What does a student learn in StateVirginia,GradeGrade 9,SubjectMathematics?

The student will determine the sine, cosine, tangent, cotangent, secant, and cosecant of the acute angles in a right triangle and use these ratios to solve for missing sides and angle measures, including application in contextual problems.

Define and represent the six triangular trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in a right triangle.

Describe the relationships between side lengths in special right triangles (30°-60°-90° and 45°-45°-90°).

Use the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines to solve contextual problems.

Represent and solve contextual problems involving right triangles, including problems involving angles of elevation and depression.

The student will find the area of any triangle and solve for the lengths of the sides and measures of the angles in a non-right triangle using the Law of Sines and the Law of Cosines.

Apply the Law of Sines, and the Law of Cosines, as appropriate, to find missing sides and angles in non-right triangles.

Recognize the ambiguous case when applying the Law of Sines and the potential for two triangle solutions in some situations.

Solve problems that integrate the use of the Law of Sines and the Law of Cosines and the triangle area formula (Area = 1 2 absinC, where a and b are triangle sides and C is the included angle) to find the area of any triangle, including those in contextual problems.

The student will represent data and convert data between different number systems.

Represent data in different number systems, including binary, decimal, and hexadecimal.

Convert data between number systems (e.g., binary to decimal, decimal to hexadecimal).

The student will differentiate between variable data types based upon their characteristics.

Describe the characteristics of different variable data types, including

Boolean;

character;

integer;

decimal (double/float); and

string.

Differentiate between variable data types to determine the data type needed based upon intended use (e.g., character versus string, integer versus double/float).

The student will represent data using appropriate data structures.

Given a specific task or problem, determine the appropriate data structure (e.g., lists, arrays, objects) to represent data.

Perform tasks related to lists or arrays (one-dimensional or two-dimensional), including

declare a list or array (one-dimensional or two-dimensional);

choose an appropriate data type for a list or an array; and

fill the list or array with data.

Access and manipulate a particular element of a list or an array.

Implement predefined objects to consolidate related information of different data types.

The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.

Translate propositional statements and compound statements into symbolic form, including negations (~𝑝, read “not p”), conjunctions (p ∧ 𝑞, read “p and q”), disjunctions (p ∨ 𝑞, read “p or q”), conditionals (p → q, read “if p then q”), and biconditionals (p ↔ q, read “p if and only if q”), including statements representing geometric relationships.

Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships.

Use Venn diagrams to represent set relationships, including union, intersection, subset, and negation.

Interpret Venn diagrams, including those representing contextual situations.

The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal.

Prove and justify angle pair relationships formed by two parallel lines and a transversal, including:

corresponding angles;

alternate interior angles;

alternate exterior angles;

same-side (consecutive) interior angles; and

same-side (consecutive) exterior angles.

Prove two or more lines are parallel given angle measurements expressed numerically or algebraically.

Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal.

The student will solve problems, including contextual problems, involving symmetry and transformation.

Locate, count, and draw lines of symmetry given a figure, including figures in context.

Determine whether a figure has point symmetry, line symmetry, both, or neither, including figures in context.

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include:

translations;

reflections over any horizontal or vertical line or the lines y = x or y = -x;

clockwise or counterclockwise rotations of 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation is limited to the origin; and

dilations, from a fixed point on a coordinate grid.

The student will perform operations on and simplify rational expressions.

The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Translate between verbal quantitative situations and algebraic expressions, including contextual situations.

Add, subtract, multiply, or divide rational algebraic expressions, simplifying the result.

Evaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator.

Justify and determine equivalent rational algebraic expressions with monomial and binomial factors. Algebraic expressions should be limited to linear and quadratic expressions.

The student will perform operations on and factor polynomial expressions in one variable.

Recognize a complex algebraic fraction and simplify it as a product or quotient of simple algebraic fractions.

Represent and demonstrate equivalence of rational expressions written in different forms.

The student will perform operations on and simplify radical expressions.

Determine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models.

Determine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to five or fewer terms (e.g., (4x + 2)(3x + 5) represents four terms and (x + 1)(2x 2 + x + 3) represents five terms).

Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.

Simplify and determine equivalent radical expressions that include numeric and algebraic radicands.

Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

Add, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator.

Represent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical).

Convert between radical expressions and expressions containing rational exponents.

The student will perform operations on polynomial expressions and factor polynomial expressions in one and two variables.

The student will derive and apply the laws of exponents.

Determine sums, differences, and products of polynomials in one and two variables.

Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

Factor polynomials completely in one and two variables with no more than four terms over the set of integers.

Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

Determine the quotient of polynomials in one and two variables, using monomial, binomial, and factorable trinomial divisors.

The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

Represent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.

Simplify and determine equivalent radical expressions involving the square root of a whole number in simplest form.

The student will perform operations on complex numbers.

Simplify and determine equivalent radical expressions involving the cube root of an integer.

Explain the meaning of i.

Add, subtract, and multiply radicals, limited to numeric square and cube root expressions.

Identify equivalent radical expressions containing negative rational numbers and expressions in a + bi form.

Generate equivalent numerical expressions and justify their equivalency for radicals using rational exponents, limited to rational exponents of 1 2 and 1 3 (e.g., √5 = 5 1 2; √8 3 = 8 1 3 = (2 3 ) 1 3 = 2).

Apply properties to add, subtract, and multiply complex numbers.

The student will investigate, analyze, and compare linear, quadratic, and exponential function families, algebraically and graphically, using transformations.

Identify graphs and equations of parent functions for linear, quadratic, and exponential function families.

Describe the transformation from the parent function given the equation or the graph of the function.

Determine and analyze whether a linear, quadratic, or exponential function best models a given representation, including those in context.

Write the equation of a linear, quadratic, or exponential function, given a graph, using transformations of the parent function.

Use a graphical or algebraic representation of a function to solve problems within a context, graphically and algebraically, when appropriate.

Graph a function given the equation of a function, using transformations of the parent function. Use technology to verify transformations of functions.

Compare and contrast linear, quadratic, and exponential functions using multiple representations (e.g., graphs, tables, equations, verbal descriptions).

The student will investigate and analyze characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions.

Determine the domain and range of a function given a graphical representation, including those limited by contexts.

Identify intervals on a graph for which a function is increasing, decreasing, or constant.

Given a graph, identify the location and value of the absolute maximum and absolute minimum of a function over the domain of a function.

Given a graph, determine the zeros and intercepts of a function.

Describe and recognize the connection between points on the graph and the value of a function.

Describe the end behavior of a function given its graph.

Identify horizontal and/or vertical asymptotes from the graph of a function, if they exist.

Describe and relate the characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions, including those in contextual situations.

The student will represent and interpret contextual situations with constraints that require optimization using linear programming techniques, including systems of linear equations or inequalities, solving graphically and when appropriate, algebraically.

Represent and interpret contextual problems requiring optimization with systems of linear equations or inequalities.

Solve systems of no more than four equations or inequalities graphically and when appropriate, algebraically.

Identify the feasible region of a system of linear inequalities.

Identify the coordinates of the vertices of a feasible region.

Determine and describe the maximum or minimum value for the function defined over a feasible region.

Interpret the validity of possible solution(s) algebraically, graphically, using technology, and in context and justify the reasonableness of the answer(s) or the solution method in context.

The student will use reasoning to develop and apply logical arguments.

Use Venn diagrams to codify and solve logic problems.

Express logical statements in symbolic form.

Represent a conditional statement as its converse, inverse, and contrapositive.

Describe how symbolic logic can be used to map the processes of computer applications.

Construct a truth table to display all possible input combinations and their outputs.

Identify the rules of inference and model basic logical statements including De Morgan’s Law.

Apply logical reasoning to model contextual situations and make decisions.

The student will apply logic and proof techniques in the construction of a sound argument.

Apply informal logical reasoning to contextual problems (e.g., predicting the behavior of software, solving puzzles).

Outline the basic structure of a proof technique (e.g., direct proof, proof by contradiction, induction).

Deduce the best type of proof for a given problem.

Use the rules of inference to construct direct proofs and proofs by contradiction.

Construct induction proofs involving summations and inequalities.

Use a truth table to prove the logical equivalence of statements.

The student will apply Boolean algebra to represent and analyze the function of logical gates and circuits.

Explain basic properties of Boolean algebra: duality, complements, and standard forms.

Represent verbal statements as Boolean expressions.

Apply Boolean algebra to prove identities and simplify expressions.

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

Explain the operation of discrete logic gates.

Describe the relationship between Boolean algebra and electronic circuits.

Analyze a combinational network using Boolean expressions.

Design simple combinational networks that use NAND (AND followed by NOT), NOR (OR followed by NOT), and XOR (exclusive-OR) gates.

The student will use mathematical induction to prove formulas and mathematical statements.

Compare and contrast inductive and deductive reasoning.

Explain the relationship between weak and strong induction.

Construct induction proofs involving a divisibility argument.

Prove the Binomial Theorem through mathematical induction.

The student will use a statistical cycle to formulate questions, describe types of data, data sources, and constraints within the context of a problem.

Define the stages of the statistical cycle and how each stage relates to the others.

Identify and explain characteristics that best lend themselves to a data driven approach to problem solving.

Formulate questions based on context.

Formulate questions and conclusions based on context.

Understand the type of data relevant to the question at hand (e.g., quantitative versus categorical).

Understand the type of data relevant to the context of the question at hand.

Compare and contrast population and sample, and parameter and statistic.

Define relationships between variables and constant relationships.

Identify and explain constraints of the statistical approach.

Create a hypothesis of interest in terms of measurable data.

The student will compare and contrast data collection methods to plan and conduct an observational study.

Define the stages of the data cycle and how each stage is related to the other.

Identify and explain constraints of the data-driven approach.

Investigate and describe sampling techniques (e.g., simple random sampling, stratified sampling, systematic sampling, cluster sampling).

Determine which sampling technique is best, given a particular context.

Design a data project plan, which is aligned with the data science cycle, that includes the following components:

Investigate and explain biased influences inherent within sampling methods and various forms of response bias.

definition of the goal of the project as it pertains to a real-world problem;

Use the statistical cycle to plan and conduct an observational study to answer a question or address a problem.

identification of the various parameters of the problem and stakeholders;

The student will utilize the principles of experimental design to plan and conduct a well-designed experiment.

a timeline for the project with deliverables;

Describe the principles of experimental design, including:

Key Performance Indicators (KPI) for the successful data project deliverables;

resource needs and tools for the project;

treatment/control groups;

blinding/placebo effects;

bias considerations for the sampling process of the project; and

limitations of the project.

experimental units/subjects; and

blocking/matched pairs and completely randomized designs.

Given the context and parameters of a problem, choose from among various sampling techniques, which may include

simple random;

Evaluate the principles of experimental design to address comparison, randomization, replication, and control within the context of the problem.

systematic;

Compare and contrast controlled experiments and observational studies and the conclusions that may be drawn from each.

Use the statistical cycle to plan and conduct a well-designed experiment to answer a question or address a problem.

stratified; and

Select a data collection method appropriate for a given context.

cluster;

identify specific examples of real-world problems that can be effectively addressed using data science.

formulate a top down plan for data collection and analysis, with quantifiable results, based on the context of a problem.

The student will identify and analyze the properties of polynomial, rational, piecewise-defined, absolute value, radical, and step functions and sketch the graphs of the functions.

Use mathematical reasoning to identify polynomial, rational, piecewise-defined, absolute value, radical, and step functions, given an equation or graph.

Given multiple representations of a polynomial, rational, piecewise-defined, absolute value, radical, and step function, analyze:

domain and range;

roots (including complex roots);

intercepts;

symmetry (including even and odd functions);

asymptotes (horizontal, vertical, and oblique/slant;

points of discontinuity;

intervals for which the function is increasing, decreasing or constant;

end behavior; and

relative and/or absolute maximum and minimum points.

Sketch the graph of a polynomial, rational, piecewise-defined, absolute value, radical, and step function.

The student will determine the limit of a function if it exists.

Verify estimates about the limit of a function using graphing technology.

Determine the limit of a function algebraically and verify with graphing technology.

Determine the limit of a function numerically and verify with graphing technology.

Use proper limit notation, including when describing the end behavior of a function.

As the variable approaches a finite number,

determine the limit of a function numerically by direct substitution;

determine the limit of a function using algebraic manipulation;

estimate the limit of a function using a table; and

determine the limit of a function from a given graph.

As the variable approaches positive or negative infinity, analyze the limit of a function to describe the end behavior.

The student will analyze and describe the continuity of functions.

Describe continuity of a function.

Use mathematical notation to communicate and describe the continuity of functions including polynomial, rational, piecewise, absolute value, radical, and step function, using graphical and algebraic methods.

Prove continuity at a point, using the definition.

Classify types of discontinuity based on which condition of continuity is violated.

The student will determine the degree and radian measure of angles; sketch angles in standard position on a coordinate plane; and determine the sine, cosine, tangent, cosecant, secant, and cotangent of an angle, given a point on the terminal side of an angle in standard position or the value of a trigonometric function of the angle.

Define a radian as a unit of angle measure and determine the relationship between the radian measure of an angle and the length of the intercepted arc in a circle.

Determine the degree and radian measure of angles to include both negative and positive rotations in the coordinate plane.

Find both positive and negative coterminal angles for a given angle.

Identify the quadrant or axis in/on which the terminal side of an angle lies.

Draw a reference right triangle when given a point on the terminal side of an angle in standard position.

Draw a reference right triangle when given the value of a trigonometric function of an angle (sine, cosine, tangent, cosecant, secant, and cotangent).

Determine the value of any trigonometric function (sine, cosine, tangent, cosecant, secant, and cotangent) when given a point on the terminal side of an angle in standard position.

Given one trigonometric function value, determine the other five trigonometric function values.

Calculate the length of an arc of a circle in radians.

Calculate the area of a sector of a circle.

The student will develop and apply the properties of the unit circle in degrees and radians.

Convert between radian and degree measure of special angles of the unit circle without the use of technology.

Define the six circular trigonometric functions of an angle in standard position on the unit circle.

Apply knowledge of right triangle trigonometry, special right triangles, and the properties of the unit circle to determine trigonometric functions values of special angles (0°, 30°, 45°, 60°, and 90°) and their related angles in degree and radians without the use of technology.

The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

Write a linear equation or inequality in one variable to represent a contextual situation.

Create an absolute value equation in one variable to model a contextual situation.

Solve multistep linear equations in one variable, including those in contextual situations, by applying the properties of real numbers and/or properties of equality.

Solve an absolute value equation in one variable algebraically and verify the solution graphically.

Solve multistep linear inequalities in one variable algebraically and graph the solution set on a number line, including those in contextual situations, by applying the properties of real numbers and/or properties of inequality.

Rearrange a formula or literal equation to solve for a specified variable by applying the properties of equality.

Create an absolute value inequality in one variable to model a contextual situation.

Determine if a linear equation in one variable has one solution, no solution, or an infinite number of solutions.

Solve an absolute value inequality in one variable and represent the solution set using set notation, interval notation, and using a number line.

Verify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

Verify possible solution(s) to absolute value equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables.

The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.

Create a quadratic equation or inequality in one variable to model a contextual situation.

Create a system of two linear equations in two variables to represent a contextual situation.

Solve a quadratic equation in one variable over the set of complex numbers algebraically.

Apply the properties of real numbers and/or properties of equality to solve a system of two linear equations in two variables, algebraically and graphically.

Determine whether a system of two linear equations has one solution, no solution, or an infinite number of solutions.

Determine the solution to a quadratic inequality in one variable over the set of real numbers algebraically.

Verify possible solution(s) to quadratic equations or inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Create a linear inequality in two variables to represent a contextual situation.

The student will solve a system of equations in two variables containing a quadratic expression.

Represent the solution of a linear inequality in two variables graphically on a coordinate plane.

Create a system of two linear inequalities in two variables to represent a contextual situation.

Represent the solution set of a system of two linear inequalities in two variables, graphically on a coordinate plane.

Create a linear-quadratic or quadratic-quadratic system of equations to model a contextual situation.

Verify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

Determine the number of solutions to a linear-quadratic and quadratic-quadratic system of equations in two variables.

The student will represent, solve, and interpret the solution to a quadratic equation in one variable.

Solve a linear-quadratic and quadratic-quadratic system of equations algebraically and graphically, including situations in context.

Verify possible solution(s) to linear-quadratic or quadratic-quadratic system of equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems.

The student will represent, solve, and interpret the solution to an equation containing rational algebraic expressions.

Determine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions.

Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Create an equation containing a rational expression to model a contextual situation.

Solve rational equations with real solutions containing factorable algebraic expressions algebraically and graphically. Algebraic expressions should be limited to linear and quadratic expressions.