# Massachusetts State Standards for Mathematics: Grade 11

Currently Perma-Bound only has suggested titles for grades K-8 in the Science and Social Studies areas. We are working on expanding this.

MA.12.P. Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

12.P.1. Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal's Triangle.

12.P.2. Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the general term and sum recursively and explicitly.

12.P.3. Demonstrate an understanding of the binomial theorem and use it in the solution of problems.

12.P.4. Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions.

12.P.5. Perform operations on functions, including composition. Find inverses of functions.

12.P.6. Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.

12.P.7. Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.

12.P.8. Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.

12.P.9. Use matrices to solve systems of linear equations. Apply to the solution of everyday problems.

12.P.10. Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods.

12.P.11. Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint and combined variation, and periodic processes.

12.P.12. Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Identify maximum and minimum values of functions in simple situations. Apply these concepts to the solution of problems.

12.P.13. Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af(b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, logarithmic, and trigonometric functions.

MA.12.G. Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

12.G.1. Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

12.G.2. Derive and apply basic trigonometric identities and the laws of sines and cosines.

12.G.3. Use the notion of vectors to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and geometrically. Use vector methods to obtain geometric results.

12.G.4. Relate geometric and algebraic representations of lines, simple curves, and conic sections.

12.G.5. Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

MA.12.M. Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

12.M.1. Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration.

12.M.2. Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense.

MA.12.D. Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

12.D.1. Design surveys and apply random sampling techniques to avoid bias in the data collection.

12.D.2. Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.

12.D.3. Apply regression results and curve fitting to make predictions from data.

12.D.4. Apply uniform, normal, and binomial distributions to the solutions of problems.

12.D.5. Describe a set of frequency distribution data by spread (i.e., variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications.

12.D.6. Use combinatorics (e.g., 'fundamental counting principle,' permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate.

12.D.7. Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities.

MA.G.G. Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

G.G.1. Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses). Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, midsegments); determine interior angles for regular polygons. Draw and label sets of points such as line segments, rays, and circles. Detect symmetries of geometric figures.

G.G.2. Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contra positive.

G.G.3. Apply formulas for a rectangular coordinate system to prove theorems.

G.G.4. Draw congruent and similar figures using a compass, straightedge, protractor, or computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments. (10.G.2)

G.G.5. Apply congruence and similarity correspondences and properties of the figures to find missing parts of geometric figures, and provide logical justification. (10.G.4)

G.G.6. Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

G.G.7. Solve simple triangle problems using the triangle angle sum property, and/or the Pythagorean Theorem. (10.G.5)

G.G.8. Use the properties of special triangles (e.g., isosceles, equilateral, 30 degrees-60 degrees-90 degrees, 45 degrees-45 degrees-90 degrees) to solve problems. (10.G.6)

G.G.9. Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

G.G.10. Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and solve problems.

G.G.11. Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the 'point-slope' or 'slope y-intercept' formulas. Explain the significance of a positive, negative, zero, or undefined slope. (10.P.2)

G.G.12. Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems. (10.G.7)

G.G.13. Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the 'point-slope' form of the equation. (10.G.8)

G.G.14. Demonstrate an understanding of the relationship between geometric and algebraic representations of circles.

G.G.15. Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems. (10.G.9)

G.G.16. Demonstrate the ability to visualize solid objects and recognize their projections and cross sections. (10.G.10)

G.G.17. Use vertex-edge graphs to model and solve problems. (10.G.11)

G.G.18. Use the notion of vectors to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and pictorially. Use vector methods to obtain geometric results. (12.G.3)

MA.G.M. Geometry: Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

MA.12.N. Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

12.N.1. Define complex numbers (e.g., a + bi) and operations on them, in particular, addition, subtraction, multiplication, and division. Relate the system of complex numbers to the systems of real and rational numbers.

12.N.2. Simplify numerical expressions with powers and roots, including fractional and negative exponents.

G.M.1. Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles. (10.M.1)

G.M.2. Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area. (10.M.2)

G.M.3. Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume. (10.M.3)

G.M.4. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. (10.M.4)

G.M.5. Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

MA.AII.N. Algebra II: Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

AII.N.1. Define complex numbers (e.g., a + bi) and operations on them, in particular, addition, subtraction, multiplication, and division. Relate the system of complex numbers to the systems of real and rational numbers. (12.N.1)

AII.N.2. Simplify numerical expressions with powers and roots, including fractional and negative exponents. (12.N.2)

MA.AII.P. Algebra II: Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

AII.P.1. Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal's Triangle. (12.P.1)

AII.P.2. Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly. (12.P.2)

AII.P.3. Demonstrate an understanding of the binomial theorem and use it in the solution of problems. (12.P.3)

AII.P.4. Demonstrate an understanding of the exponential and logarithmic functions.

AII.P.5. Perform operations on functions, including composition. Find inverses of functions. (12.P.5)

AII.P.6. Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential. (12.P.6)

AII.P.7. Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems. (12.P.7)

AII.P.8. Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, and logarithmic functions; expressions involving the absolute values; and simple rational expressions. (12.P.8)

AII.P.9. Use matrices to solve systems of linear equations. Apply to the solution of everyday problems. (12.P.9)

AII.P.10. Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods. (12.P.10)

AII.P.11. Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, and step functions, absolute values and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; logistic growth; joint, and combined variation. (12.P.11)

AII.P.12. Identify maximum and minimum values of functions in simple situations. Apply to the solution of problems. (12.P.12)

AII.P.13. Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af (b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, and logarithmic functions. (12.P.13)

MA.AII.G. Algebra II: Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

AII.G.1. Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems. (12.G.1)

AII.G.2. Derive and apply basic trigonometric identities and the laws of sines and cosines. (12.G.2)

AII.G.3. Relate geometric and algebraic representations of lines, simple curves, and conic sections. (12.G.4)

MA.AII.D. Algebra II: Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

AII.D.1. Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data. (12.D.2)

AII.D.2. Use combinatorics (e.g., 'fundamental counting principle,' permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate. (12.D.6)

MA.PC.N. Precalculus: Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

PC.N.1. Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates. Apply DeMoivre's theorem to multiply, take roots, and raise complex numbers to a power.

MA.PC.P. Precalculus: Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

PC.P.1. Use mathematical induction to prove theorems and verify summation formulas, e.g., verify.

PC.P.2. Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients.

PC.P.3. Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.

PC.P.4. Explain the identity sin squared x + cos squared x = 1. Relate the identity to the Pythagorean Theorem.

PC.P.5. Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre's theorem and use them to prove other trigonometric identities. Apply to the solution of problems.

PC.P.6. Understand, predict, and interpret the effects of the parameters a, w, b, and c on the graph of y = asin(w(x - b)) + c; similarly for the cosine and tangent. Use to model periodic processes. (12.P.13)

PC.P.7. Translate between geometric, algebraic, and parametric representations of curves. Apply to the solution of problems.

PC.P.8. Identify and discuss features of conic sections: axes, foci, asymptotes, and tangents. Convert between different algebraic representations of conic sections.

PC.P.9. Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Explain the significance of a horizontal tangent line. Apply these concepts to the solution of problems.

MA.PC.G. Precalculus: Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

PC.G.1. Demonstrate an understanding of the laws of sines and cosines. Use the laws to solve for the unknown sides or angles in triangles. Determine the area of a triangle given the length of two adjacent sides and the measure of the included angle. (12.G.2)

PC.G.2. Use the notion of vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results. (12.G.3)

PC.G.3. Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems. (12.G.5)

MA.PC.M. Precalculus: Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

PC.M.1. Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration. (12.M.1)

PC.M.2. Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

MA.PC.D. Precalculus: Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

PC.D.1. Design surveys and apply random sampling techniques to avoid bias in the data collection. (12.D.1)

PC.D.2. Apply regression results and curve fitting to make predictions from data. (12.D.3)

PC.D.3. Apply uniform, normal, and binomial distributions to the solutions of problems. (12.D.4)

PC.D.4. Describe a set of frequency distribution data by spread (variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications. (12.D.5)

PC.D.5. Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities. (12.D.7)

MA.CC.HS.N. Conceptual Category: Number and Quantity

HS.N.RN. The Real Number System

HS.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 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^ (1/3)3 to hold, so (5^1/3)^3 must equal 5.

HS.N.RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

HS.N.RN.3. Explain why the sum or product of two rational numbers are 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.

HS.N.Q. Quantities

HS.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.

HS.N.Q.2. Define appropriate quantities for the purpose of descriptive modeling.

HS.N.Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

HS.N.Q.MA.3a. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure.

HS.N.CN. The Complex Number System

HS.N.CN.1. Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.

HS.N.CN.2. Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

HS.N.CN. 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

HS.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.

HS.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 + square root of 3i)^3 = 8 because (-1 + square root of 3i) has modulus 2 and argument 120 degrees.

HS.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.

HS.N.CN.7. Solve quadratic equations with real coefficients that have complex solutions.

HS.N.CN. 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x - 2i).

HS.N.CN. 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

HS.N.VM. Vector and Matrix Quantities

HS.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|,

HS.N.VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

HS.N.VM.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

HS.N.VM.4. (+) Add and subtract vectors.

HS.N.VM.4.a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

HS.N.VM.4.b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

HS.N.VM.4.c. Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

HS.N.VM.5. (+) Multiply a vector by a scalar.

HS.N.VM.5.a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

HS.N.VM.5.b. Compute the magnitude of a scalar multiple cv using

HS.N.VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

HS.N.VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

HS.N.VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

HS.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.

HS.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.

HS.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.

HS.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.

MA.CC.HS.A. Conceptual Category: Algebra

HS.A.SSE. Seeing Structure in Expressions

HS.A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

HS.A.SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.

HS.A.SSE.1.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.

HS.A.SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

HS.A.SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

HS.A.SSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines.

HS.A.SSE.3.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

HS.A.SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1/12)^12t = 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

HS.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.

HS.A.APR. Arithmetic with Polynomials and Rational Expressions

HS.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.

HS.A.APR.MA.1a. Divide polynomials.

HS.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).

HS.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.

HS.A.APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.

HS.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.

HS.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.

HS.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.

HS.A.CED. Creating Equations

HS.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.

HS.A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HS.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.

HS.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.

HS.A.REI. Reasoning with Equations and Inequalities

HS.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.

HS.A.REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

HS.A.REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

HS.A.REI.MA.3a. Solve linear equations and inequalities in one variable involving absolute value.

HS.A.REI.4. Solve quadratic equations in one variable.

HS.A.REI.4.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.

HS.A.REI.4.b. Solve quadratic equations by inspection (e.g., for x^2 = 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.

A.REI.MA.4c. Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations.

HS.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.

HS.A.REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

HS.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 x^2 + y^2 = 3.

HS.A.REI. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.

HS.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).

HS.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).

HS.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.

HS.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.

MA.CC.HS.F. Conceptual Category: Functions

HS.F.IF. Interpreting Functions

HS.F.IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

HS.F.IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

HS.F.IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n > or = 1.

HS.F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

HS.F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

HS.F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

HS.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.

HS.F.IF.7.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

HS.F.IF.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

HS.F.IF.7.c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

HS.F.IF.7. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

HS.F.IF.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

HS.F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HS.F.IF.8.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

HS.F.IF.8.b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

HS.F.IF.MA.8c. Translate between different representations of functions and relations: graphs, equations, point sets, and tables.

HS.F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

HS.F.IF.MA.10. Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.

HS.F.BF. Building Functions

HS.F.BF.1. Write a function that describes a relationship between two quantities.

HS.F.BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

HS.F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

HS.F.BF.1. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

HS.F.BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

HS.F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

HS.F.BF.4. Find inverse functions.

HS.F.BF.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x^3 or f(x) = (x+1)/(x-1) for x does not equal 1.

HS.F.BF.4.b. (+) Verify by composition that one function is the inverse of another.

HS.F.BF.4.c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

HS.F.BF.4.d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

HS.F.BF.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

HS.F.LE. Linear, Quadratic, and Exponential Models

HS.F.LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

HS.F.LE.1.a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

HS.F.LE.1.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

HS.F.LE.1.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

HS.F.LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

HS.F.LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

HS.F.LE.4. For exponential models, express as a logarithm the solution to a b^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

HS.F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

HS.F.TF. Trigonometric Functions

HS.F.TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

HS.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.

HS.F.TF. 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for pi /3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number.

HS.F.TF. 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

HS.F.TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

HS.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.

HS.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.

HS.F.TF.8. Prove the Pythagorean identity sin^2 + cos^2 = 1 and use it find sin, cos, or tan given sin, cos, or tan and the quadrant.

HS.F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

MA.CC.HS.G. Conceptual Category: Geometry

HS.G.CO. Congruence

HS.G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

HS.G.CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

HS.G.CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

HS.G.CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

HS.G.CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HS.G.CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

HS.G.CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

HS.G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

HS.G.CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

HS.G.CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

HS.G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

HS.G.CO.MA.11a. Prove theorems about polygons. Theorems include: measures of interior and exterior angles, properties of inscribed polygons.

HS.G.CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

HS.G.CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

HS.G.SRT. Similarity, Right Triangles, and Trigonometry

HS.G.SRT.1. Verify experimentally the properties of dilations given by a center and a scale factor:

HS.G.SRT.1.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.

HS.G.SRT.1.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

HS.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.

HS.G.SRT.3. Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.

HS.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.

HS.G.SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

HS.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.

HS.G.SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.

HS.G.SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

HS.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.

HS.G.SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

HS.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).

HS.G.C. Circles

HS.G.C.1. Prove that all circles are similar.

HS.G.C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

HS.G.C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

HS.G.C.MA.3a. Derive the formula for the relationship between the number of sides and sums of the interior and sums of the exterior angles of polygons and apply to the solutions of mathematical and contextual problems.

HS.G.C.4. (+) Construct a tangent line from a point outside a given circle to the circle.

HS.G.C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

HS.G.GPE. Expressing Geometric Properties with Equations

HS.G.GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

HS.G.GPE.2. Derive the equation of a parabola given a focus and directrix.

HS.G.GPE. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

HS.G.GPE.MA.3a. (+) Use equations and graphs of conic sections to model real-world problems.

HS.G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, square root of 3) lies on the circle centered at the origin and containing the point (0, 2).

HS.G.GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

HS.G.GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

HS.G.GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

HS.G.GMD. Geometric Measurement and Dimension

HS.G.GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

HS.G.GMD. 2. (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

HS.G.GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

HS.G.GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

HS.G.MG. Modeling with Geometry

HS.G.MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

HS.G.MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

HS.G.MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

HS.G.MG.MA.4. Use dimensional analysis for unit conversions to confirm that expressions and equations make sense.

MA.CC.HS.S. Conceptual Category: Statistics and Probability

HS.S.ID. Interpreting Categorical and Quantitative Data

HS.S.ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

HS.S.ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HS.S.ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

HS.S.ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

HS.S.ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

HS.S.ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

HS.S.ID.6.a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

HS.S.ID.6.b. Informally assess the fit of a function by plotting and analyzing residuals.

HS.S.ID.6.c. Fit a linear function for a scatter plot that suggests a linear association.

HS.S.ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

HS.S.ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

HS.S.ID.9. Distinguish between correlation and causation.

HS.S.IC. Making Inferences and Justifying Conclusions

HS.S.IC.1. Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

HS.S.IC.2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

HS.S.IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

HS.S.IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

HS.S.IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

HS.S.IC.6. Evaluate reports based on data.

HS.S.CP. Conditional Probability and the Rules of Probability

HS.S.CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (''or,'' ''and,'' ''not'').

HS.S.CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

HS.S.CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

HS.S.CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

HS.S.CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

HS.S.CP.6. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

HS.S.CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

HS.S.CP. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

HS.S.CP. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

HS.S.MD. Using Probability to Make Decisions

HS.S.MD.1. (+) 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.

HS.S.MD.2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

HS.S.MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For 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.

HS.S.MD.4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For 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?

HS.S.MD.5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

HS.S.MD.5.a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

HS.S.MD.5.b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

HS.S.MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

HS.S.MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling [out] a hockey goalie at the end of a game).

MA.CC.AI. Traditional Pathway Model Course: Algebra I

AI-N.RN. Number and Quantity - The Real Number System

AI-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 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)*3 to hold, so (5^1/3)^3 must equal 5.

AI-N.RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

AI-N.RN.3. Explain why the sum or product of 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.

AI-N.Q. Number and Quantity - Quantities

AI-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.

AI-N.Q.2. Define appropriate quantities for the purpose of descriptive modeling.

AI-N.Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

AI-N.Q.MA.3a. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given the precision of the tools used to measure.

AI-A.SSE. Algebra - Seeing Structure in Expressions

AI-A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

AI-A.SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.

AI-A.SSE.1.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.

AI-A.SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

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

AI-A.SSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines.

AI-A.SSE.3.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

AI-A.SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1/12)^12t = 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

AI-A.APR. Algebra - Arithmetic with Polynomials and Rational Functions

AI-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.

AI-A.CED. Algebra - Creating Equations

AI-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.

AI-A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

AI-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.

AI-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.

AI-A.REI. Algebra - Reasoning with Equations and Inequalities

AI-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.

AI-A.REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

AI-A.REI.MA.3a. Solve linear equations and inequalities in one variable involving absolute value.

AI-A.REI.4. Solve quadratic equations in one variable.

AI-A.REI.4.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.

AI-A.REI.4.b. Solve quadratic equations by inspection (e.g., for x^2 = 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.

AI-A.REI.MA.4c. Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations.

AI-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.

AI-A.REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

AI-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 x^2 + y^2 = 3.

AI-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).

AI-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.

AI-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.

AI-F.IF. Functions - Interpreting Functions

AI-F.IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

AI-F.IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

AI-F.IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n > or = 1.

AI-F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

AI-F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

AI-F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

AI-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.

AI-F.IF.7.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

AI-F.IF.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

AI-F.IF.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

AI-F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

AI-F.IF.8.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

AI-F.IF.8.b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = 1.02^t, y = (0.97)^t, y = (1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential growth and decay.

AI-F.IF.MA.8c. Translate between different representations of functions and relations: graphs, equations point sets, and tabular.

AI-F.IF.8.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

AI-F.IF.8.MA.10. Given algebraic, numeric, and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.

AI-F.BF. Functions - Building Functions

AI-F.BF.1. Write a function that describes a relationship between two quantities.

AI-F.BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

AI-F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

AI-F.BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

AI-F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

AI-F.BF.4. Find inverse functions.

AI-F.BF.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x^3 or f(x) = (x+1)/(x-1) for x does not equal 1.

AI-F.LE. Functions - Linear, Quadratic, and Exponential Models

AI-F.LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

AI-F.LE.1.a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

AI-F.LE.1.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

AI-F.LE.1.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

AI-F.LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

AI-F.LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

AI-F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

AI-S.ID. Statistics and Probability - Interpreting Categorical and Quantitative Data

AI-S.ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

AI-S.ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

AI-S.ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

AI-S.ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

AI-S.ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

AI-S.ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

AI-S.ID.6.a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

AI-S.ID.6.b. Informally assess the fit of a function by plotting and analyzing residuals.

AI-S.ID.6.c. Fit a linear function for a scatter plot that suggests a linear association.

AI-S.ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

AI-S.ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

AI-S.ID.9. Interpret linear models. Distinguish between correlation and causation.

MA.CC.G. Traditional Pathway Model Course: Geometry

G-N.Q. Number and Quantity - Quantity

G-N.Q.2. Define appropriate quantities for the purpose of descriptive modeling.

G-N.Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

G-N.Q.MA.3a. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measures. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure.

G-G.CO. Geometry - Congruence

G-G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G-G.CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-G.CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-G.CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-G.CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-G.CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-G.CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-G.CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G-G.CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

G-G.CO.MA.11a. Prove theorems about polygons. Theorems include: measures of interior and exterior angles, properties of inscribed polygons.

G-G.CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G-G.CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-G.SRT. Similarity, Right Triangles, and Trigonometry

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

G-G.SRT.1.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.

G-G.SRT.1.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-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.

G-G.SRT.3. Use the properties of similarity transformations to establish the Angle-Angle criterion (AA) for two triangles to be similar.

G-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.

G-G.SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G-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.

G-G.SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.

G-G.SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G-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.

G-G.SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

G-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).

G-G.C. Circles

G-G.C.1. Prove that all circles are similar.

G-G.C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G-G.C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-G.C.MA.3a. Derive the formula for the relationship between the number of sides and sums of the interior and sums of the exterior angles of polygons and apply to the solutions of mathematical and contextual problems.

G-G.C.4. (+) Construct a tangent line from a point outside a given circle to the circle.

G-G.C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-G.GPE. Expressing Geometric Properties with Equations

G-G.GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

G-G.GPE.2. Derive the equation of a parabola given a focus and a directrix.

G-G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, square root of 3) lies on the circle centered at the origin and containing the point (0, 2).

G-G.GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G-G.GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G-G.GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

G-G.GMD. Geometric Measurement and Dimension

G-G.GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

G-G.GMD.2. (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

G-G.GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G-G.GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G-G.MG. Modeling with Geometry

G-G.MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-G.MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G-G.MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

G-G.MG.MA.4. Use dimensional analysis for unit conversion to confirm that expressions and equations make sense.

MA.CC.AII. Traditional Pathway Model Course: Algebra II

AII-N.RN. Number and Quantity - The Real Number System

AII-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 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^(1/3)*3 to hold, so (5^1/3)^3 must equal 5.

AII-N.RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

AII-N.CN. Number and Quantity - The Complex Number System

AII-N.CN.1. Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.

AII-N.CN.2. Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

AII-N.CN.7. Solve quadratic equations with real coefficients that have complex solutions.

AII-N.CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x - 2i).

AII-N.CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

AII-N.VM. Number and Quantity - Vector Quantities and Matrices

AII-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|,

AII-N.VM.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

AII-N.VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

AII-N.VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

AII-N.VM.12. (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

AII-A.SSE. Algebra - Seeing Structure in Expressions

AII-A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

AII-A.SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

AII-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.

AII-A.APR. Algebra - Arithmetic with Polynomials and Rational Expressions

AII-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.

AII-A.APR.MA.1a. Divide polynomials.

AII-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).

AII-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.

AII-A.APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.

AII-A.APR.5. (+) Know and apply that the Binomial Theorem gives 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.

AII-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.

AII-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.

AII-A.CED. Algebra - Creating Equations

AII-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.

AII-A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

AII-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.

AII-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.

AII-A.REI. Algebra - Reasoning with equations and inequalities

AII-A.REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

AII-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.

AII-F.IF. Functions - Interpreting Functions

AII-F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

AII-F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

AII-F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

AII-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.

AII-F.IF.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

AII-F.IF.7.c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

AII-F.IF.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

AII-F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

AII-F.IF.MA.8c. Translate between different representations of functions and relations: graphs, equations, point sets, and tables.

AII-F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

AII-F.BF. Functions - Building Functions

AII-F.BF.1. Write a function that describes a relationship between two quantities.

AII-F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

AII-F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

AII-F.BF.4. Find inverse functions.

AII-F.BF.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x does not equal 1.

AII-F.LE. Functions - Linear, Quadratic, and Exponential Models

AII-F.LE.4. For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

AII-F.TF. Functions - Trigonometric Functions

AII-F.TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

AII-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.

AII-F.TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

AII-F.TF.8. Prove the Pythagorean identity (sin A)^2 + (cos A)^2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.

AII-S.ID. Statistics and Probability - Interpreting Categorical and Quantitative Data

AII-S.ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

AII-S.IC. Statistics and Probability - Making Inferences and Justifying Conclusions

AII-S.IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

AII-S.IC.2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model?

AII-S.IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

AII-S.IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

AII-S.IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

AII-S.IC.6. Evaluate reports based on data.

MA.CC.MI. Integrated Pathway Model Course: Mathematics I

MI-N.Q. Number and Quantities - Quantities

MI-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.

MI-N.Q.2. Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling.

MI-N.Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

MI-N.Q.MA.3a. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measures.

MI-A.SSE. Algebra - Seeing Structure in Expressions

MI-A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

MI-A.SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.

MI-A.SSE.1.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.

MI-A.CED. Algebra - Creating Equations

MI-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.

MI-A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MI-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.

MI-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.

MI-A.REI. Algebra - Reasoning with Equations and Inequalities

MI-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.

MI-A.REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MI-A.REI.MA.3a. Solve linear equations and inequalities in one variable involving absolute value.

MI-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.

MI-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).

MI-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.

MI-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.

MI-F.IF. Functions - Interpreting Functions

MI-F.IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MI-F.IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MI-F.IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n > or = 1 (n is greater than or equal to 1).

MI-F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

MI-F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

MI-F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MI-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.

MI-F.IF.7.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

MI-F.IF.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

MI-F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

MI-F.IF.MA.10. Given algebraic, numeric, and/or graphical representations of functions, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.

MI-F.BF. Functions - Building Functions

MI-F.BF.1. Write a function that describes a relationship between two quantities.

MI-F.BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

MI-F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

MI-F.BF.2. Build a function that models a relationship between two quantities. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MI-F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

MI-F.LE. Functions - Linear, Quadratic, and Exponential Models

MI-F.LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

MI-F.LE.1.a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

MI-F.LE.1.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

MI-F.LE.1.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

MI-F.LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

MI-F.LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

MI-F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

MI-G.CO. Geometry - Congruence

MI-G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MI-G.CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

MI-G.CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

MI-G.CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MI-G.CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

MI-G.CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

MI-G.CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

MI-G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

MI-G.CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

MI-G.CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

MI-G.GPE. Geometry - Expressing Geometric Properties with Equations

MI-G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, square root of 3) lies on the circle centered at the origin and containing the point (0,2).

MI-G.GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

MI-G.GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

MI-S.ID. Statistics and Probability - Interpreting Categorical and Quantitative Data

MI-S.ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

MI-S.ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

MI-S.ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

MI-S.ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

MI-S.ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MI-S.ID.6.a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

MI-S.ID.6.b. Informally assess the fit of a function by plotting and analyzing residuals.

MI-S.ID.6.c. Fit a linear function for a scatter plot that suggests a linear association.

MI-S.ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

MI-S.ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

MI-S.ID.9. Distinguish between correlation and causation.

MA.CC.MII. Integrated Pathway Model Course: Mathematics II

MII-N.RN. Number and Quantity - The Real Number System

MII-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 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5(^1/3)3 to hold, so (5^1/3)^3 must equal 5.

MII-N.RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

MII-N.RN.3. Explain why the sum or product of 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.

MII-N.CN. Number and Quantity - The Complex Number System

MII-N.CN.1. Know there is a complex number i such that I^2 = -1, and every complex number has the form a + bi with a and b real.

MII-N.CN.2. Use the relation I^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MII-N.CN.7. Solve quadratic equations with real coefficients that have complex solutions.

MII-N.CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x +2i)(x - 2i).

MII-N.CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

MII-A.SSE. Algebra - Seeing Structure in Expressions

MII-A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

MII-A.SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.

MII-A.SSE.1.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.

MII-A.SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

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

MII-A.SSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines.

MII-A.SSE.3.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

MII-A.SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^ (1/12)]^(12t) = 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

MII-A.APR. Algebra - Arithmetic with Polynomials and Rational Expressions

MII-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.

MII-A.CED. Algebra - Creating Equations

MII-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.

MII-A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MII-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.

MII-A.REI. Reasoning with Equations and Inequalities

MII-A.REI.4. Solve quadratic equations in one variable.

MII-A.REI.4.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.

MII-A.REI.4.b. Solve quadratic equations by inspection (e.g., for x^2 = 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.

MII-A.REI.MA.4c. Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations.

MII-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 x^2 + y^2 = 3.

MII-F.IF. Functions - Interpreting Functions

MII-F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

MII-F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

MII-F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MII-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.

MII-F.IF.7.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

MII-F.IF.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MII-F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MII-F.IF.8.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

MII-F.IF.8.b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

MII-F.IF.MA.8c. Translate between different representations of functions and relations: graphs, equations, point sets, and tables.

MII-F.IF.10. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

MII-F.IF.MA.10a. Given algebraic, numeric, and/or graphical representations of functions, recognize functions as polynomial, rational, logarithmic, exponential or trigonometric.

MII-F.BF. Functions - Building Functions

MII-F.BF.1. Write a function that describes a relationship between two quantities.

MII-F.BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

MII-F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

MII-F.BF.1.c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

MII-F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

MII-F.BF.4. Find inverse functions.

MII-F.BF.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x does not equal 1 (x not equal to 1).

MII-F.LE. Functions - Linear, Quadratic, and Exponential Models

MII-F.LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

MII-F.TF. Functions - Trigonometric Functions

MII-F.TF.8. Prove the Pythagorean identity (sin A)^2 + (cos A)^2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.

MII-G.CO. Geometry - Congruence

MII-G.CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

MII-G.CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

MII-G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

MII-G.CO.MA.11a. Prove theorems about polygons. Theorems include: measures of interior and exterior angels, properties of inscribed polygons.

MII-G.SRT. Geometry - Similarity, Right Triangles, and Trigonometry

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

MII-G.SRT.1.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.

MII-G.SRT.1.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MII-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.

MII-G.SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

MII-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.

MII-G.SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MII-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.

MII-G.SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.

MII-G.SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

MII-G.C. Geometry - Circles

MII-G.C.1. Prove that all circles are similar.

MII-G.C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

MII-G.C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

MII-G.C.MA.3a. Derive the formula for the relationship between the number of sides and the sums of the interior and the sums of the exterior angles of polygons and apply to the solutions of mathematical and contextual problems.

MII-G.C.4. (+) Construct a tangent line from a point outside a given circle to the circle.

MII-G.C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

MII-G.GPE. Geometry - Expressing Geometric Properties with Equations

MII-G.GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

MII-G.GPE.2. Derive the equation of a parabola given a focus and directrix.

MII-G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, square root of 3) lies on the circle centered at the origin and containing the point (0, 2).

MII-G.GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

MII-G.GMD. Geometry - Geometric Measurement with Dimension

MII-G.GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

MII-G.GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

MII-S.CP. Statistics and Probability - Conditional Probability and the Rules of Probability

MII-S.CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (''or,'' ''and,'' ''not'').

MII-S.CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

MII-S.CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

MII-S.CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

MII-S.CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

MII-S.CP.6. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

MII-S.CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

MII-S.CP.8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)], and interpret the answer in terms of the model.

MII-S.CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

MII-S.MD. Statistics and Probability - Using Probability to Make Decisions

MII-S.MD.6. (+) Use probabilities to make fair decision (e.g., drawing by lots, using a random number generator).

MII-S.MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling [out] a hockey goalie at the end of a game).

MA.CC.MIII. Integrated Pathway Model Course: Mathematics III

MIII-N.CN. Number and Quantity - The Complex Number System

MIII-N.CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x +2i)(x - 2i).

MIII-N.CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

MIII-A.SSE. Algebra - Seeing Structure in Expressions

MIII-A.SSE.1. Interpret expressions that represent a quantity in terms of its context.

MIII-A.SSE.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.

MIII-A.SSE.1.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.

MIII-A.SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

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

MIII-A.APR. Algebra - Arithmetic with Polynomials and Rational Expressions

MIII-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.

MIII-A.APR.MA.1a. Divide polynomials.

MIII-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).

MIII-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.

MIII-A.APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.

MIII-A.APR.5. (+) Know and apply that the Binomial Theorem gives 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

MIII-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.

MIII-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.

MIII-A.CED. Algebra - Creating Equations

MIII-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.

MIII-A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MIII-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.

MIII-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.

MIII-A.REI. Algebra - Reasoning with Equations and Inequalities

MIII-A.REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MIII-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.

MIII-F.IF. Functions - Interpreting Functions

MIII-F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

MIII-F.IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

MIII-F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MIII-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.

MIII-F.IF.7.b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MIII-F.IF.7.c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MIII-F.IF.7.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

MIII-F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MIII-F.IF.8.a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

MIII-F.IF.8.b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

MIII-F.IF.MA.8c. Translate between different representations of functions and relations: graphs, equations, point sets, and tables.

MIII-F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

MIII-F.BF. Functions - Building Functions

MIII-F.BF.1. Write a function that describes a relationship between two quantities.

MIII-F.BF.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

MIII-F.BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

MIII-F.BF.4. Find inverse functions.

MIII-F.BF.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x not equal to 1.

MIII-F.LE. Functions - Linear, Quadratic, and Exponential Models

MIII-F.LE.4. For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

MIII-F.TF. Functions - Trigonometric Functions

MIII-F.TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

MIII-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.

MIII-F.TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

MIII-G.SRT. Geometry - Similarity, Right Triangles, and Trigonometry

MIII-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.

MIII-G.SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

MIII-G.SRT.11. (+) Apply trigonometry to general triangles. 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).

MIII-G.GMD. Geometry - Geometric Measurement and Dimension

MIII-G.GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

MIII-G.MG. Geometry - Modeling with Geometry

MIII-G.MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

MIII-G.MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

MIII-G.MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

MIII-G.MG.MA.4. Use dimensional analysis for unit conversion to confirm that expressions and equations make sense.

MIII-S.ID. Statistics and Probability - Interpreting Categorical and Quantitative Data

MIII-S.ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

MIII-S.IC. Statistics and Probability - Making Inferences and Justifying Conclusions

MIII-S.IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

MIII-S.IC.2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model.

MIII-S.IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

MIII-S.IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

MIII-S.IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

MIII-S.IC.6. Evaluate reports based on data.

MIII-S.MD. Statistics and Probability - Using Probability to Make Decisions

MIII-S.MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

MIII-S.MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling [out] a hockey goalie at the end of a game).

MA.CC.PC. Advanced Model Course: Precalculus

PC-N.CN. Number and Quantity - The Complex Number System

PC-N.CN.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

PC-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 numbers represent the same number.

PC-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 - square root of si)^3 = 8 has modulus 2 and argument 120 degrees.

PC-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.

PC-N.CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x - 2i).

PC-N.CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

PC-N.VM. Number and Quantity - Vector Quantities and Matrices

PC-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|,

PC-N.VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

PC-N.VM.3. (+) Solve problems involving velocity and the other quantities that can be represented by vectors.

PC-N.VM.4. (+) Add and subtract vectors.

PC-N.VM.4.a. (+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically no the sum of the magnitudes.

PC-N.VM.4.b. (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

PC-N.VM.4.c. (+) Understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

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

PC-N.VM.5.a. (+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx,vy) = (cvx, cvy).

PC-N.VM.5.b. (+) Compute the magnitude of a scalar multiple cv using

PC-N.VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

PC-N.VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

PC-N.VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

PC-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.

PC-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.

PC-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.

PC-N.VM.12. (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinate in terms of area.

PC-A.APR. Algebra - Arithmetic with polynomials and rational expressions

PC-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.

PC-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.

PC-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.

PC-A.REI. Algebra - Reasoning with Equations and Inequalities

PC-A.REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.

PC-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).

PC-F.IF. Functions - Interpreting Functions

PC-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.

PC-F.IF.7.d. (+) Graph rational functions, identifying zeros when suitable factorizations are available, and showing end behavior.

PC-F.BF. Functions - Building Functions

PC-F.BF.1. Write a function that describes a relationship between two quantities.

PC-F.BF.1.c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as the function of time.

PC-F.BF.4. Find inverse functions.

PC-F.BF.4.b. (+) Verify by composition that one function is the inverse of another.

PC-F.BF.4.c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

PC-F.BF.4.d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

PC-F.BF.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

PC-F.TF. Functions - Trigonometric Functions

PC-F.TF.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosines, and tangent for x, pi+x, and 2pi -x in terms of their values for x, where x is any real number.

PC-F.TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

PC-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.

PC-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.

PC-F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

PC-G.SRT. Geometry - Similarity, right triangles, and trigonometry

PC-G.SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line form a vertex perpendicular to the opposite side.

PC-G.SRT.10. (+) prove the Laws of Sines and Cosines and use them to solve problems.

PC-G.SRT.11. (+) Understand and apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles, e.g., surveying problems, resultant forces.

PC-G.C. Geometry - Circles

PC-G.C.4. (+) Construct a tangent line from a point outside a given circle to the circle.

PC-G.GPE. Geometry - Expressing Geometric Properties with Equations

PC-G.GPE.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum of difference of distances from the foci is constant.

PC-G.GPE.MA.3a. (+) Use equations and graphs of conic sections to model real-world problems.

PC-G.GMD. Geometry - Geometric Measurement and Dimension

PC-G.GMD.2. (+) Give an informal argument using Cavalieri's Principle for the formulas for the volume of a sphere and other solid figures.

PC-G.GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

MA.CC.AQR. Advanced Model Course: Advanced Quantitative Reasoning

AQR-N.VM. Number and Quantity - Vectors and Matrix Quantities

AQR-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|,

AQR-N.VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

AQR-N.VM.3. (+) Solve problems involving velocity and the other quantities that can be represented by vectors.

AQR-N.VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

AQR-N.VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

AQR-N.VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.

AQR-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.

AQR-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.

AQR-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.

AQR-N.VM.12. (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinate in terms of area.

AQR-A.APR. Algebra - Arithmetic with polynomials and rational expressions

AQR-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.

AQR-A.REI. Algebra - Reasoning with Equations and Inequalities

AQR-A.REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.

AQR-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).

AQR-F.TF. Functions - Trigonometric Functions

AQR-F.TF.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosines, and tangent for x, pi+x, and 2pi -x in terms of their values for x, where x is any real number.

AQR-F.TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

AQR-F.TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

AQR-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.

AQR-F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

AQR-G.SRT. Geometry - Similarity, right triangles, and trigonometry

AQR-G.SRT.11. (+) Understand and apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles, e.g., surveying problems, resultant forces.

AQR-G.C. Geometry - Circles

AQR-G.C.4. (+) Construct a tangent line from a point outside a given circle to the circle.

AQR-G.GPE. Geometry - Expressing Geometric Properties with Equations

AQR-G.GPE.3. (+) Derive the equations of ellipses and hyperbolas given the foci and directrices.

AQR-G.GPE.MA.3a. Use equations and graphs of conic sections to model real-world problems.

AQR-G.GMD. Geometry - Geometric Measurement and Dimension

AQR-G.GMD.2. (+) Give an informal argument using Cavalieri's Principle for the formulas for the volume of a sphere and other solid figures.

AQR-G.GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

AQR-G.MG. Geometry - Modeling with Geometry

AQR-G.MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

AQR-G.MG.MA.4. Use dimensional analysis for unit conversion to confirm that expressions and equations make sense.

AQR-S.ID. Statistics and Probability - Interpreting Categorical and Quantitative Data

AQR-S.ID.9. Distinguish between correlation and causation.

AQR-S.IC. Statistics and Probability - Making Inferences and Justifying Conclusions

AQR-S.IC.4. Use data from a sample survey to estimate a population mean or proportion; develop margin of error through the use of simulation models for random sampling.

AQR-S.IC.5. Use data from a randomized experiment to compare two treatments; sue simulations to decide if differences between parameters are significant.

AQR-S.IC.6. Evaluate reports based on data.

AQR-S.CP. Statistics and Probability - Conditional Probability and the Rules of Probability

AQR-S.CP.8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B A) = P(B) P(A B), and interpret the answer in terms of the model.

AQR-S.CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

AQR-S.MD. Statistics and Probability - Using Probability to Make Decisions

AQR-S.MD.1. (+) 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.

AQR-S.MD.2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

AQR-S.MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For 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.

AQR-S.MD.4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For 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?

AQR-S.MD.5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

AQR-S.MD.5.a. (+) Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

AQR-S.MD.5.b. (+) Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

AQR-S.MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

AQR-S.MD.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling [out] a hockey goalie at the end of a game).