What does a student learn in StateNorth Carolina,GradeGrade 9,SubjectMathematics?

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Use strategies and procedures flexibly.

Reflect on mistakes and misconceptions.

Apply properties and operations with complex numbers.

Execute procedures to add and subtract complex numbers.

Use the Fundamental Theorem of Algebra to determine the number and potential types of solutions for polynomial functions.

Rewrite algebraic expressions with integer exponents using the properties of exponents.

Explain how expressions with rational exponents can be rewritten as radical expressions.

Execute procedures to multiply complex numbers.

Rewrite expressions with radicals and rational exponents into equivalent expressions using the properties of exponents.

Apply properties and operations with matrices and vectors.

Interpret expressions that represent a quantity in terms of its context.

Use the properties of rational and irrational numbers to explain why:<ul><li>the sum or product of two rational numbers is rational;</li><li>the sum of a rational number and an irrational number is irrational;</li><li>the product of a nonzero rational number and an irrational number is irrational.</li></ul>

Execute procedures of addition, subtraction, multiplication, and scalar multiplication on matrices.

Identify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including terms, factors, coefficients, and exponents.

Interpret expressions composed of multiple parts by viewing one or more of their parts as a single entity to give meaning in terms of a context.

Execute procedures of addition, subtraction, and scalar multiplication on vectors

Know there is a complex number i such that i² = -1, and every complex number has the form a + bi where a and b are real numbers.

Use the structure of an expression to identify ways to write equivalent expressions.

Write an equivalent form of an exponential expression by using the properties of exponents to transform expressions to reveal rates based on different intervals of the domain.

Understand and apply the Remainder Theorem.

Understand the relationship among factors of a polynomial expression, the solutions of a polynomial equation and the zeros of a polynomial function.

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).

Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.

Add and subtract two rational expressions, a(x) and b(x), where the denominators of both a(x) and b(x) are linear expressions.

Multiply and divide two rational expressions.

Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.

Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.

Create systems of equations and/or inequalities to model situations in context.

Justify a solution method for equations and explain each step of the solving process using mathematical reasoning.

Solve and interpret one variable rational equations arising from a context, and explain how extraneous solutions may be produced.

Extend an understanding that the x-coordinates of the points where the graphs of two equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using a graphing technology or successive approximations with a table of values.

Interpret expressions that represent a quantity in terms of its context.

Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents.

Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression.

Write an equivalent form of a quadratic expression-ax² + bx + c, where a is an integer, by factoring to reveal the solutions of the equation or the zeros of the function the expression defines.

Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions.

Interpret expressions that represent a quantity in terms of its context.

Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function.

Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents.

Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context.

Create equations and inequalities in one variable that represent linear, exponential, and quadratic relationships and use them to solve problems.

Write an equivalent form of a quadratic expression by completing the square, where a is an integer of a quadratic expression, ax² + bx + c, to reveal the maximum or minimum value of the function the expression defines.

Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities.

Extend the understanding that operations with polynomials are comparable to operations with integers by adding, subtracting, and multiplying polynomials.

Create systems of linear equations and inequalities to model situations in context.

Solve for a quantity of interest in formulas used in science and mathematics using the same reasoning as in solving equations.

Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric relationships and use them to solve problems.

Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning.

Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.

Create systems of linear, quadratic, square root, and inverse variation equations to model situations in context.

Solve linear equations and inequalities in one variable.

Solve for the real solutions of quadratic equations in one variable by taking square roots and factoring.

Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning.

Explain why replacing one equation in a system of linear equations by the sum of that equation and a multiple of the other produces a system with the same solutions.

Use tables, graphs, or algebraic methods (substitution and elimination) to find approximate or exact solutions to systems of linear equations and interpret solutions in terms of a context.

Solve and interpret one variable inverse variation and square root equations arising from a context, and explain how extraneous solutions may be produced.

Understand that the graph of a two variable equation represents the set of all solutions to the equation.

Solve for all solutions of quadratic equations in one variable.

Understand that the quadratic formula is the generalization of solving ax² + bx + c by using the process of completing the square.

Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values.

Represent the solutions of a linear inequality or a system of linear inequalities graphically as a region of the plane.

Explain when quadratic equations will have non-real solutions and express complex solutions as a ± bi for real numbers a and b.

Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations, and interpret the solutions in terms of a context.

Extend the understanding that the x-coordinates of the points where the graphs of two square root and/or inverse variation equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values.

Apply properties of function composition to build new functions from existing functions.

Execute algebraic procedures to compose two functions.

Execute a procedure to determine the value of a composite function at a given value when the functions are in algebraic, graphical, or tabular representations.

Apply properties of trigonometry to solve problems.

Translate trigonometric expressions using the reciprocal and Pythagorean identities

Implement the Law of Sines and the Law of Cosines to solve problems.

Interpret key features (amplitude, period, phase shift, vertical shifts, midline, domain, range) of models using sine and cosine functions in terms of a context

Apply the properties and key features of logarithmic functions

Execute properties of logarithms to rewrite expressions and solve equations algebraically

Implement properties of logarithms to solve equations in contextual situations.

Interpret key features of a logarithmic function using multiple representations.

Understand the properties and key features of piecewise functions.

Translate between algebraic and graphical representations of piecewise functions (linear, exponential, quadratic, polynomial, square root)

Construct piecewise functions to model a contextual situation.

Understand how to model functions with regression

Construct regression models of linear, quadratic, exponential, logarithmic, & sinusoidal functions of bivariate data using technology to model data and solve problems

Compare residuals and residual plots of non-linear models to assess the goodness-of-fit of the model

Apply operations with matrices and vectors.

Implement procedures of addition, subtraction, multiplication, and scalar multiplication on matrices.

Implement procedures of addition, subtraction, and scalar multiplication on vectors.

Implement procedures to find the inverse of a matrix.

Understand matrices to solve problems.

Organize data into matrices to solve problems.

Interpret solutions found using matrix operations including Leslie Models and Markov Chains, in context.

Represent a system of equations as a matrix equation.

Use inverse matrices to solve a system of equations with technology.

Understand set theory to solve problems.

Recognize sets, subsets, and proper subsets.

Implement set operations to find unions, intersections, complements and set differences with multiple sets.

Represent properties and relationships among sets using Venn diagrams.

Interpret Venn diagrams to solve problems.

Understand statements related to number theory and set theory.

Use the Euclidean Algorithm to determine greatest common factor and least common multiple.

Use the Fundamental Theorem of Arithmetic to solve problems.

Conclude that sets are equal using the properties of set operations.

Explain theorems related to greatest common factor, least common multiple, even numbers, odd numbers, prime numbers, and composite numbers.

Apply recursively-defined relationships to solve problems.

Implement procedures to find the nth term in an arithmetic or geometric sequence using spreadsheets.

Represent the sum of a sequence using sigma notation.

Implement procedures to find the sum of a finite sequence.

Implement procedures to find the sum of an infinite sequence and determine if the series converges or diverges.

Interpret the solutions to arithmetic and geometric sequences and series problems, in context.

Apply combinatorics concepts to solve problems.

Implement the Fundamental Counting Principle to solve problems.

Implement procedures to calculate a permutation or combination.

Understand graph theory to model relationships and solve problems.

Represent real world situations with a vertex-edge graph, adjacency matrix, and vertex-edge table.

Test graphs and digraphs for Euler paths, Euler circuits, Hamiltonian paths, or Hamiltonian circuits.

Interpret a complete digraph to determine rank.

Apply graph theory to solve problems.

Implement critical path analysis algorithms to determine the minimum project time.

Implement the brute force method, the nearest-neighbor algorithm, and the cheapest-link algorithm to find solutions to a Traveling Salesperson Problem.

Implement vertex-coloring techniques to solve problems.

Implement Kruskal and Prim's algorithms to determine the weight of the minimum spanning tree of a connected graph.

Evaluate mathematical logic to model and solve problems.

Construct truth tables that encode the truth and falsity of two or more statements.

Critique logic arguments (e.g., determine if a statement is valid or whether an argument is a tautology or contradiction).

Check 1s and 0s to determine whether a statement is true or false using Boolean logic circuits.

Judge whether two statements are logically equivalent using truth tables.

Extend the concept of a function by recognizing that trigonometric ratios are functions of angle measure.

Use function notation to evaluate piecewise defined functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Build an understanding 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 by recognizing that:<ul><li>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.</li><li>the graph of f is the graph of the equation y = f(x).</li></ul>

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities.

Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.

Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function.

Extend the concept of a function to include geometric transformations in the plane by recognizing that:<ul><li>the domain and range of a transformation function f are sets of points in the plane;</li><li>the image of a transformation is a function of its pre-image.</li></ul>

Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums.

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.

Write a function that describes a relationship between two quantities.

Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes.

Build polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically.

Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.

Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.

Build a new function, in terms of a context, by combining standard function types using arithmetic operations.

Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior.

Use equivalent expressions to reveal and explain different properties of a function.

Extend an understanding of the effects on the graphical and tabular representations of a function when replacing f(x) with k ∙ f(x), f(x) + k, f(x + k) to include f(k ∙ x) for specific values of k (both positive and negative).

Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Rewrite a quadratic function to reveal and explain different key features of the function

Find an inverse function.

Interpret and explain growth and decay rates for an exponential function.

Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

Understand the inverse relationship between exponential and logarithmic, quadratic and square root, and linear to linear functions and use this relationship to solve problems using tables, graphs, and equations.

Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function f with k ∙ f(x), f(x) + k, f(x + k) for specific values of k (both positive and negative).

Compare key features of two functions (linear, quadratic, or exponential) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Determine if an inverse function exists by analyzing tables, graphs, and equations.

If an inverse function exists for a linear, quadratic and/or exponential function, f, represent the inverse function, f¹, with a table, graph, or equation and use it to solve problems in terms of a contex-

Write a function that describes a relationship between two quantities.

Compare the end behavior of functions using their rates of change over intervals of the same length to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a polynomial function.

Build linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table).

Use logarithms to express the solution to ab^ct = d where a, c, and d are numbers and evaluate the logarithm using technology.

Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication.

Translate between explicit and recursive forms of arithmetic and geometric sequences and use both to model situations.

Understand radian measure of an angle as:<ul><li>The ratio of the length of an arc on a circle subtended by the angle to its radius.</li><li>A dimensionless measure of length defined by the quotient of arc length and radius that is a real number.</li><li>The domain for trigonometric functions.</li></ul>

Identify situations that can be modeled with linear and exponential functions, and justify the most appropriate model for a situation based on the rate of change over equal intervals.

Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions.

Interpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate.

Compare the end behavior of linear, exponential, and quadratic functions using graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

Interpret the cosine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.

Interpret the parameters a and b in a linear function f(x) = ax + b or an exponential function g(x) = ab^x in terms of a context.

Use technology to investigate the parameters, a, b, and h of a sine function, f(x) = a ∙ sin(b ∙ x) + h, to represent periodic phenomena and interpret key features in terms of a context.

Create statistical investigations to make sense of real-world phenomena

Construct statistical questions to guide explorations of data in context.

Design sample surveys and comparative experiments using sampling methods to collect and analyze data to answer a statistical question.

Organize large datasets of real-world contexts (i.e. datasets that include 3 or more measures and have sample sizes >200) using technology (e.g., spreadsheets, dynamic data analysis tools) to determine: types of variables in the data set, possible outcomes for each variable, statistical questions that could be asked of the data, and types of numerical and graphical summaries could be used to make sense of the data.

Interpret non-standard data visualizations from the media or scientific papers to make sense of real-world phenomena.

Apply informal and formal statistical inference to make sense of, and make decisions in, meaningful real-world contexts.

Design a simulation to make a sampling distribution that can be used in making informal statistical inferences.

Construct confidence intervals of population proportions in the context of the data.

Implement a one proportion z-test to determine if an observed proportion is significantly different from a hypothesized proportion.

Apply probability distributions in making decisions in uncertainty.

Implement discrete probability distributions to model random phenomena and make decisions (e.g., expected value of playing a game, etc.).

Implement the binomial distribution to model situations and make decisions

Recognize from simulations of sampling distributions of sample means and proportions that a normal distribution can be used as an approximate model in certain situations.

Implement the normal distribution as a probability distribution to determine the likelihood of events occurring.

Use technology to represent data with plots on the real number line (histograms, and box plots).

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. Interpret differences in shape, center, and spread in the context of the data sets.

Examine the effects of extreme data points (outliers) on shape, center, and/or spread.

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

Fit a least squares regression line to linear data using technology. Use the fitted function to solve problems.

Assess the fit of a linear function by analyzing residuals.

Fit a function to exponential data using technology. Use the fitted function to solve problems.

Interpret in context the rate of change and the intercept of a linear model. Use the linear model to interpolate and extrapolate predicted values. Assess the validity of a predicted value.

Analyze patterns and describe relationships between two variables in context. Using technology, determine the correlation coefficient of bivariate data and interpret it as a measure of the strength and direction of a linear relationship. Use a scatter plot, correlation coefficient, and a residual plot to determine the appropriateness of using a linear function to model a relationship between two variables.

Distinguish between association and causation.

Understand the process of making inferences about a population based on a random sample from that population.

Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on known information about the population.

Recognize the purposes of and differences between sample surveys, experiments, and observational studies and understand how randomization should be used in each.

Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of other events.

Use simulation to understand how samples can be used to estimate a population mean or proportion and how to determine a margin of error for the estimate.

Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.

Develop and understand independence and conditional probability.

Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B's outcomes that also belong to event A.

Evaluate articles and websites that report data by identifying the source of the data, the design of the study, and the way the data are graphically displayed.

Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B. That is P(A|B)=P(A).

Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations

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 context.

Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in context.

Apply the general Multiplication Rule P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in context. Include the case where A and B are independent: P(A and B) = P(A) P(B).

Experiment with transformations in the plane.<ul><li>Represent transformations in the plane.</li><li>Compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations).</li><li>Understand that rigid motions produce congruent figures while dilations produce similar figures.</li></ul>

Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry.

Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image.

Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other.

Use the properties 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.

Use congruence in terms of rigid motion. Justify the ASA, SAS, and SSS criteria for triangle congruence. Use criteria for triangle congruence (ASA, SAS, SSS, HL) to determine whether two triangles are congruent.

Prove theorems about lines and angles and use them to prove relationships in geometric figures including:<ul><li>Vertical angles are congruent.</li><li>When a transversal crosses parallel lines, alternate interior angles are congruent.</li><li>When a transversal crosses parallel lines, corresponding angles are congruent.</li><li>Points are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment.</li><li>Use congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle.</li></ul>

Prove theorems about triangles and use them to prove relationships in geometric figures including:<ul><li>The sum of the measures of the interior angles of a triangle is 180º.</li><li>An exterior angle of a triangle is equal to the sum of its remote interior angles.</li><li>The base angles of an isosceles triangle are congruent.</li><li>The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.</li></ul>

Use coordinates to solve geometric problems involving polygons algebraically<ul><li>Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.</li><li>Use coordinates to verify algebraically that a given set of points produces a particular type of triangle or quadrilateral.</li></ul>

Verify experimentally the properties of dilations with given center and scale factor:

Verify experimentally properties of the centers of triangles (centroid, incenter, and circumcenter).

Use coordinates to prove the slope criteria for parallel and perpendicular lines and use them to solve problems.<ul><li>Determine if two lines are parallel, perpendicular, or neither.</li><li>Find the equation of a line parallel or perpendicular to a given line that passes through a given point.</li></ul>

Use coordinates to find the midpoint or endpoint of a line segment.

Prove theorems about parallelograms.<ul><li>Opposite sides of a parallelogram are congruent.</li><li>Opposite angles of a parallelogram are congruent.</li><li>Diagonals of a parallelogram bisect each other.</li><li>If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.</li></ul>

When a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not pass through the center of dilation, the line segment and its image are parallel.

The length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor.

Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems.

The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image.

Understand and apply theorems about circles.<ul><li>Understand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles.</li><li>Understand and apply theorems about relationships with line segments and circles including, radii, diameter, secants, tangents and chords.</li></ul>

Dilations preserve angle measure.

Using similarity, demonstrate that the length of an arc, s, for a given central angle is proportional to the radius, r, of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the circle, s/r. Find arc lengths and areas of sectors of circles.

Understand similarity in terms of transformations.

Determine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other.

Use the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent.

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.

Use transformations (rigid motions and dilations) to justify the AA criterion for triangle similarity.

Use similarity to solve problems and to prove theorems about triangles. Use theorems about triangles to prove relationships in geometric figures.<ul><li>A line parallel to one side of a triangle divides the other two sides proportionally and its converse.</li><li>The Pythagorean Theorem</li></ul>

Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.

Verify experimentally that the side ratios in similar right triangles are properties of the angle measures in the triangle, due to the preservation of angle measure in similarity. Use this discovery to develop definitions of the trigonometric ratios for acute angles.

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

Use trigonometric ratios and the Pythagorean Theorem to solve problems involving right triangles in terms of a context.

Apply geometric concepts in modeling situations <ul><li>Use geometric and algebraic concepts to solve problems in modeling situations:</li> <li>Use geometric shapes, their measures, and their properties, to model real-life objects.</li> <li>Use geometric formulas and algebraic functions to model relationships.</li> <li>Apply concepts of density based on area and volume.</li> <li>Apply geometric concepts to solve design and optimization problems.</li></ul>

Develop properties of special right triangles (45-45-90 and 30-60-90) and use them to solve problems.