MILWAUKEE PUBLIC SCHOOLS

Mathematics Curriculum Guide Algebra II/Trigonometry

Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

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Mathematical Practice Standards

Mathematics – High School Algebra II/ Trigonometry: Introduction

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

Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course, organized into four units, are as follows:

Critical Area 1: This area develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Critical Area 2: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Critical Area 3: Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Critical Area 4: Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data—including sample surveys, experiments, and simulations—and the role that randomness and careful design play in the conclusions that can be drawn. In this course rational functions are limited to those whose numerators are of degree at most 1 and

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denominators of degree at most 2; radical functions are limited to square roots or cube roots of at most quadratic polynomials. Forms of quadratic expressions, in particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions – absolute value, step, and those that are piecewise-defined.

*NOTE: The misconceptions listed within the template are only samples. You will encounter additional misconceptions as you work with students.

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The assessments and strategies for learning listed on this page are not meant to be an “end all”. Additionally, these are repeated for every grade level curriculum guide with the intent that our students experience some of the same strategies year to year, teacher to teacher. On the following pages, you will find specific usage for some of these. Classroom Assessment For Learning Literacy Connections Assessment for learning provides students with insight to improve Comprehensive literacy is the ability to use reading, writing, speaking, achievement and helps teachers diagnose and respond to student listening, viewing and technological skills and strategies to access and needs. (Stiggins, Rick, Judith Arter, Jan Chappuis, Steve Chappuis. communicate information effectively inside and outside of the Classroom Assessment for Student Learning. Educational Testing classroom and across content areas. (CLP, p.11) Service, 2006.) Formative Assessment tools include:  Constructed response items  Descriptive Feedback (oral/written)  Effective use of questioning  Exit slips  Milwaukee Math Partnership (MMP) CABS  Observational checklists and anecdotal notes  Portfolio items  Projects (PBL)  Student journals  Student self-assessment  Students analyze strong and weak work samples  Use of Learning Intentions  Use of rubrics with students  Use of talk formats and talk moves, p. 55 of the CMSP  Use of Success Criteria

Literacy strategies can help students learn mathematics:  Concept mapping  Graphic organizers  Journaling  K-W-L  Literature  RAFT  Reciprocal teaching  Talk moves  Think Aloud strategy  Think, Pair, Share strategy  Three-minute pause  Two column note taking  Vocabulary strategies (e.g. Marzano’s 6 steps; Frayer model)

Summative Assessment:  Unit Tests  WKCE/WAA Progress Monitoring Tool  Measure of Academic Progress (MAP)

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Standards for Mathematical Practice

Mathematics – High School Algebra II/Trigonometry Course

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

Modeling Modeling Standards Modeling is best interpreted in relation to other standards, not as a collection of isolated topics. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*).

The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on the relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situations, and then either improving the model or, it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.

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Mathematics – High School Algebra II/Trigonometry Course

Standards for Mathematical Practice 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

Number and Quantity The standards identified are built on the development from the previous courses of Algebra and Geometry. The Complex Number System N-CN Perform arithmetic operations with complex numbers 1. Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. 2. Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Represent complex numbers and their operations on the complex plane 4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same numbers. (note: polar form is not required for this course) Use complex numbers in polynomial identities and equations 7. Solve quadratic equations with real coefficients that have complex solutions.

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Mathematics – High School Algebra II/Trigonometry Course Number and Quantity Essential/Enduring Understandings: Assessments  Expanding the number system to include complex numbers allows for the  Have students create quadratic equations to represent a variety of solution of quadratic (and higher degree) equations possible roots in the real and complex number system 

There are deep connections between geometry and complex numbers

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Have students explain: “How are the representations of complex conjugates on the complex plan related?”

Common Misconceptions/Challenges Expanding the number system to include complex numbers allows for the solution of quadratic (and higher degree) equations There are deep connections between geometry and complex numbers  Students have a hard time visualizing the imaginary numbers within the number system (where is i?). To address this misconception, graph imaginary numbers on a plane. See the instructional practice below. Instructional Practices N-CN: The Complex Number System  Create problems such that the solution will not be a real number for students to experience numbers beyond the real number system Example: Graph and solve y = x2 + 4x + 5, after students graph and solve. Ask students; “in your own words, how would you describe complex numbers?” Complex numbers consist of real numbers, imaginary numbers, and their sums. 

Although complex numbers don’t fit into the real number line, they can be represented on a plane.

Example: You cannot graph a complex number, such as 3 + 4i, on a real number line, but you can graph it on a complex plane, where the horizontal axis is the real axis and the vertical axis is the imaginary axis. In the graph, 3 + 4i is located at the point with coordinates (3,4). Any complex number a + bi has (a, b) as its coordinates on the complex plane.

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Remind students that in previous experiences there were problems that they weren’t able to solve and now that their understanding of the number system has expanded, they are able to find their solution. Example: For any complex number in the form a + bi, a is the real part and bi is the imaginary part. The set of the complex numbers contains all real numbers and all imaginary numbers. This diagram shows the relationship between these numbers and some other sets you may be familiar with, as well as examples of numbers within each set.

Differentiation Expanding the number system to include complex numbers allows for the solution of quadratic (and higher degree) equations There are deep connections between geometry and complex numbers  Use dynamic software to explore the complex number system  Have students use a graphic organizer to summarize all the methods for solving quadratic equations, with the requirements, advantages, and disadvantages of each. Provide the graph of a parabola with no x-intercepts to make the concept of imaginary numbers. Literacy Connections Academic Vocabulary  Research the history of complex numbers  Associative Properties  Students use the Frayer Models to surface their conceptual understanding  Commutative Properties of imaginary numbers (for the teachers goal of getting to the  Complex numbers understanding that imaginary numbers are no more imaginary than real  Distributive Properties numbers)  Imaginary numbers  Polynomial Functions  Quadratic Functions Resources  Dynamic software such as Geometer’s Sketchpad  Resources on the history of complex numbers: Berlinghoff & Gouvêa, Math Through the Ages, A Gentle History for Teachers and Others Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

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Standards for Mathematical Practice

Mathematics – High School Algebra II/Trigonometry Course

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

Algebra The standards identified are built on the development from the previous courses of Algebra and Geometry. Seeing Structure in Expressions Arithmetic with Polynomials and Rational Reasoning with Equations and Inequalities Expressions A-SSE A-APR A-REI Write expressions in equivalent forms to solve Understand the relationship between zeros and Solve equations and inequalities in one variable problems factors of polynomials 4. Solve quadratic equations in one variable 4. Derive the formula for the sum of a finite 2. Know and apply the Remainder Theorem: For a b. Solve quadratic equations by inspection (e.g., for x2 geometric series (when the common ratio is not polynomial p(x) and a number a, the remainder on = 49), taking square roots, completing the square, the 1), and use the formula to solve problems. For division by x – a is p(a), so p(a) = 0 if and only if (x – quadratic formula, and factoring, as appropriate to example, calculate mortgage payments.* a) is a factor of p(x). the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write Note: the earlier standards in this domain must 3. Identify zeros of polynomials when suitable them as a + bi for real numbers a and b. now be interpreted in the complex number factorization are available and use the zeros to system. construct a rough graph of the function defined by Represent and solve equations and inequalities the polynomial. graphically. 11. Explain why the x-coordinate of the points where Use polynomial identities to solve problems. the graphs of the equations y = f(x), and y = g(x) 4. Prove polynomial identities and use them to intersect are the solutions of the equation f(x) = g(x); describe numerical relationships. For example, the find the solutions approximately, e.g., using 2 2 2 2 2 2 2 polynomial identity (x + y ) = (x – y ) + (2xy) can technology to graph the functions, make tables of Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

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be used to generate Pythagorean triples. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. Rewrite rational expressions. 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

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values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponents, and logarithmic functions.* 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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Mathematics – High School Algebra II/Trigonometry Course Algebra Essential/Enduring Understandings: Assessments  Algebraic expressions can be written in infinitely many equivalent forms;  Given a contextual problem, students will be able to choose and justify different forms may be useful for answering different questions about the the best representation or form of their solution context of the problem Common Misconceptions/Challenges Algebraic expressions can be written in infinitely many equivalent forms; different forms may be useful for answering different questions about the context of the problem

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Students have a difficult time predicting the characteristics of the graph by looking at the equation Students have a difficult time translating between multiple representations of an algebraic equation Students misapply formal algebraic rules without understanding the meaning of the operation that it carries out

Instructional Practices A-SSE: Seeing Structure in Expressions  Have students visualize the graph before graphing the equation  Have students determine and apply logarithmic and polynomial functions to solve and interpret real-world situations, distinguishing relevant from irrelevant information in selecting an appropriate model, and identifying, finding, or estimating missing information A-APR: Arithmetic with Polynomials and Rational Expressions  When students perform arithmetic operations on polynomials compare to the operations of integers. Example: When students divide (x2 + 7x + 12) by (x + 4). Show that the solution (x + 3) multiplied by (x + 4) will result in the original polynomial. A-REI: Reasoning with Equations and Inequalities  When students are solving algebraic equations, continue to ask: “what does this value mean?”, “does this make sense?”, “why are you doing this next step?”  Plan for the Math Practice Standard – Reason Abstractly and Quantitatively for students to have a better conceptual understanding of applying algebraic rules to expressions

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Differentiation Algebraic expressions can be written in infinitely many equivalent forms; different forms may be useful for answering different questions about the context of the problem  Use algebra tiles to model algebraic expressions and deepen their understanding of factoring.  Use dynamic software to explore polynomials  Use the area model for multiplication Literacy Connections  Use Talk Moves for students to construct viable arguments and critique the reasoning of others on the best representation for a contextual problem.

Academic Vocabulary  Logarithmic Function  Rational Functions  Roots  Vertex

Resources  Dynamic software and graphing calculators to represent multiple representations of an equation  Algebra tiles

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Standards for Mathematical Practice

Mathematics – High School Algebra II/Trigonometry Course

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

Functions The standards identified are built on the development from the previous courses of Algebra and Geometry. Building Functions Linear, Quadratic, and Exponential Models* Trigonometric Functions F-BF F-LE F-TF Build new functions from existing functions Construct and compare linear, quadratic, and Extend the domain of trigonometric functions 3. Identify the effect on the graph of replacing f(x) by exponential models and solve problems using the unit circle f(x) + k, k f(x), f(kx), and f(x + k) for specific values of 4. For exponential models, express as a 1. Understand radian measure of an angle as the k (both positive and negative); find the value of k logarithm the solution to abct = d where a, c, length of the arc on the unit circle subtended by given the graphs. Experiment with cases and and d are numbers and the base b is 2, 10, or the angle. illustrate an explanation of the effects on the graph e; evaluate the logarithm using technology. using technology. Include recognizing even and odd 2. Explain how the unit circle in the coordinate functions from their graphs and algebraic Interpret expressions for functions in terms plane enables the extension of trigonometric expressions for them. of the situation they model functions to all real numbers, interpreted as 5. Interpret the parameters in a linear or radian measures of angles traversed 4. Find inverse functions exponential function in terms of a context. counterclockwise around the unit circle. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x 1. b. (+) verify by composition that one function is the inverse of another. Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4, and /6, and use the unit circle to express the values of sine, cosine, and tangent for -x, +x, and 2 -x in terms of their values for x, where x is any real number.

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c. (+) Read values of an inverse function from a graph or a table given that the function has an inverse. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

4. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* Model periodic phenomena with trigonometric functions 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* Prove and apply trigonometric identities 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

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Mathematics – High School Algebra II/Trigonometry Course Functions Essential/Enduring Understandings Assessments  Functions come in families  Students will be able to model situations in which they collect data,  Functions can be used to model real world phenomena; in particular choose an appropriate function family, and a function from that family trigonometric functions can be used to model periodic phenomena to model the situation and then justify their reasoning Common Misconceptions/Challenges Functions come in families  Students have a difficult time translating between verbal, algebraic, tabular, and graphical representations of functions. To address this misconception have students explore the effects of parameters using multiple representations. For example, when comparing the graph of the parabolas y = x2 and y = x2 + 7, students should see that the graph did not change shape, but only translated 7 units vertically. Students should recognize that in the table, the yvalues each increase by 7. Functions can be used to model real world phenomena; in particular trigonometric functions can be used to model periodic phenomena  Students have a difficult time predicting the characteristics of the graph by looking at the function. For a trigonometric function, have

students construct a paddle wheel to help them explore characteristics of periodic functions. Have students use the paddle wheel to collect data and create a trigonometric graph.  

Students believe that “function” is synonymous with “formula”, rather than seeing a function as an input/output When faced with a problem presented algebraically, students do not automatically think of making a graphical representation

Instructional Practices F-BF: Building Functions  Represent functions in various ways, including tabular, graphic, symbolic (explicit and recursive), visual, and verbal; making decisions about which representations are most helpful in problem solving circumstances; and moving flexibly among those representations.  Use a general representation of a function in a given family to analyze the effects of varying coefficients or other parameters F-LE: Linear, Quadratic, and Exponential Models*  When students are exploring relationships between two variables, allow them to interpret characteristic behaviors of the data to determine if it models exponential or linear relationships and justify their reasoning

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F-TF: Trigonometric Functions  Give students real-world phenomena (i.e. Using a Paddle Wheel vs. a Ferris Wheel, tidal waves, etc) to model trigonometric situations Differentiation Functions come in families Functions can be used to model real world phenomena; in particular trigonometric functions can be used to model periodic phenomena  

Have students use graphing calculators to organize the data that students collected Use dynamic software to model trigonometric functions

Literacy Connections  Students will justify their reasoning from the function models they chose from their data

Academic Vocabulary  Functions  Function Family  Periodic Functions  Phenomena  Trigonometric Functions

Resources  Graphing Calculators  Geometer’s Sketchpad  Dynamic Software

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Standards for Mathematical Practice

Mathematics – High School Algebra II/Trigonometry Course

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

Geometry The standards identified are built on the development from the previous courses of Algebra and Geometry Similarity, Right Triangles, and Trigonometry Expressing Geometric Properties with Equations G-SRT G-GPE Prove triangles involving similarity Translate between the geometric description and the equation for a conic 5. Use congruence and similarity criteria for triangles to solve problems and to section prove relationships in geometric figures 2. Derive the equation of a parabola given a focus and directrix

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Mathematics – High School Algebra II/Trigonometry Course Geometry Essential/Enduring Understandings: Assessments  Pythagorean Theorem in the guise of the distance formula can be used to  Have students journal about their previous understanding of parabolas derive the equations for circles and parabolas before beginning work in conic sections and then continue adding to their journal regarding their understanding of a parabola. Common Misconceptions/Challenges Pythagorean Theorem in the guise of the distance formula can be used to derive the equations for circles and parabolas  Students may not see the connection between the previous work they’ve done with parabolas defined with the equation and their expanded definition of a parabola. Explicitly connect the familiar y = x2 to the new expanded definition. Instructional Practices G-SRT: Similarity, Right Triangles, and Trigonometry G-GPE: Expression Geometric Properties with Equations  Students should be drawing pictures and use paper folding to support their reasoning. Example: Fold the patty paper parallel to one edge to form the directrix for a parabola. Mark a point on the larger portion of the paper to serve as the focus for your parabola. Fold the paper so that the focus lies on the directrix. Unfold, and then fold again, so that the focus is at another point on the directrix. Repeat this many times. The creases from these folds should create a parabola. Lay the patty paper on top of a sheet of graph paper. Identify the coordinates of the focus and the equation of the directrix, and write an equation for your parabola. Differentiation Pythagorean Theorem in the guise of the distance formula can be used to derive the equations for circles and parabolas  Have students explore the vocabulary by folding a parabola (see instructional practice) to deepen their understanding  Use reading strategies to support understanding of the contextual problem Literacy Connections

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Academic Vocabulary

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Students use graphic organizers to organize their thinking

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Auxiliary Line Derive Law of Cosine Law of Sine

Resources  Geometer’s Sketchpad  Dynamic Software

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Standards for Mathematical Practice

Mathematics – High School Algebra II/Trigonometry Course

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

Statistics and Probability The standards identified are built on the development from the previous courses of Algebra and Geometry Interpret Categorical and Quantitative Data Making Inferences and Justifying Conclusions Conditional Probability and the Rules of S-ID S-IC Probability S-CP Summarize, represent, and interpret data on a single count or measurement variable 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 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, and tables to estimate areas under the normal curve. Summarize, represent, and interpret data on two categorical and quantitative variables 5. Summarize categorical data for two categories

Understand and evaluate random processes underlying statistical experiments 1. Understand statistics as a process for making inferences about population parameter’s based on a random sample from that population. 2. Decide if a specified model is consistent with results form a given data-generated 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? Make inferences and justify conclusions form sample surveys, experiments, and observational studies 3. Recognize the purposes of and differences among sample surveys, experiments, and observational

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Understand independence and conditional probability and use them to interpret data 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. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Use the rules of probability to compute probabilities of compound events in a uniform probability model

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in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by plotting and analyzing residuals.

studies; explain how randomization relates to each. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. 6. Evaluate reports based on data.

7. Apply the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(AIB) = P(B)P(AIB), and interpret the answer in terms of the model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Interpret linear models 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. 9. Distinguish between correlation and causation.

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This Curriculum Guide is to be used in conjunction with the Pacing Guide and Instructional Design Lesson Plan. The Instructional Design will help with lesson planning using Launch-ExploreSummarize-Apply.

Mathematics – High School Algebra II/Trigonometry Course Statistics and Probability Essential/Enduring Understandings: Assessments  Real world data is messy but often conforms to general patterns if data  Have students design a Statistical Study: can be recognized as conforming to the general shape of a function 1. Formulate a question family, it can be modeled with an appropriate function from that family. 2. Design an experiment 3. Collect data  The probability of an event is the measure of its likelihood; the 4. Analyze data mathematical laws of probability allow us to compute the probabilities of complex events in terms of known probabilities of simple events. Common Misconceptions/Challenges Real world data is messy but often conforms to general patterns if data can be recognized as conforming to the general shape of a function family, it can be modeled with an appropriate function from that family.  Students are misled by outliers and how they fit into the big picture of the data; students have a difficult time distinguishing trends from outliers. To address this misconception, allow students the opportunity to discuss why a particular outlier may have happened in the context of the problem. The probability of an event is the measure of its likelihood; the mathematical laws of probability allow us to compute the probabilities of complex events in terms of known probabilities of simple events.  Some students may believe that experimental probabilities cannot be accurate. To address this misconception, plan on finding probabilities based on trial and observation on situations where one can determine the theoretical probability. Students should be familiar with the law of large numbers. Instructional Practices S-ID: Interpreting Categorical and Quantitative Data  To reinforce vocabulary: Example: Assign each student a statistical vocabulary word. Have students bring in an article that highlights that particular concept. For example, students find an article on normal distribution. Students would present their article to the class to reinforce their understanding of the concept. 

Have students do a project on Correlation Vs. Causation: Example: Think of a relationship someone claims involves causation, but you think might involve only correlation. Your claim can be about anything – science, popular beliefs, and sociology – but it must be something that can be tested. First, research data related to the claim and determine whether or not the data seem to show a correlation between the two variables. Then, think about whether or not one event really causes the other. What other factors might be involved? Might the data you found be misleading in some way? If you can, find the data for any other factors and see how these data are related to your claim. Write a report on your findings.

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S-IC: Making Inferences and Justifying Conclusions  Give students examples of populations along with an experiment and allow students to critique their sampling technique. S-CP: Conditional Probability  Have students reason through contextual problems before you give them the formula. The difference between independent and dependent events is crucial for applying probability. Provide several examples of both types of events and have students act them out. Differentiation Real world data is messy but often conforms to general patterns if data can be recognized as conforming to the general shape of a function family, it can be modeled with an appropriate function from that family.  If students are having a difficult time finding an equation that models data, allow them to sketch the shape of the data and explain the trend as a way of assessing their baseline understanding. Allow students to use technology to explore possible equations that models the data. The probability of an event is the measure of its likelihood; the mathematical laws of probability allow us to compute the probabilities of complex events in terms of known probabilities of simple events.  Use manipulative to model probability properties.  Allow students to use tree diagrams if they are not comfortable with the formulas. Literacy Connections Academic Vocabulary  Interpret and critique the statistics reported within an article (Standards  Causation for Mathematical Practice – Construct viable arguments and critique the  Combination reasoning of others)  Complex Numbers  Conditional Probability  Correlation  Dependent Events  Frequency Tables  Independent Events  Outliers  Permutation  Quartiles  Standard Deviation  Spread Resources  GAISE Report, New York Time can be used to interpret and critique a statistical report  Manipulative (dice, spinners, counters, etc) can be used to help students experience probability  Making Sense of Statistical Studies by Roxy Peck and Daren Starnes with Henry Kranendonk and June Morito Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

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This Curriculum Guide is to be used in conjunction with the Pacing Guide and Instructional Design Lesson Plan. The Instructional Design will help with lesson planning using Launch-ExploreSummarize-Apply.

Math_Curriculum Guide Algebra II/Trigonometry_07.28.11_v1

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This Curriculum Guide is to be used in conjunction with the Pacing Guide and Instructional Design Lesson Plan. The Instructional Design will help with lesson planning using Launch-ExploreSummarize-Apply.