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PRE CAL CCRS Standard 1. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Mathematics CCRS Standards and Alabama COS Standard ID The Complex Number System Perform arithmetic operations with complex numbers. Number/Quantity N-CN.3 Evidence of Student Attainment Students: Given a complex number, Teacher Vocabulary Find the conjugate and the modulus. Knowledge Students know: Given a quotient of complex numbers, Franklin County Schools Multiply by 1 by multiplying the numerator and denominator by the conjugate of the denominator to write the indicated quotient as a single complex number. Understanding Resources Students understand that: Conjugate Complex number Modulus/Moduli Skills The definition of the conjugate of a complex number. A complex Students are able to: number divided by itself Find the equals 1. product of two The complex product of numbers. a complex number and its conjugate is a real number (the square of the modulus). The modulus of a complex number correspond s to the magnitude of a vector and, therefore, is useful in the geometric representa Click below to tion of access all ALEX complex resources aligned numbers. to this standard. Mathemati cal ALEX convention Resources is that radical expression s are not left in denominat ors to facilitate numerical approxima tions; therefore, since the i is equal to the square root of -1, convention al form PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding 2. (+) 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 The Complex Number System Represent complex numbers and their operations on the complex. Number/Quantity N-CN.4 Franklin County Schools Students: Given any complex number, Graphically represent the number on the complex plane in Students know: In the complex plane the horizontal axis is the real axis (a) and the vertical Students are able to: Use trigonomet ry to find the measures of angles and says that i does not appear in the denominat or of a fraction. Different forms of a complex number quotient (indicated quotient, single complex number) may be more useful for various purposes. Students understand that: Resources Click below to access all ALEX resources aligned A complex to this standard. number (a +bi) can ALEX be Resources graphed in a rectangula PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment represent the same number. rectangular and polar form. Justify that the rectangular (a+bi) and polar forms (z = r(cosθ +isinθ)) are equivalent Teacher Vocabulary Knowledge 3. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. Example: (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and The Complex Number System Represent complex numbers and their operations on the complex. Number/Quantity N-CN.5 Franklin County Schools axis is the imaginary axis (b). Trigonome tric techniques for finding measures of angles and coordinate s on the unit circle. The characteris tics of the polar coordinate system. Technique s for plotting polar coordinate s. Students know: Complex numbers are represente d geometrica lly in the complex plane with the real part Skills coordinate s on the unit circle. Use the Pythagore an Theorem to find the lengths of sides of a right triangle. Convert between polar and rectangula r forms. Plot polar coordinate s. Students are able to: Add, subtract, and multiply the component parts of complex numbers to find Understanding Resources r coordinate system as (a, b). A complex number may be represente d in the plane using equivalent polar and rectangula r coordinate s. Different representa tions of a complex number may be more useful for various purposes. Students understand that: Click below to Different access all ALEX representa resources aligned to this standard. tions of mathemati cal ALEX concepts Resources (e.g., algebraic and geometric PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge argument 120°. Franklin County Schools measured on the xaxis and the imaginary is represente d on the yaxis. Complex numbers can be added or subtracted by combining the real parts and the imaginary parts or by using vector procedures geometrica lly (end-toend, parallelogr am rule). The product of complex numbers in polar form may be found by multiplying the magnitude s and adding the Skills sums, differences , and products. Identify the conjugate of a complex number and use this as a computatio nal aid, e.g., to find a quotient of complex numbers. Represent complex numbers in the complex plane. To add and subtract complex numbers geometrica lly. Multiply complex numbers in polar form. Understanding representa tion of complex numbers) reveal different features of the concept and each may facilitate computatio n and sense making in different settings. Mathemati cs is a coherent whole and structure within mathemati cs allows for procedures from one area to be used in another (e.g., coordinate geometry and the complex plane, vectors and complex numbers, Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding arguments . 4. (+) 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. The Complex Number System Represent complex numbers and their operations on the complex. Number/Quantity N-CN.6 5. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama) The Complex Number System Represent complex numbers and their operations on the complex. Number/Quantity Franklin County Schools or plotting of a conjugate of a complex number in transforma tional geometry). Students understand that: Students know: Students are able Complex to: numbers can be Add and represente subtract d complex geometrica numbers. lly. Find the Complex modulus of numbers a complex can be number. added and Represent subtracted complex either numbers geometrica geometrica lly or lly. algebraical ly. Students know: How to graph the families of functions. Resources Students are able to: Identify continuity and discontinui Representi ng complex numbers on a rectangula r coordinate system allows the use of techniques developed for real numbers to find distances and midpoints. Students understand: That functions have various Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX resources aligned to this standard. ALEX PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID N-CN.10 Apply limits in problems involving convergence and divergence. (Alabama) 6. (+) 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). Evidence of Student Attainment Teacher Vocabulary Knowledge Vector and Matrix Quantities Perform operations on vectors. Number/Quantity N-VM.1 Franklin County Schools How to factor, simplify, and rationalize functions. How to evaluate functions at a given input value. Students know: The difference between a ray and a directed line segment. When drawing a vector on the xyplane, magnitude is represente d by a distance. Skills ty (if discontinu ous, identify the type). Calculate limits. Students are able to: Locate vectors on the xyplane. Use the unit circle to find trigonomet ric ratios for values in all four quadrants. Use the Pythagore an Theorem to find the lengths of sides of a right Understanding types of continuity and discontinui ty. That limits can be determine d numericall y, graphically , and algebraicall y. Resources Resources Students understand: Vectors represent magnitude and direction. Click below to access all ALEX When resources aligned vectors are to this standard. represente d in ALEX component Resources form, ensuing computatio ns and application s can be accomplish ed. PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources triangle. 7. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Vector and Matrix Quantities Perform operations on vectors. Number/Quantity N-VM.2 Students understand: Students know: 8. (+) Solve problems involving velocity and other quantities that can be represented by vectors. Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.3 Franklin County Schools If a vector is transposed in the xyplane, it retains its magnitude and direction. Students know: The interpretati on of forces as vectors. Vectors can be Students are able to: Transpose a vector from one position to another position in the xyplane. Students are able to: Write and represent a given force as a vector. Combine Vectors having the same magnitude and direction are equivalent Click below to regardless access all ALEX of where resources aligned they are in to this standard. the xyplane. ALEX Resources Vectors in standard position have a terminal point that is equal to the component s of the vector. Students understand: Click below to access all ALEX resources aligned The result to this standard. of combining ALEX multiple Resources vector forces PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills used to represent forces. 9. (+) Add and subtract vectors. (+) Add vectors endto-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. (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. (+) Understand vector subtraction v – w as v + Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.4 Franklin County Schools Understanding vector quantities. Resources creates a net magnitude and direction. Students understand: Students know: The aspects of end-toend, componen twise, and the parallelogr am rule involving vectors. The additive inverse of a vector has the same magnitude but the opposite direction (θ + 180⁰). Students are able to: Draw and find the diagonal of a parallelogr am. Represent vectors on an xyplane. Find the component s of a vector given the direction and magnitude . Find the additive inverse of a vector. There are multiple ways to find the sum and difference of a pair of vectors. The magnitude Click below to of the sum access all ALEX of two resources aligned vectors will to this standard. not be the same as ALEX the sum of Resources the magnitude s unless the vectors are in the same direction. The vector with the larger magnitude will have the greatest effect on PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding (–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 componentwise. the result. 10. (+) Multiply a vector by a scalar. Vector and Matrix Quantities Perform operations on matrices and use (+) matrices in Represent applications. scalar multiplication Number/Quantity graphically by N-VM.5 scaling vectors and possibly reversing their direction; perform Franklin County Schools Resources Students are able to: Students know: The representa tion of vectors graphically on the xyplane. Find the component s of a vector. Find the magnitude and direction of a vector. Students understand: Scalar multiplicati on results in a dilation of the original vector where a scalar greater than 1 Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). (+) 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). 11. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. Understanding Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.6 Franklin County Schools Students know: The aspects of a matrix with regard to entries, rows, columns, Students are able to: Translate data into a matrix. would increase the magnitude and a scalar from 0 to 1 would decrease the magnitude . A negative scalar would reverse the direction (θ + 180⁰). The absolute value of the scalar has the resulting affect on the magnitude . Students understand: Resources That a matrix is a tool that can help to organize, manipulate , and Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding dimension s, elements, and subscript notations. 12. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 13. (+) Add, subtract, and multiply matrices of appropriate dimensions. Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.7 Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.8 Franklin County Schools interpret data. Students are able to: Students know: The aspects of a matrix with regard to entries, rows, columns, dimension s, elements, and subscript notations. Students know: The aspects of a matrix with regard to entries, rows, Resources Write percents as decimals. Increase or decrease an amount by multiplying by a percent (i.e., increase of 10% would multiply by 1.1). Students are able to: Strategicall y choose and apply appropriat e representa Students understand: Multiplying a matrix by a scalar affects every Click below to element in access all ALEX the matrix resources aligned equally. to this standard. Scalar multiplicati ALEX on is a tool Resources that allows all elements of a matrix to be changed in a simple manner. Students understand: Matrix addition and subtraction may be performed Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills columns, dimension s, elements, and subscript notations. tions of matrices on which arithmetic operations can be performed. Understanding 14. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity Franklin County Schools Students know: Commutati ve and associative properties of Students are able to: Add, subtract, and multiply only if the matrices have the same dimensions . Matrix multiplicati on can be performed only when the number of columns in the first matrix equal the number of rows in the second matrix. There are many contextual situations where arithmetic operations on matrices allow us to solve problems. Students understand: Resources Click below to access all ALEX resources aligned Properties to this standard. that apply to other ALEX number PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID operation, but still N-VM.9 satisfies the associative and distributive properties. Evidence of Student Attainment Teacher Vocabulary Knowledge 15. (+) 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. Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.10 Understanding matrices. Resources systems do not necessarily apply with matrices. Resources Students understand: Students know: Franklin County Schools multiplicati on. Distributiv e property. The aspects of a square matrix. Skills The additive Students are able and to: multiplicati ve identity Find the properties determina for real nt of a numbers. square The matrix. aspects of Find the the zero multiplicati and ve inverse identity of a matrices. square A matrix matrix. multiplied Add and by its multiply multiplicati matrices. ve inverse equals the identity matrix. Identity properties that apply to other number systems apply to matrices. The multiplicati ve inverse property that applies to other number systems applies to matrices. A matrix with a determina nt equal to zero does not have a multiplicati ve inverse analogous Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding 16. (+) 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. Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.11 Students know: Franklin County Schools Conditions under which matrix multiplicati on is defined. Technique s for adding and multiplying matrices. Technique s for scalar multiplicati on. Technique s for performing translation s, rotations, reflections, Students are able to: Write a vector in matrix notation. Determine if matrix multiplicati on is defined for a given product of a matrix by a vector. Multiply a vector by a matrix. Add, multiply, and perform scalar multiplicati Resources to zero in the real number system not having a multiplicati ve inverse. Division by zero in the real number system is undefined. Students understand: Mathemati cal representa tions such as vectors and matrices may be used in a wide variety of settings to model and solve real world problems and this modeling is facilitated by fluent use of techniques for Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills and dilations. Understanding on on matrices. 17. (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. Vector and Matrix Quantities Perform operations on matrices and use matrices in applications. Number/Quantity N-VM.12 Students know: Franklin County Schools Conditions under which matrix multiplicati on is defined. Technique s for adding and multiplying matrices. Technique Students are able to: Find the area of a parallelogr am in a coordinate plane by composing or decomposi ng shapes. Multiply a vector by a matrix. Resources working with these representa tions. Transform ations of multiple vectors may be accomplish ed by the use of matrix multiplicati on as a means of transformi ng all vectors at the same time in a similar manner. Students understand: Multiplicati Click below to on of a vector by a access all ALEX resources aligned 2X2 matrix is a to this standard. transforma tion on the ALEX coordinate Resources plane and this transforma tion can be represente d in matrix PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge 18. (+) Represent a system of linear equations as a single matrix equation in a vector variable. Reasoning with Equations & Inequalities Solve systems of equations. Algebra Franklin County Schools s for performing translation s and rotations. Procedures for finding the value of the determina nt of a 2 X 2 matrix. Students know: The structure of a matrix equation is Skills Understanding Find the value of the determina nt of a 2 X 2 matrix. Students are able to: Transform a system of linear form or algebraicall y by performing the matrix multiplicati on. When a polygon is transforme d by 2 X 2 matrix multiplicati on in the coordinate plane, the area of the image polygon may be found by multiplying the absolute value of the determina nt of the transforma tion matrix by the area of the original polygon. Students understand that: Resources Multiple representa tions of Click below to access all ALEX resources aligned to this standard. ALEX PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge A-REI.8 19. (+) 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 × 3 or greater). Reasoning with Equations & Inequalities Solve systems of equations. Algebra A-REI.9 the product of the coefficient matrix on the left and the vector variable on the right is equal to the constant vector. Students know: Franklin County Schools Skills Procedures for converting systems of equations to matrix equations. Conditions that determine the inverse of a matrix exists. Technique s for determinin g the Understanding equations to a matrix equation. Students are able to: Perform row operations on a matrix to find the inverse of the matrix. Efficiently and accurately find the inverse of a matrix using technology Resources linear systems are needed to facilitate work with more complex systems. Matrix representa tion allows for solutions with systems containing any number of variables. Resources Students understand that: The solutions to a matrix equation are the solutions to the system of equations that produced the matrix equation. Solving a matrix equation is analogous Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills inverse of a matrix (including using technology ). Process for using the inverse of a matrix to find the solution to a matrix equation (2 x 2). Matrix multiplicati on is not commutati ve, specifically Understanding . Resources to solving a linear equation. Technolog y is a useful tool that facilitates investigati on and, once the initial process is understoo d, helps find solutions of more complex problems. A-1AX = A-1B not A-1AX = BA-1. 20. Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. (Alabama) Example: Graph x2 6x + y2 - 12y + 41 = 0 or y2 - 4x + 2y + 5 = 0. Conic Sections Understand the graphs and equations of conic sections. (Alabama) Functions F-IF.7D Formulate equations of Franklin County Schools Students: Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewisedefined functions, polynomial, exponential, logarithmic, trigonometric, and (+) rational), Produce an Students know: Vertex form of a parabola. Standard form of a circle. Vertex and axis of symmetry of a parabola. Completin Students are able to: Create graphs of parabolas from second degree equations. Create graphs of hyperbolas from Students understand: Click below to The access all ALEX determinin resources aligned g to this standard. characteris tics of ALEX conic Resources section graphs. The parameter s that PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID conic sections from their determining characteristic s. (Alabama) Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4. Answer: (x 5)/ (y + 3)/ 25 + 4 = 1. Evidence of Student Attainment accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function, Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions: Franklin County Schools quadratic or linear (intercepts, maxima, and minima), square root, cube root, and piecewisedefined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the Teacher Vocabulary Knowledge g the square. Factoring a quadratic function. Skills second degree equations. Create graphs of ellipses from second degree equations. Create graphs of circles from second degree equations. Create graphs of degenerat e conics from second degree equations. Formulate equations of conic sections given a set of determinin g characteris tics. Understanding determine the graph. Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment 21. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Interpreting Functions Analyze functions using different representations. (Logarithmic and trigonometric functions.) Franklin County Schools Teacher Vocabulary Knowledge Skills Understanding Resources function and those that are not), polynomial (zeros when suitable factorization s are available, end behavior), (+) rational (zeros and asymptotes when suitable factorization s are available, end behavior), exponential and logarithmic (intercepts and end behavior), trigonometri c functions (period, midline, and amplitude). Students: Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewisedefined functions, Students know: Technique s for graphing, Key features of Students are able to: Identify the type of function from the Students understand that: Key features are different Click below to access all ALEX resources aligned to this standard. ALEX PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Functions F-IF.7E Evidence of Student Attainment polynomial, exponential, logarithmic, trigonometric, and (+) rational), Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function, Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions: Franklin County Schools quadratic or linear (intercepts, maxima, and minima), square root, cube root, and piecewisedefined functions, including step functions and absolute value Teacher Vocabulary Knowledge Skills graphs of functions. symbolic representa tion, Manipulate expression s to reveal important features for identificati on in the function, Accurately graph any relationshi p. Understanding depending on the function, Identifying key features of functions aid in graphing and interpretin g the function. Resources Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Franklin County Schools functions (descriptive features such as the values that are in the range of the function and those that are not), polynomial (zeros when suitable factorization s are available, end behavior), (+) rational (zeros and asymptotes when suitable factorization s are available, end behavior), exponential and logarithmic (intercepts and end behavior), trigonometri c functions (period, midline, and amplitude). Teacher Vocabulary Knowledge Skills Understanding Resources PRE CAL Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID 22. (+) Compose functions. 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. Building Functions Build a function that models a relationship between two quantities. Functions F-BF.1C Evidence of Student Attainment Students: Given a contextual situation containing two quantities, Express a relationship between the quantities through an explicit expression using function notation, recursive process, or steps for calculation, Explain and justify how the expression or process models the relationship between the given quantities, Create a new function by using standard function types and arithmetic operations to combine the original functions to model the relationship of the given quantities, (+) standards not covered. Teacher Vocabulary Explicit expression Recursive process Knowledge Students know: Decaying exponential Franklin County Schools Technique s for expressing functional relationshi ps (explicit expression ,a recursive process, or steps for calculation ) between two quantities. Technique s to combine functions using arithmetic operations . Technique s to compose functions using algebraic operations . Notation for function compositio n. Skills Understanding Resources Students are able to: Accurately develop a model that shows the functional Students relationshi understand that: p between two Relationshi quantities. ps can be Accurately modeled create a by several new methods function (e.g., through Click below to explicit arithmetic expression access all ALEX operations resources aligned or of other recursive to this standard. functions. process). Accurately ALEX Arithmetic create a Resources combinatio new ns and/or function compositio through n of compositio functions n of other may be functions. used to When improve functions the fit of a are model. combined to create a new function, present an argument to show how the function PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources models the relationshi p between the quantities. 23. Determine the inverse of a function and a relation. (Alabama) Building Functions Build new functions from existing functions. Functions F-BF.5A Students: Given a contextual situation containing two quantities, Express a relationship between the quantities through an explicit expression using function notation, recursive process, or steps for calculation, Explicit expression Recursive process Decaying exponential Students know: Explain and justify how the expression or process models the relationship between the given quantities, Create a new function by using standard function types and arithmetic operations to combine the original functions to model the relationship of the given quantities, (+) standards not covered. 24. (+) Verify by composition that one Building Functions Build new functions Franklin County Schools Students: Given an invertible Students understand: Inverse The Students are able domain and range to: of a given relation or Find the function. inverse of Algebraic relation/fu properties. nction. Symmetry about the line y = x. Students know: Students are able to: The graphical and algebraic relationshi p between a function and its inverse. The process of finding the inverse of a function. Students understand that: Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID function is the inverse from existing of another. functions. Functions F-BF.4B Evidence of Student Attainment function in algebraic form, Use algebraic properties to find the inverse of the given function, (+) standards not covered. Teacher Vocabulary Knowledge Franklin County Schools Algebraic properties of equality (Table 4). Technique s for composing functions. The compositio n of a function and its inverse is the identity function. When (x,y) is a point on an invertible function, (y,x) is a point on the inverse. In order for a function to have an inverse function, the original function must have a one-toone correspon Skills Accurately perform algebraic properties to find the inverse. Accurately identify restrictions on a noninvertible function that allow it to be invertible. Accurately find the compositio n of two functions. Understanding Resources resources aligned to this standard. The inverse of ALEX a function interchang Resources es the input and output values from the original function. The inverse of a function must also be a function to exist and the domain may need to be restricted to make this occur. PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources dence. 25. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. Building Functions Build new functions from existing functions. Functions F-BF.4C Students: Given an invertible function in algebraic form, Use algebraic properties to find the inverse of the given function, (+) standards not covered. Inverse Students know: Franklin County Schools Algebraic properties of equality Students (Table 4). understand that: Technique Students are able s for The to: composing inverse of functions. a function Accurately The interchang perform compositio es the algebraic n of a input and properties function output to find the and its values inverse. inverse is from the Accurately the original identify identity function. restrictions function. The on a nonWhen inverse of invertible (x,y) is a a function function point on must also that allow an be a it to be invertible function to invertible. function, exist and Accurately (y,x) is a the find the point on domain compositio the may need n of two inverse. to be functions. In order restricted for a to make function to this occur. have an inverse function, the original function Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources must have a one-toone correspon dence. 26. (+) Produce an invertible function from a non-invertible function by restricting the domain. Building Functions Build new functions from existing functions. Functions F-BF.4D Students: Given an invertible function in algebraic form, Use algebraic properties to find the inverse of the given function, (+) standards not covered. Inverse Students know: Franklin County Schools Algebraic Students properties understand that: of equality Students are able (Table 4). The to: Technique inverse of s for a function Accurately composing interchang perform functions. es the algebraic The input and properties compositio output to find the n of a values inverse. function from the Accurately and its original identify inverse is function. restrictions the The on a nonidentity inverse of invertible function. a function function When must also that allow (x,y) is a be a it to be point on function to invertible. an exist and Accurately invertible the find the function, domain compositio (y,x) is a may need n of two point on to be functions. the restricted inverse. to make In order this occur. for a function to have an inverse Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources function, the original function must have a one-toone correspon dence. 27. (+) Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. Building Functions Build new functions from existing functions. Functions F-BF.5 Students Know: Franklin County Schools Definition of Logarithm. If bx = y, then logb y =x Restriction s on domain and range of both functions (b > 0, b not equal to 1, y > 0). Technique s for creating graphs of exponentia l and logarithmic functions. Situations that can be modeled Students are able to: Re-write the equivalent inverse function from an exponentia l function or logarithm function in various forms (graphs, tables, or equations) . Model contextual situations using logarithmic and exponentia l functions. Solve exponentia l and Students understand that: Equivalent forms of a function, specifically logarithmic and exponentia l, may be Click below to useful at access all ALEX different resources aligned times to to this standard. solve problems. ALEX Although Resources the algebraic representa tions look different, the logarithmic form and exponentia l form of the same relationshi p are PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills by exponentia l and logarithmic functions. 28. Compare effects of parameter changes on graphs of transcendental functions. (Alabama) Example: Explain the relationship of the graph y = ex-2 to the graph y = ex. Building Functions Build new functions from existing functions. Functions F-BF.5B logarithmic equations by isolating the variable. Resources equivalent. Students are able to: Students know: Franklin County Schools Understanding Transform ations (shift, stretch, compress, reflect). Exponentia l functions. Logarithmi c functions. Analyze and describe the transforma tion of the graphs, equations, Students and tables understand: of values Click below to of The access all ALEX transcende transforma resources aligned ntal tion of to this standard. functions transcende based on ntal ALEX parameter functions Resources changes. based on Describe parameter the changes. differences in the domain, range, intercepts, asymptote s, end behavior, and increasing and decreasing PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources intervals as a result of parament changes in transcende ntal functions. 29. Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. (Alabama) Trigonometric Functions Recognize attributes of trigonometric functions and solve problems involving trigonometry. (Alabama) Functions F-TF.10 Students are able to: Students know: How to graph trigonomet ric functions including amplitude, midline, and period. How to graph the inverses of sine, cosine, and tangent functions. Franklin County Schools Find amplitude, period, phase shift, domain, and range of the sine and cosine functions. Find period, phase shift, domain and range of the tangent, cotangent, secant, and cosecant functions. Find the domain and range of the trigonomet ric inverse Students understand: Amplitude, period, phase shift, domain, and range of the sine and cosine functions. Period, phase shift, domain and range of the tangent, cotangent, secant, and cosecant functions. The interval defined by the domain and range of the trigonomet Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding functions. 30. Use the sum, difference, and halfangle identities to find the exact value of a trigonometric function. (Alabama) Trigonometric Functions Recognize attributes of trigonometric functions and solve problems involving trigonometry. (Alabama) Functions F-TF.11 a. b. Trigonometric Functions Recognize attributes of trigonometric functions and solve problems involving Solve application- trigonometry. (Alabama) based problems Functions involving F-TF.12 parametric equations. (Alabama) Solve applied problems that include sequences Franklin County Schools ric inverse functions. Students know: 31. Utilize parametric equations by graphing and by converting to rectangular form. (Alabama) Resources How to simplify Students are able Students radicals. to: How to understand: rationalize radicals. Evaluate The sum trigonomet How to identity. ric use the The functions unit circle difference using sum, to identity. difference, determine The halfand halftrigonomet angle angle ric values identity. identities. at specific angles. Special right triangles. Students know: How to model motion using quadratic functions. Algebraic manipulati on of equations. Students are able to: Develop parametric equations for a given situation. Graph the parametric equations. Answer questions in the context of Click below to access all ALEX resources aligned to this standard. ALEX Resources Students understand: Parametric equations can be used to model and evaluate trajectory of projectiles. Parametric equations can be Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills with recurrence relations. (Alabama) Understanding the parametric problem. 32. (+) 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 Extend the domain of trigonometric functions using the unit circle. Functions F-TF.3 Students are able to: Students know: The relationshi p between the lengths of the sides of a 4545-90 and 30-60-90 triangle. The basic trig ratios. Franklin County Schools Find the value of sine, cosine, and tangent of π/3, π/4 and π/6 using special triangles. Locate an angle in standard position in the unit circle. Convert between degrees and radians. Resources used to model and evaluate range of projectiles. Which situtations paramatric modeling is appropriat e. Students understand that: For an angle in standard position, the point where the terminal ray intersects the unit circle has an xcoordinate which is the value of the cosine of the angle and a ycoordinate which is the value of the sine of the angle. Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding 33. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. Functions F-TF.4 Trigonometric Functions Franklin County Schools Patterns that can be identified on the unit circle allow for application of right triangle trigonomet ry to angles of all sizes. Students understand that: Students Know: 34. (+) Understand that restricting a Resources The characteris tics of the unit circle. The characteris tics of even and odd functions. Students Know: Students are able to: Transpose coordinate s from the unit circle to a graph on the xyplane. Use the unit circle to verify even and odd trig identities. Students are able to: Basic trig functions can be classified as either even or odd. The repetitive nature of the unit circle creates periodic functions which is displayed in their graphs. Students understand that: Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX PRE CAL CCRS Standard trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. Mathematics CCRS Standards and Alabama COS Standard ID Model periodic phenomena with trigonometric functions. Functions F-TF.6 Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Characteri stics of an always increasing or always decreasing function. When a function has an inverse that is also a function. Trigonometric Functions Model periodic phenomena with trigonometric functions. Functions F-TF.7 Franklin County Schools Resources resources aligned 35. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in Understanding Students Know: Periodic situations are best modeled by trig Identify intervals where the function is increasing or decreasing . Identify the domain of a function that will produce an inverse. Produce different models of functions and their resulting inverse. Students are able to: Solve a trig equation. Interpret Due to the to this standard. periodic ALEX nature of trig Resources functions, domains have to be restricted in order to produce a one-to-one relationshi p. While there are many different intervals that are always increasing or always decreasing , there are convention al choices for the restricted domain of the trig functions. Students understand that: Click below to access all ALEX resources aligned There are to this standard. periodic phenomen ALEX a in real PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID terms of the context.★ Evidence of Student Attainment Teacher Vocabulary Knowledge 36. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Trigonometric Functions Prove and apply trigonometric identities. Functions F-TF.9 Franklin County Schools functions. Solutions that are mathemati cally possible may not be physically possible in the context of the problem. Determine when technology is appropriat e to finding a solution. Students know: Finding the ratio of a the sum or difference Skills the meaning of the solution. Interpret a domain in a contextual situation. Use technology to solve a trig equation. Translate a computed solution to an equivalent solution that fits in the physical domain (i.e., If -30 degrees is a solution, then another solution can be 330 degrees). Students are able to: Simplify expression s with trig ratios. Understanding life that can be modeled by trig functions. From the many solutions that result from a periodic function, only some may be logical given the context. Students understand that: Resources Knowing the exact value of the trig Resources Click below to access all ALEX resources aligned to this standard. ALEX PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge of two angles is more complex than just adding or subtractin g their ratios. Procedures for proving formulas. Know the exact values of the trig ratios of the special angles using the unit circle. Skills When given a trig ratio for an angle and the quadrant, determine any other needed trig ratio using a reference angle. Convert radians to degrees and degrees to radians. Understanding Franklin County Schools ratios of the special angles, along with the addition and subtraction formulas, allow for determinat ion of the exact values for the trig ratios of other angles. Creating equivalent trig expression s often involves the use of trig identities and other trig ratios where the resulting expression doesn't have a similar appearanc e to the original expression . Proof is necessary Resources Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources to establish that a conjecture about a relationshi p in mathemati cs is always true and may provide insight into the mathemati cs being addressed. 37. Derive the equation of a parabola given a focus and directrix. Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. Geometry G-GPE.2 Students: Given the focus and directrix of a parabola, Use the distance formula to write an equation for all points which are equidistant from a given focus and directory, Use logical repeated reasoning to generalize to a general point and general directrix, developing the formula for the equation of a Franklin County Schools Parabola Focus Students know: Directrix The distance formula for points in the coordinate plane and methods of finding the distance between a point and a line, Properties of equality (Table 4). Students are able to: Create and justify a formula through algebraic manipulati on after setting the distances between two points and between a point and a line equal to each Students understand that: The points that satisfy the Click below to equation access all ALEX of a resources aligned parabola to this standard. represent all of the ALEX points that Resources are equidistant from a given point and a given line. PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills parabola, Understanding Resources other. Compare and contrast parabola equations for horizontal and vertical directrices. 38. (+) 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. Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. Geometry G-GPE.3 Students are able to: Students know: Franklin County Schools The characteris tics of an ellipse and a hyperbola. Distance formula for finding the distance between two points on the coordinate plane. Pythagore an Theorem. Create and justify a formula through algebraic manipulati on after finding the sum of the distances from the foci to a point on the ellipse and setting the sum equal to a constant. Create and justify a formula through algebraic manipulati on after finding the difference of the distances from the foci to a Students understand: The points that satisfy the equation of the ellipse are all the points where the sum of the Click below to distances access all ALEX from the resources aligned foci is to this standard. constant. The points ALEX that satisfy Resources the equation of the hyperbola are all the points where the absolute value of the difference of the distances from the foci is PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding point on the hyperbola and setting the absolute value of the difference equal to a constant. 39. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Geometric Measurement & Dimension Explain volume formulas and use them to solve problems. Geometry G-GMD.2 always a constant. Students understand that: Students know: Franklin County Schools Resources Attributes of solid Students are able figures. to: Volume formulas Make an for a cone informal and a argument. cylinder Perform and other geometric solid transforma figures. tions. The aspects of Cavalieri's principle. There are relationshi ps between the sizes of various solid figures (i.e. cone and sphere, pyramid and prism, etc.). Cavalieri’s principle applies for all types of solids. As long as they have the same height and the same cross sectional Click below to access all ALEX resources aligned to this standard. ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources area, they will have the same volume. 40. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. Using Probability to Make Decisions Calculate expected values and use them to solve problems. Statistics & Probability S-MD.1 Students understand that: Students know: The sum of all the probabilitie s in the sample space will equal 1. The difference between experimen tal and theoretical probabilitie s. Students are able to: Determine the probabilitie s of any event in the sample space. Create a data distribution for a sample space and show it graphically . Franklin County Schools Probabilitie s can be calculated using a theoretical model, experiment , or simulation, as well as using the Click below to results access all ALEX from data resources aligned collection. to this standard. When the number of ALEX trials Resources approache s infinity, the experiment al probability will approach the theoretical probability (Law of Large Numbers). When symmetry PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding Resources or a trend exist in a probability distribution , it may aid in interpretati on. 41. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. Using Probability to Make Decisions Calculate expected values and use them to solve problems. Statistics & Probability S-MD.2 Students know: 42. (+) Develop a probability distribution for a random variable defined for a sample Using Probability to Make Decisions Calculate expected values and use them Franklin County Schools The sum of all the probabilitie s in the sample space will equal 1. In a probability distributio n, the probabilitie s of values of the random variable are weighted by their relative frequency of occurrence . Students know: Distinguish Students are able to: Find the probability Students distribution understand that: for a sample The mean space. Click below to of the Find the probability access all ALEX expected distribution resources aligned value of a to this standard. is a probability weighted distribution ALEX average in as the sum which each Resources of each outcome is value of weighted the by its random probability. variable multiplied by its associated probability. Students are able to: Students understand that: Click below to access all ALEX resources aligned to this standard. PRE CAL Mathematics CCRS Standards and Alabama COS CCRS Standard Standard ID space in which theoretical probabilities can be calculated; find the expected value. Example: Find the 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. to solve problems. Statistics & Probability S-MD.3 Evidence of Student Attainment Teacher Vocabulary Knowledge ing characteris tics of empirical and theoretical probabilitie s. Theoretical probabilitie s can be mathemati cally calculated to obtain an exact value, not an approxima tion. The expected value of a probability distributio n is the mean of that probability distributio n. Skills Franklin County Schools Determine probabilitie s from contexts that can be modeled by probability distribution s which may be uniform, binomial, etc. Describe a probability distribution via a function, graph, or table for a random variable defined on a sample space where the theoretical values are obtainable. Find the expected value of a probability distribution . Identify which values of a Understanding The rules for finding the probability of a compound event may be extended to determine the probability of being above or below a certain criterion in a probability distribution . The expected value is the anticipated average outcome if the event is repeated a large number of times. Probabilitie s obtained by a simulation or experiment Resources ALEX Resources PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge Skills Understanding probability distribution fit into certain intervals (e.g., If a grade of 70% is needed to pass, what is the probability of passing when guessing on a true/false test?). 43. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: Find a current data 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? Using Probability to Make Decisions Calculate expected values and use them to solve problems. Statistics & Probability S-MD.4 Franklin County Schools Students know: Distinguish ing characteris tics of empirical and theoretical probabilitie s. Empirical probabilitie s are approxima tions to the theoretical probability . The expected Students are able to: Determine empirical probabilitie s. Determine the expected value of a probability distribution . Extrapolat e the expected value of a random variable to a different sized Resources are not necessarily equal to the theoretical probability. Students understand that: Probabilitie s obtained by a simulation Click below to or access all ALEX experiment resources aligned are not to this standard. necessarily equal to ALEX the Resources theoretical probability. Values obtained using the expected value and proportion s are PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID Evidence of Student Attainment Teacher Vocabulary Knowledge value of a probability distributio n is the mean of that probability distributio n. 44. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values. a. b. Using Probability to Make Decisions Use probability to evaluate outcomes of decisions. Statistics & Probability Find the expected S-MD.5 payoff for a game of chance. Examples: Find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. Evaluate and compare strategies on the basis of expected values. Example: Compare a highdeductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a Franklin County Schools Skills population. Students know: Understanding Interpretat Students are able to: ion of probabilitie s and odds Determine associated the with a probability game of of an chance. event and A situation expected in which a value of a decision random has to be variable. made has Find or an research expected the payoff that probability is based and on the expected probability value of an of each event. outcome Communic and the ate the value of reasoning the behind associated decisions. outcome. Multiple factors may Resources merely estimates of actual values. Students understand that: Objective decisions can be based on mathemati cal informatio n obtained by using Click below to probabilitie access all ALEX s and resources aligned expected to this standard. values. Multiple ALEX factors Resources may ultimately determine the decision one makes other than the probability of events, such as ethical constraints PRE CAL CCRS Standard Mathematics CCRS Standards and Alabama COS Standard ID major accident. Evidence of Student Attainment Teacher Vocabulary Knowledge ultimately determine the given probability of an event. Skills Understanding Resources , social policy, or feelings of others. Click below to access all ALEX resources aligned to this standard. Franklin County Schools ALEX Resources