1. All students solve equations and inequalities involving absolute value.

2. All students solve systems of linear equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices.

3. All students are adept at operations on polynomials, including long division.

4. All students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

5. All students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

6. All students add, subtract, multiply, and divide complex numbers.

7. All students add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complicated fractions including fractions with negative exponents in the denominator.

8. All students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. All students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.

9. All students demonstrate and explain the effect changing a coefficient has on the graph of quadratic functions. That is, all students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2 + c.

10. All students graph quadratic functions and determine the maxima, minima, and zeros of the function.

11. All students prove simple laws of logarithms.

• All students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.
  
• All students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
  

12. All students know the laws of exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

13. All students use the definition of logarithms and the product formula for logs to translate between logarithms in any bases.

14. All students understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values.

15. All students determine if a specific algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true.

16. All students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.

17. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, all students can use the method of completing the square to put the equation into standard form and can recognize whether its graph is a circle, ellipse, parabola, or hyperbola. All students can then graph the equation.

18. All students use fundamental counting principles to compute combinations and permutations.

19. All students use combinations and permutations to compute probabilities.

20. All students know the Binomial Theorem and use it to expand binomial expressions which are raised to positive integer powers.

21. All students apply the method of mathematical induction to prove general statements about the positive integers.

22. All students find the general term and the sums of arithmetic series and both finite and infinite geometric series.

23. All students derive the summation formulas for arithmetic series and both finite and infinite geometric series.

24. All students solve problems involving functional concepts such as composition, inverse, and arithmetic operations on functions.

25. All students use properties from number systems to justify steps in combining and simplifying functions.