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Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 1.10, 3.3 Knowledge: (MA) 2 MACLE: See below NETS: 3d; 4c; 6b DOK: 1,2 • Functions • Trigonometric Functions (F-TF) • Extend the domain of trigonometric functions using the unit circle (F-TF.1-3) Standards Learning Targets F-TF.1-3 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle 3. Use special triangles to determine geometrically the values of sine, ! ! ! cosine, and tangent for ! , ! , and ! , and use the unit circle to express the values of sine, cosine, and tangent for ! − !, ! + !, and 2! − ! in terms of their values for x, where x is any real number Unit A: Trigonometric Functions – Unit Circle Approach: Know and understand radian measure and how it relates to the unit circle. Develop the trigonometric functions from the special angles on the unit circle • • • • • Sketch angles in standard position in degrees and radian Name coterminal angle degrees and radians Convert degrees to radians and vice versa Find the length of a circular arc given the radius and subtended angle Find the linear and angular speeds of an object traveling in a circular motion CCSS: F-TF.1 MACLE: N/A • • • Find coordinates associated with “familiar” angles Find values of the six trig functions using the unit circle and a calculator Find values of the six trig functions for points on terminal rays of an angle CCSS: F-TF.1-3 MACLE: N/A Approved 7-15-13 Revised 2013 1 Math Analysis/Trigonometry Instructional Strategies • • • • • • • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Demonstrations: e.g., Twisty-Tie activity to explain the concept of radians Problem solving such as finding: • the distance between cities • linear and angular speeds of rotating objects Reflective discussion Class discussion Computer assisted instruction Games: e.g., “Unit Circle Jeopardy” to practice finding trignonometric values Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Quizzes: Function Friday Quiz #1 • Homework assignments: See attached pacing guide • Formal common assessment: Unit A test Mastery Level: 80% Instructional Resources/Tools • • • Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website(s): Finding trig values practice: http://www.mathmistakes.info/facts/TrigFacts/index.html Graphing calculator Approved 7-15-13 Revised 2013 2 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster • • • • Functions Trigonometric Functions (F-TF) Extend the domain of trigonometric functions using the unit circle (F-TF.4) Model periodic phenomena with trigonometric functions (F-TF.5) Alignments: CCSS: See below Performance: 1.6, 1.10 Knowledge: (MA) 2,4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1,2 Standards F-TF.4, 5 4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline Learning Targets Unit B: Graphing the Six Basic Trigonometric Functions: Graph and interpret graphs of the six basic trigonometric functions. Understand periodicity, amplitude, and midline • • • • • Graph the six basic trigonometric functions Determine the domain, range, period, and amplitude for the six basic trigonometric functions Graph the six trigonometric functions with transformations Determine the transformations for any trigonometric function Write the equation of a sinusoidal function given a graph or specified information CCSS: F-TF.4,5 MACLE: N/A Approved 7-15-13 Revised 2013 3 Math Analysis/Trigonometry Instructional Strategies • • • • • • • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Demonstrations: Ann Fibian activity – students will create a sinusoid function using collected data Problem solving: e.g., Ferris Wheel problem Reflective discussion Class discussion Computer assisted instruction Station review activity Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Class activities: • Biorhythms lab: Students will determine their biorhythms for the current month and make predictions • Stayed – Tuned lab (optional: if time allows) • Quizzes: Function Friday quizzes #2 and #3 • Homework assignments: see attached pacing guide • Formal common assessment: Unit B test Mastery Level: 80% Instructional Resources/Tools Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website(s): • This java applet graphs functions based on unit circle values and the students can explore the effects of “a” values • http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php • Graphing calculator • • Approved 7-15-13 Revised 2013 4 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 3.5 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1-3 • Functions • Trigonometric Functions (F-TF) • Prove and apply trigonometric identities. (F-TF.8, 9) Standards Learning Targets F-TF.8, 9 8. Prove the Pythagorean identity !"#! ! + !"# ! ! = 1 and use it to find !"# ! , !"# ! , !" !"# ! given !"# ! , !"# ! , !" !"# ! and the quadrant of the angle 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems Unit C: Analytic Trigonometry: Develop and use trigonometric identities to simplify trigonometric expressions. Verify trigonometric identities algebraically • • • • Use trigonometric identities to simplify expressions Verify/prove trigonometric identities algebraically Use sum/difference, double angle, and half angle identities to determine exact values of trig functions Use trigonometric identities and algebraic techniques to solve trigonometric equations CCSS: F-TF.8,9 MACLE: N/A Instructional Strategies • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Approved 7-15-13 Revised 2013 5 Math Analysis/Trigonometry Demonstrations: • Sum and Difference Identity activity • “Sinbad and Cosette” story • Problem solving: e.g., Use identities to find exact values of non-familiar angles • Reflective discussion • Class discussion • Computer assisted instruction • Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Identities Matching activity • Quizzes: Function Friday quiz #4 and #5 • Homework assignments: See attached pacing guide • Formal common assessment: Unit C test Mastery Level: 80% Instructional Resources/Tools • • • Textbook: Prentice Hall Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website(s): e.g., Trig Identities: http://calculustricks.com/lessons/trig-identities/ Graphing calculator Approved 7-15-13 Revised 2013 6 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 1.10, 3.7 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1-3 • Functions • Trigonometric Functions (F-TF) • Model periodic phenomena with trigonometric functions. (F-TF.6, 7) Standards F-TF.6, 7 6. Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed 7. Use inverse functions to solve trigonometric equations that arise in modeling context; evaluate the solutions using technology, and interpret them in terms of the context Learning Targets Unit D: Solving Equations and Inequalities: Solve trigonometric equations and inequalities. Apply the concepts of reference angles, principal values, and general solutions to solve trigonometric equations within a modeling context • • • • • Solve trigonometric equations over one period, and solve trigonometric equations or inequalities for all real solutions Determine the reference angle for any given angle Use trigonometric identities and algebraic techniques to solve trigonometric equations Find the exact values of the six trigonometric functions given a point on the terminal side of an angle, then find the angle and its reference angle Use a calculator to find the approximate value of an inverse trigonometric function CCSS: F-TF.7 MACLE: N/A • Evaluate inverse trig expressions and give principal and/or general exact values CCSS: F-TF.6 MACLE: N/A Approved 7-15-13 Revised 2013 7 Math Analysis/Trigonometry Instructional Strategies Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet • Drill and guided practice • Demonstration – Graphing: • sin -1 x • cos -1 x • tan -1 x on a calculator to visualize principal values • Problem solving: • Yo-Yo problem • Plant germination • Reflective discussion • Class discussion • Computer assisted instruction • Activity: • Mission Impossible – Students will work collaboratively to solve application problems • Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Quizzes • Homework assignments: See attached pacing guide • Formal common assessment: Unit D test Mastery Level: 80% Instructional Resources/Tools • • • Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website: http://mathworld.wolfram.com/InverseTrigonometricFunctions.html Graphing calculator Approved 7-15-13 Revised 2013 8 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 1.10, 3.7 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1-3 • • • • • • Geometry Number and Quantity Similarity, Right Triangles, and Trigonometry (G-SRT) Vector and Matrix Quantities (N-VM) Apply trigonometry to general triangles. (G-SRT.9-11) Represent and model with vector quantities. (N-VM.1-3) Standards G-SRT.9-11 ! 9. Derive the formula ! = ! !"#$% ! for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side 10. Prove the Laws of Sines and Cosines and use them to solve problems 11. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces) N-VM.1-3 1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, ! , ! , v) 2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point 3. Solve problems involving velocity and other quantities that can be represented by vectors Learning Targets Unit E: Solving Triangles: Solve right and non-right triangles using the Law of Sines or Cosines. Recognize and use vector quantities to solve problems • • • Solve right triangles using trigonometry as it applies to real world situations Use the Law of Sines and Cosines to solve oblique triangles Apply the Law of Sines and Cosines to real world situations CCSS: G-SRT.10,11 MACLE: N/A • Determine the area of triangles CCSS: G-SRT.9 MACLE: N/A • • Represent and use vectors appropriately Find vector components given coordinates of initial and terminal points Approved 7-15-13 Revised 2013 9 Math Analysis/Trigonometry • Solve real world problems that can be represented by vectors CCSS: N-VM.1-3 MACLE: N/A Instructional Strategies Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet • Drill and guided practice Demonstration: Show how two triangles can be formed from 2 sets of congruent sides and a set of non-included congruent angles (S-S-A) • Problem solving – Find: • heights of objects directly and indirectly • resultant vectors in situations • Reflective discussion • Class discussion • Computer assisted instruction • Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Projects with scoring guides: • Flag Pole lab – using direct and indirect measurement to determine heights of objects • Trig theme based problems: (optional- if time allows) Students will create three word problems using: • SOH-CAH-TOA • Law of Sines • Law of Cosines • Quizzes: Appropriate Function Friday (with inverses) quizzes • Homework assignments: see attached pacing guide • Formal common assessment: Unit E test Mastery Level: 80% Approved 7-15-13 Revised 2013 10 Math Analysis/Trigonometry Instructional Resources/Tools • • • Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website(s): • This website has an interactive demonstration of the law of cosines – http://www.mathwarehouse.com/trigonometry/law-of-cosinesformula-examples.php • This website uses “vector addition” and “find resultant vector” (magnitude and direction) – http://www.youtube.com/watch?v=8iMix7klXbs Graphing calculator Approved 7-15-13 Revised 2013 11 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 3.5 Knowledge: (MA) 4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1-3 • • • • • • Numbers and Quantity Algebra Vector and Matrix Quantities (N-VM) Reasoning with Equations and Inequalities (A-REI) Perform operations on matrices and use matrices in applications (N-VM.6-10) Solve systems of equations (A-REI.9) Standards Learning Targets N-VM.6-10 6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network 7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled 8. Add, subtract, and multiply matrices of appropriate dimensions. 9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties 10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse Unit F: Matrices: Perform basic operations on matrices, understand which basic properties are satisfied with matrices, and use matrices to represent and manipulate data. Find and use an inverse matrix to solve a system of equations A-REI.9 9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater) • Use matrices to represent real world data and problem solve CCSS: N-VM.6,7 MACLE: N/A • Add, subtract, and multiply matrices and understand which properties of equality are satisfied or not satisfied with each operation CCSS: N-VM.8,9 MACLE: N/A • Understand and use zero and identity matrices, and determinants appropriately with technology CCSS: N-VM.10 MACLE: N/A Approved 7-15-13 Revised 2013 12 Math Analysis/Trigonometry • • Use matrices to represent a system of linear equations Use inverse matrices to solve systems of linear equations CCSS: A-REI.9 MACLE: N/A Instructional Strategies • • • • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Reflective discussion Class discussion Computer assisted instruction Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Quizzes • Homework assignments: see attached pacing guide • Formal common assessment: Unit F test Mastery Level: 80% Instructional Resources/Tools • • • Textbook: Algebra 2 – Common Core (Pearson) SMART Board resource – “Matrices and Vectors” downloadable at: http://exchange.smarttech.com/search.html?q=matrices%20and%20vectors&lang=en Graphing calculator Approved 7-15-13 Revised 2013 13 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster Alignments: CCSS: See below Performance: 1.6, 1.10 Knowledge: (MA) 2,4 MACLE: See below NETS: 3d; 4c; 6b DOK: 1-3 • Geometry • Expressing geometric properties with equations (G-GPE) • Translate between the geometric description and the equation for a conic section (G-GPE.1-3) Standards G-GPE.1-3 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation 2. Derive the equation of a parabola given a focus and directrix 3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distance from the foci is constant Learning Targets Unit G: Conics: Identify and graph conic sections • • • Use the distance formula, midpoint formula and complete the square Write the equation of the perpendicular bisector Identify, graph, and write the equation of circles, ellipses, hyperbolas, and parabolas CCSS: G-GPE.1-3 MACLE: N/A Approved 7-15-13 Revised 2013 14 Math Analysis/Trigonometry Instructional Strategies • • • • • • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Demonstrations – Websites demonstrating the constructions of: • conic sections • wax paper folding Reflective discussion Class discussion Computer assisted instruction “Sorting Conics” activity: Students will take equations in general form and determine which conic it is Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Quizzes • Homework assignments: see attached pacing guide • Formal common assessment: Unit G test Mastery Level: 80% Instructional Resources/Tools • • • Textbook: Prentice Hall, Precalculus Graphing and Data Analysis, Prentice Hall, Inc., Sullivan and Sullivan 2001. Website: Conics: http://mathworld.wolfram.com/ConicSection.html Graphing calculator Approved 7-15-13 Revised 2013 15 Math Analysis/Trigonometry Conceptual Category(s) Domain Cluster • • • • • • • • • • Alignments: CCSS: See below Performance: 1.2, 3.1, 3.5 Knowledge: (MA) 3 MACLE: See below NETS: 1d; 3d; 4c DOK: 1-3 • • Statistics and Data Analysis Statistics and Probability (8.SP) Conditional Probability and the Rules of Probability (S-CP) Making Inferences and Justifying Conclusions (S-IC) Interpreting Categorical and Quantitative Data (S-ID) Investigate patterns of association in bivariate data (8.SP.4) Understand independence and conditional probability and use them to interpret data (S-CP.1-5) Use the rules of probability to compute probabilities of compound events in a uniform probability model (S-CP.6-9) Understand and evaluate the random processes underlying statistical experiments (S-IC.1-2) Make inferences and justify conclusions from sample surveys, experiments, and observational studies (S-IC.3) Summarize, represent, and interpret data on a single count or measurement variable (S-ID.4) Summarize, represent, and interpret data on two categorical and quantitative variables (S-ID.5) Standards 8.SP.4 4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables S-CP.1-5 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”) Learning Targets Unit H: Statistics and Data Analysis: Know and understand both the experimental and theoretical probability of an event. Determine whether events are dependent or independent. Recognize normally distributed data and work with sample surveys and biases • • • • Determine the experimental probability of an event Use experimental probability to make predictions Determine the theoretical probability of an event Convert between probabilities and odds CCSS: S-CP.1 MACLE: D.3.A Approved 7-15-13 Revised 2013 16 Math Analysis/Trigonometry 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent 3. Understand the conditional probability of A given B as P(A and B)|P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A 4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations S-CP.6-9 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model 7. Apply the Addition Rule, P(AorB)= P(A) + P(B)- P(AandB), and interpret the answer in terms of the model 8. Apply the general Multiplication Rule in a uniform probability model, P(AandB)= P(A)|P(B|A)=P(B)P(A|B), and interpret the answer in terms of the model 9. Use permutations and combinations to compute probabilities of compound events and solve problems S-IC.1-3 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population 2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. would a result of 5 tails in a row cause you to question the model? • • Solve problems involving the Fundamental Counting Principle Solve problems involving permutations and combinations CCSS: S-CP.9 MACLE: D.4.B • • Find probabilities of overlapping events (Addition Rule) Find the probability of (disjoint) mutually exclusive events (Addition Rule) CCSS: S-CP.7,8 MACLE: D.4.B • • • • • Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified Determine whether events are independent or dependent Find the probability of independent events (Multiplication Rule) Find the probability of dependent events (Multiplication Rule) Find and understand the conditional probability of A given B CCSS: S-CP.2-6 MACLE: D.4.B • • • • • Recognize normally distributed data Use the characteristics of the normal distribution to solve problems Use tables to estimate areas under normal curves Recognize data sets that are not normal Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified CCSS: S-ID.4,5 MACLE: D.1.A Approved 7-15-13 Revised 2013 17 Math Analysis/Trigonometry 3. Recognize the purposes and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each S-ID.4-5 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets 5. Summarize categorical data from two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and a conditional relative frequency). Recognize possible associations and trends in the data • • • • • Understand and interpret simulations Describe sample surveys and their biases Describe experiments and their biases Describe observational studies and their biases Explain how randomization relates to each CCSS: S-IC.1-3 MACLE: N/A Instructional Strategies • • • • • • • • • • Lecture enhanced with: • SMART Notebook • PowerPoint • the Internet Drill and guided practice Demonstrations – The teacher will use a: • spinner • die • deck of cards • computer to model simulations of probability Problem solving – Students will determine: • types of sampling given real-life problem situations • if any biases exist while gathering data Reflective discussion Class discussion Computer assisted instruction Simulations Technology enhanced: TI-graphing calculator Game: Plinko – www.math.psu.edu/dlittle/java/probability/plinko/index.html Approved 7-15-13 Revised 2013 18 Math Analysis/Trigonometry Assessments/Evaluations The students will be assessed on the concepts taught using a variety of modalities: • Direct teacher observations • Quizzes • Homework assignments: see attached pacing guide • Formal common assessment: Unit H test Mastery Level: 80% Instructional Resources/Tools Textbook(s): • Geometry and Algebra 2 Common Core (primary source) • On-Core Mathematics: • Geometry • Algebra 2 • Website: www.wolframalpha.com • Graphing calculator • Approved 7-15-13 Revised 2013 19