I. Content Standard: Number, Number Sense and Operations Standard

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1 Course Description: Honors Precalculus is the study of advanced topics in functions, algebra, geometry, and data analysis including the conceptual underpinnings of Calculus. The course is an in-depth study of trigonometric and circular functions including modeling, graphing, and connecting to polar coordinates, complex numbers, and series. All concepts will be taught with an emphasis on the graphing calculator. A graphing calculator is required. Note: The above description is the same as Precalculus. Honors Precalculus will move at a faster pace, resulting in a deeper study of topics, i.e emphasis on proof and verification, along with more of an introduction to Calculus topics. I. Content Standard: Number, Number Sense and Operations Standard Students demonstrate number sense, including an understanding of number systems and reasonable estimates using paper and pencil, technology-supported and mental methods. Benchmarks (Grade Band) A. Demonstrate that vectors and matrices are systems having the same properties of the real number system. B. Develop an understanding of properties of and representations for addition and multiplication of vectors and matrices. Honors Precalulus Indicators 1. Determine what properties hold for vector addition and multiplication, and for scalar multiplication. 1. Determine what properties hold for vector addition and multiplication, and for scalar multiplication. 2. Model using the coordinate plane, vector addition and scalar multiplication. C. Apply factorials and exponents, including fractional exponents, to solve practical problems. D. Demonstrate fluency in operations with real numbers, vectors and matrices, using mental computation or paper and pencil calculations for simple cases, and technology for more complicated cases. E. Represent and compute with complex numbers. 1. Use factorial notation and computations to represent and solve problem situations involving arrangements. 1. Use vector addition and scalar multiplication to solve problems. 1. Represent complex numbers on the complex plane. 2. Compute sums, differences, products and quotients of complex numbers. 1

2 II. Content Standard: Patterns, Functions and Algebra Standard Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations. Benchmarks (Grade Band) A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and global behavior. Honors Precalculus Indicators 1. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, radical and rational functions; e.g. general shape, number of roots, domain and range, asymptotic behavior. 2. Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology. 3. Recognize, analyze, and graph parent graphs and their transformations. (DI) 4. Represent the inverse of a function symbolically and graphically as a reflection about y=x. 5. Describe the characteristics of the graphs of conic sections. 6. Recognize, analyze, and graph conic sections. 7. Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. 8. Analyze the behavior of the arithmetic and geometric sequences and series as the number of terms increases. 9. Translate between the numeric and symbolic form of a sequence or series. 10. Recognize and use the relationship between the unit circle, right triangles, and the six trigonometric functions using radian and degree measure. 11. Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior. 2

3 12. Represent the inverse of a transcendental function symbolically and graphically. 13. Make mathematical arguments using the concept of limit. 14. Translate freely between polar and Cartesian coordinate systems. 15. Use the concept of limit to find instantaneous rate of change for a point on a graph as the slope of a tangent at a point. B. Use the quadratic formula to solve quadratic equations that have complex roots. C. Use exponential functions to model and solve problems. D. Apply algebraic methods to represent and generalize problem situations involving vectors and matrices. 1. Solve equations involving radical expressions and complex roots. 1. Identify and describe problem situations involving exponential growth and decay. 2. Use the model to predict behavior. 1. Model and solve problems with matrices and vectors. 3

4 III. Content Standard: Geometry and Spatial Sense Standard Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems. Benchmarks (Grade Band) A. Use trigonometric relationships to determine solutions in problem situations. Honors Precalculus Indicators 1. Use trigonometric relationships to determine lengths and angle measures; i.e., right triangle ratios, Law of Sines and Law of Cosines. 2. Derive and apply the basic trigonometric identities; i.e., Pythagorean, sum and difference, double and half angle. 4

5 IV. Content Standard: Mathematical Processes The benchmarks for mathematical processes articulate what students should demonstrate in problem solving, representation, communication, reasoning and connections at key points in their mathematics program. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution. B. Apply mathematical knowledge and skills routinely in other content areas and practical situations. C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities. D. Apply reasoning processes and skills to construct logical verifications or counterexamples to test conjectures and to justify and defend algorithms and solutions. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. F. Use precise mathematical language and notations to represent problem situations and mathematical ideas. G. Write clearly and coherently about mathematical thinking and ideas. H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner. 5

6 Mathematics Targets Pre-Calculus Big Idea: Patterns, relations, and functions help to represent and understand quantitative relationships and analyze change. Essential Learning: Analysis of trigonometric, exponential, logarithmic, polynomial, radical, rational and other algebraic equations Target 1 Various functions, patterns, and relationships will be used to analyze, model, and solve problems. I can manipulate, solve, and simplify trigonometric, exponential, logarithmic, polynomial, radical, and rational equations. create and use tables of values for any function and relate it back to the graph. recognize and analyze any conic from an equation. find an inverse of a function and prove inverse relationships by composing functions. verify trigonometry identities. apply the laws of exponents. create and use a pattern in a sequence or series, including factorials. analyze rates of change as slope (derivative) of a graph. Dublin City Schools August 2007

7 Mathematics Targets Pre-Calculus Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations are used to analyze mathematical situations and solve problems. Essential Learning: Analysis of parent graphs Target 1 Spatial and graphical representations will be used to interpret, analyze, solve, and create various patterns and models. I can solve triangles using the appropriate methods including Right Triangle Trigonometry, Law of Sines, and Law of Cosines. Analyze and manipulate graphs to find patterns and characteristics including domain, range, asymptotes, maximum, minimum, inverses, reflections, vertical and horizontal shifts and stretches, and zeros (roots). For trig functions this also includes amplitude, period, and phase shift. translate between polar and rectangular graphs. recognize the parent graphs and graph many elementary functions with and without a graphing calculator, including conic sections. find an inverse graphically. model and solve a problem using vectors. Dublin City Schools August 2007

8 Mathematics Targets Pre-Calculus Big Idea: Number, number systems and operations relate to one another and build to computational fluency. Essential Learning: Measurement and manipulation of mathematical relationships Target 1 Mathematical relationships can be measured, estimated, and manipulated by the use of numbers, systems and appropriate tools/units. I can identify domain and range for parent graphs and their transformations by testing regions (using a number line). convert between radians and degrees, units involving velocity, and between polar and rectangular coordinates, including complex numbers. evaluate trig expressions with and without a calculator. (Special angles without a calculator.) Dublin City Schools August 2007