New Jersey Student Learning Standards for Mathematics:

Transcription

1 Honors Pre-Calculus Course Description and Philosophy The Pre-Calculus course presented at the high school is an accelerated course intended to serve those students who have successfully completed the course Algebra 2 Honors. The purpose of the course is to provide advanced analysis of functions including topics in algebra, geometry, trigonometry and limit theory. Much attention is given to the understanding and nature of proof. The course is geared at preparing the student for their first course in Calculus or other collegiate mathematics courses. Pre-Calculus Honors challenges students to master mathematical concepts required as a foundation for the future study of Calculus and encourages students to become critical and independent mathematical problem solvers and to reason mathematically using higher order thinking. Further, students will use current technology such as the graphing calculator as well as other available computer software such as desmos and geogebra. The applications in this course will reinforce skills students will carry through to adulthood and utilize in a society increasingly dominated by technology and quantitative methods. Teacher Resources: New Jersey Student Learning Standards for Mathematics: NJSLS Mathematics Crosswalk: State of New Jersey Department of education Curricular Frameworks for Mathematics: Additional resources provided by the Common Core Content Standards that may be utilized: Illustrative Mathematics Content Standards: High School The Teaching Channel (Common Core Math Channel) The Mathematics Common Core Toolbox ( has both sample scope and sequence documents as well grades 4-12 PARRC assessment tasks. Reference: Larson, Ron. Pre-Calculus with Limits, 2014, 2011, 2007 Brooks/Cole, Cengage Learning, Boston MA/ USA 1

2 Chapters 1-3: (Review) The first three chapters were covered and a part of the Algebra II and Algebra II Honor curriculum the prior year. The units will serve as a review in a summer assignment packet and reviewed in class the first week of school to serve as a needs assessment. Chapter 1- Functions and Their Graphs 1.1 Rectangular Coordinates : A-CED A. 1., F-IF C 7, FIF-C Graphs of Equations: A-CED A2, A-SSE, F-IF A-1, F-IF A 2, F-IF C7, 1.3 Linear Equations in Two Variables: A-CED-A2, F-IF B 4, F- LE-1, F-LE Functions: F-IF A 1, F- IF A-2, F- IF B-2, F-IF B 5, 1.5 Analyzing Graphs of Functions: A- SSE-1, A-APR B3, F-IF, B-4, F-IF B6, F-BF B A Library of Parent Functions: F- BF 3, F-IF-C9, F-IF C 9, F-IF C 7a,b. 1.7 Transformation of Functions: F- BF B3, 1.8 Combinations of Functions: Composite Function: F-BFB 4b, A- APR, 1.9 Inverse Functions: F-BF4, F-BF 1.10 Mathematical Modeling and Variation: FLE 1 Chapter 2- Polynomial and Rational Functions 2.1 Quadratic Function and Models ASSE- 1, F-IF 7, A-REI- 4, 2.2 Polynomial Functions of Higher Degree ASSE-B3, A-APR b2,3., F-IF- B4, F-IF C 7, 2.3 Polynomial and Synthetic Division A-REI A, F-IF 7, 2.4 Complex numbers N-CN-1, N-CN-2 (Common Core) 2.5 Zeros of Polynomial Functions A-APR B3, F-IF 7c 2.6 Rational Functions A-APR D6, A-APR D7, F-IF C 7c, 2.7 Nonlinear Inequalities A-REI B3, A-REI C6, A-REI D Chapter 3- Exponential and Logarithmic Functions 3.1 Exponential Functions and Their Graphs: F-IF C 7 e, F-IF C 8b., F-LE 2. F-LE Logarithmic Functions and Their Graphs; F-IF C 7 e, F-IF C 8b., F-LE A c, 3.3 Properties of Logarithmic functions F-IF- 7e. 3.4 Exponential Functions and Logarithmic Equations F-BF B5, F- LE Exponential and Logarithmic Models F-LE 2, F-LE-3.

3 Essential Question: Unit I (REVIEW FUNCTIONS AND THEIR GRAPHS) Why are relations and functions represented in multiple ways? Unit II (Review Polynomial and Rational Functions) How do polynomial functions model real-world problems and their solutions? Unit III (Review Exponential and Logarithmic Functions) How do I use exponential models to solve problems? (ie.compound interest, Rate of Growth and Decay). Objectives: Unit 1 Students will be able to Represent functions and relations numerically, graphically, and algebraically. Describe and apply properties of functions and relations. Perform a variety of operations and geometrical transformations on functions and relations. Use numerical, algebraic, and graphical representations of functions and relations in order to solve realworld problems Objectives: Unit 2 Students will be able to Analyze and graph polynomial and rational functions Identify key characteristics while creating graphs of quadratic and polynomial functions. Use long and synthetic division to find real, imaginary and complex roots. Find asymptotes, intercepts and holes associated with rational functions. Solve nonlinear inequality problems. Objectives: Unit 3 Students will be able to Write, graph and interpret exponential and logarithmic functions. Recognize the basic characteristics of exponential and logarithmic functions. Model real world problems Solve equations involving logarithmic and exponential problems. Assessments Resources Instructional Method Tech Infusion NJ Student Learning Standards for Mathematics Units I III (Review) Class Shown above 3

4 Differentiated Learning Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery. Ethical Decision Making/Character Education Discuss the history of the quintic solution including the issue of plagiarism regarding Ferrari, Cardano, and Tartaglia. 21st Century Skills Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Applications that may be utilized. Data Analysis: Mail Population Statistics College Enrollment Cost, Revenue, and Profit Digital Music Sales Fluid Flow Fuel Use Consumer Awareness Diesel Mechanics Path of a Diver Tree Growth Lyme Disease Impedance Candle Making Nuclear Reactor Accident Human Memory Model Sound Intensity Trees per Acre Earthquakes 4

5 Unit IV: Trigonometric Functions Essential Question: How do trigonometric and circular functions model real- world problems and their solutions? How are the six trigonometric and circular functions related to each other? Objectives: Students will be able to Evaluate and graph trig functions, their inverses, and reciprocals. Find trigonometric ratios in a right triangle and on the unit circle. Apply the problems in real life examples. Lesson 4.1 Radian and Degree Measure Convert degree measure to radian measure. Measure indirectly using trigonometric relationships. Determine the angular and linear velocities of an object moving at a constant speed on a circular path Describe and apply the relationship between the radian measure of a central angle of a circle and its intercepted arc. Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core) F-TF A1 (e.g. (e.g. (e.g. (e.g. 5

6 Evaluate a trigonometric expression using radian measure. (e.g. Lesson 4.2 Trigonometric Functions the Unit Circle Describe and apply the relationship between the trigonometry of the right triangle and the unit circle. Interpret, evaluate and solve problems involving trigonometric functions. Use domain and period to evaluate sine and cosine functions Lesson 4.3 Right Triangle Trigonometry Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core) F-TF A2, F- TF A3, F-TF A4 (e.g. (e.g. (e.g. Assessment Materials Instructional Method Tech Infusion NJSLSFM Common Core F-TF A 3, F-TF C 8 G- SRT C 8 6

7 Define and graph the six circular functions. Apply the fundamental trigonometric identities. Solve trigonometric equations. Interpret and solve problems involving trigonometric functions. (e.g. (e.g. (e.g. (e.g. Chapter 4.4 Trigonometric Functions of Any Angle Evaluate trigonometric functions of any angle Find the reference angles on the unit circle Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core) F-TF A 2, F-TF A3, G- SRT C 6 7 (e.g.

8 Evaluate trigonometric functions of real numbers Chapter 4.5 Graphs of Sine and Cosine Functions Sketch the graphs of basic sine and cosine functions Determine the period, amplitude, phase shift, and/or vertical shift of a trigonometric function represented graphically or algebraically. Describe the effect of transformations on the graphs of trigonometric functions. Use sine and cosine functions to model real life situations (e.g. (e.g. Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core) FTF B 5 Class (e.g. (e.g. (e.g. (e.g. 8

9 Chapter 4.6 Graphs of Other Trigonometric Functions Sketch the graphs of tangent, cotangent, secant, and cosecant functions Sketch the graphs of damped trigonometric functions Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core0 F-TF- B5 (e.g. (e.g. Chapter 4.7 Inverse Trig Functions Assessment Materials Instructional Method Tech Infusion NJSLSFM (Common Core) F-TF- B6 F-TF- B7 Write the value of an inverse trigonometric expression in radians Evaluate the graph of inverse and sine functions (e.g. (e.g. 9

10 Evaluate the compositions of trigonometric functions (e.g. Chapter 4.8 Applications and Models Solve real-life problems involving right triangles Solve real life problems involving directional bearing Solve real-life problems involving harmonic motion Assessment Materials Instructional Method Tech Infusion NJSLSFM (CCCS) F-TF- B5 G-SRT C8 (e.g. (e.g. Assessment Materials Instructional Method Tech Infusion Differentiated Instruction: Strategic Learner: Use visual models and manipulatives. Use graphing calculator and. Advanced Learner: Emphasize proof and foundations. Ethical Decision Making/Character Education Review Honor Code and its implication for Midterm exams. Discuss/review policy on group work vs. individual. 21st Century Skills Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: 10

11 Applications that may be utilized: Television coverage Respiratory cycles Skateboarding ramps Waterslide design Meteorology Unit V: Analytic Trigonometry Essential Question: How are the algebraic properties related to the trigonometric functions? Objectives: Students will be able to Apply strategies for simplifying expressions and solving equations by using trigonometric identities. Solve trigonometric equations written in quadratic form and equations containing more than one angle. Rewrite trigonometric expressions that contain functions of multiple or half-angles and that involve squares or products of trigonometric functions. Lesson 5.1 Using Fundamental Identities Recognize and Write the Fundamental Trig Identities Use the fundamental trig identities to evaluate trig functions, simplify trig expressions and rewrite trig expressions. Assessment Materials/Resources Instructional Method Tech Infusion F-TF - C 11

12 Lesson 5.2 Verifying Trigonometric Identities Applying Techniques for verifying and solving trigonometric identities. Assessment Materials/Resources Instructional Method Tech Infusion F-TF - C Lesson 5.3 Solving Trigonometric Equations Using standard algebraic techniques to solve trigonometric equations Solving trigonometric equations of quadratic types. Solve trigonometric equations involving multiple angles. Assessment Materials/Resources Instructional Method Tech Infusion F-TF C 8,9 A-REI-A1 F-TF-B7 12

13 Using inverse trigonometric functions to trigonometric equations Lesson 5.4 Sum and Difference Formulas Using sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations Assessment Materials/Resources Instructional Method Tech Infusion F-TF-B7 Lesson 5.5 Multiple Angle and Product-to-Sum Formulas Use multiple-angle formulas to rewrite and evaluate trigonometric functions Use power-reducing formulas to rewrite and evaluate trigonometric functions Assessment Materials/Resources Instructional Method Tech Infusion FTF- C9 13

14 Use half-angle formulas to rewrite and evaluate trigonometric functions Use trigonometric formulas to rewrite real life models Differentiated Learning : Use the graphic calculator and smartboard to explore various graphs and manipulations of the graphs. 21st Century Skills: Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Standing waves Honey comb cell Friction Projectile Motion Shadow length Unit VI: Additional Topics in Trigonometry Essential Questions: What are the applications of the Law of Sines and Law of Cosines, vectors and dot products? To what extent can right triangle trigonometry be generalized to all triangles? Objectives: Students will be able to Apply trigonometry to oblique triangles, vectors and complex numbers Find side lengths, angles, and areas of oblique triangles Use trigonometry to solve real world problems Write and perform operations on complex numbers in trigonometric form. 14

15 Lesson 6.1 Law of Sines Use the Law of Sines to solve oblique triangles Find the areas of oblique triangles Use the Law of Sines to solve real life problems Assessment Materials/Resources Instructional Method Tech Infusion G-SRT-D- 10,11 Lesson 6.2 Law of Cosines Use the Law of Cosines to solve oblique triangles Use the Law of Cosines to solve real life problems Use Heron s Area formula to find the area of the triangle. Assessment Materials/Resources Instructional Method Tech Infusion G-SRT-D- 10,11 15

16 Lesson 6.3 Vectors in the Plane Represent vectors as directed line segments Write the component form of vectors Perform basic vector operations and represent them graphically Write vectors as linear combinations of unit vectors Find the direction angles of vectors Solved applied problems involving vectors Assessment Materials/Resources Instructional Method Tech Infusion N-VM-A- 1,2,3, N-VM-B-4,5 16

17 Lesson 6.4 Vectors and Dot Products Find the dot product of two vectors and use the properties of the dot product Find the angle between two vectors and determine whether two vectors are orthogonal. Write a vector as the sum of two vector components Use vectors to find the work done by a force Assessment Materials/Resources Instructional Method Tech Infusion N-VM-A- 1,2,3, N-VM-B-4,5 Lesson 6.5 Trigonometric Forms of a Complex Number Plot complex numbers in the complex plane and find absolute values of complex numbers Assessment Materials/Resources Instructional Method Tech Infusion N-CN-A N-CN-B 17

18 Write the trigonometric form of complex numbers Multiply and divide complex numbers written in trigonometric form Use DeMoive s Theorem to find power of complex numbers Find the nth roots of complex numbers Differentiated Learning Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery. 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Navigation Surveying Electrical engineering Engine design Applications that may be utilized: 18

19 Unit VII: Systems of Equations and Inequalities Essential Questions: How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? Objectives: Students will be able to Solve systems of equations and inequalities using various methods. Find the solution of a system of inequalities and use linear programming to optimize a linear function. Lesson 7.1 Linear and Non Linear Systems of Equations Use the method of substitution to solve systems of linear and non linear equations in two variables Use a graphical approach to solve systems of equations of two variables Solve applied problems using systems of equations Lesson 7.2 Two-Variable Linear Systems Use elimination to solve systems of linear equations in two variables Assessment Materials/Resources Instructional Method Tech Infusion A-CED- A- 2,3 A-REI-C 19 Assessment Materials/Resources Instructional Method Tech Infusion A-CED-A- 2,3 A-REI-C

20 Interpret graphically the number of solutions of systems of linear equations in two variables Solve applied problems using systems of linear equations in two variables Lesson 7.3 Multivariable Linear Systems Use back-substitution to solve linear systems in row-echelon form Solve nonsquare systems of linear equations Solve applied problems using systems of linear equations in three or more variables Lesson 7.4 Partial Fractions Recognize partial fraction decompositions of rational expressions Assessment Materials/Resources Instructional Method Tech Infusion A-CED-A- 2,3 A-REI-C 20 Assessment Materials/Resources Instructional Method Tech Infusion A-APR-D- 6,7

21 Find partial fraction decompositions of rational expressions Lesson 7.5 Partial Fractions Sketch the graphs of inequalities in two variables Solve systems of inequalities Solve applied problems using systems of inequalities in two variables Assessment Materials/Resources Instructional Method Tech Infusion A-CED-A-3 A-REI-C Lesson 7.6 Linear Programming Solve linear programming problems Assessment Materials/Resources Instructional Method Tech Infusion 21 A-CED-A-3 A-REI-D

22 Solve applied problems using linear programming Differentiated Learning Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery. 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Global positioning satellite systems Traffic flow Nutrition Fuel Mixture Applications that may be utilized: Unit VIII: Matrices and Determinants Essential Question: What are the similarities and differences between matrices and real numbers and how can they be used to represent data? Objectives: Students will be able to Write matrices, perform matric operations, use matrices to solve problems. Solve a system of equations by writing and operating on the system in matrix form Find the inverse of a matrix and use the inverse to solve systems of equations Find the determinant of a square matrix to apply matrices to applications including Cramer s Rule, finding area of triangles and encoding messages. 22

23 Lesson 8.1 Matrices and Systems of Equations Write matrices and identify their orders Perform elementary row operations on matrices Use matrices and elimination methods to solve systems of linear equations Lesson 8.2 Operations with Matrices Decide whether two matrices are equal Add and subtract matrices and multiply matrices by scalars Multiply two matrices Assessment Materials/Resources Instructional Method Tech Infusion NVM-C A-REI-C5 23 Assessment Materials/Resources Instructional Method Tech Infusion NVM-C- 6,7,8,9

24 Solve applied problems using matrix operations Lesson 8.3 The Inverse of a Square Matrix Verify that two matrices are inverses of each other Use elimination methods to find the inverses of matrices Find the inverses of a 2 x 2 matrices Use inverse matrices to solve systems of linear equations Assessment Materials/Resources Instructional Method Tech Infusion NVM-C- 6,7,8,9 A-REI- C-9 24

25 Lesson 8.4 The Determinant of a Square Matrix Find the determinant of 2 x 2 matrix Find minors and cofactors of square matrices Find the determinants of square matrices Assessment Materials/Resources Instructional Method Tech Infusion NVM-C Lesson 8.5 Applications of Matrices and Determinants Use Cramer s Rule to solve systems of linear equations Use determinants to find the areas of a triangle Assessment Materials/Resources Instructional Method Tech Infusion NVM- C 25

26 Use a determinant to test for collinear points and find an equation of a line passing through two points Use a matrices to encode and decode messages Differentiated Learning : Strategic Learner: Encourage students to make each calculation aloud as they work the problems. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Students may create their own tutorials via Youtube to help peers and to demonstrate their own mastery. 21st Century Skills: Problem-Based Assignment: Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Sudoku Data encryption Health and wellness Number of calories burned in exercise Unit IX: Sequences, Series and Probability Essential Question: What are the types of real-life situations where sequences & series that can be used as models and prediction tools? Objectives: Students will be able to Analyze sequences and series, expand binomials and determine the probability of events Model and find sums of arithmetic and geometric sequences Expand binomials Apply probability to count possible outcomes and determine the probability of multiple events 26

27 Lesson 9.1 Sequences and Series Use sequence notation to write the terms of sequences Use factorial notation Use summation notation to write sums and find the sums of series Solve applied problems using sequences and series Assessment Materials/Resources Instructional Method Tech Infusion S-CP-A,B Lesson 9.2 Arithmetic Sequences and Partial Sums Recognize, write, and find the nth terms of arithmetic sequences Assessment Materials/Resources Instructional Method Tech Infusion F-IF-3 FBF-2 27

28 Find nth partial sums of arithmetic sequences Solve applied problems using arithmetic sequences Lesson 9.3 Geometric Sequences and Series Recognize, write and find the nth terms of geometric sequences Find the sum of a finite geometric sequence and infinite geometric series Solve applied problems using geometric sequences Assessment Materials/Resources Instructional Method Tech Infusion F-IF-3 FBF-2 28

29 (Not in the NJSLSFM) Lesson 9.4 Mathematical Induction Use mathematical induction to prove statements involving a positive integer n. Use pattern recognition and mathematical induction to write a formula for the nth term of a sequence Find the sums of powers of integers Find finite differences of sequences Lesson 9.5 The Binomial Theorem Use the Binomial Theorem to calculate binomial coefficients Use Pascals Triangle to calculate binomial coefficients Assessment Materials/Resources Instructional Method Tech Infusion 29 Assessment Materials/Resources Instructional Method Tech Infusion A-APR-C-5

30 Use Binomial coefficients to write binomial expansions Lesson 9.6 The Counting Principals Solve simple counting problems Use the Fundamental Counting Principal to solve counting problems Use permutation and combinations to solve counting problems Lesson 9.7 Probability Find the probabilities of events Find the probabilities of mutually exclusive and independent events Assessment Materials/Resources Instructional Method Tech Infusion S-CP-B 30 Assessment Materials/Resources Instructional Method Tech Infusion S-CP- A,B

31 Find the probability of the complement of any event Differentiated Learning Strategic Learner: Encourage students to write out factorial in their calculations until they have mastered the basics. Advanced Learner: Investigate various computer software that implement the unit demonstrate lesson the class. 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Health Cases Dominoes Electricity Tossing Die Population Modeling Applications that may be utilized: Unit X: Topics and Analytic Geometry Essential Question: How can the understanding of conic sections make more sense of the constructions and designs in our world? Objectives: Students will be able to Understand conic sections and equations in parametric and polar form Solve problems involving conic sections Understand parametric and polar forms for writing and graphing equations Using polar coordinates to represent and solve problems involving conic sections 31

32 Lesson 10.1 Lines Find the inclination of a line Find the angle between two lines Find the distance between a point an a line Lesson 10.2 Introduction to Conics: Parabolas Recognize a conic as the intersection of a plane and a double-napped cone Write equations of parabolas in standard form Solve applied problems using reflective properties of parabolas Assessment Materials/Resources Instructional Method Tech Infusion F-IF-C G-GPE-B 32 Assessment Materials/Resources Instructional Method Tech Infusion G-GPE-A-2 A-REI-B4

33 Lesson 10.3 Ellipses Write equations of ellipses in standard form and graph ellipses Solve applied problems using properties of ellipses Find eccentricities of ellipses Assessment Materials/Resources Instructional Method Tech Infusion G-GPE-A-3 Lesson 10.4 Hyperbolas Write equations of hyperbolas in standard form Find asymptotes of and graph hyperbolas Assessment Materials/Resources Instructional Method Tech Infusion G-GPE-3 33

34 Solve applied problems using properties of hyperbolas Classify conics from their general equations Lesson 10.6 Parametric Equations Evaluate sets of parametric equations for given values of the parameter Sketch curves that are represented by sets of parametric equations Rewrite sets of parametric equations as single rectangular equations by eliminating the parameter Find sets of parametric equations for graphs Assessment Materials/Resources Instructional Method Tech Infusion 34

35 Lesson 10.7 Polar Coordinates Plot points in the polar coordinate system Convert points from rectangular to polar form and vice versa Convert equations from rectangular to polar form and vice versa Assessment Materials/Resources Instructional Method Tech Infusion N-CN-B-4 Lesson 10.8 Graphs of Polar Equations Graph Polar Equations by point plotting Use symmetry, zeros, and maximum r-values to sketch graphs of polar equations Assessment Materials/Resources Instructional Method Tech Infusion N-CN-B-4 35

36 Recognize special polar graphs Lesson 10.8 Graphs of Polar Equations Graph Polar Equations by point plotting Use symmetry, zeros, and maximum r-values to sketch graphs of polar equations Recognize special polar graphs Lesson 10.9 Polar Equations of Conics Define conics in terms of eccentricity, and write and graph equations of conics in polar form Assessment Materials/Resources Instructional Method Tech Infusion N-CN-B-4 Assessment Materials/Resources Instructional Method Tech Infusion N-CN-B-4 36

37 Solve applied problems using equations of conics in polar form Differentiated Learning Strategic Learner: Break problems into distinct steps, verbalizing the reasoning process. Encourage the use of the graphing calculator. Students should refer to Khan tutorials for additional support. Advanced Learner: Pose journal problems. Discuss ill-posed problems. Students may create their own tutorials to help peers and to demonstrate their own mastery. 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Radio telescopes Nuclear cooling towers Satellite orbit Radio waves from space Unit XI: Analytical Geometry in Three Dimensions Applications that may be utilized: Essential Questions: What are the characteristics of a three-dimensional coordinate system and how do you use it to solve problems. Objectives: Students will be able to Describe and analyze points, vectors, lines, and planes in three-dimensional space. Analyze the relationship between two or three vectors in space by calculating quantities such as the dot or cross products Study parametric and symmetric equations and compare lines and planes in space. 37

38 Lesson 11.1 The Three Dimensional Coordinate System Plot points in the three dimensional coordinate system Find distances between points in space and find midpoints of line segments joining points in space Write equations of spheres in standard form and find traces of surfaces in space Assessment Materials/Resources Instructional Method Tech Infusion Lesson 11.2 Vectors in Space Find the component forms of the unit vectors in the same direction of, the magnitude of, the dot product, and the angles between vectors in space Determine whether vectors in space are parallel Assessment Materials/Resources Instructional Method Tech Infusion N-VM-A,B 38

39 Solve applied problems using vectors in space Lesson 11.3 The Cross Product of Two Vectors Find the cross products of vectors in space Use geometric properties of cross products of vectors in space Use triple scalar products to find volumes of parallelepipeds Assessment Materials/Resources Instructional Method Tech Infusion N-VM-A,B Lesson 11.4 Lines and Planes in Space Find parametric and symmetric equations of lines in space Assessment Materials/Resources Instructional Method Tech Infusion A-CED-A 39

40 Find equations of and sketch planes in space Find distance between points and planes in space Differentiated Learning Strategic Learner: Students that can construct a three dimensional system and refer to it as they work through examples. Advanced Learner: Integrate problems into graphic designs on the computer. 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Product design Physical forces such as tension in cables Torque on the crank of a bicycle brake Model the spherical shape of the earth. Applications that may be utilized: Unit XII: Limits and Introduction to Calculus Essential Questions: a. What is a limit? b. How do we represent limits graphically? c. What techniques are most appropriate to find a limit? 40

41 Objectives: Students will be able to Find tangent lines of a function and the area of a region. Find the slope of a graph at a given point and calculate a derivative of a function. Find the limit of functions at infinity and the limits of sequences. Lesson 12.1 Introduction to Limits Use the definition of limit to estimate limits Determine whether limits of functions exists Use Properties of Limits and Direct Substitution to evaluate limits Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B, C Lesson 12.2 Techniques for Evaluating Limits Use the dividing out technique to evaluate limits of functions. Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B,C 41

42 Use the rationalizing technique to evaluate limits of functions Evaluate limits of functions graphically and numerically Evaluate one sided limits of functions Evaluate limits from calculus Lesson 12.3 The Tangent Line Problem Use a tangent line to approximate the slope of a graph at a point. Use the limit definition of slope to find the exact slope of graphs Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B,C 42

43 Find the derivative of functions and use derivative to find the slopes of graphs Lesson 12.4 Limits at Infinity and Limits of Sequences Evaluate limits of functions at infinity Find limits of sequences Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B,C Lesson 12.5 The Area Problem Find limits of summations Use rectangle to approximate areas of plane regions Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B,C 43

44 Use limits of summations to find areas of plane regions Differentiated Learning Strategic Learner: Students find the limits numerically through the use of a table. Advanced Learner: Students will identify the limit graphically of different graphs 21st Century Skills: Problem-Based Assignment Students use graphing calculators, computer algebra systems, or both, to assist with some applications recommended below: Land area Maximum volume Velocity of free-falling object Path a ball travels Applications that may be utilized: Unit V: Advanced Calculus Essential Question: After learning how to calculate derivatives, how can we use them to solve problems involving rates of change and the approximation of functions? Objectives: Students will be able to: Present the derivative graphically, numerically, and analytically Interpret the derivative as an instantaneous rate of change Define the derivative as the limit of the difference quotient Understand the difference between differentiability and continuity 44

45 Find the slope of a curve at a point Find the tangent line to a curve Recognize instantaneous rate of change as the limit of average rate of change Approximate rate of change from graphs and tables Describe the corresponding characteristics of the graphs of f and f Optimize models; Model rates of change, including related rates Use implicit differentiation to find the derivative of an inverse function Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Compute derivatives of basic functions Apply the rules for the derivative of sums, products, and quotients of functions Use the chain rule and differentiate implicitly Calculus The Area Problem Derivatives Derivatives of Trig Functions Assessment Materials/Resources Instructional Method Tech Infusion F-IF A,B,C 45

46 The Chain Rule Implicit Differentiation Linearization and Differentials Differentiated Learning Strategic Learner: Use Java applets to illustrate concepts involving differentiation. Advanced Learner: Develop proofs for several theorems on differentiation. Analyze functions whose derivatives are not continuous. 46