Students learn the meaning and notation of recursive sequences in a modeling setting.
Students use recursive sequences to model and answer problems.
Students create equations and inequalities to solve a modeling problem.
Students represent constraints by equations and inequalities and interpret solutions as viable or non-viable options in a modeling context.

Students create equations and inequalities in one variable and use them to solve problems.
Students create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
Students represent constraints by inequalities and interpret solutions as viable or non-viable options in a modeling context.

Students examine sequences and are introduced to the notation used to describe them.

Students compare the rate of change for simple and compound interest and recognize situations in which a quantity grows by a constant percent rate per unit interval.

Students describe and analyze exponential decay models; they recognize that in a formula that models exponential decay, the growth factor b is less than 1; or, equivalently, when b is greater than 1, exponential formulas with negative exponents could also be used to model decay.

Students understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain to exactly one element of the range and understand that if f is a function and x is an element of its domain, then f(x) denotes the output of corresponding to the input x.
Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Students understand set builder notation for the graph of a real-valued function: {(x, f(x)) | x ∈ D}.
Students learn techniques for graphing functions and relate the domain of a function to its graph.

Students understand the meaning of the graph of y = f(x), namely {(x,y) | x ∈ D and y = f(x)}.
Students understand the definitions of when a function is increasing or decreasing.

Students create tables and graphs of functions and interpret key features including intercepts, increasing and decreasing intervals, and positive and negative intervals.

Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Students create functions that represent a geometric situation and relate the domain of a function to its graph and to the relationship it describes.