Number and Quantity - Category
N-RN Real Number System - Domain
Extend the properties of exponents to rational exponents.

CC_N-RN-1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)*3 to hold, so 5(1/3)*3 must equal 5.
CC_N-RN-2  Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Use properties of rational and irrational numbers.

CC_N-RN-3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N -Q  Quantities ★ - Domain
Reason quantitatively and use units to solve problems.

CC_N-Q-1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
CC_N-Q-2 Define appropriate quantities for the purpose of descriptive modeling.
CC_N-Q-3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N -CN The Complex Number System - Domain
Perform arithmetic operations with complex numbers.

CC_N-CN-1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
CC_N-CN-2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CC_N-CN-3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane.

CC_N-CN-4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
CC_N-CN-5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°.
CC_N-CN-6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Use complex numbers in polynomial identities and equations.

CC_N-CN-7 Solve quadratic equations with real coefficients that have complex solutions.
CC_N-CN-8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
CC_N-CN-9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N -VM Vector and Matrix Quantities - Domain
Represent and model with vector quantities.

CC_N-VM-1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
CC_N-VM-2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CC_N-VM-3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

CC_N-VM-4 (+) Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

CC_N-VM-5 (+) Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Perform operations on matrices and use matrices in applications.

CC_N-VM-6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
CC_N-VM-7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
CC_N-VM-8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
CC_N-VM-9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
CC_N-VM-10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
CC_N-VM-11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
CC_N-VM-12 (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.


Algebra - Category
A-SSE Seeing Structure in Expressions - Domain
Interpret the structure of expressions

CC_A-SSE-1 Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

CC_A-SSE-2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Write expressions in equivalent forms to solve problems

CC_A-SSE-3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

CC_A-SSE-4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★

A-APR Arithmetic with Polynomials and Rational Expressions - Domain
Perform arithmetic operations on polynomials

CC_A-APR-1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Understand the relationship between zeros and factors of polynomials

CC_A-APR-2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
CC_A-APR-3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Use polynomial identities to solve problems

CC_A-APR-4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
CC_A-APR-5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

Rewrite rational expressions

CC_A-APR-6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
CC_A-APR-7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A-CED Creating equations ★ - Domain
Create equations that describe numbers or relationships

CC_A-CED-1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
CC_A-CED-2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
CC_A-CED-3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
CC_A-CED-4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

A-REI Reasoning with Equations and Inequalities - Domain
Understand solving equations as a process of reasoning and explain the reasoning

CC_A-REI-1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
CC_A-REI-2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable.
CC_A-REI-3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
CC_A-REI-4 Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2=49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Solve systems of equations

CC_A-REI-5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
CC_A-REI-6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
CC_A-REI-7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
CC_A-REI-8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
CC_A-REI-9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). Represent and solve equations and inequalities graphically
CC_A-REI-10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
CC_A-REI-11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★
CC_A-REI-12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.



Categories: Geometry SRT / Modeling ★

Geometry -Category

G-SRT Similarity, right triangles, and trigonometry  

Understand similarity in terms of similarity transformations

CC_G-SRT-1 Verify experimentally the properties of dilations given by a center and a scale factor:

a.  A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b.  The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

CC_G-SRT-2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
CC_G-SRT-3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

CC_G-SRT-4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

CC_G-SRT-5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles

CC_G-SRT-6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
CC_G-SRT-7 Explain and use the relationship between the sine and cosine of complementary angles.
CC_G-SRT-8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

★ Apply trigonometry to general triangles

CC_G-SRT-9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
CC_G-SRT-10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
CC_G-SRT-11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).




Categories: Functions IF and TF / Geometry SRT / Modeling ★

Functions - Category

F-IF Interpreting functions

Analyze functions using different representations

CC_F-IF-7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ (modeling)
  e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F-TF Trigonometric Functions

Extend the domain of trigonometric functions using the unit circle

CC_F-TF-1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
CC_F-TF-2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
CC_F-TF-3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
CC_F-TF-4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions

CC_F-TF-5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★ (modeling)
CC_F-TF-6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
CC_F-TF-7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★ (modeling)

Prove and apply trigonometric identities

CC_F-TF-8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
CC_F-TF-9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Geometry -Category

G-SRT Similarity, right triangles, and trigonometry  

Define trigonometric ratios and solve problems involving right triangles

CC_G-SRT-6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
CC_G-SRT-7 Explain and use the relationship between the sine and cosine of complementary angles.
CC_G-SRT-8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

★ Apply trigonometry to general triangles

CC_G-SRT-9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
CC_G-SRT-10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
CC_G-SRT-11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

(+) All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students.
(★) Making mathematical models is a Standard for  Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ( ★ ).


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Page last modified on January 03, 2011, at 03:27 AM