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Ohio’s Learning Standards – Clear Learning Targets Precalculus F-BF.1 Build a function that models Essential Understanding Academic Vocabulary/Language a relationship between two quantities - Students should be able to identify and describe characteristics of parent functions. Students should develop understanding of translations/transformations and their applications to all functions. -zeros, roots, transformation, translation reflections, dilation, parent square root, constant identity, quadratic cubic, reciprocal absolute value step, greatest integer function, composition Write a function that describes a relationship between two quantities. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. I Can Statements I can use graphs of functions to estimate function values. I can identify, graph, and describe parent functions. I can identify and graph transformations of parent functions. I can perform operations with functions. I can find compositions of functions. Tier 2 Vocabulary Extended Understanding - describe, compose, transform - Students can translate piecewise functions or functions that they have not yet seen. - Students may find one function when given the composition of the functions and the other function. Prior Knowledge Future Learning - transform/translate linear, quadratic, and exponential functions. - Use compositions and transformations to find inverses of relations and functions both algebraically and graphically. - identify domain and range of linear, quadratic, and exponential functions. - evaluate functions. - operations on functions. Columbus City Schools Clear Learning Targets Precalculus 2015-2016 1 Instructional Strategies Utilize graphing technology so that students can investigate behaviors of transformations on different families of functions. Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles when a function for the cost of each (given the number of miles driven) is known. Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual models to generate sequences of numbers that can be explored and described with both recursive and explicit formulas. Emphasize that there are times when one form to describe the function is preferred over the other. Common Misconceptions and Challenges Students may not consider order when transforming functions. Students may misunderstand function notation for composition of functions to represent multiplication (e.g., f(g(x)) means to multiply the f and g function values). Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Lessons 1-2, 1-5, 1-6 Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/BF/B/4 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Students will research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and interest rate). They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact hybrid). Once they choose a vehicle, they will use their evaluations to show why they chose the vehicle. Their research will include interviewing automotive professionals, visiting dealerships, and navigating company websites. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 2 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-BF.4 Build new functions from Essential Understanding Academic Vocabulary/Language existing functions - Students should be able to find inverse functions algebraically and graphically. - Students should be able to evaluate whether two functions are inverses by examining their graphs. - set notation, interval notation, implied domain, relevant domain, piecewise function, continuous, limit, discontinuous, infinite, jump, point, removable discontinuity, nonremovable discontinuity, end behavior, inverse relation, inverse function, one-to-one Find inverse functions. 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) = 2x3 for x>0 or f(x) = (x+1)/(x-1) for x ≠1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or table, given that the function has an inverse. d. (+) Produce an invertible function from a noninvertible function by restricting the domain. I Can Statements I can describe subsets of real numbers. I can identify and evaluate functions and state their domain. I can identify odd and even functions. I can use limits to determine the continuity of a function. I can use limits to describe the end behavior of functions. I can use the horizontal line test to determine whether a function has an inverse. I can find inverse functions algebraically and graphically. Columbus City Schools Extended Understanding - Students can evaluate whether piecewise functions have inverses. - Investigate inverses of even and odd functions. Tier 2 Vocabulary - compose, construct, evaluate, produce Prior Knowledge Future Learning - Find domain and range of functions. - Analyze graphs of polynomial and rational functions. - Find composition of functions. - Evaluate functions. - use limits to determine function continuity. - use limits to define end behavior. Clear Learning Targets Precalculus 2015--‐2016 3 Instructional Strategies Provide examples of inverses that are not purely mathematical to introduce the idea. For example, given a function that names the capital of a state, f(Ohio) = Columbus. The inverse would be to input the capital city and have the state be the output, such that f --1 (Denver) = Colorado. Use real-world examples of functions and their inverses. For example, students might determine that folding a piece of paper in half 5 times results in 32 layers of paper, but that if they are given that there are 32 layers of paper, they can solve to find how many times the paper would have been folded in half. Students should also recognize that not all functions have inverses. Again using a nonmathematical example, a function could assign a continent to a given country’s input, such as g(Singapore) = Asia. However, g-1 (Asia) does not have to be Singapore (e.g., it could be China). Exchange the x and y values in a symbolic functional equation and solve for y to determine the inverse function. Recognize that putting the output from the original function into the input of the inverse results in the original input value. Common Misconceptions and Challenges Students may believe that all functions have inverses and need to see counter examples, as well as examples in which a non-invertible function can be made into an invertible function by restricting the domain. For example, f(x) = x 2 has an inverse ( f -1 (x) = x ) provided that the domain is restricted to x ≥ 0. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Lessons 1-1, 1-2, 1-3, 1-7 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSF/TF/A/4 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Aerospace engineers Perform a variety of engineering work in designing, constructing, and testing aircraft, missiles, and spacecraft. May conduct basic and applied research to evaluate adaptability of materials and equipment to aircraft design and manufacture. May recommend improvements in testing equipment and techniques. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 4 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-IF.7 Analyze functions using Essential Understanding Academic Vocabulary/Language different representations - Students should be able to solve rational equations algebraically, finding extraneous solutions. - Students should be able to analyze rational functions, determining whether there are any asymptotes. - Students should be able to graph rational functions, including asymptotes. - rational functions, asymptote, vertical asymptote, horizontal asymptote, oblique asymptote, holes Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Tier 2 Vocabulary - analyze, construct, evaluate Extended Understanding - Students can construct a rational function when given intercepts and asymptotes or when given a graph. I Can Statements I can analyze and graph rational functions. I can find zeros of rational functions. I can solve rational equations. I can find asymptotes of rational functions and explain discontinuities. I can describe end behavior of rational functions using limit notation. Columbus City Schools Prior Knowledge Future Learning - Identify points of discontinuity and end bahavior of graphs of funtions using limits. - Write partial fraction decompositions of rational expressions. Clear Learning Targets Precalculus 2015--‐2016 5 Instructional Strategies Important features of rational functions are vertical asymptotes, or boundaries that the function gets closer and closer to, but never actually reaches. Vertical asymptotes exist when the denominator of the function is 0 and can be found by computing the zeros of the denominator of the function. If the highest-order x terms in the numerator and denominator have the same exponent, then there will also be a horizontal asymptote when their coefficients are divided. https://www.khanacademy.org/math/algebra2/rational-expressions/rational-function-graphing/e/graphs-of-rational-functions http://betterlesson.com/common_core/browse/640/ccss-math-content-hsf-if-c-7d-graph-rational-functions-identifying-zeros-andasymptotes-when-suitable-factorizations-are-availab Common Misconceptions and Challenges Students may believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that they will come to understand the usefulness of these skills in the context of examining characteristics of functions. Students may forget to factor the numerator when finding asymptotes. When using graphing technology, students may not consider the window they are viewing the function through Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Lesson 2-2 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Chemical engineers Design chemical plant equipment and devise processes for manufacturing chemicals and products, such as gasoline, synthetic rubber, plastics, detergents, cement, paper, and pulp, by applying principles and technology of chemistry, physics, and engineering. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 6 Ohio’s Learning Standards – Clear Learning Targets Precalculus CN.3 Perform arithmetic Essential Understanding Academic Vocabulary/Language operations with complex numbers - It is not possible in the realm of real numbers to find the square root of a negative number. - The square root of -1 is referred to as i. - Rational Zero Theorem, Descartes’ Rule of Signs, Fundamental Theorem of Algebra, Linear Factorization Theorem, complex conjugate (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Extended Understanding - The modulus of a complex number is always a real number and in fact it will never be negative. I Can Statements I can find the conjugate of a complex number. I can find the moduli of complex numbers. I can find the quotient of complex numbers. Columbus City Schools Tier 2 Vocabulary - perform, find, understand Prior Knowledge Future Learning - Student learned in Integrated Mathematics II that the square root of -1 is called i. - Students will learn to build new functions from current functions. - Students performed addition, subtraction, multiplication, and division on complex numbers. Clear Learning Targets Precalculus 2015--‐2016 7 Instructional Strategies Refer here for some notes on this topic: http://tutorial.math.lamar.edu/Extras/ComplexPrimer/ConjugateModulus.aspx Remind students of what they learned about the Fundamental Theorem of Algebra: A polynomial function of degree n can have at most n real zeroes. Refer to section 2-1 in the book. Go over examples 1-4. Have students use synthetic division to determine roots. Common Misconceptions and Challenges Students may need a reminder as to how to do synthetic division. You can use this puzzle to review (page 54). Remind students that when they set up their synthetic division steps, they must change the sign on the divisor. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Chapter 2-1 Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-N-CN#HSN-CN.A.3 Gizmo activity: http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=147 Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-cn-3.html Career Connections Many careers that deal with imaginary number revolve around electronics and engineering. Electrical engineers and electronic engineers use complex numbers to model AC with dealing with electrical issues. Other fields that use complex numbers include physicists, astronomers, audio technicians, and audiologists. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 8 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-BF.5 Build new functions from Essential Understanding Academic Vocabulary/Language existing functions -Students should identify domain, range, and end behavior of exponential and logarithmic functions. -Students should use the properties of exponents and logarithms to solve exponential and logarithmic functions. -Students should use function models to predict and make decisions and judgments. - Algebraic function transcendental function, exponential function, natural base, continuous compound interest, APY, APR, principal, logarithmic function, base, common logarithm, natural logarithm, exponential growth, exponential decay Extended Understanding - analyze, predict, judge Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Tier 2 Vocabulary - Analyze real world data that can be modeled by exponential and logarithmic functions. I Can Statements I can evaluate, analyze, and graph exponential functions. I can solve problems involving exponential growth and decay. I can evaluate expressions involving logarithms. I can sketch and analyze graphs of logarithmic functions. I can apply properties of logarithms. I can evaluate logarithms. I can use properties of exponential functions to solve equations. I can use properties of logarithmic functions to solve equations. Columbus City Schools Prior Knowledge Future Learning - Identify, graph, and describe different parent functions. - describe parent functions symbolically and graphically. - identify domains and ranges of functions. - determine the domain and range of functions using graphs, tables, and symbols. - understand properties of exponents. - solve algebraic equations. Clear Learning Targets Precalculus 2015--‐2016 - use regression to determine the appropriateness of an exponentail, logarithmic, logisteic, ubic, quartic, or quadratic model. 9 Instructional Strategies Students need to recognize that exponential and logarithmic functions are inverses of one another and use these functions to solve real-world problems. Nonmathematical examples of functions and their inverses can help students to understand the concept of an inverse and determining whether a function is invertible. Provide applied examples of exponential and logarithmic functions, such as the use of a logarithm to determine pH or the strength of an earthquake on the Richter Scale. Both pH and Richter Scale values are powers of 10 and are, therefore, logarithms. For example, the magnitude of an earthquake, M, on the Richter Scale can be calculated using the formula M = log10A , where A represents the amplitude of measured seismic waves. Common Misconceptions and Challenges Students may confuse exponential and polynomial functions. While both have an exponent, 𝑦 = 𝑥 ! is a polynomial function while 𝑦 − 2! is an exponential function. Students may confuse the exponents in logarithmic and exponential forms. Students my confuse properties of logarithms with properties of exponents and incorrectly represent the logarithm Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 3 Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/BF/B/5 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htmhttps://www.ixl.com/standards/ohio/m ath/high-school Career Connections Exponential and logarithmic functions are used for the Richter scale, the pH scale, to model populations, carbon date artifacts, determine time of death, and compute investments. Some of the career fields in which exponential and logarithmic functions are used include Economists, Bankers, Financial Advisors, Insurance Risk Assessors, Biologists, Engineers, Computer Programmers, Chemists, Physicists, Geographers, Sound Engineers, Statisticians, Mathematicians, and Geologists. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 10 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-TF.3 Extend the domain of Essential Understanding Academic Vocabulary/Language trigonometric functions using the unit circle - Students should be able to evaluate and graph inverse trig functions and find compositions of trig functions. -Students should be able to use the Laws of Sines and Cosines to solve and find area of oblique triangles. - arcsine function, arccosine function, arctangent function, oblique triangles, Law of Sines, ambiguous case, Law of Cosines, Heron’s Formula Extended Understanding - solve, evaluate, compare Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number Tier 2 Vocabulary -Students can compare translations of inverse functions -Students can find area of irregular figures I Can Statements I can evaluate and graph inverse trigonometric functions. I can find compositions of trigonometric functions I can solve oblique triangles by using the Law of Sines or the Law of Cosines. I can find areas of oblique triangles Columbus City Schools Prior Knowledge Future Learning - Find and graph inverses of relations and functions - Solve trig equations - Solve right triangles using trig functions Clear Learning Targets Precalculus 2015--‐2016 - Verify trig identities 11 Instructional Strategies Students can use what they know about 30-60-90 triangles and right isosceles triangles to determine the values for sine, cosine, and tangent for π/3, π/4, and π/6. In turn, they can determine the relationships between, for example, the sine of π/6, 7π/6, and 11π/6, as all of these use the same reference angle and knowledge of a 30-60-90 triangle. http://betterlesson.com/common_core/browse/686/ccss-math-content-hsf-tf-a-3-use-special-triangles-to-determine-geometrically-the- valuesof-sine-cosine-tangent-for-3-4-and-6-an?from=domain_core Common Misconceptions and Challenges Students may believe that there is no need for radians if one already knows how to use degrees. Students need to be shown a rationale for how radians are unique in terms of finding function values in trigonometry since the radius of the unit circle is 1. Students may also believe that all angles having the same reference values have identical sine, cosine and tangent values. They will need to explore in which quadrants these values are positive and negative. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Lesson 4-6, 4-7 Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/TF/A/3 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Drafters prepare the drawings used to build everything from spacecraft to bridges. Using rough sketches done by others, they produce detailed technical drawings with specific information to create a finished product. Drafters use handbooks, tables, calculators, and computers to do their work. Many specialize in architecture, electronics, or aeronautics. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 12 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-TF.4 Extend the domain of Essential Understanding Academic Vocabulary/Language trigonometric functions using the unit circle - Students should find trig values for any angle. -Students should find values of trig functions using the unit circle. - quadrantal angle, reference angle, unit circle, circular function, periodic function, period Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Extended Understanding - Explore relationships between equivalent values of sine and cosine on the unit circle. I Can Statements I can find the values of trigonometric functions for any angle. I can construct the unit circle I can find values of trigonometric functions using the unit circle. Columbus City Schools Tier 2 Vocabulary - explore, construct Prior Knowledge Future Learning - find values of trig functions for acute angels uisng ratios in right triangles - graph transformations of sine and cosine functions Clear Learning Targets Precalculus 2015--‐2016 - graph tangent and reciprocal trig functions 13 Instructional Strategies Provide students with real-world examples of periodic functions. One good example is the average high (or low) Ohio Department of Education, March 2015 Page 17 Mathematics Model Curriculum temperature in a city in Ohio for each of the 12 months. These values are easily located at weather.com and can be graphed to show a periodic change that provides a context for exploration of these functions. Allow plenty of time for students to draw – by hand and with technology – graphs of the three trigonometric functions to examine the curves and gain a graphical understanding of why, for example, cos (π/2) = 0 and whether the function is even (e.g., cos(-x) = cos(x)) or odd (e.g., sin(-x) = -sin(x)). Similarly students can generalize how function values repeat one another, as illustrated by the behavior of the curves. http://betterlesson.com/common_core/browse/687/ccss-math-content-hsf-tf-a-4-use-the-unit-circle-to-explain-symmetry-odd-and-even-andperiodicity-of-trigonometric-functions?from=domain_core_container Common Misconceptions and Challenges Students may also believe that all angles having the same reference values have identical sine, cosine and tangent values. They will need to explore in which quadrants these values are positive and negative. Students may ignore negative values of trig functions in the unit circle. Students may confuse the coordinates of the ordered pair on the unit circle. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Lesson 4-3 Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/TF/A/4 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Civil engineers perform engineering duties in planning, designing, and overseeing construction and maintenance of building structures and facilities, such as roads, railroads, airports, bridges, harbors, channels, dams, irrigation projects, pipelines, power plants, water and sewage systems, and waste disposal units. Includes architectural, structural, traffic, ocean, and geo-technical engineers. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 14 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-TF.7 & 9 Model periodic Essential Understanding Academic Vocabulary/Language phenomena with trigonometric functions - Students should be able to graph transformations of sine and cosine functions. - Students should be able to use sine and cosine functions to model data. Students should be able to graph the tangent and reciprocal trig functions and damped trig functions. - sinusoid, amplitude, frequency, phase shift, vertical shift, midline, damped trigonometric function, damping factor, damped oscillation, damped wave, damped harmonic motion Prove and apply trigonometric identities Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of context. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. I Can Statements I can graph tangent and reciprocal trigonometric functions I can graph transformations of the sine and cosine function. I can use a graphing calculator to graph the sine functions and its inverse. I can use sinusoidal functions to solve problems. I can graph and examine the periods and sums and differences of sinusoids. I can graph damped trigonometric functions. Columbus City Schools Tier 2 Vocabulary - model, evaluate, prove Extended Understanding - Write functions to represent graphs and when given characteristics of functions. Prior Knowledge Future Learning - Students should be aware that sine is odd and cosine is even. Students should know the relations between sine, cosine and tangent and should know the relation between trigonometric values of complementary angles. - Evaluate and graph inverse trig functions. Clear Learning Targets Precalculus 2015--‐2016 15 Instructional Strategies Students can explore the inverse trigonometric functions, recognizing that the periodic nature of the functions depends on restricting the domain. These inverse functions can then be used to solve real-world problems involving trigonometry with the assistance of technology. Students can explore other trigonometric identities, such as the half-angle, double-angle, and addition/subtraction formulas to extend on the Pythagorean relationship. Formulas should be proven and then used to determine exact values when given an angle measure, to prove identities, and to solve trigonometric equations. For example, by dividing the formula sin2 (θ) + cos2 (θ) = 1 by cos2 (θ), a new formula is generated ( tan2 (θ) +1= sec 2 (θ) ). Common Misconceptions and Challenges Students may believe that sin-1 A = 1/sin A, thus confusing the ideas of inverse and reciprocal functions. Additionally, students may not understand that when sin A = 0.4, the value of A represents an angle measure and that the function sin-1 (0.4) can be used to find the angle measure. Students may believe that sin(A +B) = sinA + sinB and need specific examples to disprove this assumption. Students may have difficulty in remembering which of the trig functions pass through the origin. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Lessons 4-4, 4-5 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSF/TF/B https://www.illustrativemathematics.org/content-standards/HSF/TF/C/9 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career Connections Civil engineers perform engineering duties in planning, designing, and overseeing construction and maintenance of building structures and facilities, such as roads, railroads, airports, bridges, harbors, channels, dams, irrigation projects, pipelines, power plants, water and sewage systems, and waste disposal units. Includes architectural, structural, traffic, ocean, and geo-technical engineers. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 16 Ohio’s Learning Standards – Clear Learning Targets Precalculus F-TF.8 Prove and apply Essential Understanding Academic Vocabulary/Language Trigonometric Functions - Students should be able to use trig identities to find trig values, simplify expressions, and solve equations. -Students should verify trig identities and determine whether equations are trig identities. - identity, trig identity, cofunction, odd-even identities, Pythagorean identities, Prove the Pythagorean identity sin2 (θ) + cos2 (θ) = 1 and use it to calculate trigonometric ratios. Tier 2 Vocabulary - verify, prove Extended Understanding - Students can solve trig inequalities I Can Statements I can identify and use trig identities to find trig values. I can use trig identities to simplify trig expressions. I can verify trig identities. I can determine whether equations are trig identities. I can solve trig equations using trig identities. Columbus City Schools Prior Knowledge Future Learning - Students should be able to find trig values using the unit circle. - Students will use trig identities to transform expresisons into forms that will be used for integration and differentiation. - Students should be able to solve right triangles. - Students should be able to find values of trig functions for any angle. Clear Learning Targets Precalculus 2015--‐2016 - Students will use trig substitution for integration. 17 Instructional Strategies In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse is always 1, the Pythagorean relationship sin2 (θ) + cos2 (θ) = 1 is always true. Students can make a connection between the Pythagorean Theorem in geometry and the study of trigonometry by proving this relationship. In turn, the relationship can be used to find the cosine when the sine is known, and vice-versa. Provide a context in which students can practice and apply skills of simplifying radicals. Drawings of the unit circle can be useful in showing why the Pythagorean relationship must be true. Dynamic geometry software, such as Geometer’s Sketchpad or Geogebra, can be used to demonstrate that, regardless of the angle measure, the Pythagorean relationship always holds in the unit circle. http://betterlesson.com/common_core/browse/694/ccss-math-content-hsf-tf-c-8-prove-the-pythagorean-identity-sin2-cos2-1-and-use-it-to-find-sin-cos-or-tan-given- sin-cos-or-tana?from=domain_core Common Misconceptions and Challenges Students may believe that there is no connection between the Pythagorean Theorem and the study of trigonometry. Students may also believe that there is no relationship between the sine and cosine values for a particular angle. The fact that the sum of the squares of these values always equals 1 provides a unique way to view trigonometry through the lens of geometry. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Lesson 5-1, 5-2, 5-3 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8 Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career connections Physicists study the matter that makes up the universe. They also study forces of nature such as gravity and nuclear interaction. They use their studies to design medical equipment, electronic devices, and lasers. Astronomers study the moon, sun, planets, galaxies, and stars. Their knowledge is used in space flight and navigation. Many teach in colleges and universities. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 18 Ohio’s Learning Standards – Clear Learning Targets Precalculus REI.8 Reasoning with Equations & Essential Understanding Academic Vocabulary/Language Inequalities - Gaussian elimination is a method that is used to solve systems of equations where row operations are performed. - multi-variable linear system, reduce row-echelon, Gaussian elimination, augmented matrix, coefficient matrix, Gauss-Jordan elimination. Extended Understanding Tier 2 Vocabulary - The solution of a system solved by reduced row-echelon method is an ordered triple labeled (r, s, t). - represent (+) Represent a system of linear equations as a single matrix equation in a vector variable. I Can Statements I can perform operations with matrices. I can use matrices to represent a system of linear equations. Columbus City Schools Prior Knowledge Future Learning - Students have solved systems of equations in two variables before. They have also solved in three variables but not by these methods. - Students will perform various operations on matrices. Clear Learning Targets Precalculus 2015--‐2016 19 Instructional Strategies Remind students how to solve a system of three equations with three unknowns by elimination. You can refer to this spreadsheet to help instruct students on how elimination methods work. Show the following Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSA-A-REI#HSA-REI.C.8 Common Misconceptions and Challenges If students make a mistake early in a problem, chances are the rest of the problem will be incorrect. Teachers may wish to start with problems that are integers and have students ask if the first variable they solve for is correct before moving on with the rest of the problem. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 6-1 Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-8.html Solve a system of equations using augmented matrices (Algebra 1) Solve a system of equations using augmented matrices: word problems (Algebra 1) Solve a system of equations using augmented matrices (Algebra 2) Solve a system of equations using augmented matrices: word problems (Algebra 2) Career Connections Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 20 Ohio’s Learning Standards – Clear Learning Targets Precalculus N-VM.6 Perform operations on Essential Understanding Academic Vocabulary/Language matrices and use matrices in applications. - Data can be organized into matrices. When this is done, data can be manipulated easily. - transformations, translations, reflections, rotations, dilations, vertex matrix, translation matrix, pre-image, image (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. Extended Understanding - Students can see where matrices can be used in real world applications. I Can Statements I can use matrices to represent data. I can use matrices to manipulate data. I can use matrices in real life applications. Columbus City Schools Tier 2 Vocabulary - perform, operation, use Prior Knowledge Future Learning - Students just used matrices in Gaussian elimination to solve a system of three equations with three unknows. - Student will perform scalar operations on matrices. Clear Learning Targets Precalculus 2015--‐2016 21 Instructional Strategies Complete the Focus activity on page 384 of the textbook, having students draw a triangle and labeling the matrix formed by its vertices. Show the Khan Academy video at: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.C.6 Common Misconceptions and Challenges Remind students that when translating, right and up imply positive change where down and left imply negative change. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 6-1, 6-6 Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-6.html Sophia: https://www.sophia.org/ccss-math-standard-9-12nvm6-pathway http://www.ct4me.net/Common-Core/hsnumber/hsn-vector-matrix-quantities.htm Career Connections Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its customers for claims. Some actuaries are self-employed and work as consultants. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 22 Ohio’s Learning Standards – Clear Learning Targets Precalculus N-VM.7&8 Perform operations on Essential Understanding Academic Vocabulary/Language matrices and use matrices in applications - Students will add, subtract, and multiply matrices by a scalar and should be able to interpret the resulting data. -Students will multiply matrices if possible. -Students will determine whether the properties of real numbers hold under operations on matrices. - matrix, element, dimensions, spreadsheet, row, cell, column, scalar, scalar multiplication Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Add, subtract, and multiply matrices of appropriate dimensions. Tier 2 Vocabulary - evaluate, model Extended Understanding - Students can use transformations on matrices to model motion. I Can Statements I can perform algebraic operations with matrices. I can multiply matrices. I can use the properties of matrix multiplication. Columbus City Schools Prior Knowledge Future Learning -Students should be able to organize data into matrices. Students will use matrices to solve systems of equations. Clear Learning Targets Precalculus 2015--‐2016 23 Instructional Strategies Point out the existence of more than one type of matrix multiplication. Scalar multiplication refers to the multiplication of a matrix by a constant (a scalar) to produce another matrix of the same size. This is similar to multiplying a number by a scale factor to increase or decrease its value in proportion to its original value. The scalar multiplication is performed by multiplying each element of a matrix by the same constant. To help students understand the operations with matrices, review properties of real number operations (commutative, associative, identity and inverse for addition and multiplication) prior to introducing operations with matrices. Matrix addition (subtraction) and multiplication are similar to real number addition (subtraction) and multiplication in many instances, but there are some important differences. Begin with defining equality of matrices and emphasize the importance of the same size. Two matrices are equal if they have the same size and their corresponding elements are equal. The sum of two matrices of the same size is a matrix with elements that are the sums of the corresponding elements of the two given matrices. Addition is not defined for matrices of different sizes. Because two matrices are added by adding their corresponding elements, it follows from the properties of real numbers that matrices of the same size are commutative and associative relative to addition. When matrices are used in a context, they should not be added or subtracted unless their labels match. Provide students with contextual applications to demonstrate the proper use of matrix addition (subtraction). Common Misconceptions and Challenges A side effect of treating matrices as an isolated area of study with many symbol manipulation rules similar to operations with real numbers is that students may overestimate the similarity between operations with matrices and operations with real numbers. For example, some students may believe that: matrix multiplication is commutative; a product of matrices with the same sizes is a product of their corresponding elements; the identity matrix has all elements that are ones; and the inverse matrices have all entrees that are reciprocals of the elements of the original matrix. Teachers can remedy these misconceptions by offering students a wide range of applications that show connections between matrices, transformation, vectors, and systems of linear equations. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Lesson 6-4, 6-5 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSN/VM/C/7 https://www.illustrativemathematics.org/content-standards/HSN/VM/C/8 Achieve the Core Modules, Resources: http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio: https://www.ixl.com/standards/ohio/math/high-school Career connections Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its customers for claims. Some actuaries are self-employed and work as consultants. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 24 Ohio’s Learning Standards – Clear Learning Targets Precalculus N-VM.9-10 Perform operations on matrices and use matrices in applications. VM.9: (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 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. I Can Statements I can understand that the multiplication of square matrices is not commutative. I can understand that multiplication of square matrices works with the associative property. I can understand that the multiplication of square matrices works with the distributive property. Columbus City Schools Essential Understanding Academic Vocabulary/Language - Multiplication of square matrices is not commutative. - Multiplication of square matrices does hold true for associative and distributive properties. - identity matrix, inverse matrix, inverse, invertible, singular matrix, deteminant Extended Understanding - perform, operation, use Tier 2 Vocabulary - The determinant of a square matrix is nonzero only if the matrix has a multiplicative inverse. Prior Knowledge Future Learning - Students should already know how to find the determinant of matrices and how to multiply matrices. - Students will study conic sections next. Clear Learning Targets Precalculus 2015--‐2016 25 Instructional Strategies Show students how multiplying two numbers works either way; that is, 4 x 8 is the same as 8 x 4. Explain that this is always true for numbers. Does it work for square matrices? Show the students an example of a 4x4 matrix being multiplied by another 4x4 matrix. Then, reverse the two matrices and multiply again. Are the results the same? Are they as expected? Why do they think the products are different? Multiply a matrix and its inverse together to obtain a zero matrix. Show students how the determinant is 0. Common Misconceptions and Challenges Students may think that given Matrix A and Matrix B, that A x B and B x A yield the same result. Use an example as described above to prove this is not the case. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 6-2, 6-2 (Extend), 6-3 Matrix operation rules (Algebra 1) Matrix operation rules (Algebra 2) Properties of matrices (Algebra 2) Matrix operation rules (Precalculus) Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-NVM#HSN-VM.C.9 Determinant of a matrix (Algebra 2) Is a matrix invertible? (Algebra 2) Inverse of a matrix (Algebra 2) Identify inverse matrices (Algebra 2) Solve matrix equations using inverses (Algebra 2) Determinant of a matrix (Precalculus) Is a matrix invertible? (Precalculus) Inverse of a 2 x 2 matrix (Precalculus) Inverse of a 3 x 3 matrix (Precalculus) Identify inverse matrices (Precalculus) Solve matrix equations using inverses (Precalculus) Career Connections Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its customers for claims. Some actuaries are self-employed and work as consultants. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 26 Ohio’s Learning Standards – Clear Learning Targets Precalculus GPE.3 Translate between the Essential Understanding Academic Vocabulary/Language geometric description and the equation for a conic section. - Students will be able to derive the equation of an ellipse. - Students will be able to derive the equation of a hyperbola. - hyperbola, conic section, ellipse, foci, constant, transverse axis, conjugate axis Tier 2 Vocabulary (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. I Can Statements I can derive the equation of an ellipse given the foci. I can derive the equation of a hyperbola give the foci. I can understand that the sum or difference of distances from the foci is constant. Columbus City Schools Extended Understanding - translate, derive, using facts - Students will realize that the sum or difference of distances from the foci is constant. Prior Knowledge Future Learning - Students learned in GPE.2 how to derive the equation of a parabola given a focus and diretrix. - Students will be able to explain volume formulas. Clear Learning Targets Precalculus 2015--‐2016 27 Instructional Strategies You can remind students of how they derived the equation of a parabola in earlier lessons. Go over section 7-1 and 7-2 from the book, following the examples. You can use the idea of roller coasters to spark interest: Why might elliptical-shaped loops be safer than circle-shaped loops on roller coasters? Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSG-G-GPE#HSG-GPE.A.3 Common Misconceptions and Challenges Make sure students keep the terms ellipse and hyperbola straight. They often confuse the two. Review the form of the equation of each. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 7-1, 7-2 Shmoop: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-3.html Find properties of ellipses (Precalculus) Find the eccentricity of an ellipse (Precalculus) Write equations of ellipses in standard form (Precalculus) Find properties of hyperbolas (Precalculus) Find the eccentricity of a hyperbola (Precalculus) Write equations of hyperbolas in standard form (Precalculus) Convert equations of conic sections from general to standard form (Precalculus) Career Connections There are many careers that involve upper-level geometry. Many jobs can be found in the medicine field, transportation and construction industries, arts and architecture fields, and engineering. For more information, read the article found on the link below: http://work.chron.com/careers-require-geometry-10361.html Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 28 Ohio’s Learning Standards – Clear Learning Targets Precalculus Explain volume formulas GMD.2 and use them to solve problems. Essential Understanding Academic Vocabulary/Language - Students can use explain how Cavalieri's principle can be used to find the volume of a sphere. - volume, Cavalieri’s principle, solid figure Tier 2 Vocabulary (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Extended Understanding - explain, use - Student can apply Cavalieri’s principle to find the volume of a sphere. I Can Statements I can I can give an informal argument using Cavalieri’s principle for the formula for the volume of a sphere. I can I can give an informal argument using Cavalieri’s principle for the formula for the volume of other solid figures. Columbus City Schools Prior Knowledge Future Learning - Students should already know the formula for finding the volume of a sphere. - Students will explore vectors. Clear Learning Targets Precalculus 2015--‐2016 29 Instructional Strategies The textbook does not specifically discuss Cavalieri’s principle. More information is coming out all the time, and this topic will be updated. Here is one explanation: http://www.shmoop.com/common-core-standards/ccss-hs-g-gmd-2.html https://en.wikipedia.org/wiki/Cavalieri's_principle Video: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=14&cad=rja&uact=8&ved=0CGoQtwIwDWoVChMI5ZnBoZDFxwIVCTMCh3jbALs&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DCYp824FeJ9s&ei=TNfcVeXDE4nm-AHj2YngDg&usg=AFQjCNGCY65n9Mvf1VMzovGakPZKhjqu0Q&sig2=3a9n1TYLy5Mp7nYI3o5gNA Common Misconceptions and Challenges None at this time. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 7-3 https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=9&ved=0CFUQFjAIahUKEwjlmcGhkMXHAhUJMz4KHeNsAuw&url=http%3A%2F%2Fwww.mat h.was hington.edu%2Fnwmi%2Fmaterials%2FCavalieri.pptx.pdf&ei=TNfcVeXDE4nmAHj2YngDg&usg=AFQjCNFszUSk67KEmQV1gzuUJLsymAuY_g&sig2=zrwV_VTBUYpSIWMMF6_zrw Career Connections The following list briefly describes work associated with some upper-level mathematics-related professions : actuary, mathematics teacher, operations research analyst, statistician, physician, research scientist, computer scientist, inventory strategist, air traffic control, analyst, attorney, economist, environmental mathematician, robotics engineer, geophysical mathematician, design ecologist, photogrammetrist, civil engineer, geomatics engineer. https://www.math.ucdavis.edu/~kouba/MathJobs.html Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 30 Ohio’s Learning Standards – Clear Learning Targets Precalculus VM.1-2 Represent and model with Essential Understanding Academic Vocabulary/Language vector quantities. - Vectors have magnitude and direction. - Vector quantities can be represented by directed line segments. - vector, magnitude, direction, directed line segment, initial point, terminal point. Extended Understanding - represent, model 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 (e.g., v, |v|, ||v||, v). VM.2: (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. I Can Statements I can understand that vector quantities have both magnitude and direction. I can represent vector quantities by directed line segments. I can use appropriate symbols for vectors and their magnitude. I can find the components of a vector by subtraction. Columbus City Schools Tier 2 Vocabulary - Students will find the components of a vector by using subtraction of coordinates. Prior Knowledge Future Learning - Students are already knowledgable about the Coordinate plane. They should be comfortable with positive directions going up and right, while negative directions go down and left. - Students will perform multiple operation on vectors, including several different ways to add them. - Students know the distance formula and Pythagorean Theorem. Clear Learning Targets Precalculus 2015--‐2016 31 Instructional Strategies Explore the links in the support section. There are several for Geometry as well as for Precalculus. You may wish to review the Pythagorean Theorem and Distance Formulas. Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-1.html Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-1.html OpenEd: https://www.opened.com/homework/n-vm-1-recognize-vector-quantities-as-having-both-magnitude-and/3689205 Khan Academy: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.A.1 Common Misconceptions and Challenges When students create a triangle on a graph and try to determine a missing side, students often mis-label the ‘long side’ (hypotenuse). Sometimes students put an entire ordered pair in each ( ) in the magnitude formula. Instead of having (x2 – x1), for instance, they will do (x1y1). Remind students that you are finding the length of each side, and each side is represented by the same variable. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 8-1, 8-4, 8-6 Find the magnitude of a vector (Geometry) Find the magnitude of a vector (Precalculus) Find the direction angle of a vector (Precalculus) Find the magnitude of a three-dimensional vector (Precalculus) Career Connections Management occupations Engineering and natural sciences managers Farmers, ranchers, and agricultural managers Industrial production managers Medical and health services managers Property, real estate, and community association managers Purchasing managers, buyers, and purchasing agents Columbus City Schools Find the component form of a vector (Geometry) Find the component form of a vector given its magnitude and direction angle (Geometry) Find the component form of a vector (Precalculus) Find the component form of a vector from its magnitude and direction angle (Precalculus) Find the component form of a three-dimensional vector (Precalculus) Computer and mathematical occupations Actuaries Computer software engineers Mathematicians Statisticians Engineers Aerospace engineers Chemical engineers Clear Learning Targets Precalculus 2015--‐2016 32 Ohio’s Learning Standards – Clear Learning Targets Precalculus Perform operations on vectors. VM.4-5 VM.4a: Add vectors end-to-end, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. VM.4b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. VM.4c: Understand vector subtraction v - w 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. VM.5a: Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). Essential Understanding - Students will add vectors in a variety of ways including end-toend and the parallelogram rule. - Students will perform scalar multiplication. Extended Understanding Academic Vocabulary/Language - component form, unit vector, linear combination, cross product, dot product, magnitude Tier 2 Vocabulary - perform, understand, determine, represent - Students can represent scalar multiplication graphically. VM.5b: Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). I Can Statements I can add vectors end-to-end. I can add vectors component-wise. I can add vectors by the parallelogram rule. I can determine the magnitude and direction of the sum of two vectors. I can understand vector subtraction. I can multiply a vector by a scalar. I can represent scalar multiplication graphically. I can perform scalar multiplication component-wise. I can compute the magnitude of a scalar multiple. Columbus City Schools Prior Knowledge Future Learning - Student previously defined vectors and learned that vectors have magnitude and direction. They performed subtraction on vectors. - Students will multiply a vector by a matrix. Clear Learning Targets Precalculus 2015--‐2016 33 Instructional Strategies Use the links below to help you teach this topic. Multiply a vector by a scalar (Precalculus) Scalar multiples of three-dimensional vectors (Precalculus) Find the magnitude or direction of a vector scalar multiple (Precalculus) Common Misconceptions and Challenges Student confuse the unit vector i with the imaginary number i. The unit vector is denoted by a bold, nonitalic letter i. The imaginary number is denoted by a bold italic letter i. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 8-1, 8-2, 8-3, 8-4, 8-5 Add and subtract vectors (Precalculus) Graph a resultant vector using the triangle method (Precalculus) Graph a resultant vector using the parallelogram method (Precalculus) Add and subtract three-dimensional vectors (Precalculus) Career Connections Management occupations Engineering and natural sciences managers Farmers, ranchers, and agricultural managers Industrial production managers Medical and health services managers Property, real estate, and community association managers Purchasing managers, buyers, and purchasing agents Columbus City Schools Find the magnitude and direction of a vector sum (Precalculus) Add and subtract vectors (Precalculus) Add and subtract three-dimensional vectors (Precalculus) Computer and mathematical occupations Actuaries Computer software engineers Mathematicians Statisticians Engineers Aerospace engineers Chemical engineers Clear Learning Targets Precalculus 2015--‐2016 34 Ohio’s Learning Standards – Clear Learning Targets Precalculus VM.11-12 Perform operations on Essential Understanding Academic Vocabulary/Language matrices and use matrices in applications. - Vectors can be multiplied by other matrices to create a new matrix. - cross product, torque, parallelepiped, triple scalar product VM.11: (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. VM.12: (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. I Can Statements I can multiply a vector by a matrix to produce another vector. I can work with 2x2 matrices as a transformation of the plane. I can interpret the absolute value of the determinant in terms of area. Columbus City Schools Extended Understanding - Students will interpret the absolute value of the determinant in terms of area. Tier 2 Vocabulary - perform, use, produce, interpret Prior Knowledge Future Learning - Students have already add vectors in several ways and performed scalar multiplication of vectors. - Students will continue their study of complex numbers. Clear Learning Targets Precalculus 2015--‐2016 35 Instructional Strategies Watch the Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.C.11 OpenEd: https://www.opened.com/homework/n-vm-11-multiply-a-vector-regarded-as-a-matrix-with-one-column/3689775 Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-11.html Common Misconceptions and Challenges Review the characteristics of triangles so students do not confuse them: A right triangle has a 90 degree angle and two sides that are perpendicular to each other. An isosceles triangle has two sides that have the same length. And equilateral triangle has three sides that have the same length. A scalene triangle has no sides that are the same length. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill: Chapter 8-3, 8-4 (Extend), 8-5 Coming soon: https://www.superteacherworksheets.com/common-core/n.vm.11.html Career Connections Physical scientists Atmospheric scientists Chemists and materials scientists Environmental scientists and hydrologists Physicists and astronomers Metal workers and plastic workers Computer control programmers and operators Machinists Columbus City Schools Drafters and engineering technicians Drafters Engineering technicians Life scientists Biological scientists Medical scientists Electrical and electronic equipment mechanics, installers, and repairers Electronic home entertainment equipment installers and repairers Clear Learning Targets Precalculus 2015--‐2016 Engineers Aerospace engineers Chemical engineers Civil engineers Electrical engineers Environmental engineers Industrial engineers Nuclear engineers 36 Ohio’s Learning Standards – Clear Learning Targets Precalculus N-CN.4,5,&6 Summarize, represent, Essential Understanding Academic Vocabulary/Language and interpret data on a single count or measurement variable - Students should be able to graph polar coordinates, converting between polar and rectangular forms. -Students should be able to convert between polar and rectangular forms of complex numbers and perform operations in polar form. - polar coordinate system, pole, polar axis, polar coordinates, polar equation, polar graph, limacon, cardioid, rose, lemniscuses, spiral of Archimedes, complex plane, real axis, imaginary axis, Argand plane, absolute value of a complex number, polar form, trig form, modulus, argument, pth roots of unity 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. 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 - √3i)3 = 8 because (1 - √3i) has modulus 2 and argument 120˚. 6. 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. I Can Statements I can graph points with polar coordinates I can graph polar equations I can convert between polar and rectangular coordinates I can convert between polar and rectangular equations I can identify polar equations of conics I can write and graph the polar equation of a conic I can convert complex numbers from rectangular to polar from and vice versa I can find products, quotients, powers and roots of complex numbers in polar form. Columbus City Schools Extended Understanding - The distance between numbers in the complex plane is the modulus of the difference. Tier 2 Vocabulary - calculate, summarize, represent, interpret (data), use (properties). Prior Knowledge Future Learning -Students should know how to find distance between points on the rectangular coordinate plane. -Students will find the area of a region ounded by a urve whose equationis given in polar form. -Students should be able to analyze and draw the graph of conic equations in the rectangular coordinate plane. -Students will solve a system of polar equations. -Students should be able to define, simplify, add, multiply, and solve equations with imaginary numbers. -Students should be abel to add and subtract complex numbers and find their conjugates. Clear Learning Targets Precalculus 2015--‐2016 37 Instructional Strategies A complex plane is obtained by associating the ordered pairs of real numbers with points in a rectangular coordinate system. Students should see a correspondence between complex numbers written in rectangular form (a +bi) and a unique ordered pair of real numbers (a, b), where a represents a real number and b represents an imaginary number. They also need to view a pair of conjugate complex numbers (a - bi) and (a + bi) as a reflection of the point (a, b) to the point (a, -b) over the x- axis. Similar to a vector, a complex number is characterized by length and direction. A complex number can be written in the alternative polar form, such that an arbitrary point P representing a number is associated with polar coordinates (r, θ), where r is a modulus and θ is an argument. Point out that, similar to a complex number in a coordinate plane associated with the ordered pair of real numbers, each geometric vector in the standard position on a coordinate plane is associated with the ordered pair of real numbers that are coordinates of its terminal point. Students need to realize that vectors and complex numbers as systems have some common properties and operations. For example, addition, subtraction or multiplication of complex numbers by a real number is performed by the same rules as addition, subtraction or multiplication of vectors by a real number. Multiplication of two complex numbers results in the cross product of two vectors. Common Misconceptions and Challenges Students may believe that complex numbers, as an area of study of mathematics, is completely isolated from other areas of study, such as vectors and matrices, and only has a connection to quadratic equations with negative discriminants. The variety of approaches to connect complex numbers, vectors and matrices can help students develop their understanding of important concepts of all three overlapping areas. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Chapter 9 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSN/CN/B Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career connections Computer control programmers and operators use special computer-controlled machines to cut and shape products such as car parts, machine parts, and compressors. They use lathes, spindles, and milling machines following blueprints from engineers. While they work they must constantly monitor readouts from the computer to make sure parts are being made properly. Because of the machinery they use, this can be a dangerous job. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 38 Ohio’s Learning Standards – Clear Learning Targets Precalculus S-MD.1,2,3,4,5 Summarize, represent, and interpret data on a single count or measurement variable 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 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. 5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. I Can Statements I can represent the sum of a series using sigma notation. I can find the nth term of arithmetic series and sequences. I can find the nth term of geometric series and sequences. I can use Pascal’s Triangle and/or the Binomial Theorem to write binomial expansion. I can use a power series to represent a rational function. I can use approximate values of transcendental functions. Columbus City Schools Essential Understanding Academic Vocabulary/Language - Students will be able to use sigma notation to represent and calculate sums of series, find nth terms of arithmetic/geometric sequences and series. -Students will be able to use Pascal’s Triangle or the Binomial Theorem to write binomial expansions. -Students will be able to use a power series to represent a rational function. - principle of mathematical induction, anchor step, inductive hypothesis, inductive step, extended principle of mathematical induction, binomial coefficient, Pascal’s triangle, Binomial Theorem, power series, exponential series, Euler’s Formula Tier 2 Vocabulary - compare, interpret, observe Extended Understanding - Students should be able to assign probabilities to payoff values. Prior Knowledge Future Learning -Students should be able to use funcions to generate ordered pairs and use graphs to analyze the end behavior of functions and write usning limit notation. -Students will find limits of convergent functions -Students should be able to perfom operaitons ith imaginary and complex number. -Students will represent functions as series for use in differrntial equations. -Studetn wil proie interavl estimates from sample proportions to estimate population proportions. -students should be able to graph and analyze rational funtions. Clear Learning Targets Precalculus 2015--‐2016 39 Instructional Strategies Have students practice their understanding of the different types of graphs for categorical and numerical variables by constructing statistical posters. Note that a bar graph for categorical data may have frequency on the vertical (student’s pizza preferences) or measurement on the vertical (radish root growth over time - days). Measures of center and spread for data sets without outliers are the mean and standard deviation, whereas median and interquartile ranges are better measures for data sets with outliers. Introduce the formula of standard deviation by reviewing the previously learned MAD (mean absolute deviation). The MAD is very intuitive and gives a solid foundation for developing the more complicated standard deviation measure. Informally observing the extent to which two boxplots or two dot plots overlap begins the discussion of drawing inferential conclusions. Don’t shortcut this observation in comparing two data sets. As histograms for various data sets are drawn, common shapes appear. To characterize the shapes, curves are sketched through the midpoints of the tops of the histogram’s rectangles. Of particular importance is a symmetric unimodal curve that has specific areas within one, two, and three standard deviations of its mean. It is called the Normal distribution and students need to be able to find areas (probabilities) for various events using tables or a graphing calculator. Common Misconceptions and Challenges That a bar graph and a histogram are the same. A bar graph is appropriate when the horizontal axis has categories and the vertical axis is labeled by either frequency (e.g., book titles on the horizontal and number of students who like the respective books on the vertical) or measurement of some numerical variable (e.g., days of the week on the horizontal and median length of root growth of radish seeds on the vertical). A histogram has units of measurement of a numerical variable on the horizontal (e.g., ages with intervals of equal length). That the lengths of the intervals of a boxplot (min,Q1), (Q1,Q2), (Q2,Q3), (Q3,max) are related to the number of subjects in each interval. Students should understand that each interval theoretically contains one-fourth of the total number of subjects. Sketching an accompanying histogram and constructing a live boxplot may help in alleviating this misconception. That all bell-shaped curves are normal distributions. For a bell-shaped curve to be Normal, there needs to be 68% of the distribution within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations. Common Core Support and Curriculum Resources Integrated Math 4, McGraw Hill, Chapter 10 and 11 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/HSS/ID/A Achieve the Core Modules, Resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12 The Common Core in Ohio https://www.ixl.com/standards/ohio/math/high-school Career connections Using national statistics, underwriters decide whether a person applying for insurance is a good risk. They also help the company decide how much to charge. If they set the rates too low, the company will lose money. Too high, and the company will lose business to competitors. They answer questions such as: How much should an insurance company charge for insurance? Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 40 References Many of these standards can be found on Shmoop.com: http://www.shmoop.com Others can be found at OpenED: http://www.OpenEd.com Some of the topics cite Illustrative Mathematics. However, others are still being added to their website. Much of the material, especially in later sections of this course, is still being written. As they are found, they will be added to this Clear Learning Target. Click here for Santa Fe’s explanation of some of these standards. Columbus City Schools Clear Learning Targets Precalculus 2015--‐2016 41