Precalculus, Quarter 3, Unit 3.1 Trigonometric Equations and Inverses Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Solve simple trigonometric equations. Solve trigonometric equations by substituting equivalent trigonometric expressions into the trigonometric equations. Solve complex trigonometric equations through the use of previously learned trigonometric identities and problem-solving strategies. Construct the inverse of a trigonometric function algebraically, incorporating domain restrictions. Verify solutions to equations through the use of technology. Mathematical practices to be integrated Look for and make use of structure. Evaluate work and make modifications or try a new approach, if necessary. Make sense of problems and persevere in solving them. Think about simpler problems to help solve more complex problems. Check the reasonableness of the solution; check answers by using multiple methods. Use technology to solve problems. Justify and explain solutions. Persevere do not give up. Model with mathematics. Interpret results in the context of a problem. Essential questions What are the similarities and differences between solving trigonometric equations and verifying trigonometric identities? Why is it necessary to incorporate basic trigonometric identities in solving complicated trigonometric equations, and which identities are most useful in accomplishing this? Why is it necessary to restrict the domain of a trigonometric function when producing its inverse. What are the similarities and differences among solutions of trigonometric equations and other functions previously studies? What real-life situations employ the use of trigonometric equations? 35
Precalculus, Quarter 1, Unit 3.1 Trigonometric Equations and Inverses (10 days) Written Curriculum Common Core State Standards for Mathematical Content Trigonometric Functions F-TF Model periodic phenomena with trigonometric functions F-TF.6 F-TF.7 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (+) 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. Common Core Standards for Mathematical Practice 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. 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, two-way 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. 36
Precalculus, Quarter 1, Unit 3.1 Trigonometric Equations and Inverses (10 days) 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 well remembered 7 5 + 7 3, 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 2 + 7. 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. Clarifying the Standards Prior Learning In grades 1 and 2, students demonstrated conceptual understanding of equality by finding values that make open sentences true using models, verbal explanations, or written equations. At this time, it was limited to one operation using addition or subtraction. In grade 3, multiplication was added as an operation. In grade 4, students began to simplify numerical expressions where left-to-right computations were modified only by parentheses. Solving one-step linear equations of the form ax = c or x ± b = c, where a, b, and c are whole numbers was introduced. In grade 5, x = c (where a 0)-type equations were a added, as well as equations of the form ax ± b = c, using selections from a replacement set. In grade 6, solving multistep linear equations ax ± b = c, without a replacement set, were undertaken. In grade 7, equations with variables and constants on both sides were introduced, and ax ± b = cx ± d, and x ± b = c were solved. In addition, translating a problem-solving situation into an equation consistent with a the above equations was introduced. In grade 8, solving formulas for a variable using one transformation was explored. Solving multistep equations with integer coefficients was introduced. The commutative, associative, and distributive properties were learned; order of operations or substitution and informally solving systems of linear equations in a context were processes used to solve equations. In grades 9 and 10, symbolic, graphical, algebraic, analytic, and verbal approaches were used to solve equations. Models and representations were employed throughout. In grades 11 and 12, processes included factoring, completing the square, quadratic formula, synthetic division, domain restriction and graphing to solve multistep linear and nonlinear equations, expressions, and inequalities. These included the study of rational and radical functions in addition to all other parent functions. Current Learning Students in advanced mathematics classes solve equations and verify identities involving trigonometric expressions using graphs, tables, equations, and algebraic manipulation. Units parallel to this have logarithmic and exponential expressions at the core of learning. The Intermediate Value Theorem is applied to find exact or approximate solutions of equations or zeroes of continuous functions. 37
Precalculus, Quarter 1, Unit 3.1 Trigonometric Equations and Inverses (10 days) Future Learning In calculus, the derivative concept will be applied to the trigonometric functions. A more rigorous approach to the end and asymptotic behavior of these functions will be analyzed in a variety of representations. Trigonometric applications will be investigated via the continuity and limit concept. sin x will be studied as the basic premise underlying the Sandwich/Squeeze Theorem. x Additional Findings Beyond Numeracy states that many people remember trigonometric identities indicating the unconditional equality of one complicated trigonometric expression for another. However only a few identities are crucial; the most important is sin 2 x + cos 2 x = 1 (pp. 250 256). 38
Precalculus, Quarter 3, Unit 3.2 Rectangular and Polar Representations and the Complex Coordinate Plane Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Find the conjugate of a complex number and express it in a + bi form. Represent a complex number in a + bi form graphically on the complex coordinate plane. Perform basic operations on complex numbers and plot solutions on the complex coordinate plane. Compute conversions between rectangular and polar coordinates. Construct the moduli and quotients of complex numbers using the conjugates of said number. Represent complex numbers on the complex coordinate plane in rectangular and polar form to include real and imaginary numbers. Prove that different representations of the same number written in complex and polar form signify the same value graphically and algebraically. Calculate the distance between numbers in the complex plane as the modulus of the difference. Calculate the midpoint of a segment as the average of the numbers of its endpoints, using complex numbers graphed in the complex coordinate plane. Mathematical practices to be integrated Use appropriate tools strategically. Use appropriate tools, including technology, to help solve problems. Use technology to help visualize results. Determine specific tools that will be helpful. Make sense of problems and persevere in solving them. Think about simpler problems to help solve more complex problems. Work among different representations. Use technology to solve problems. Justify and explain solutions. Persevere do not give up. Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Use labels of axes and units of measure correctly. Calculate and compute accurately (including technology). 39
Precalculus, Quarter 1, Unit 3.2 Rectangular and Polar Representations and the Coordinate Plane (10 days) Essential questions Why is it important to represent a complex number in polar or rectangular form? What situations arise that determine which representation of a complex number is better? What changes need to take place before using the calculator to graph polar equations? Why are these calculator changes necessary? What impact does a polar equation have on the symmetry of a graph? What are the similarities and differences among numbers represented in complex form, rectangular form, and polar form? What real-life situations employ the use of numbers represented in polar form? In the equation of the form r = asinbθ or r = acosbθ, what effect does changing the values of a and b have on the graph? What examples can be found in nature that display the characteristics of a polar graph? In what ways do they accomplish this? Written Curriculum Common Core State Standards for Mathematical Content The Complex Number System N-CN Perform arithmetic operations with complex numbers. N-CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the complex plane. N-CN.4 N-CN.5 N-CN.6 (+) 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. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( 1 + 3 i) 3 = 8 because ( 1 + 3 i) has modulus 2 and argument 120. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 40
Precalculus, Quarter 1, Unit 3.2 Rectangular and Polar Representations and the Coordinate Plane (10 days) Common Core Standards for Mathematical Practice 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. 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, two-way 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. 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. 41
Precalculus, Quarter 1, Unit 3.2 Rectangular and Polar Representations and the Coordinate Plane (10 days) Clarifying the Standards Prior Learning In algebra 2, students learned the operations using complex numbers. They also used complex numbers as solutions to polynomial equations. Current Learning Students manipulate complex numbers using rectangular and polar coordinates. Future Learning The study of the rectangular and polar coordinate systems will lead to the development of parametric equations, which is useful for studying motion in calculus. The derivatives of parametric, polar, and rectangular equations will be emphasized in calculus, with emphasis on related rates and real-world applications. Additional Findings Science for All Americans indicates that rectangular, polar, and parametric equations are the patterns in math models for wave behavior and planetary motion (pp. 52 56, 124, and 145 151). Principles and Standards for School Mathematics indicates that polar coordinates are used to analyze geometric situations. The polar coordinate representation can be simpler and may be more useful for solving certain problems (pp. 308, 314). 42
Precalculus, Quarter 3, Unit 3.3 Sequences and Series Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Compute partial sums of infinite arithmetic and geometric sequences. Determine when and if a geometric series converges. Find the sum of an infinite geometric series. Connect arithmetic and geometric sequences to linear and exponential functions, respectively. Mathematical practices to be integrated Look for and express regularity in repeated reasoning. Look for general methods, patterns, repeated calculations, and shortcuts. Look for patterns to find generalizations. Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including using technology). Model with mathematics. Relate what has been learned in mathematics to everyday life. Essential questions What algebraic concept relates to the common difference of an arithmetic sequence? What are the similarities and differences between arithmetic and geometric sequences defined explicitly or recursively? What real-world applications are modeled by arithmetic and geometric series? What is the primary distinction between a sequence and a series? 43
Precalculus, Quarter 1, Unit 3.3 Sequences and Series (15 days) Written Curriculum Common Core State Standards for Mathematical Content None for this unit of study. Common Core Standards for Mathematical Practice 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, two-way 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. 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. 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. 44
Precalculus, Quarter 1, Unit 3.3 Sequences and Series (15 days) Clarifying the Standards Prior Learning In kindergarten, students identified patterns to find the next one, two, or three elements. In grade 1, patterns were represented in models, tables, or sequences. Students also found missing elements in a numerical pattern. In grade 4, students studied nonlinear patterns and wrote rules in words or symbols to find the next case. In grade 5, patterns were studied in problem situations. In grade 6, students wrote patterns in expression or equation form, using words or symbols to express the generalization of a linear relationship. In grade 7, students extended this writing of patterns to nonlinear relationships. In grade 8, students continued their study of these concepts in more depth. In grade 9 and 10, students solved problems involving patterns, and in grade 11, they studied first, second, and third differences. Also in grade 11, students derived the formula for the sum of a finite geometric series and solved problems. Current Learning Students identify arithmetic and geometric sequences and find the nth term. Students determine when an infinite geometric series converges and find its sum. Students connect concepts to linear and exponential functions. They also compute sums and partial sums of arithmetic and geometric series. Future Learning The study of sequences and series ultimately leads to an analysis of limits, integral calculus, engineering calculus, and applications of higher calculus. Students will compute limits for various functions, the integral as a net accumulator, and Taylor series for various functions. In addition, students will calculate the area under a curve and the volume of a curve using partitioning, slicing, and cross-sections. Additional Findings According to Beyond Numeracy, infinite series and their applications constitute an important area of mathematical analysis. Used informally by mathematicians long before they were completely understood, series appeal to our intuitions about numbers and infinity. Annuities provide a practical application of infinite geometric series. Geometric series also arise in determining the quantity of medicine in the blood of a person on a long-term regimen of the medicine. Finding derivatives, integrals, solving differential equations, and working with complex and imaginary numbers are simplified and represented by power series (pp. 221 224). 45
Precalculus, Quarter 1, Unit 3.3 Sequences and Series (15 days) 46