Math Algebra

Math: Algebra Active component (content knowledge)

Comprehend • • • • • • • • • •

When it is possible (or not possible) to simplify, solve, substituted or evaluate equations and expression The properties of rational exponents, integer exponents and roots and apply these properties to simplify algebraic expressions How the concept of a function has a specific definition beyond being a type of algebraic expression Algebraic language, notation, and properties for a variety of functions (e.g., polynomial, rational, exponential, logarithmic, and trigonometric) Basic forms of the equation of a straight line without the aid of a calculator Basic shape of a quadratic function and the relationships between the roots of the quadratic and zeros of the function Basic shape of the graph of exponential and log functions, including exponential decay Formal notations (e.g., sigma and factorial notation) Arithmetic and geometric progressions and series Basic algebraic concepts such as: Exponents, roots and their properties ○ Basic theorems of exponents and roots ○ Properties and theorems of logarithms Properties and representation of functions by: ○ Recognizing whether relationship given in symbolic or graphical form is a function ○ Determining the domain of a function represented in either symbolic or graphical form ○ Understanding functional notation and evaluating a function at a specified point in its domain ○ Combining functions by composition as well as by addition, subtraction, multiplication and division ○ Identifying whether a function has an inverse and when functions are inverses of each other

Learner outcomes Demonstrate by solving:

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Linear equations and absolute value equations

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Systems linear equations (two or three variables) and inequalities using algebraic and graphical methods (e.g., substitution, elimination, addition and graphing)

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Quadratic equations using various appropriate methods while recognizing real solutions by: factoring, completing the square, the quadratic formula

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Polynomial equations

College Readiness Standards

Equations involving several variables for one variable in terms of the others

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Linear inequalities and absolute value inequalities

Add, subtract, multiply polynomials Divide low degree polynomials (e.g., long division) Factor polynomials (e.g., difference of squares, perfect square trinomials, difference of two cube and trinomials) Add, subtract, multiply, divide and simplify rational expressions Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified values of their variables

Develop ability to: •

Represent functions, patterns and relationships in different ways (e.g., statements, formulas, and graphs)

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Recognize which type of expression best fits the context of a basic application

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Use distributive property to multiply polynomials Compose and decompose functions and how to find inverses of basic functions 22082231 6/29/2009

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Knowing that the inverse of an exponential function is a logarithm, prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems

The relationship between the coefficients of a linear equation and the slope and x- and y- intercepts of a graph The relationship between a solution of a system of two linear equations in two variables and the graphs of the corresponding lines

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Simplify and perform operations on rational expressions, including finding common denominators

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Derive and use the formula for general term and summation of finite arithmetic and geometric series

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Graph a variety of equations and inequalities ○ ○ ○ ○ ○

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Graph a linear equation and demonstrate that it has a constant rate of change Graph the solution set of a linear inequality and identify whether the solution set is an open or a closed halfplane Graph the solution set of a system of two or three linear inequalities Graph a quadratic function and understand the relationship between its real zeros and the x-intercepts of its graph Ellipses and hyperbolas whose axes are parallel to the x and y axes and demonstrate understanding of the relationship between their standard algebraic form and their graphical characteristics Exponential functions and identify their key characteristics

Read information and draw conclusions from graphs Identify properties of a graph that provide useful information about the original problem Recognize and solve problems algebraically that can be modeled using: ○ ○ ○ ○ ○ ○

A linear equation in one variable A system of two equations in two variables A quadratic equation An exponential function An exponential function whose solution requires facility with logarithms Finite geometric series

(Conley, 2003; 2005; 2007) (The American Diploma Project, 2004)

College Readiness Standards

22082231 6/29/2009

Standards for Success (S4S) Math: Algebra Successful students: A. B. C. D. E.

Know and apply basic algebraic concepts Use various appropriate techniques to solve basic equations and inequalities Distinguish between and among expression, formulas, equations and functions Understand the relationship between equations and graphs Understand algebra well enough to apply it procedurally and conceptually to a range of common problems F. Demonstrate the ability to work with formulas and symbols albebraically G. Demonstrate active participation in the process of learning mathematics

Cognitive Strategies Emphasized •

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Habits of the mind such as: ○ Time management – budgeting time to complete reading tasks ○ Understanding expectations of readings ○ Academic persistence Critical thinking skills such as: ○ Ability to discuss materials in-depth by asking engaging questions ○ Problem solving Understanding the connection between reading comprehension skills and disciplines: writing, speaking and research Self-analysis – learning from constructive criticism and feedback Developing comfort with ambiguity of readings and assignments

Bibliography Conley, D. T. (2005). College Knowledge. San Francisco: Jossey-Bass. Conley, D. T. (2003). Understanding University Success: A Project of the Association of American Universities and The Pew Charitable Trusts. Eugene: Center for Educational Policy Research. Conley, D. (2007). Towards a More Comprehensive Comprehension of College Readiness. Eugene, OR: Educational Policy Improvement Center. The American Diploma Project. (2004). Ready or Not: Creating a High School Diploma that Counts. Achieve, Inc.

College Readiness Standards

22082231 6/29/2009