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PreCalculus Correlation of the ALEKS course PreCalculus to the Common Core State Standards for Fourth Courses (2010) Number and Quantity = ALEKS course topic that addresses the standard TD = Teacher Directed N-CN: The Complex Number System N-CN.3: (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Dividing complex numbers N-CN.4: (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. Plotting complex numbers Writing a complex number in standard form given its trigonometric form Writing a complex number in trigonometric form: Special angles Writing a complex number in trigonometric form: Decimal answers N-CN.5: (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For 3 example, (-1 + √3 i) = 8 because (-1 + √3 i) has modulus 2 and argument 120°. De Moivre's Theorem: Answers in standard form Finding the nth roots of a number: Problem type 1 Finding the nth roots of a number: Problem type 2 N-CN.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. TD N-VM: Vector & Matrix Quantities N-VM.1: (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). Magnitude of a vector given in ai+bj form Magnitude of a vector given in component form Finding the magnitude and direction of a vector given its graph Writing a vector given its magnitude and direction angle N-VM.2: (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Writing a position vector in ai+bj form given its graph Writing a vector in ai+bj form given its initial and terminal points ALEKS - Copyright © 2017 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 1/5 Writing a vector in component form given its initial and terminal points N-VM.3: (+) Solve problems involving velocity and other quantities that can be represented by vectors. Writing a vector to represent a force pushing or pulling an object Finding magnitudes of forces related to an object suspended by cables N-VM.4: (+) Add and subtract vectors. N-VM.4.a: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Vector addition and scalar multiplication: ai+bj form Vector addition and scalar multiplication: Component form Vector addition: Geometric approach N-VM.4.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Finding the magnitude and direction angle of the resultant force of two vectors N-VM.4.c: 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. Vector addition and scalar multiplication: ai+bj form Vector addition and scalar multiplication: Component form Vector subtraction: Geometric approach N-VM.5: (+) Multiply a vector by a scalar. N-VM.5.a: Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v v ) = (cv cv ). x' y x' y Multiplication of a vector by a scalar: Geometric approach N-VM.5.b: 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). Finding the magnitude and direction of a vector given its graph N-VM.6: (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. Word problem involving multiplication of matrices N-VM.7: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Scalar multiplication of a matrix N-VM.8: (+) Add, subtract, and multiply matrices of appropriate dimensions. Scalar multiplication of a matrix Addition or subtraction of matrices Linear combination of matrices Squaring and multiplying 2x2 matrices Multiplication of matrices: Basic N-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. Squaring and multiplying 2x2 matrices ALEKS - Copyright © 2017 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 2/5 N-VM.10: (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. Finding the inverse of a 2x2 matrix Finding the inverse of a 3x3 matrix N-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. TD N-VM.12: (+) Work with 2 x 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. TD Algebra = ALEKS course topic that addresses the standard A-REI: Reasoning with Equations & Inequalities A-REI.8: (+) Represent a system of linear equations as a single matrix equation in a vector variable. Finding the inverse of a matrix to solve a 2x2 system of linear equations Using the inverse of a matrix to solve a 3x3 system of linear equations A-REI.9: (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater). Finding the inverse of a 2x2 matrix Finding the inverse of a 3x3 matrix Finding the inverse of a matrix to solve a 2x2 system of linear equations Using the inverse of a matrix to solve a 3x3 system of linear equations Functions = ALEKS course topic that addresses the standard F-IF: Interpreting Functions F-IF.7: F-IF.7.d: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Finding the asymptotes of a rational function: Constant over linear Finding the asymptotes of a rational function: Linear over linear Finding horizontal and vertical asymptotes of a rational function: Quadratic numerator or denominator Finding the asymptotes of a rational function: Quadratic over linear Graphing a rational function: Constant over linear Graphing a rational function: Linear over linear Transforming the graph of a rational function Graphing a rational function: Quadratic over linear ALEKS - Copyright © 2017 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 3/5 Graphing rational functions with holes Graphing a rational function with more than one vertical asymptote F-BF: Building Functions F-BF.1: Write a function that describes a relationship between two quantities.* F-BF.1.c: (+) 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. Introduction to the composition of two functions Composition of two functions: Basic Composition of a function with itself Expressing a function as a composition of two functions Composition of two functions: Domain and range Composition of two functions: Advanced Composition of two rational functions Word problem involving composition of two functions F-BF.4: Find inverse functions. F-BF.4.b: (+) Verify by composition that one function is the inverse of another. Determining whether two functions are inverses of each other F-BF.4.c: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. Inverse functions: Linear, discrete F-BF.4.d: (+) Produce an invertible function from a non-invertible function by restricting the domain. Inverse functions: Quadratic, square root F-BF.5: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Converting between logarithmic and exponential equations Converting between natural logarithmic and exponential equations Evaluating logarithmic expressions Solving an equation of the form log a = c b Solving a multi-step equation involving a single logarithm: Problem type 1 Solving a multi-step equation involving a single logarithm: Problem type 2 Solving a multi-step equation involving natural logarithms Solving an equation involving logarithms on both sides: Problem type 1 Solving an equation involving logarithms on both sides: Problem type 2 Solving an exponential equation by using logarithms: Decimal answers, basic Solving an exponential equation by using natural logarithms: Decimal answers Solving an exponential equation by using logarithms: Decimal answers, advanced Solving an exponential equation by using logarithms: Exact answers in logarithmic form F-TF: Trigonometric Functions F-TF.3: (+) 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. Trigonometric functions and special angles: Problem type 1 Special right triangles: Exact answers ALEKS - Copyright © 2017 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 4/5 F-TF.4: (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Even and odd properties of trigonometric functions F-TF.6: (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. Values of inverse trigonometric functions F-TF.7: (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* Using trigonometry to find angles of elevation or depression in a word problem Solving a basic trigonometric equation using a calculator Solving a trigonometric equation modeling a real-world situation F-TF.9: (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Sum and difference identities: Problem type 1 Sum and difference identities: Problem type 2 Sum and difference identities: Problem type 3 Sum and difference identities: Problem type 4 Solving a trigonometric equation using sum and difference identities Geometry = ALEKS course topic that addresses the standard TD = Teacher Directed G-GPE: Expressing Geometric Properties with Equations G-GPE.3: (+) 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. Writing an equation of an ellipse given the foci and the major axis length Writing an equation of a hyperbola given the foci and the vertices Writing an equation of a hyperbola given the foci and the asymptotes: Basic Writing an equation of a hyperbola given the foci and the asymptotes: Advanced G-GMD: Geometric Measurement & Dimension G-GMD.2: (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. TD Note: The Statistics and Probability standards are not covered in ALEKS PreCalculus. ALEKS - Copyright © 2017 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 5/5