**Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.**

**1.0** Students 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:

** 1.1** Students use properties of numbers to demonstrate whether assertions are true or false.

** 2.0** Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

**3.0 **Students solve equations and inequalities involving absolute values.

**4.0** Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.

**5.0** Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

**6.0** Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

**7.0** Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

**8.0** Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

**9.0** Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

**10.0** Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

**11.0** 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.

**12.0** Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

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

**14.0** Students solve a quadratic equation by factoring or completing the square.

**15.0** Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

**16.0** Students 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.

**17.0** Students 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.

**18.0** Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

**19.0** Students know the quadratic formula and are familiar with its proof by completing the square.

**20.0** Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

**21.0** Students graph quadratic functions and know that their roots are the x-intercepts.

**22.0** Students 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.

**23.0** Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

**24.0** Students use and know simple aspects of a logical argument:

** 24.1** Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

**24.2** Students identify the hypothesis and conclusion in logical deduction.

**24.3** Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

**25.0** Students 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:

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

** 25.2** Students 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.

**25.3** Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.