1. All students identify and use the arithmetic properties of subsets of integers, rational, irrational and real numbers. This includes closure properties for the four basic arithmetic operations where applicable

• All students use properties of numbers to demonstrate that assertions are true or false.
  

2. All students understand and use such operations as taking the opposite, reciprocal, raising to a power, and taking a root. This includes the understanding and use of the rules of exponents.

3. All students solve equations and inequalities involving absolute values.

4. All students simplify expressions prior to solving linear equations and inequalities in one variable such as 3(2x-5) + 4(x-2) = 12. 5. All students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable, with justification of each step.

6. All 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., sketch the region defined by 2x + 6y < 4).

7. All students verify that a point lies on a line given an equation of the line. All students are able to derive linear equations using the point-slope formula.

8. All students understand the concepts of parallel and perpendicular lines and how their slopes are related. All students are able to find the equation of a line perpendicular to a given line that passes through a given point.

9. All students solve a system of two linear equations in two variables algebraically, and are able to interpret the answer graphically. All students are able to use this to solve a system of two linear inequalities in two variables, and to sketch the solution sets.

10. All students add, subtract, multiply and divide monomials and polynomials. All students solve multistep problems, including word problems, using these techniques.

11. All students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor to all of the terms in a polynomial and recognizing the difference of two squares, and recognizing perfect squares of binomials.

12. All students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing to lowest terms.

13. All students add, subtract, multiply, and divide rational expressions and functions. All students solve both computationally and conceptually challenging problems using these techniques.

14. All students solve a quadratic equation by factoring or completing the square.

15. All students apply algebraic techniques to rate problems, work problems, and percent mixture problems.

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

18. All students determine whether a relation defined by a graph, a set of ordered pairs, or symbolic expression is a function and justify the conclusion.

19. All students know the quadratic formula and are familiar with its proof by completing the square.

20. All students use the quadratic formula to find the roots of a second degree polynomial and to solve quadratic equations.

21. All students graph quadratic functions and know that their roots are the x-intercepts.

22. All students use the quadratic formula and/or factoring techniques to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

23. All students apply quadratic equations to physical problems such as the motion of an object under the force of gravity.

24. All students use and know simple aspects of a logical argument.

• All students explain the difference between inductive and deductive reasoning and identify and provide examples of each
  
• All students identify the hypothesis and conclusion in logical deduction
  
• All students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
  

25. All 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

• All students use properties of numbers to construct simple valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions
  
• All students judge the validity of an argument based on whether the properties of the real number system and order of operations have been applied correctly at each step
  
• Given a specific algebraic statement involving linear, quadratic or absolute value expressions, equations or inequalities, all students determine if the statement is true sometimes, always, or never.