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SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Number and The Complex Number Perform arithmetic 3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of N-CN.3 Precalculus, Unit 2, Activity 2-3: Complex Quantity System N -CN operations with complex complex numbers. Polynomial Roots and Inequalities numbers. Grades 9-12 Number and The Complex Number Represent complex 4. Represent complex numbers on the complex plane in rectangular and polar form (including N-CN.4 Precalculus, Unit 5, Activity 5-6: DeMoivre’s Quantity System N -CN numbers and their real and imaginary numbers), and explain why the rectangular and polar forms of a given Theorem operations on the complex number represent the same number. complex plane. Grades 9-12 Number and The Complex Number Represent complex 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers N-CN.5 Precalculus, Unit 2, Getting Ready Quantity System N -CN numbers and their geometrically on the complex plane; use properties of this representation for computation. For operations on the example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°. complex plane. Grades 9-12 Number and The Complex Number Represent complex 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers N-CN.5 Precalculus, Unit 2, Activity 2-3: Complex Quantity System N -CN numbers and their geometrically on the complex plane; use properties of this representation for computation. For Polynomial Roots and Inequalities operations on the example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°. complex plane. Grades 9-12 Number and The Complex Number Represent complex 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers N-CN.5 Precalculus, Unit 2, Unit Practice Quantity System N -CN numbers and their geometrically on the complex plane; use properties of this representation for computation. For operations on the example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°. complex plane. Grades 9-12 Number and The Complex Number Represent complex 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers N-CN.5 Precalculus, Unit 5, Getting Ready Quantity System N -CN numbers and their geometrically on the complex plane; use properties of this representation for computation. For operations on the example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°. complex plane. Grades 9-12 Number and The Complex Number Represent complex 5. Represent addition, subtraction, multiplication, and conjugation of complex numbers N-CN.5 Precalculus, Unit 5, Activity 5-6: DeMoivre’s Quantity System N -CN numbers and their geometrically on the complex plane; use properties of this representation for computation. For Theorem operations on the example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°. complex plane. Grades 9-12 Number and The Complex Number Represent complex 6. Calculate the distance between numbers in the complex plane as the modulus of the N-CN.6 Precalculus, Unit 5, Activity 5-6: DeMoivre’s Quantity System N -CN numbers and their difference, and the midpoint of a segment as the average of the numbers at its endpoints. Theorem operations on the complex plane. Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their to Vectors magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their Two and Three Dimensions magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their Equations Revisited magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their Equations and Vectors magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Unit Reflection Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 1. Recognize vector quantities as having both magnitude and direction. Represent vector N-VM.1 Precalculus, Unit 6, Math Standards Review Quantity Quantities N -VM vector quantities. quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v , |v |, ||v ||, v ). Grades 9-12 Number and Vector and Matrix Represent and model with 2. Find the components of a vector by subtracting the coordinates of an initial point from the N-VM.2 Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vector quantities. coordinates of a terminal point. to Vectors Grades 9-12 Number and Vector and Matrix Represent and model with 2. Find the components of a vector by subtracting the coordinates of an initial point from the N-VM.2 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vector quantities. coordinates of a terminal point. Two and Three Dimensions Grades 9-12 Number and Vector and Matrix Represent and model with 2. Find the components of a vector by subtracting the coordinates of an initial point from the N-VM.2 Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vector quantities. coordinates of a terminal point. Equations Revisited Grades 9-12 Number and Vector and Matrix Represent and model with 2. Find the components of a vector by subtracting the coordinates of an initial point from the N-VM.2 Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vector quantities. coordinates of a terminal point. Equations and Vectors Grades 9-12 Number and Vector and Matrix Represent and model with 2. Find the components of a vector by subtracting the coordinates of an initial point from the N-VM.2 Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vector quantities. coordinates of a terminal point. Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 4, Activity 4-5: Law of Quantity Quantities N -VM vector quantities. Cosines Page 1 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 4, Activity 4-6: Law of Quantity Quantities N -VM vector quantities. Sines Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 4, Unit Practice Quantity Quantities N -VM vector quantities. Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vector quantities. to Vectors Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vector quantities. Two and Three Dimensions Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vector quantities. Equations Revisited Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vector quantities. Equations and Vectors Grades 9-12 Number and Vector and Matrix Represent and model with 3. Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.3 Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vector quantities. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. N-VM.4.a Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vectors. Understand that the magnitude of a sum of two vectors is typically not the to Vectors sum of the magnitudes. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. N-VM.4.a Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vectors. Understand that the magnitude of a sum of two vectors is typically not the Two and Three Dimensions sum of the magnitudes. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. N-VM.4.a Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vectors. Understand that the magnitude of a sum of two vectors is typically not the Equations Revisited sum of the magnitudes. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. N-VM.4.a Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vectors. Understand that the magnitude of a sum of two vectors is typically not the Equations and Vectors sum of the magnitudes. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. N-VM.4.a Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vectors. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. b. Given two vectors in magnitude and direction form, determine the N-VM.4.b Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vectors. magnitude and direction of their sum. to Vectors Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. b. Given two vectors in magnitude and direction form, determine the N-VM.4.b Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vectors. magnitude and direction of their sum. Two and Three Dimensions Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. b. Given two vectors in magnitude and direction form, determine the N-VM.4.b Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vectors. magnitude and direction of their sum. Equations Revisited Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. b. Given two vectors in magnitude and direction form, determine the N-VM.4.b Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vectors. magnitude and direction of their sum. Equations and Vectors Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. b. Given two vectors in magnitude and direction form, determine the N-VM.4.b Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vectors. magnitude and direction of their sum. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. c. Understand vector subtraction v – w as v + (–w ), where –w is the N-VM.4.c Precalculus, Unit 6, Activity 6-2: Introduction Quantity Quantities N -VM vectors. additive inverse of w , with the same magnitude as w and pointing in the to Vectors opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component- wise. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. c. Understand vector subtraction v – w as v + (–w ), where –w is the N-VM.4.c Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vectors. additive inverse of w , with the same magnitude as w and pointing in the Two and Three Dimensions opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component- wise. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. c. Understand vector subtraction v – w as v + (–w ), where –w is the N-VM.4.c Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vectors. additive inverse of w , with the same magnitude as w and pointing in the Equations Revisited opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component- wise. Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. c. Understand vector subtraction v – w as v + (–w ), where –w is the N-VM.4.c Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vectors. additive inverse of w , with the same magnitude as w and pointing in the Equations and Vectors opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component- wise. Page 2 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Number and Vector and Matrix Perform operations on 4. Add and subtract vectors. c. Understand vector subtraction v – w as v + (–w ), where –w is the N-VM.4.c Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vectors. 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. Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly N-VM.5.a Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vectors. reversing their direction; perform scalar multiplication component-wise, e.g., Two and Three Dimensions as c (v x, v y) = (cv x, cv y). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly N-VM.5.a Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vectors. reversing their direction; perform scalar multiplication component-wise, e.g., Equations Revisited as c (v x, v y) = (cv x, cv y). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly N-VM.5.a Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vectors. reversing their direction; perform scalar multiplication component-wise, e.g., Equations and Vectors as c (v x, v y) = (cv x, cv y). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly N-VM.5.a Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vectors. reversing their direction; perform scalar multiplication component-wise, e.g., as c (v x, v y) = (cv x, cv y). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. b. Compute the magnitude of a scalar multiple c v using ||c v || = |c |v . N-VM.5.b Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM vectors. Compute the direction of c v knowing that when |c |v ≠ 0, the direction of Two and Three Dimensions c v is either along v (for c > 0) or against v (for c < 0). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. b. Compute the magnitude of a scalar multiple c v using ||c v || = |c |v . N-VM.5.b Precalculus, Unit 6, Activity 6-4: Parametric Quantity Quantities N -VM vectors. Compute the direction of c v knowing that when |c |v ≠ 0, the direction of Equations Revisited c v is either along v (for c > 0) or against v (for c < 0). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. b. Compute the magnitude of a scalar multiple c v using ||c v || = |c |v . N-VM.5.b Precalculus, Unit 6, EA 6-1: Parametric Quantity Quantities N -VM vectors. Compute the direction of c v knowing that when |c |v ≠ 0, the direction of Equations and Vectors c v is either along v (for c > 0) or against v (for c < 0). Grades 9-12 Number and Vector and Matrix Perform operations on 5. Multiply a vector by a scalar. b. Compute the magnitude of a scalar multiple c v using ||c v || = |c |v . N-VM.5.b Precalculus, Unit 6, Unit Practice Quantity Quantities N -VM vectors. Compute the direction of c v knowing that when |c |v ≠ 0, the direction of c v is either along v (for c > 0) or against v (for c < 0). Grades 9-12 Number and Vector and Matrix Perform operations on 6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence N-VM.6 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM matrices and use relationships in a network. Two and Three Dimensions matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a N-VM.7 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM matrices and use game are doubled. Two and Three Dimensions matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 8. Add, subtract, and multiply matrices of appropriate dimensions. N-VM.8 Precalculus, Unit 6, Activity 6-3: Vectors in Quantity Quantities N -VM matrices and use Two and Three Dimensions matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is N-VM.9 Algebra 2, Unit 1, Activity 1-6: Matrix Quantity Quantities N -VM matrices and use not a commutative operation, but still satisfies the associative and distributive properties. Operations matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is N-VM.9 Algebra 2, Unit 1, Activity 1-7: Matrix Quantity Quantities N -VM matrices and use not a commutative operation, but still satisfies the associative and distributive properties. Properties and Equations matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 10. Understand that the zero and identity matrices play a role in matrix addition and multiplication N-VM.10 Algebra 2, Unit 1, Activity 1-6: Matrix Quantity Quantities N -VM matrices and use similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero Operations matrices in applications. if and only if the matrix has a multiplicative inverse. Grades 9-12 Number and Vector and Matrix Perform operations on 10. Understand that the zero and identity matrices play a role in matrix addition and multiplication N-VM.10 Algebra 2, Unit 1, Activity 1-7: Matrix Quantity Quantities N -VM matrices and use similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero Properties and Equations matrices in applications. if and only if the matrix has a multiplicative inverse. Grades 9-12 Number and Vector and Matrix Perform operations on 11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions N-VM.11 Geometry, Unit 5, EA 5-3: Matrices, Quantity Quantities N -VM matrices and use to produce another vector. Work with matrices as transformations of vectors. Transformations, and Vectors matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions N-VM.11 Geometry, Unit 5, Activity 5-7: Quantity Quantities N -VM matrices and use to produce another vector. Work with matrices as transformations of vectors. Transformations with Matrices matrices in applications. Page 3 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Number and Vector and Matrix Perform operations on 12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of N-VM.12 Geometry, Unit 5, EA 5-3: Matrices, Quantity Quantities N -VM matrices and use the determinant in terms of area. Transformations, and Vectors matrices in applications. Grades 9-12 Number and Vector and Matrix Perform operations on 12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of N-VM.12 Geometry, Unit 5, Activity 5-7: Quantity Quantities N -VM matrices and use the determinant in terms of area. Transformations with Matrices matrices in applications. Grades 9-12 Algebra Reasoning with Solve systems of 8. Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.8 Algebra 2, Unit 1, Activity 1-6: Matrix Equations and equations Operations Inequalities A -REI Grades 9-12 Algebra Reasoning with Solve systems of 8. Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.8 Algebra 2, Unit 1, Activity 1-7: Matrix Equations and equations Properties and Equations Inequalities A -REI Grades 9-12 Algebra Reasoning with Solve systems of 9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using A-REI.9 Algebra 2, Unit 1, Activity 1-6: Matrix Equations and equations technology for matrices of dimension 3 × 3 or greater). Operations Inequalities A -REI Grades 9-12 Algebra Reasoning with Solve systems of 9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using A-REI.9 Algebra 2, Unit 1, Activity 1-7: Matrix Equations and equations technology for matrices of dimension 3 × 3 or greater). Properties and Equations Inequalities A -REI Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Unit Overview F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Activity 2-3: Complex F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Polynomial Roots and Inequalities Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Activity 2-4: Rational F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Functions Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Activity 2-5: Rational F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Functions Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, EA 2-2: Rational F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Functions Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Activity 2-8: F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Transformations of Functions Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Activity 2-9: Effects of F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Transformations Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, EA 2-3: Transformed F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Functions Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Unit Practice F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Grades 9-12 Functions Interpreting Functions Analyze functions using 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple d. Graph rational functions, identifying zeros and asymptotes when suitable F-IF.7.d Precalculus, Unit 2, Math Standards Review F-IF different representations cases and using technology for more complicated cases. factorizations are available, and showing end behavior. Grades 9-12 Functions Building Functions F- Build a function that 1. Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature in the F-BF.1.c Precalculus, Unit 2, Activity 2-7: Logarithms BF models a relationship atmosphere as a function of height, and h(t) is the height of a weather between two quantities 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. Grades 9-12 Functions Building Functions F- Build a function that 1. Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature in the F-BF.1.c Precalculus, Unit 2, Activity 2-8: BF models a relationship atmosphere as a function of height, and h(t) is the height of a weather Transformations of Functions between two quantities 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. Grades 9-12 Functions Building Functions F- Build a function that 1. Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature in the F-BF.1.c Precalculus, Unit 2, Activity 2-9: Effects of BF models a relationship atmosphere as a function of height, and h(t) is the height of a weather Transformations between two quantities 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. Grades 9-12 Functions Building Functions F- Build a function that 1. Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature in the F-BF.1.c Precalculus, Unit 2, EA 2-3: Transformed BF models a relationship atmosphere as a function of height, and h(t) is the height of a weather Functions between two quantities 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. Page 4 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Functions Building Functions F- Build a function that 1. Write a function that describes a relationship between two quantities. c. Compose functions. For example, if T(y) is the temperature in the F-BF.1.c Precalculus, Unit 7, Activity 7-1: Modeling BF models a relationship atmosphere as a function of height, and h(t) is the height of a weather Functions between two quantities 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. Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. b. Verify by composition that one function is the inverse of another. F-BF.4.b Precalculus, Unit 2, Activity 2-7: Logarithms BF existing functions Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. b. Verify by composition that one function is the inverse of another. F-BF.4.b Precalculus, Unit 2, Activity 2-8: BF existing functions Transformations of Functions Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. b. Verify by composition that one function is the inverse of another. F-BF.4.b Precalculus, Unit 2, Activity 2-9: Effects of BF existing functions Transformations Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. b. Verify by composition that one function is the inverse of another. F-BF.4.b Precalculus, Unit 2, EA 2-3: Transformed BF existing functions Functions Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. c. Read values of an inverse function from a graph or a table, given that the F-BF.4.c Precalculus, Unit 3, Getting Ready BF existing functions function has an inverse. Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. c. Read values of an inverse function from a graph or a table, given that the F-BF.4.c Precalculus, Unit 3, Activity 3-6 Inverse BF existing functions function has an inverse. Trigonometric Functions Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. d. Produce an invertible function from a non-invertible function by restricting F-BF.4.d Precalculus, Unit 3, Getting Ready BF existing functions the domain. Grades 9-12 Functions Building Functions F- Build new functions from 4. Find inverse functions. d. Produce an invertible function from a non-invertible function by restricting F-BF.4.d Precalculus, Unit 3, Activity 3-6 Inverse BF existing functions the domain. Trigonometric Functions Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Activity 2-6: Exponential BF existing functions relationship to solve problems involving logarithms and exponents. Functions Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Activity 2-7: Logarithms BF existing functions relationship to solve problems involving logarithms and exponents. Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Activity 2-8: BF existing functions relationship to solve problems involving logarithms and exponents. Transformations of Functions Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Activity 2-9: Effects of BF existing functions relationship to solve problems involving logarithms and exponents. Transformations Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, EA 2-3: Transformed BF existing functions relationship to solve problems involving logarithms and exponents. Functions Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Unit Practice BF existing functions relationship to solve problems involving logarithms and exponents. Grades 9-12 Functions Building Functions F- Build new functions from 5. Understand the inverse relationship between exponents and logarithms and use this F-BR.5 Precalculus, Unit 2, Unit Reflection BF existing functions relationship to solve problems involving logarithms and exponents. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Activity 3-3: Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Trigonometric Functions and the Unit Circle using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Activity 3-4: Graphs of Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , the form y = A sin[B(x – C)] + D using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Activity 3-5: Graphs of Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Other Trigonometric Functions using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, EA 3-1: Angles, Unit Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Circle, Trigonometric Graphs using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Activity 3-6 Inverse Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Trigonometric Functions using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Activity 3-7: Solving Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Simple Trigonometric Equations using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, EA 3-2: Inverse Trig Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , Functions and Equations using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, F-TF.3 Precalculus, Unit 3, Unit Practice Functions F-TF trigonometric functions π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x , using the unit circle π+x , and 2π–x in terms of their values for x , where x is any real number. Grades 9-12 Functions Trigonometric Extend the domain of 4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric F-TF.4 Precalculus, Unit 3, Activity 3-4: Graphs of Functions F-TF trigonometric functions functions. the form y = A sin[B(x – C)] + D using the unit circle Page 5 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Functions Trigonometric Extend the domain of 4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric F-TF.4 Precalculus, Unit 3, Activity 3-5: Graphs of Functions F-TF trigonometric functions functions. Other Trigonometric Functions using the unit circle Grades 9-12 Functions Trigonometric Model periodic 6. Understand that restricting a trigonometric function to a domain on which it is always F-TF.6 Precalculus, Unit 3, Activity 3-6 Inverse Functions F-TF phenomena with increasing or always decreasing allows its inverse to be constructed. Trigonometric Functions trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 6. Understand that restricting a trigonometric function to a domain on which it is always F-TF.6 Precalculus, Unit 3, Activity 3-7: Solving Functions F-TF phenomena with increasing or always decreasing allows its inverse to be constructed. Simple Trigonometric Equations trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 6. Understand that restricting a trigonometric function to a domain on which it is always F-TF.6 Precalculus, Unit 3, EA 3-2: Inverse Trig Functions F-TF phenomena with increasing or always decreasing allows its inverse to be constructed. Functions and Equations trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 6. Understand that restricting a trigonometric function to a domain on which it is always F-TF.6 Precalculus, Unit 3, Unit Practice Functions F-TF phenomena with increasing or always decreasing allows its inverse to be constructed. trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 7. Use inverse functions to solve trigonometric equations that arise in modeling contexts; F-TF.7 Precalculus, Unit 3, Activity 3-7: Solving Functions F-TF phenomena with evaluate the solutions using technology, and interpret them in terms of the context. Simple Trigonometric Equations trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 7. Use inverse functions to solve trigonometric equations that arise in modeling contexts; F-TF.7 Precalculus, Unit 3, EA 3-2: Inverse Trig Functions F-TF phenomena with evaluate the solutions using technology, and interpret them in terms of the context. Functions and Equations trigonometric functions Grades 9-12 Functions Trigonometric Model periodic 7. Use inverse functions to solve trigonometric equations that arise in modeling contexts; F-TF.7 Precalculus, Unit 3, Unit Practice Functions F-TF phenomena with evaluate the solutions using technology, and interpret them in terms of the context. trigonometric functions Grades 9-12 Functions Trigonometric Prove and apply 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to F-TF.9 Precalculus, Unit 4, Activity 4-3: Multiple Functions F-TF trigonometric identities solve problems. Angle Identities Grades 9-12 Functions Trigonometric Prove and apply 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to F-TF.9 Precalculus, Unit 4, EA 4-1: Trigonometric Functions F-TF trigonometric identities solve problems. Equations and Identities Grades 9-12 Functions Trigonometric Prove and apply 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to F-TF.9 Precalculus, Unit 4, Unit Practice Functions F-TF trigonometric identities solve problems. Grades 9-12 Functions Trigonometric Prove and apply 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to F-TF.9 Precalculus, Unit 4, Math Standards Review Functions F-TF trigonometric identities solve problems. Grades 9-12 Geometry Expressing Geometric Translate between the 3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or G-GPE.3 Precalculus, Unit 5, Activity 5-2: Ellipses and Properties with geometric description and difference of distances from the foci is constant. Hyperbolas Equations G-GPE the equation for a conic section Grades 9-12 Geometry Expressing Geometric Translate between the 3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or G-GPE.3 Precalculus, Unit 5, Unit Practice Properties with geometric description and difference of distances from the foci is constant. Equations G-GPE the equation for a conic section Grades 9-12 Geometry Geometric Explain volume formulas 2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a G-GMD.2 Geometry, Unit 6, Activity 6-3: Volume Measurement and and use them to solve sphere and other solid figures. Dimension G-GMD problems Grades 9-12 Statistics and Using Probability to Calculate expected values 1. Define a random variable for a quantity of interest by assigning a numerical value to each S-MD.1 Algebra 2, Unit 6, Activity 6-3: Normal Probability Make Decisions S-MD and use them to solve event in a sample space; graph the corresponding probability distribution using the same Distribution problems graphical displays as for data distributions. Grades 9-12 Statistics and Using Probability to Calculate expected values 2. Calculate the expected value of a random variable; interpret it as the mean of the probability S-MD.2 Algebra 2, Unit 6, Activity 6-3: Normal Probability Make Decisions S-MD and use them to solve distribution. Distribution problems Grades 9-12 Statistics and Using Probability to Calculate expected values 3. Develop a probability distribution for a random variable defined for a sample space in which S-MD.3 Algebra 2, Unit 6, Activity 6-3: Normal Probability Make Decisions S-MD and use them to solve theoretical probabilities can be calculated; find the expected value. For example, find the Distribution problems theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, Activity 7-1: Probability Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Experiments problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, EA 7-1: Counting and Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Probability problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Page 6 of 7 SpringBoard Precalculus to Common Core Common Core State Standards Initiative Grade(s) CCS CCS Standard CCS Standard CCS Standard Level 4 CCS Standard Level 5 Standard SpringBoard Activity Standard Level 2 Level 3 ID Level 1 Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, Activity 7-2: Dependent Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data and Independent Events problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, EA 7-2: Compound Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Events, Probability, problems distribution on the number of TV sets per household in the United States, and calculate the Simulation expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, Activity 7-3: Dependent Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Compound Events problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Geometry, Unit 7, Activity 7-4: Geometric Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Probability problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Calculate expected values 4. Develop a probability distribution for a random variable defined for a sample space in which S-MD.4 Algebra 2, Unit 4, Activity 4-7: Binomial Probability Make Decisions S-MD and use them to solve probabilities are assigned empirically; find the expected value. For example, find a current data Probability problems distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Grades 9-12 Statistics and Using Probability to Use probability to 5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and a. Find the expected payoff for a game of chance. For example, find the S-MD.5.a MSM3, Unit 5, Activity 5-1: Probability Probability Make Decisions S-MD evaluate outcomes of finding expected values. expected winnings from a state lottery ticket or a game at a fast-food decisions restaurant. Grades 9-12 Statistics and Using Probability to Use probability to 5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and b. Evaluate and compare strategies on the basis of expected values. For S-MD.5.b MSM3, Unit 5, Activity 5-1: Probability Probability Make Decisions S-MD evaluate outcomes of finding expected values. example, compare a high-deductible versus a low-deductible automobile decisions insurance policy using various, but reasonable, chances of having a minor or a major accident. Page 7 of 7