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A.1
Students expand and deepen their understanding of real and complex numbers by comparing expressions and performing arithmetic computations, especially those involving square roots and exponents. They use the properties of real numbers to simplify algebraic expressions and equations, and convert between different measurement units using dimensional analysis.
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A.1.1
Know equivalent forms of real numbers (including integer exponents and radicals, percents, scientific notation, absolute value, rational numbers, irrational numbers).
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A.1.2
Compare real number expressions.
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A.1.3
Simplify real number expressions using the laws of exponents.
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A.1.4
Perform operations on real numbers (including integer exponents, radicals, percents, scientific notation, absolute value, rational numbers, and irrational numbers) using multi-step and real-world problems.
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A.1.5
Use dimensional (unit) analysis to perform conversions between units of measure, including rates.
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A.1.6
Identify the real and imaginary parts of complex numbers and perform basic operations.
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A.1.7
Represent complex numbers geometrically.
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A.1.8
Use the zero product property of real numbers in a variety of contexts to identify solutions to equations.
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A.2
Draw and interpret graphs of relations. Understand the notation and concept of a function, find domains and ranges, and link equations to functions.
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A.2.1
Create a graph to represent a real-world situation.
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A.2.2
Interpret a graph representing a real-world situation.
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A.2.3
Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions.
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A.2.4
Determine the domain and range of a relation.
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A.2.5
Graph absolute value equations and inequalities in two variables.
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A.2.6
Identify and graph common functions (including but not limited to linear, rational, quadratic, cubic, radical, absolute value).
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A.2.7
Perform operations (addition, subtraction, division and multiplication) of functions algebraically, numerically, and graphically.
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A.2.8
Determine the composition of functions.
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A.2.9
Recognize, interpret, and graph functions defined piece-wise, with and without technology.
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A.2.10
Describe and graph transformations of functions.
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A.2.11
Solve problems involving functions and their inverses.
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A.2.12
Solve problems using direct, inverse, and joint variations.
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A.2.13
Solve real-world problems involving relations and functions.
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A.3
Solve linear equations and inequalities.
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A.3.1
Solve linear equations in one variable that include simplifying algebraic expressions.
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A.3.2
Identify and apply the distributive, associative, and commutative properties of real numbers and the properties of equality.
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A.3.3
Solve literal equations for a specified variable.
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A.3.4
Solve and graph simple and compound inequalities in one variable and be able to justify each step in a solution.
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A.3.5
Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
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A.3.6
Solve and graph the solutions of absolute value equations and inequalities with one variable.
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A.3.7
Rewrite equations of a line into slope-intercept form and standard form.
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A.3.8
Graph a line given any of the following information: a table of values, the x- and y-intercepts, two points, the slope and a point, the equation of the line in slope-intercept form, standard form, or point-slope form.
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A.3.9
Determine the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.
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A.3.10
Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
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A.3.11
Write an equation of a line that models a data set and use the equation or the graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
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A.3.12
Graph a linear equation or inequality in two variables with and without graphing technology. Write an equation or inequality represented by a given graph.
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A.3.13
Use a graph to approximate the solution of a system of linear equations or inequalities in two variables with and without technology.
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A.3.14
Solve systems of linear equations and inequalities in two and three variables using graphical, substitution, and elimination methods.
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A.3.15
Solve real-world problems involving systems of linear equations and inequalities in two and three variables.
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A.4
Perform operations on polynomials. Find factors of polynomials, learning special techniques for factoring quadratics. Understand the relationships among the solutions of polynomial equations, the zeros of a polynomial function, the x-intercepts of a graph, and the factors of a polynomial.
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A.4.1
Simplify monomials and monomial expressions using the laws of integral exponents.
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A.4.2
Add, subtract, and multiply polynomials.
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A.4.3
Factor polynomial expressions.
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A.4.4
Divide polynomials by monomials and polynomials with various techniques, including synthetic division.
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A.4.5
Graph polynomial functions with and without technology and describe end behavior.
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A.4.6
Use theorems of polynomial behavior (including but not limited to the Fundamental Theorem of Algebra, Remainder Theorem, the Rational Root Theorem, Descartes' Rule of Signs, and the Conjugate Root Theorem) to find the zeros of a polynomial function.
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A.4.7
Write a polynomial equation for a given set of real and/or complex roots.
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A.4.8
Describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression, with and without technology.
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A.4.9
Use graphing technology to find approximate solutions for polynomial equations.
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A.4.10
Use polynomial equations to solve real-world problems.
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A.4.11
Solve a polynomial inequality by examining the graph with and without the use of technology.
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A.4.12
Apply the Binomial Theorem.
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A.5
Simplify rational expressions and solve rational equations using what they have learned about factoring polynomials.
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A.5.1
Simplify algebraic ratios.
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A.5.2
Add, subtract, multiply, and divide rational expressions.
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A.5.3
Simplify complex fractions.
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A.5.4
Solve algebraic proportions.
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A.5.5
Solve rational equations.
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A.5.6
Identify removable and non-removable discontinuities and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the function.
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A.5.7
Solve real-world problems involving rational equations (mixture, distance, work, interest, and ratio).
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A.6
Simplify and perform operations on radical expressions and equations. Rationalize square root expressions and understand and use the concepts of negative and rational exponents. Add, subtract, multiply, divide, and simplify radical expressions and expressions with rational exponents. Solve radical equations and equations with terms that have rational exponents.
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A.6.1
Simplify radical expressions.
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A.6.2
Add, subtract, multiply and divide radical expressions (square roots and higher).
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A.6.3
Simplify expressions using properties of rational exponents.
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A.6.4
Convert between rational exponent and radical forms of expressions.
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A.6.5
Solve equations that contain radical expressions.
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A.7
Draw graphs of quadratic functions. Solve quadratic equations and solve these equations by factoring, completing the square and by using the quadratic formula. Use graphing calculators to find approximate solutions of quadratic equations.
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A.7.1
Graph quadratic equations with and without graphing technology.
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A.7.2
Solve quadratic equations over the real numbers by factoring, and by using the quadratic formula.
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A.7.3
Solve quadratic equations over the real numbers by completing the square.
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A.7.4
Use the discriminant to determine the nature of the roots of a quadratic equation.
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A.7.5
Solve quadratic equations over the complex number system.
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A.7.6
Identify the axis of symmetry, vertex, domain, range and intercept(s) for a given parabola.
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A.7.7
Solve non-linear systems of equations with and without using technology.
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A.7.8
Use quadratic equations to solve real-world problems.
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A.7.9
Solve optimization problems.
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A.7.10
Use graphing technology to find approximate solutions of quadratic equations.
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A.8
Understand the concepts of logarithmic and exponential functions. Graph exponential functions and solve problems of growth and decay. Understand the inverse relationship between exponents and logarithms and use it to prove laws of logarithms and to solve equations. Convert logarithms between bases and simplify logarithmic expressions.
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A.8.1
Define exponential and logarithmic functions and determine their relationship.
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Exponential functions: Evaluate an exponential function (Algebra - X.1)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from graphs (Algebra - CC.1)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from tables (Algebra - CC.2)
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Functions: linear, quadratic, exponential: Write linear, quadratic, and exponential functions (Algebra - CC.3)
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A.8.2
Define and use the properties of logarithms to simplify logarithmic expressions and to find their approximate values.
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A.8.3
Graph exponential and logarithmic functions.
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A.8.4
Prove laws of logarithms.
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A.8.5
Solve logarithmic and exponential equations.
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A.8.6
Use the change of base formula.
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A.8.7
Solve applications of exponential growth and decay.
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A.9
Write equations and draw graphs of conic sections (circle, ellipse, parabola, and hyperbola), thus relating an algebraic representation to a geometric one.
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A.9.1
Write the equations of conic sections in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.).
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A.9.2
Graph conic sections with and without using graphing technology.
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A.9.3
Solve real-world problems involving conic sections.
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A.10
In a general sense, all of mathematics is problem solving. In all of mathematics, use problem-solving skills, choose how to approach a problem, explain the reasoning, and check the results
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A.10.1
Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guessing- and-checking, solving a simpler problem, writing an equation, working backwards, and creating a table.
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A.10.2
Decide whether a solution is reasonable in the context of the original situation.
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A.10.3
Decide whether a given statement is always, sometimes, or never true (statements involving linear or quadratic expressions, equations, or inequalities rational or radical expressions or logarithmic or exponential functions).
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A.10.4
Use counterexamples to show that statements are false.
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