Topic/Unit: Linear Functions Big Ideas/Enduring Understandings: Patterns can be represented numerically, graphically, symbolically, and verbally and provide insights into potential relationships. A linear equation can be expressed in many forms. Real world situations can be modeled by linear graphs and equations. Coordinate geometry can be used to represent and verify algebraic relationships on the coordinate plane. Essential Questions: How do linear patterns and functions help us describe data about the world around us and make predictions about future data? What connections exist between algebra and coordinate geometry when looking at the graph of a line or line segment on the coordinate plane? What does the equation of a line reveal about its graph and its relation to other lines on the same coordinate plane? PA Academic Standards (PA Common Core): CC.2.2.HS.D.1, CC.2.2.HS.D.2, CC.2.2.HS.D.7, CC.2.2.HS.D.8, CC.2.2.HS.D.9, CC.2.2.HS.D.10, CC.2.2.HS.C.2, CC.2.2.HS.C.3, CC.2.2.HS.C.6, CC.2.4.HS.B.3 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: coordinates, x-axis, y-axis, origin, quadrants, x-coordinate, y-coordinate, x-intercept (zero, root), y-intercept, linear equation, slope, general form, slope-intercept form, point-slope form, intercept form, parallel, perpendicular, distance, midpoint Concepts: Competencies: Instructional Practices: Assessments: The student will know: The various forms of a linear equation A system of equations has no, one, or many solution(s) The student will be able to: Find the equation of a line given certain properties Find the intersection of two lines Khan Academy video and practice review of linear functions and equations Test on review of linear functions Application problems where students use slope, distance, and midpoint to determine characteristics of quadrilaterals and triangles Page 1
A segment s length, midpoint, and slope are calculated using its endpoints Whether two lines are parallel, perpendicular, or neither Real world problems can be modeled with linear functions Find the length & midpoint of a segment Find the slope of a line Determine whether two lines are parallel, perpendicular, or neither Model real-world situations with linear functions Graphic organizer in which students analyze the linear review concepts in which they are proficient and with which they still struggle Page 2
Topic/Unit: Quadratic Functions Big Ideas/Enduring Understandings: All numbers are represented within the complex number system. The basic operations of addition, subtraction, multiplication, and division can be applied to all complex numbers. Quadratic equations can always be solved even if the quadratic is not factorable. Real world situations can be modeled by quadratic functions. Essential Questions: What are complex numbers and how do they relate to quadratic equations? What variety of techniques can be used when solving quadratic equations and how do these solutions relate to their graphs? How do quadratic equations and their graphs help us interpret events that occur in the world around us? PA Academic Standards (PA Common Core): CC.2.1.HS.F.6, CC.2.1.HS.F.7, CC.2.2.HS.D.1, CC.2.2.HS.D.3, CC.2.2.HS.D.4, CC.2.2.HS.D.5 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: Real number, irrational number, complex number, imaginary number, conjugate, quadratic equation, root, discriminant, double root, extraneous root, axis of symmetry, vertex of parabola Concepts: Competencies: Instructional Practices: Assessments: The student will know: Every number they come across is part of the complex number system. The student will be able to: Add, subtract, multiply, and divide complex numbers. Worksheet to practice the various methods of solving a quadratic equation. Quiz on the four operations with complex numbers. Test on solving and graphing quadratic equations. Page 3
i2 = -1 is the most basic complex number. Which method is best to use in solving a quadratic equation based on its characteristics. The general shape and main components of parabola and its connection to its equation. Solve quadratic equations using various methods. Define and graph quadratic functions. Model real-world situations using quadratic functions. Student created tutorials on the different methods of solving a quadratic equation and the characteristics of a the graph of a quadratic function. Paired practice with application problems that model real world data. NCTM Illuminations activity in which students evaluate, interpret, and graph real world data that models a quadratic function. Real world problems can be modeled with quadratic functions. Page 4
Topic/Unit: Polynomial Functions & Inequalities Big Ideas/Enduring Understandings: Polynomial functions produce smooth curves that cross the x-axis at the function s zeros. The equation of a polynomial function tells us about the behavior of its graph. The Fundamental Theorem of Algebra describes the nature of the roots of a polynomial function. The Remainder Theorem and the Factor Theorem are derived from polynomial division. Finding critical values and testing intervals are necessary in solving polynomial inequalities. Real world situations can be modeled by polynomial equations and inequalities. Essential Questions: What is a polynomial equation? What are the traits associated with polynomial equations of various degrees? How does a polynomial equation differ from a polynomial inequality? How can graphing technology help us solve polynomial equations and inequalities? Why is linear programming a useful tool when making decisions about real world situations that that involve several parameters? PA Academic Standards (PA Common Core): CC.2.1.HS.F.7, CC.2.2.HS.D.3, CC.2.2.HS.D.4, CC.2.2.HS.D.5, CC.2.2.HS.D.7, CC.2.2.HS.D.10 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: Polynomial, term, coefficient, leading term, leading coefficient, degree, root, zero, cubic function, quartic function, double root, triple root, local maximum, local minimum, synthetic substitution, synthetic division, polynomial inequality, linear programming, feasible region Page 5
Concepts: Competencies: Instructional Practices: Assessments: The student will know: If a function is polynomial. The Remainder and Factor Theorems and their usefulness in solving and graphing polynomial equations. The Fundamental Theorem of Algebra and how it connects the graph of a polynomial function to its equation. The graphs of polynomial inequalities in two variables are closely related to the graphs of polynomial functions. Real world problems can be modeled with polynomial functions and inequalities. The student will be able to: Identify a polynomial function, evaluate a polynomial function using synthetic substitution, and determine the zeros of a polynomial function. Use synthetic division and apply the remainder and factor theorems. Graph a polynomial function and determine an equation for the graph. Model real-world situations using polynomial functions. Solve polynomial equations by various methods. Graph polynomial inequalities in one and two variables. Video and guided note taking the form and graph of a polynomial equation. Paired activity in which students are given the graph of a polynomial equation and must describe its characteristics. Grapher activity in which students explore linear programming in terms of maximizing profit. Quiz on synthetic division and synthetic substitution. Students create their own polynomial equation by working backward from the roots. Peers must then solve and graph these equations. Test on linear and polynomial inequalities. Exit card using a polynomial equation and graphing technology to maximize or minimize area or volume. Students are given a linear programming application problem and work in pairs to develop the parameters, graph the inequalities, test the vertices of the bounded region, and find the desired solution. Apply polynomial inequalities to linear programming models. Page 6
Topic/Unit: Function Operations & Transformations Big Ideas/Enduring Understandings: Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations. Operations and transformations apply to all functions. Changes in the equation of a function cause changes in its graph. Functions help to model and explain real world situations. Essential Questions: When is a relation a function? How are the domain, range, and zeros of a function determined from its graph? What happens to a parent function when you transform its graph? How do you determine the composition of two functions? Why don t all functions have inverses? How can we distinguish those that do from those that don t? What makes a function even or odd? PA Academic Standards (PA Common Core): CC.2.2.HS.D.1, CC.2.2.HS.D.2, CC.2.2.HS.D.7, CC.2.2.HS.D.8, CC.2.2.HS.D.10, CC.2.2.HS.C.1, CC.2.2.HS.C.2, CC.2.2.HS.C.3, CC.2.2.HS.C.4, CC.2.2.HS.C.6 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: Relation, function, domain, range, zeros, function mapping, independent variable, dependent variable, composite function, line of reflection, axis of symmetry, point of symmetry, inverse function, one-to-one function Page 7
Concepts: Competencies: Instructional Practices: Assessments: The student will know: Why some relations are functions and some are not. Function notation and notation for operations, composition, and inverses. Families of functions and their characteristics. What changes occur to the graph of a function when its equation is altered. The student will be able to: Identify a function and determine its domain, range, and zeros. Perform operations on functions and determine the domain and range of the resulting function. Graph functions, use symmetry to reflect graphs, stretch and shrink graphs vertically and horizontally, and translate graphs on the coordinate plane. Determine if a function has an inverse and find the equation of the inverse. Khan Academy practice in function operations. Grapher software activity in which students explore transformations of families of functions. NCTM Illumination s Activity on Function Matching. Matching activity in which students pair a function graph to its appropriate domain and range. Sketching activity in which students discover the relationship between a function and its inverse. Exit card on function operations including composition. Students complete a teacher made assessment connecting function equations to their graphs. They will be given two options for this assessment. 1) Given the graphs of transformed functions, students will identify their equations. 2) Given the equations of transformed functions, students will sketch their graphs. Short essay discussing function transformations and inverses. Determine if a function is even or odd given certain traits. Page 8
Topic/Unit: Exponential & Logarithmic Functions Big Ideas/Enduring Understandings: Rational exponents follow the same properties as integer exponents. Exponential and logarithmic functions are inverses of each other. Properties of exponents and logarithms can be used to simplify expressions and solve equations. The characteristics of exponential and logarithmic functions and their representations are useful in solving real world problems. Essential Questions:. What are the similarities and differences in working with integral and rational exponents? Why are exponents and logarithms considered inverses? How can exponential and logarithmic functions be applied to solve real world problems? What are the Laws of Logarithms and the Change of Base Formula? How can they help us solve exponential equations? PA Academic Standards (PA Common Core): CC.2.1.HS.F.1, CC.2.1.HS.F.2, CC.2.2.HS.D.2, CC.2.2.HS.D.8, CC.2.2.HS.D.9, CC.2.2.HS.C.3, CC.2.2.HS.C.5, CC.2.2.HS.C.6 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: exponent, integral exponent, rational exponent, exponential function, exponential growth, exponential decay, exponential equation, natural exponential function, compound interest, common logarithm, natural logarithm Page 9
Concepts: Competencies: Instructional Practices: Assessments: The student will know: The laws of exponents and their extension to rational exponents. Whether a real world situation is displaying traits of exponential growth or decay. The inverse relationship between exponents and logarithms. The form of an exponential function and the characteristics of its graph. The number e and its connection to limits. The laws of logarithms and their usefulness in simplifying expressions and solving equations. The student will be able to: Apply integral and rational exponents. Use exponential and natural exponential functions. Graph exponential functions on the coordinate plane. Apply logarithms and prove laws of logarithms. Solve exponential equations using laws of logarithms and the change of base formula. Compare linear, quadratic, and exponential models. Socrative.com space race where students work in small groups to practice properties of integral and rational exponents. NCTM Illuminations activity in which students explore a medicine s half life through a series of data points, graph its curve, and develop its exponential equation. NCTM Illumination s Compound Interest Simulator. Student created online flashcards for the various formulas used in evaluating exponential growth and decay? Worksheet to practice solving exponential equations using Laws of Logarithms and the Change of Base Formula. Quiz on laws of integral and rational exponents. Constructed response questions pertaining to exponential growth and decay, half life, and compound interest. Khan Academy practice on Laws of Logarithms. Page 10
Topic/Unit: Trigonometric Functions Big Ideas/Enduring Understandings: The six basic trigonometric functions are connected through the right triangle as they all represent the ratio of two of its sides. Trigonometry can be discussed in the context of a right triangle as well as in the context of the Unit Circle. There are many similarities among the graphs of the trigonometric functions. The inverse trigonometric functions result from restricting the domain of the original function. They are an essential component in solving trigonometric equations. Essential Questions: What is trigonometry? What are the six basic trigonometric functions? Why is the Unit Circle important in our study of trigonometry? How does it change our understanding of an angle and how it is measured? How does our approach to finding the exact value of a trigonometric function differ from finding its approximation? What are the major attributes of the graphs of the six trigonometric functions? Why is it important to restrict the domain of a trigonometric function when determining its inverse? How do reciprocal and inverse trigonometric functions differ? Why are they often confused? PA Academic Standards (PA Common Core): CC.2.1.HS.F.5, CC.2.2.HS.D.1, CC.2.2.HS.D.7, CC.2.2.HS.D.8, CC.2.2.HS.C.4, CC.2.2.HS.C.7, CC.2.2.HS.C.8 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: trigonometry, initial ray, terminal ray, revolution, degree, minute, second, radian, standard position, quadrantal angle, co-terminal angles, sector, apparent size, sine, cosine, tangent, secant, cosecant, cotangent, reference angle, inverse trigonometric function, unit circle Page 11
Concepts: Competencies: Instructional Practices: Assessments: The student will know: Angles can be measured in degrees or radians. The Unit Circle and its usefulness in evaluating trigonometric expressions. Radian measures for common angle measures that appear on the Unit Circle and in the special right triangles. The ratios for the six basic trigonometric functions. When to use reference angles v. calculator approximations when evaluating a trigonometric expression. Characteristics of a trigonometric graph, including its general appearance, domain, and range The student will be able to: Find the measure of an angle in either degrees or radians and to find coterminal angles Find the arc length and area of a sector Define the six basic trigonometric functions and evaluate them by various means Graph sine, cosine, tangent, secant, cosecant, and cotangent functions Find values of inverse trigonometric functions and solve simple trigonometric equations Student created infographic of the six trigonometric functions and their properties Sine Wave Tracer activity on Geometer s Sketchpad Group activity in which students practice evaluating trigonometric expressions to find their exact and approximate solutions Grapher activity in which students compare the domain and range of a trigonometric function with its inverse Exit card on converting between radian and degree measure and finding coterminal angles Geometer s Sketchpad exploration activity on the Unit Circle Test on evaluating trigonometric expressions and solving trigonometric equations Graphing calculator activity in which students graph given trigonometric equations and must determine their domain and range Restriction of the domain of a trigonometric function allows its inverse to be constructed Page 12
Topic/Unit: Trigonometric Applications Big Ideas/Enduring Understandings: Periodic functions repeat with a predictable pattern. Changes in the equation of a trigonometric function cause changes in its graph. The characteristics of trigonometric functions are useful in solving real world problems. The trigonometric identities can be proven and are often useful when solving trigonometric equations. The Law of Sines and the Law of Cosines extend trigonometry to geometric applications. The sum, difference, double angle, and half angle formulas allow us to calculate exact values that can only be approximated with the use of a calculator. Essential Questions: How do period and amplitude connect the graph of a trigonometric function to its equation? How do transformations affect the parent graph of a trigonometric function? What types of real world situations display periodic behavior? Why do trigonometric graphs help us describe their behavior? When is it best to solve a trigonometric equation algebraically and when is it best to do so graphically? How is proving or verifying a trigonometric identity different than solving a trigonometric equation? What is an alternate means of finding area of a triangle? How is the Law of Sines derived from it? When is it necessary to use the Law of Sines or Law of Cosines to solve a triangle? How can we find the exact sine, cosine, and tangent values of angles besides those in the special right triangles or on the Unit Circle? PA Academic Standards (PA Common Core): CC.2.1.HS.F.5, CC.2.2.HS.D.1, CC.2.2.HS.D.7, CC.2.2.HS.D.8, CC.2.2.HS.C.4, CC.2.2.HS.C.7, CC.2.2.HS.C.8, CC.2.2.HS.C.9 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: angle of inclination, period, amplitude, co-functions Page 13
Concepts: Competencies: Instructional Practices: Assessments: The student will know: The relationship between a line s slope and the tangent ratio What changes occur to the graph of a trigonometric function when its equation is altered Real world situations can be modeled with trigonometric functions The basic trigonometric identities for sine, cosine, and tangent When to use Law of Sines v. Law of Cosines to solve a triangle The student will be able to: Find a line s slope given its angle of inclination Find the period, amplitude, translations, and equation of various sine and cosine curves Use trigonometric functions to model real world situations that mimic periodic behavior Simplify trigonometric expressions and prove trigonometric identities Solve complex trigonometric equations using trigonometric identities, algebraic means, and graphing technologies Geometer s Sketchpad activity exploring the connection between the slope of a line and the tangent ratio Law of Sines Geometer s Sketchpad activity exploring the ambiguous case SSA Worksheet to practice using the Law of Sines and Law of Cosines to solve a triangle Student-created tutorial explaining the sum, difference, double angle, and half angle formulas Constructed response questions in which students model real word situations such as high and low tide using trigonometric functions Student generated trigonometric art activity Test on simplifying and proving trigonometric identities and solving trigonometric equations Solving real word application problems pertaining to area of land and distance traveled by means of Law of Sines and Law of Cosines Graded assignment on sum, difference, double angle, and half angle formulas The sum, difference, double angle, and half angle formulas for sine, cosine, and tangent Find area of a triangle using two sides and an included angle Use Law of Sines and Law of Cosines to find unknown parts of a triangle Page 14
Derive and apply sum, difference, double angle, and half angle formulas for sine and cosine Page 15
Topic/Unit: The Polar Coordinate System Big Ideas/Enduring Understandings: A complex number can be represented in both rectangular and polar form, but its value does not change. Polar equations may be more useful in certain situations than rectangular equations. De Moivre s Theorem enables us to perform operations with complex numbers. Essential Questions: What are polar coordinates? How do they differ from rectangular coordinates? How can we view the coordinate plane in a different light when graphing polar coordinates? Why do polar graphs not follow the rules of functions? PA Academic Standards (PA Common Core): CC.2.1.HS.F.1, CC.2.1.HS.F.6, CC.2.1.HS.F.7, CC.2.2.HS.D.2 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: pole, polar axis, polar coordinates, rectangular form, polar form, polar equation, Argand diagram, complex plane Concepts: Competencies: Instructional Practices: Assessments: The student will know: Polar coordinates and how they differ from rectangular coordinates. The formulas for conversion between polar and rectangular coordinates. The student will be able to: Convert between polar and rectangular coordinates. Graph polar equations. Video tutorial and practice on converting between polar and rectangular coordinates. Graphing calculator group activity exploring polar graphing. Students will create and sketch an original design utilizing polar equations. Test on converting between polar and rectangular coordinates and finding powers and roots of complex numbers. Page 16
Types of curves associated with polar equations. De Moivre s Theorem and its connection to powers and roots of complex numbers. Write complex numbers in polar form and find products. in polar form Use De Moivre s Theorem to find powers of complex numbers. Find roots of complex numbers. Paired practice finding powers and roots of complex numbers Page 17
Topic/Unit: Sequences & Series Big Ideas/Enduring Understandings: An arithmetic sequence is created by repeatedly adding a constant to an initial number. A geometric sequence is created by repeatedly multiplying an initial number by a constant. A series is the sum of the numbers in a sequence. Sequences and series model real world problems and their solutions. Essential Questions: How do you determine the difference between an arithmetic and geometric sequence? What is the difference between a recursive and an explicit formula? How do you calculate the sum of an infinite series? How do sequences and series model real world situations? PA Academic Standards (PA Common Core): CC.2.2.HS.D.2, CC.2.2.HS.C.3 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: sequence, arithmetic sequence, geometric sequence, series, initial condition, recursive definition, explicit definition, limit, converge, diverge, sigma notation Page 18
Concepts: Competencies: Instructional Practices: Assessments: The student will know: The difference between a sequence and a series. Whether a sequence is arithmetic or geometric. The difference between a recursive and an explicit definition of a sequence. Formulas for the sums of finite arithmetic and geometric sequences. The student will be able to: Identify an arithmetic or geometric sequence and find a formula for its nth term. Use sequences defined recursively to solve problems. Find the sum of the first n terms of arithmetic or geometric series. Find the sum of an infinite geometric series. Investigating sequences activity using the graphing calculator. Khan Academy practice on finding the next term in a sequence and selecting the appropriate explicit formula. Student created online flashcards for the sequence and series formulas. Short essay explaining the difference between a sequence and a series, how one distinguishes between an arithmetic and a geometric sequence, and how one finds the sum of an infinite series Test on developing the explicit formula from an arithmetic or geometric sequence and finding the sum of a series. Sigma and factorial notation. Represent series using sigma notation. Page 19