FOCUS: Students will be encouraged to develop an understanding of functions and relations and apply this knowledge to solve analytical problems and systems of equations and inequalities. They will also appreciate the diverse nature of algebraic techniques and their value in solving problems.

Understand determining a root, simplifying a root (e.g. \sqrt 8 = 2\sqrt 2 surds) and the meaning of a fractional power (rational exponents) and understand and use the rules of exponents (Index laws).

Solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Understand the concepts of parallel lines and perpendicular lines and how those slopes are related and are able to find the equation of a line passing through a point and parallel or perpendicular to a given line.

Understand both algebraically and geometrically the three possible conclusions of a system of two linear equations in two variables (inconsistent, infinite and one solution).

Solves real-world problems involving the solution of a system of two linear equations in two variables, including rate problems, work problems, and percent mixture problems.

Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials (see also 1AL 5.2)

Add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.

Use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

Understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.

Determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

Identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.

Use and know simple aspects of a logical argument:

Explain the difference between inductive and deductive reasoning and identify and provide examples of each.

Identify the hypothesis and conclusion in logical deduction.

Use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

Determine whether the statement is true sometimes, always, or never, given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities.

Use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:

Use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

Judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.