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ALGEBRA STRAND

11^TH	MM3A1. Students will analyze graphs of polynomial functions of higher degree. a. Graph simple polynomial functions as translations of the function b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros. c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior. MM3A2. Students will explore logarithmic functions as inverses of exponential functions. a. Define and understand the properties of roots. b. Extend properties of exponents to include rational exponents. c. Define logarithmic functions as inverses of exponential functions. d. Understand and use properties of logarithms by extending laws of exponents. e. Investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, and rate of change. f. Graph functions as transformations of g. Explore real phenomena related to exponential and logarithmic functions including half-life and doubling time. MM3A3. Students will solve a variety of equations and inequalities. a. Find real and complex roots of higher degree polynomial equations using the theorem of algebra, incorporating complex and radical conjugates. b. Solve polynomial, exponential, and logarithmic equations analytically, graphically, and using appropriate technology. c. Solve polynomial, exponential, and logarithmic inequalities analytically, graphically, and using appropriate technology. Represent solution sets of inequalities using interval notation. d. Solve a variety of types of equations by appropriate means choosing among mental calculation, pencil and paper, or appropriate technology. MM3A4. Students will perform basic operations with matrices. a. Add, subtract, multiply, and invert matrices, when possible, choosing appropriate methods, including technology. b. Find the inverses of two-by-two matrices using pencil and paper, and find inverses of larger matrices using technology. c. Examine the properties of matrices, contrasting them with properties of real numbers. MM3A5. Students will use matrices to formulate and solve problems. a. Represent a system of linear equations as a matrix equation. b. Solve matrix equations using inverse matrices. c. Represent and solve realistic problems using systems of linear equations. MM3A6. Students will solve linear programming problems in two variables. a. Solve systems of inequalities in two variables, showing the solutions graphically. b. Represent and solve realistic problems using linear programming. MM3A7. Students will understand and apply matrix representations of vertex-edge graphs. a. Use graphs to represent realistic situations. b. Use matrices to represent graphs, and solve problems that can be represented by graphs.

12^TH	MM4A1. Students will explore rational functions. a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. b. Find inverses of rational functions, discussing domain and range, symmetry, and function composition. c. Solve rational equations and inequalities analytically, graphically, and by using appropriate technology. MM4A2. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured in degrees and radians, including but not limited to 0°, 30°, 45°, 60°, 90°, their multiples, and equivalences. b. Understand and apply the six trigonometric functions as functions of general angles in standard position. c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position. d. Understand and apply the six trigonometric functions as functions of arc length on the unit circle. e. Find values of trigonometric functions using the unit circle. MM4A3. Students will investigate and use the graphs of the six trigonometric functions. a. Understand and apply the six basic trigonometric functions as functions of real numbers. b. Determine the characteristics of the graphs of the six basic trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. d. Apply graphs of trigonometric functions in realistic contexts involving periodic phenomena. MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise. b. Investigate transformations of functions. c. Investigate characteristics of functions built through sum, difference, product, quotient, and composition. MM4A5. Students will establish the identities below and use them to simplify trigonometric expressions and verify equivalence statements. MM4A6. Students will solve trigonometric equations both graphically and algebraically. a. Solve trigonometric equations over a variety of domains, using technology as appropriate. b. Use the coordinates of a point on the terminal side of an angle to express x as rcosθ and y as rsinθ d. Apply the law of sines and the law of cosines. MM4A7. Students will verify and apply to find the area of a triangle. MM4A8. Students will investigate and use inverse sine, inverse cosine, and inverse tangent functions. a. Find values of the above functions using technology as appropriate. b. Determine characteristics of the above functions and their graphs. MM4A9. Students will use sequences and series a. Use and find recursive and explicit formulae for the terms of sequences. b. Recognize and use simple arithmetic and geometric sequences. c. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series. d. Use summation notation to explore finite series. MM4A10. Students will understand and use vectors. a. Represent vectors algebraically and geometrically. b. Convert between vectors expressed using rectangular coordinates and expressed using magnitude and direction. c. Add, subtract, and compute scalar multiples of vectors. d. Use vectors to solve realistic problems.

Global Descriptions

GRADE
3^rd	Students will understand how to express relationships as mathematical expressions.
4^th	Students will investigate and represent mathematical relationships between quantities using mathematical expressions in problem-solving situations.
5^th	Students will represent and investigate mathematical expressions algebraically by using variables.
6^th	Students will investigate relationships between two quantities. They will write and solve proportions and simple one-step equations that result from problem situations.
7^th	Students will demonstrate an understanding of linear relations and fundamental algebraic concepts.
8^th	Students will use linear algebra to represent, analyze and solve problems. They will use equations, tables, and graphs to investigate linear relations and functions, paying particular attention to slope as a rate of change.
9^th	Students will explore functions and solve simple equations. Students will simplify and operate with radical, polynomial, and rational expressions.
10^th	Students will investigate piecewise, exponential, and quadratic functions, using numerical, analytical, and graphical approaches, focusing on the use of these functions in problem-solving situations. Students will solve equations and inequalities and explore inverses of functions.
11^th	Students will investigate exponential, logarithmic and polynomial functions of degree higher than 2. Students will understand matrices and use them to solve problems.
12^th	Students will analyze rational and trigonometric functions. Students will investigate and apply sequences and series and will understand and use vectors.

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