Algebra 1

 L1 REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS L1.1 Number Systems and Number Sense L1.1.1 Know the different properties that hold in different number systems and recognize that the applicable properties change in the transition from the positive integers to all integers, to the rational numbers, and to the real numbers. L1.1.2 Explain why the multiplicative inverse of a number has the same sign as the number, while the additive inverse of a number has the opposite sign. L1.1.3 Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations. L1.1.4 Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1. L1.1. Justify numerical relationships. L1.2 Representations and Relationships L1.2.2 Interpret representations that reflect absolute value relationships. L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media. L2 CALCULATION, ALGORITHMS, AND ESTIMATION L2.1 Calculation Using Real and Complex Numbers L2.1.1 Explain the meaning and uses of weighted averages. L2.1.2 Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents. L2.1.4 Know that the imaginary number i is one of two solutions to x2 = -1. A1 EXPRESSIONS, EQUATIONS, AND INEQUALITIES A1.1 Construction, Interpretation, and Manipulation of Expressions A1.1.1 Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables. A1.1.2 Know the properties of exponents and roots and apply them in algebraic expressions. A1.1.3 Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities. A1.2 Solutions of Equations and Inequalities A1.2.1 Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve. A1.2.2 Associate a given equation with a function whose zeros are the solutions of the equation. A1.2.3 Solve linear and quadratic equations and inequalities including systems of up to three linear equations with three unknowns. Justify steps in the solution, and apply the quadratic formula appropriately. A1.2.4 Solve absolute value equations and inequalities and justify steps in the solution. A1.2.6 Solve power equations and equations including radical expressions; justify steps in the solution, and explain how extraneous solutions may arise. A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution. A2 FUNCTIONS A2.1 Definitions, Representations, and Attributes of Functions A2.1.1 Determine whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range. A2.1.2 Read, interpret, and use function notation and evaluate a function at a value in its domain. A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations. A2.1.4 Recognize that functions may be defined by different expressions over different intervals of their domains; such functions are piecewise-defined. A2.1.5 Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions. A2.1.6 Identify the zeros of a function, the intervals where the values of a function are positive or negative, and describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations. A2.1.7 Identify and interpret the key features of a function from its graph or its formula(s). A2.2 Operations and Transformations with Functions A2.2.1 Combine functions by addition, subtraction, multiplication, and division. A2.2.2 Apply given transformations to parent functions and represent symbolically. A2.2.3 Determine whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs. A2.3 Representations of Functions A2.3.1 Identify a function as a member of a family of functions based on its symbolic or graphical representation; recognize that different families of functions have different asymptotic behavior. A2.3.2 Describe the tabular pattern associated with functions having a constant rate of change (linear); or variable rates of change. A2.3.3 Write the general symbolic forms that characterize each family of functions. A2.4 Models of Real-World Situations Using Families of Functions A2.4.1 Identify the family of function best suited for modeling a given real-world situation. A2.4.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. A2.4.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled. A3 FAMILIES OF FUNCTIONS A3.1 Lines and Linear Functions A3.1.1 Write the symbolic forms of linear functions (standard, point-slope, and slope-intercept) given appropriate information and convert between forms. A3.1.2 Graph lines (including those of the form x = h and y = k) given appropriate information. A3.1.3 Relate the coefficients in a linear function to the slope and x- and y- intercepts of its graph. A3.1.4 Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give -1. A3.2 Exponential and Logarithmic Functions A3.2.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information. A3.2.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and how base affects the rate of growth or decay. A3.2.5 Relate exponential functions to real phenomena, including half-life and doubling time. A3.3 Quadratic Functions A3.3.1 Write the symbolic form and sketch the graph of a quadratic function given appropriate information. A3.3.2 Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function. A3.3.3 Convert quadratic functions from standard to vertex form by completing the square. A3.3.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function. A3.3.5 Express quadratic functions in vertex form to identify their maxima or minima and in factored form to identify their zeros. A3.4 Power Functions A3.4.1 Write the symbolic form and sketch the graph of power functions. A3.4.2 Express directly and inversely proportional relationships as functions and recognize their characteristics. A3.4.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry. A3.5 Polynomial Functions A3.5.1 Write the symbolic form and sketch the graph of simple polynomial functions. A3.5.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2. A3.5.3 Determine the maximum possible number of zeroes of a polynomial function and understand the relationship between the x-intercepts of the graph and the factored form of the function. S2 BIVARIATE DATA-EXAMINING RELATIONSHIPS S2.1 Scatterplots and Correlation S2.1.1 Construct a scatterplot for a bivariate data set with appropriate labels and scales. S2.1.2 Given a scatterplot, identify patterns, clusters, and outliers. Recognize no correlation, weak correlation, and strong correlation. S2.1.3 Estimate and interpret Pearson’s correlation coefficient for a scatterplot of a bivariate data set. Recognize that correlation measures the strength of linear association. S2.1.4 Differentiate between correlation and causation. Know that a strong correlation does not imply a cause-and-effect relationship. Recognize the role of lurking variables in correlation. S2.2 Linear Regression S2.2.1 For bivariate data that appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions.