Curriculum Map Precalculus Saugus High School Saugus Public Schools


 Rachel Simmons
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1 Curriculum Map Precalculus Saugus High School Saugus Public Schools
2 The Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on the following two sets of important processes and proficiencies, each of which has longstanding importance in mathematics education: The NCTM process standards o problem solving o reasoning and proof o communication o representation o connections The strands of mathematical proficiency specified in the National Research Council s report Adding It Up o adaptive reasoning o strategic competence o conceptual understanding (comprehension of mathematical concepts, operations, and relations) o procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately) o productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one s own efficacy) 1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically, and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meanings of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
3 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the wellremembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square, and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
4 Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word understand are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential points of intersection between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. Foreword This Curriculum Map outlines all of the topics and standards expected of a Precalculus class. These concepts should be introduced, studied, and assessed at a level that matches each student s abilities. The level of academic rigor should match both the course level and the students ability levels. The standards have been outlined based on the Massachusetts Curriculum Framework for Mathematics that incorporated the for Mathematics. The Honors level class will address advanced concepts. These concepts will be denoted by an asterisk, as time allows, or part of supplementary units located at the end of each term.
5 Precalculus ( 343 and 346) Term One Week 1 F.BF.1. Write a function that describes a relationship between two quantities. F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases. Review of all Summer Assignment Concepts* Number patterns (F.BF.1, F.IF.7) Unit Zero Basic Algebraic Concept work with number patterns, sequences and equations. Chapter One Lessons Chapter One Practice Worksheets Summer Packet*: To be given prior to summer vacation. This will be collected during the 2 nd or 3 rd day of classes and selected problems will be graded as a Quiz. Review: All concepts from the Summer Packet will be reviewed during the first 3 days of school. Quiz: on concepts involving from the Homework Summer Packet during the week. Baseline Assessment: The Baseline Assessment focused on Precalculus Concepts will be given the 1 st week of classes. What is the connection between linear functions and arithmetic sequences? Investigating Sequences Activity on Graphing Calculator
6 Precalculus ( 343 and 346) Term One Week 2 A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (10.P.6) A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (10.P.6) Review of all Summer Assignment Concepts* Equations (A.CED.1, A.REI.3 ) Inequalities (A.CED.1, A.REI.3 ) Unit Zero Basic Algebraic Concepts solve linear and quadratic equation. solve basic compound inequalities. Chapter Two Lessons Chapter Two Practice Worksheets Quiz: on concepts involving Number Patterns, Equations and Inequalities during the week. Test: on concepts involving Basic Algebraic Concepts at the end of the week. What are the similarity and differences between solving a linear equation compared to a linear inequality? Exploring Inequalities Activity Graphically on Graphing Calculator As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Basic Algebraic Concepts..
7 Precalculus ( 343 and 346) Term One Week 3 F.BF.1. Write a function that describes a relationship between two quantities. F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases. Introduction to Functions (F.BF.1) Graphs of Functions (F.BF.1, F.IF.7) Quadratic Functions (F.BF.1, F.IF.7) Unit One Functions and Their Graphs determine whether a function is a relation. find the domain of a function. determine a graph to be a function. identify parts if a parabola based on the equation of the function. convert from one form of a quadratic function to another. Chapter Three Lessons Chapter Three Practice Worksheets Quiz: on concepts involving Introduction to Functions and Graphs during the week. How is the domain and range of a function, determined from the graph of the function? Investigating Quadratic Functions Activity on Graphing Calculator.
8 Precalculus ( 343 and 346) Term One Week 4 F.BF.1. Write a function that describes a relationship between two quantities. F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases. Graphs and Transformations (F.BF.1, F.IF.7) Operations on Functions (F.BF.1, F.IF.7) Unit One Functions and Their Graphs determine parent functions. What happens to a parent function when you transform its graph? transform graphs of parent functions. form sum, difference and product and quotient functions and find their domain. form composite functions and find their domain. determine whether a function is even, odd or neither. Chapter Three Lessons Chapter Three Practice Worksheets Quiz: on concepts involving Graphs and Operations on Functions during the week. Investigating Parabolas Activity on Graphing Calculator regarding Parent functions and transformations.
9 Precalculus ( 343 and 346) Term One Week 5 F.BF.1. Write a function that describes a relationship between two quantities. F.BF.4. Find inverse functions. F.BF.4b. Verify by composition that one function is an inverse of another. F.BF.4c. Read values of an inverse function from a graph or table, given that the function has an inverse. F.BF.4d. Produce an invertible function from a noninvertible function by restricting a domain. Unit One Functions and Their Graphs Inverse Functions (F.BF.4) OnetoOne Functions (F.BF.4, F.IF.4b, F.IF.4c) Rates of Changes (F.BF.1) define inverse relations and functions. find inverse relations from tables, graphs and equations. determine whether an inverse relation is a function. find the average rate of change of a function over an interval. work with and solve various problems involving the average rate of change. Chapter Three Lessons Chapter Three Practice Worksheets Quiz: on concepts involving Quiz on Inverse Functions and OnetoOne Functions during the week. Test: on concepts involving Functions and Their Graphs. What is the algebraic process that is used to find the inverse of a function? Exploring Functions and Their Inverses Activity on Graphing Calculator. Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Functions and Their Graphs.
10 Precalculus ( 343 and 346) Term One Week 6 A.APR.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. A.APR.7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions. Unit Two Polynomial and Rational Functions Polynomial Functions (A.APR.6, A.APR.7) The Division Algorithm (A.APR.6, A.APR.7) Remainder and Factor Theorems (A.APR.6, A.APR.7) The Rational Zero Test and Finding Real Zeros (A.APR.6, A.APR.7) define and divide polynomials. apply the Remainder and Factor Theorems and make connections between remainders and factors. determine the maximum number of zeros of a polynomial. find all rational zeros of a polynomial function. factor a polynomial completely. recognize and describe the graphs of various polynomial functions. identify the properties of general polynomial functions. Chapter Four Lessons Chapter Four Practice Worksheets Quiz: on concepts involving Dividing Polynomials and the Remainder and Factor Theorems during the week. What is the procedure that is used to find real zeros of a polynomial function? Mountain Problem on Investigating Quadratic Functions Activity on Graphing Calculator. Activity to identify the properties of general polynomial functions on Graphing Calculator.
11 Precalculus ( 343 and 346) Term One Week 7 F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases. F.IF.7d. Graph rational functions, identifying zeros when suitable factorizations are available, and showing end behavior. Unit Two Polynomial and Rational Functions Graphs of Polynomial Functions (F.IF.7, F.IF.7d) Rational Functions (F.IF.7, F.IF.7d) Vertical Asymptotes and Holes (F.IF.7, F.IF.7d) Graphing Rational Functions (F.IF.7, F.IF.7d) find the domain of a rational function. Why does every nonconstant polynomial have a zero in the complex number system? find intercepts, asymptotes, and holes. describe the end behavior of a function. write and perform arithmetic operations on complex numbers. find the number of zeros of a polynomial. give the complete factorization of polynomial expressions. Chapter Four Lessons Chapter Four Practice Worksheets Quiz: on concepts involving Graphs of Polynomial Functions during the week. Activity to Explore Vertical and Horizontal Asymptotes on Graphing Calculator.
12 Precalculus ( 343 and 346) Term One Week 8 N.CN.3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. N.CN.8. Extend polynomial identities to the complex numbers. For example, rewrite 4 as 2 2. N.CN.9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Unit Two Polynomial and Rational Functions Complex Numbers (N.CN.3, N.CN.8) Fundamental Theorem of Algebra (N.CN.9) find the conjugate of a complex number. How do you find the conjugate of a complex number? simplify square roots of negative numbers. use the Fundamental Theorem of Algebra. Chapter Four Lessons Chapter Four Practice Worksheets Quiz: on concepts involving Complex Numbers during the week. Test: on concepts involving Polynomial and Rational Functions. Exploring Powers of i Activity. Exploring Zeros of Functions Activity on Graphing Calculator and Relating that to the Fundamental Theorem of Algebra. Group Work Practice or Play Jeopardy or similar game on Smartboard to Review for Test on Polynomial and Rational Functions.
13 Precalculus ( 343 and 346) Term Two Week 1 F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Solving Exponential Equations (F.BF.5) Unit Three Exponential and Logarithmic Functions solve exponential equations. How are exponential and logarithmic models used along with polynomial models to solve real world situations? Chapter Five Lessons Chapter Five Practice Worksheets Graphing Calculator Activity Investigating Exponential Functions Activity on Compound Interest Formula Quiz: on concepts involving Solving Exponential Equations during the week.
14 Precalculus ( 343 and 346) Term Two Week 2 F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Unit Three Exponential and Logarithmic Functions Solving Logarithmic Equations (F.BF.5) Modeling with Exponential and Logarithmic Functions (F.BF.5) solve logarithmic equations. solve a variety of application problems by using exponential and logarithmic equations. Chapter Five Lessons Chapter Five Practice Worksheets Quiz: on concepts involving Solving Logarithmic Equations during the week. Test: on concepts involving Exponential and Logarithmic Functions. How are exponential and logarithmic models used along with polynomial models to solve real world situations? School Flu Activity using Logarithms Graphing Calculator Activity on Solving Exponential Inequalities Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Exponential and Logarithmic Functions.
15 Precalculus ( 343 and 346) Term Two Week 3 F.TF.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, cosines, and tangent for,, and 2 in terms of their values for x, where x is any real number. Unit Four Trigonometry Right Triangle Trigonometry (F.TF.3) Trigonometric Ratios (F.TF.3) Conversion Between Decimal and DMS Form (F.TF.3) Special Angles (F.TF.3) Solving Right Triangles (F.TF.3) Right Triangle Real World Situations (F.TF.3) Indirect Measurement (F.TF.3) define the six trigonometric ratios of an acute angle in terms of a right triangle. evaluate trigonometric ratios using triangles and/or calculators. solve triangles using trigonometric ratios. solve applications using triangles. Chapter Six Lessons Chapter Six Practice Worksheets Quiz: on concepts involving Trigonometric Ratios during the week. Quiz: on concepts involving Right Triangle Trigonometry during the week. How is right triangle trigonometry used to solve right triangles? Flash Card Activity for help with SOH CAHTOA Angle of Elevation and Angle of Depression Activity
16 Precalculus ( 343 and 346) Term Two Week 4 F.TF.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, cosines, and tangent for,, and 2 in terms of their values for x, where x is any real number. Unit Four Trigonometry Extending Angle Measures and Coterminal Angles (F.TF.3) Radian Angle Measure and Conversion between Degrees and Radians (F.TF.3) Arc Length (F.TF.3) Trigonometric Functions and Trigonometric Ratios in the Coordinate Plane (F.TF.3) use a rotating ray to extend the definition of angle measure to negative angles and greater than 180 degrees. define radian measure and convert angle measures between degrees and radians. How is the unit circle used to describe trigonometric functions? Chapter Six Lessons Chapter Six Practice Worksheets Quiz: on concepts involving Coterminal Angles during the week. Quiz: on concepts involving Radian Angle Measure and Conversion between Degrees and Radians during the week. Benchmark Assessment 1: The Benchmark Assessment will focus on all Precalculus Concepts covered to date. Flash Card Activity for help with Unit Circle on Changing Radians to Degrees and Vice Versa Unit Circle Game Activity Online for Practice on Smartboard Activity on Arc Length As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for The Benchmark Assessment.
17 Precalculus ( 343 and 346) Term Two Week 5 F.TF.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, cosines, and tangent for,, and 2 in terms of their values for x, where x is any real number. F.TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. G.C.4. Construct a tangent line from a point outside a given circle to the circle.* Unit Four Trigonometry Unit Circle and Reference Angles (F.TF.3, F.TF.4) Finding Trigonometric Values (F.TF.3, F.TF.4) define the trigonometric ratios in the coordinate plane. define trigonometric functions in terms of the unit circle. construct a tangent line from a point outside a given circle to the circle. * Chapter Six Lessons Chapter Six Practice Worksheets Quiz: on concepts involving Unit Circle and Reference Angles during the week. Test: on concepts involving Trigonometry. Describe how do you convert between radians and degrees? Flash Card Activity for help with Unit Circle on Changing Radians to Degrees and Vice Versa Unit Circle Game Activity Online for Practice on Smartboard Activity on Linear Speed and Angular Speed
18 Precalculus ( 343 and 346) Term Two Week 6 F.TF.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Quotient Identities (F.TF.9) Reciprocal Identities (F.TF.9) Pythagorean Identities (F.TF.9) Periodicity Identities (F.TF.9) Negative Angle Identities (F.TF.9) Unit Four Trigonometry develop basic trigonometric ratios. prove and work with basic trigonometric identities. Chapter Six Lessons Chapter Six Practice Worksheets Quiz: on concepts involving Quotient, Reciprocal, and Pythagorean Identities during the week. Test: on concepts involving Trigonometry. What are the relationships between the Pythagorean Identities for Trigonometry? Exploring on Calculator Activity Involving Graphing and Basic Identities Identity Cloud Activity for Practice help with Identities on Smartboard Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Trigonometry.
19 Precalculus ( 343 and 346) Term Two Week 7 F.TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F.TF.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. Unit Five Trigonometric Graphs Graphs of the Sine Functions (F.TF.4, F.TF.6) Graphs of the Cosine Functions (F.TF.4, F.TF.6) Graphs of the Cosecant Functions (F.TF.6) Graphs of the Secant Functions (F.TF.6) graph the basic trigonometric functions. graph the sine, cosine, cosecant, and secant functions. state the domain and range of these trigonometric functions. graph transformations of these basic trigonometric functions. use the unit circle to explain symmetry (odd/even) functions. Chapter Seven Lessons Chapter Seven Practice Worksheets How do you graph the basic trigonometric functions on the coordinate plane? Exploring Basic Trigonometric Graphs Activity on Graphing Calculator. Quiz: on concepts involving Graphing the Sine, Cosine, Cosecant, and Secant functions during the week.
20 Precalculus ( 343 and 346) Term Two Week 8 F.TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F.TF.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. Unit Five Trigonometric Graphs Graphs of the Tangent Functions (F.TF.4, F.TF.6) Graphs of the Cotangent Functions (F.TF.6) Periodic Graphs (F.TF.4, F.TF.6) Amplitude (F.TF.4, F.TF.6) Amplitude and Period (F.TF.4, F.TF.6) graph the sine, cosine, tangent, cosecant, secant and tangent functions. graph the transformations of the sine, cosine, tangent, cosecant, secant and tangent functions. state the period and amplitude of basic trigonometric functions. sketch the graphs of these basic trigonometric functions using period and amplitude. How do transformations affect the trigonometric graphs of each function? Such as, how do you determine the period and amplitude of a trigonometric function without looking at the graph of the function? Chapter Seven Lessons Chapter Seven Practice Worksheets Quiz: on concepts involving Trigonometric Graphs during the week. MidYear Exam: This will cover all of the concepts from semester one. Biorhythm Graphing Activity related to Sine Graphs Jeopardy Review Practice, or similar game on Smartboard to begin Review for Mid Year
21 Precalculus ( 343 and 346) Term Three Week 1 F.TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. F.TF.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. Unit Five Trigonometric Graphs Periodic Graphs (F.TF.4, F.TF.6) Amplitude (F.TF.4, F.TF.6) Amplitude and Period (F.TF.4, F.TF.6) Vertical Shifts (F.TF.4, F.TF.6) Phase Shifts (F.TF.4, F.TF.6) Combined Transformations (F.TF.4, F.TF.6) Graphs and Identities (F.TF.4, F.TF.6) state the period and amplitude of basic trigonometric functions. sketch the graphs of these basic trigonometric functions using period and amplitude. state the period, amplitude, vertical shift and phase shift of basic trigonometric functions. use graphs to determine whether an equation could possibly be an identity. How do you use graphs or trigonometric functions to determine trigonometric identities? Chapter Seven Lessons Chapter Seven Practice Worksheets Test: on concepts involving Trigonometric Graphs. Begin Catch Some Rays Activity related to Cosine Graphs. Work on computers. Exploring Trigonometric Graphs and Transformations Activity on Graphing Calculator. Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Trigonometric Graphs.
22 Precalculus ( 343 and 346) Term Three Week 2 F.TF.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. F.TF.7. 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. Unit Six Solving Trigonometric Equations Basic Trigonometric Equations (F.TF.6, F.TF.7) Solving Trigonometric Equations Graphically (F.TF.6, F.TF.7) Inverse Trigonometric Functions (F.TF.7) solve trigonometric equations graphically. state the complete solution to a trigonometric equation. define the domain and range of inverse trigonometric functions. Chapter Eight Lessons Chapter Eight Practice Worksheets How do you graphically solve a trigonometric equation? Continue Catch Some Rays Activity relate Activity Solving Trigonometric Equations Graphically on Graphing Calculator. Quiz: on concepts involving Solving Trigonometric Equations Graphically and Basic Trigonometric Equations during the week.
23 Precalculus ( 343 and 346) Term Three Week 3 F.TF.7. 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. Unit Six Solving Trigonometric Equations Inverse Trigonometric Functions (F.TF.7) Properties of Inverse Trigonometric Functions (F.TF.7) Algebraic Solutions of Trigonometric Equations (F.TF.7) state the complete solution to a trigonometric equation. define the domain and range of inverse trigonometric functions. use inverse trigonometric notation. solve trigonometric equations algebraically. work with a variety of techniques to solve trigonometric equations. What is the difference between sine function and the restricted sine function and why is it important when working with the inverse sine function? Chapter Eight Lessons Chapter Eight Practice Worksheets Test: on concepts involving Solving Trigonometric Equations. Present Catch Some Rays Projects related to Cosine Graphs. Exploring The Inverse Relation Activity of the Sine Cosine and Tangent Graphs. Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Solving Trigonometric Equations.
24 Precalculus ( 343 and 346) Term Three Week 4 F.TF.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Unit Seven Trigonometric Identities Identities and Proof (F.TF.9) Basic Trigonometric Identities (F.TF.9) Strategies for Proving Trigonometric Identities (F.TF.9) identify possible identities by using graphs. apply strategies to prove identities. How is proving or verifying a trigonometric identity different then solving a trigonometric equation? Chapter Nine Lessons Chapter Nine Practice Worksheets Quiz: on concepts involving Basic Trigonometric Identities during the week. Benchmark Assessment 2: The Benchmark Assessment will focus on all Precalculus Concepts covered to date. Exploring on Calculator Activity Involving Graphing and Proving Identities As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for The Benchmark Assessment.
25 Precalculus ( 343 and 346) Term Three Week 5 F.TF.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Unit Seven Trigonometric Identities Addition and Subtraction Identities for Sine and Cosine Functions (F.TF.9) Addition and Subtraction Identities for the Tangent Function (F.TF.9) Co Function Identities (F.TF.9) use addition and subtraction identities for sine, cosine, and tangent functions. use the co function identities. What is the difference between the reciprocal and co functional relationships for trigonometric functions? Chapter Nine Lessons Chapter Nine Practice Worksheets Activity Proving the Difference Identity for Cosine Co Function Activity. Quiz: on concepts involving Addition and Subtraction Identities for Sine, Cosine and Tangent Functions during the week.
26 Precalculus ( 343 and 346) Term Three Week 6 F.TF.9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Other Trigonometric Identities (F.TF.9) Using Trigonometric Identities (F.TF.9) Unit Seven Trigonometric Identities use double angle identities. use power reducing identities. use half angle identities. use producttosum identities. use sumtoproduct identities. use the appropriate identity rule to solve trigonometric identities. Chapter Nine Lessons Chapter Nine Practice Worksheets Quiz: on concepts involving Using Double Angle Identities during the week. How can the doubleangle identity for sine be used calculate a distance? Architecture Application Activity to help prove the HalfAngle and DoubleAngle Identities As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Trigonometric Identities. Test: on concepts involving Trigonometric Identities.
27 Precalculus ( 343 and 346) Term Three Week 7 G.SRT.9. Derive the formula sin for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.10. Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11. Understand and apply the Laws of Sines and Cosines to find unknown measurements in right and nonright triangles, e.g., surveying problems, resultant forces. Unit Eight Trigonometric Applications The Law of Cosines (G.SRT.10, G.SRT.11) Solving of Triangle Using the Law of Cosines (G.SRT.11) Applications using the Law of Cosines (G.SRT.11) The Law of Sines (G.SRT.10, G.SRT.11) Supplementary Angle Identity (G.SRT.11) Area of a Triangle Using the Law of Sines (G.SRT.9) Heron s Formula (G.SRT.9) solve oblique triangles using the Law of Cosines. solve real world problems using the Law of Cosines. solve oblique triangles using the Law of Sines. solve real world problems using the Law of Sines. use area formulas to find area of triangles. Chapter Ten Lessons Chapter Ten Practice Worksheets Quiz: on concepts involving The Law of Sines and the Law of Cosines during the week. When is it necessary to use the Law of Sines to solve a triangle? Law of Cosines Activity Using Graphs to Explore Solutions on Graphing Calculator. Law of Sines Activity Exploring SSA Information and The Ambiguous Case
28 Precalculus ( 343 and 346) Term Three Week 8 A.APR.5. Know and apply the Binomial Theorem for the expansion of 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. G.SRT.9. Derive the formula sin for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Unit Eight Trigonometric Applications Area of a Triangle Using the Law of Sines (G.SRT.9) Heron s Formula (G.SRT.9) Binomial Theorem (A.APR.5) use area formulas to find area of triangles. expand a power of a binomials using the Binomial Theorem. find the coefficient of a given term of a binomial expansion. Chapter Ten Lessons Chapter Ten Practice Worksheets Quiz: on concepts involving Finding Area of Triangles during the week. How can you use the Binomial Theorem to expand binomials? Area of Triangle Activity As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Trigonometric Applications. Test: on concepts involving Trigonometric Applications.
29 Precalculus ( 343 and 346) Term Four Week 1 N.CN.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 given complex numbers represent the same number. N.CN.5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, has modulus 2 and argument120. Unit Eight Trigonometric Applications The Complex Plane (N.CN.4, N.CN.5) Polar Form for Complex Numbers (N.CN.4, N.CN.5) Demoivre s Theorem (N.CN.4, N.CN.5) Formulas and Rules for the nth Roots of Complex Numbers* (N.CN.5) Multiplication of Polynomials (N.CN.5) graph a complex number in the complex plane. find the absolute value of a complex number. express a complex number in polar form. perform polar multiplication and division. calculate power and roots of complex numbers. find and graph roots of unity. How is a complex number converted to polar form? Chapter Ten Lessons Chapter Ten Practice Worksheets Quiz: on concepts involving The Complex Plane and Polar Form for Complex Numbers during the week. Activity on Converting Between Complex Form and Polar Form. Activity on Polar Multiplication and Division.
30 Precalculus ( 343 and 346) Term Four Week 2 N.VM.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. N.VM.2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N.VM.3. Solve problems involving velocity and the other quantities that can be represented by vectors. N.VM.5a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; performs scalar multiplication componentwise. N.VM.5b. Compute the magnitude of a scalar multiple cv by using!!. Compute the direction of cv knowing that when! # 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). N.VM.4a. Add vectors endtoend, component wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. N.VM.4b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. N.VM.4c. Understand vector subtraction vw as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component wise. Unit Eight Trigonometric Applications Vectors in the Plane (N.VM.1, N.VM.2, N.VM.5a, N.VM.5b) Properties of Vectors (N.VM.4a, N.VM.4b, N.VM.4c) Applications of Vectors in the Plane (N.VM.3) The Dot Product *(N.VM.3) Properties of the Dot Product*(N.VM.3) find the components and magnitude of a vector. What is the difference between vectors and rays? perform scalar multiplication of vectors, vector addition, and vector subtraction. How do you find the dot product of two vectors? perform operations with linear combinations of vectors. determine the direction angle of a vector. determine resultant forces in physical applications. find the dot product of two vectors and the angle between two vectors. determine projection and component vectors and use them in physical applications. Chapter Ten Lessons Chapter Ten Practice Worksheets prepared for class, participate in class activities and actively engage in class discussion. Quiz: on concepts involving Vectors during the week. Test: on concepts involving Trigonometric Applications. Activity on Graphing Vectors As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for Test on Trigonometric Applications.
31 Precalculus ( 343 and 346) Term Four Week 3 G.CPE.3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum of difference of distances from the foci is constant. G.CPE.3a. Use equations and graphs of conic sections to model realworld problems. Unit Nine Analytic Geometry Standard Equation of an Ellipse Centered at the Origin (G.CPE.3, G.CPE.3a) Characteristics of Ellipses (G.CPE.3, G.CPE.3a) Standard Equation of a Hyperbola Centered at the Origin (G.CPE.3, G.CPE.3a) Characteristics of Hyperbolas (G.CPE.3, G.CPE.3a) define and write the equation of an ellipse. identify important characteristics and graph ellipses. define and write the equation of a hyperbola. identify important characteristics and graph hyperbolas. Chapter Eleven Lessons Chapter Eleven Practice Worksheets Quiz: on concepts involving Ellipses and Hyperbolas during the week. How does the concept of distance relate to the concepts of ellipses and hyperbolas? Ellipse ActivityCreating your own Ellipse on a Paper Plate and identifying important characteristics of the Ellipse
32 Precalculus ( 343 and 346) Term Four Week 4 G.CPE.3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum of difference of distances from the foci is constant. G.CPE.3a. Use equations and graphs of conic sections to model realworld problems. Unit Nine Analytic Geometry Standard Equation of a Parabola with Vertex at the Origin (G.CPE.3a) Characteristics of Parabolas (G.CPE.3a) Horizontal and Vertical Shifts (G.CPE.3a) Standard Equations of Conic Sections (G.CPE.3a) Graphs of SecondDegree Equations* (G.CPE.3a) define and write the equation of a parabola. identify important characteristics and graph parabolas. graph and write the equation of a translated conic. determine the shape of a translated conic without graphing. Chapter Eleven Lessons Chapter Eleven Practice Worksheets Quiz: on concepts involving Parabolas during the week. Benchmark Assessment 3: The Benchmark Assessment will focus on all Precalculus Concepts covered to date. How do you determine the shape of a translated conic section with graphing? Activity to determine the shape of a translated conic without graphing. As time allows, Group Work Practice or Jeopardy Review Practice, or similar game on Smartboard to Review for The Benchmark Assessment.
33 Precalculus ( 343 and 346) Term Four Week 5 N.CN.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 number. The Polar Coordinate System (N.CN.4) Coordinate Conversion Formulas (N.CN.4) Polar Graphs (N.CN.4) Polar Equations of Conics * (N.CN.4) Unit Nine Analytic Geometry locate points in a polar coordinate system. convert between coordinates in rectangular and polar systems. create graphs of equations in polar coordinates. recognize the equations of a cardioids, rose, circle, lemniscates, and limacon. define eccentricity of a ellipse, parabola, and a hyperbola. develop and use the general polar equation of a conic section. Chapter Eleven Lessons Chapter Eleven Practice Worksheets Test: on concepts involving Analytic Geometry. How is the procedure of parameterization of conic sections used to solve real world problems? Activity on SmartBoard to Graph the Given Coordinates, Then Give Three More Polar Coordinates that Represent the Same Point. Polar Graphing Activity on Paper and Graphing Calculator Mickey Mouse Activity on Calculator creating graphs of equations in polar coordinates
34 Precalculus ( 343 and 346) Term Four Week 6 A.REI.11 Explain why the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (AI.P.4) N.VM.6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N.VM.7. Multiply matrices by scalars to produce new matrices, e.g., as when all payoffs in a game are doubled. N.VM.8. Adds, subtracts, and multiplies matrices of appropriate dimensions. N.VM.9. Understand that unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associate and distributive properties. N.VM.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 Ten Systems of Equations Solving Systems of Equations (N.VM.6, A.REI.11) Matrices (N.VM.7, N.VM.8, N.VM.9, N.VM.10) Matrix Operations (N.VM.6, N.VM.7, N.VM.8, N.VM.9, N.VM.10) solve a system of equations. solve systems using matrices. add and subtract matrices. multiply using a scalar. multiply two matrices. use matrix multiplication to solve problems. define n x n identity matrix. find the inverse of an invertible matrix. solve square systems of equations using inverse matrices. Chapter Twelve Lessons Chapter Twelve Practice Worksheets Quiz: on concepts involving Operations of Matrices during the week. What is an example of where you would use matrices? Solving Systems Activity Graphically on Paper and Graphing Calculator and Algebraically Solving 2 x 2 Systems Using Matrices on Graphing Calculator Activity Exploring Secret Codes Using Matrices and Inverses of Matrices
35 Precalculus ( 343 and 346) Term Four Week 7 N.VM.6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N.VM.7. Multiply matrices by scalars to produce new matrices, e.g., as when all payoffs in a game are doubled. N.VM.8. Add, subtract, and multiply matrices of appropriate dimensions. N.VM.9. Understand that unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associate and distributive properties. N.VM.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. A.REI.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 3 x 3 or greater). Unit Ten Systems of Equations Matrix Operations (continued) (N.VM.6, N.VM.7, N.VM.8, N.VM.9, N.VM.10.) Matrix Methods for Square Systems (N.VM.9) Solve a System of Equations Using Matrices (N.VM.6, N.VM.8, N.VM.9, and A.REI.9) Type of System and Number of Solutions * (N.VM.6, N.VM.7, N.VM.8, N.VM.9, N.VM.10) Consistent and Inconsistent Systems* (N.VM.6, N.VM.7, N.VM.8, N.VM.9, N.VM.10) multiply two matrices. use matrix multiplication to solve problems. define n x n identity matrix. find the inverse of an invertible matrix. solve square systems of equations using inverse matrices. solve systems of equations by graphing, substitution and elimination. recognize consistent and inconsistent systems. solve applications using systems. Chapter Twelve Lessons Chapter Twelve Practice Worksheets Quiz: on concepts involving Solving Systems of Matrices during the week. How do you determine whether two matrices can be added, subtracted, or multiplied or solved? Activity Exploring Systems and Solving Systems of Three Equations Using Matrices on Graphing Calculator
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