New York 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.
NY.3. Integrated Algebra: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
A.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
A.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
A.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A.PS.6. Use a variety of strategies to extend solution methods to other problems.
A.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
A.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A.PS.9. Interpret solutions within the given constraints of a problem.
A.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
A.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
A.RP.2. Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture.
A.RP.3. Recognize when an approximation is more appropriate than an exact answer.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
A.RP.4. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A.RP.5. Construct logical arguments that verify claims or counterexamples that refute them.
A.RP.6. Present correct mathematical arguments in a variety of forms.
A.RP.7. Evaluate written arguments for validity.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
A.RP.8. Support an argument by using a systematic approach to test more than one case.
A.RP.9. Devise ways to verify results or use counterexamples to refute incorrect statements.
A.RP.10. Extend specific results to more general cases.
A.RP.11. Use a Venn diagram to support a logical argument.
A.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
A.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
A.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A.CM.4. Explain relationships among different representations of a problem.
A.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
A.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A.CM.8. Reflect on strategies of others in relation to one's own strategy.
A.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
A.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A.CM.11. Represent word problems using standard mathematical notation.
A.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
A.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
A.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A.CN.5. Understand how quantitative models connect to various physical models and representations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
A.CN.6. Recognize and apply mathematics to situations in the outside world.
A.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A.CN.8. Develop an appreciation for the historical development of mathematics.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
A.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A.R.2. Recognize, compare, and use an array of representational forms.
A.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
A.R.4. Select appropriate representations to solve problem situations.
A.R.5. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
A.R.6. Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground).
A.R.7. Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales).
A.R.8. Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x to the power of 2 and y = -x to the power of 2).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
A.N.1. Number Theory: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2. Operations: Simplify radical terms (no variable in the radicand).
A.N.3. Operations: Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form.
A.N.4. Operations: Understand and use scientific notation to compute products and quotients of numbers.
A.N.5. Operations: Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation.
A.N.6. Operations: Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s).
A.N.7. Operations: Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting.
A.N.8. Operations: Determine the number of possible arrangements (permutations) of a list of items.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
A.A.1. Variables and Expressions: Translate a quantitative verbal phrase into an algebraic expression.
A.A.2. Variables and Expressions: Write a verbal expression that matches a given mathematical expression.
A.A.3. Equations and Inequalities: Distinguish the difference between an algebraic expression and an algebraic equation.
A.A.4. Equations and Inequalities: Translate verbal sentences into mathematical equations or inequalities.
A.A.5. Equations and Inequalities: Write algebraic equations or inequalities that represent a situation.
A.A.6. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable.
A.A.7. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables.
A.A.8. Equations and Inequalities: Analyze and solve verbal problems that involve quadratic equations.
A.A.9. Equations and Inequalities: Analyze and solve verbal problems that involve exponential growth and decay.
A.A.10. Equations and Inequalities: Solve systems of two linear equations in two variables algebraically (See A.G.7).
A.A.11. Equations and Inequalities: Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
A.A.12. Variables and Expressions: Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only.
A.A.13. Variables and Expressions: Add, subtract, and multiply monomials and polynomials.
A.A.14. Variables and Expressions: Divide a polynomial by a monomial or binomial, where the quotient has no remainder.
A.A.15. Variables and Expressions: Find values of a variable for which an algebraic fraction is undefined.
A.A.16. Variables and Expressions: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms.
A.A.17. Variables and Expressions: Add or subtract fractional expressions with monomial or like binomial denominators.
A.A.18. Variables and Expressions: Multiply and divide algebraic fractions and express the product or quotient in simplest form.
A.A.19. Variables and Expressions: Identify and factor the difference of two perfect squares.
A.A.20. Variables and Expressions: Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF).
A.A.21. Equations and Inequalities: Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.
A.A.22. Equations and Inequalities: Solve all types of linear equations in one variable.
A.A.23. Equations and Inequalities: Solve literal equations for a given variable.
A.A.24. Equations and Inequalities: Solve linear inequalities in one variable.
A.A.25. Equations and Inequalities: Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26. Equations and Inequalities: Solve algebraic proportions in one variable which result in linear or quadratic equations.
A.A.27. Equations and Inequalities: Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots.
A.A.28. Equations and Inequalities: Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
A.A.29. Patterns, Relations, and Functions: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form.
A.A.30. Patterns, Relations, and Functions: Find the complement of a subset of a given set, within a given universe.
A.A.31. Patterns, Relations, and Functions: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets).
A.A.32. Coordinate Geometry: Explain slope as a rate of change between dependent and independent variables.
A.A.33. Coordinate Geometry: Determine the slope of a line, given the coordinates of two points on the line.
A.A.34. Coordinate Geometry: Write the equation of a line, given its slope and the coordinates of a point on the line.
A.A.35. Coordinate Geometry: Write the equation of a line, given the coordinates of two points on the line.
A.A.36. Coordinate Geometry: Write the equation of a line parallel to the x- or y-axis.
A.A.37. Coordinate Geometry: Determine the slope of a line, given its equation in any form.
A.A.38. Coordinate Geometry: Determine if two lines are parallel, given their equations in any form.
A.A.39. Coordinate Geometry: Determine whether a given point is on a line, given the equation of the line.
A.A.40. Coordinate Geometry: Determine whether a given point is in the solution set of a system of linear inequalities.
A.A.41. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ).
A.A.42. Trigonometric Functions: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides.
A.A.43. Trigonometric Functions: Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle.
A.A.44. Trigonometric Functions: Find the measure of a side of a right triangle, given an acute angle and the length of another side.
A.A.45. Trigonometric Functions: Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
A.G.1. Shapes: Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
A.G.2. Shapes: Use formulas to calculate volume and surface area of rectangular solids and cylinders.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
A.G.3. Coordinate Geometry: Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations.
A.G.4. Coordinate Geometry: Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions.
A.G.5. Coordinate Geometry: Investigate and generalize how changing the coefficients of a function affects its graph.
A.G.6. Coordinate Geometry: Graph linear inequalities.
A.G.7. Coordinate Geometry: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10).
A.G.8. Coordinate Geometry: Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9. Coordinate Geometry: Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
A.G.10. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41 ) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
A.M.1. Units of Measurement: Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail).
A.M.2. Units of Measurement: Solve problems involving conversions within measurement systems, given the relationship between the units.
3.27. Measurement Strand: Understand that all measurement contains error and be able to determine its significance.
A.M.3. Error and Magnitude: Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
A.S.1. Organization and Display of Data: Categorize data as qualitative or quantitative.
A.S.2. Organization and Display of Data: Determine whether the data to be analyzed is univariate or bivariate.
A.S.3. Organization and Display of Data: Determine when collected data or display of data may be biased.
A.S.4. Organization and Display of Data: Compare and contrast the appropriateness of different measures of central tendency for a given data set.
A.S.5. Organization and Display of Data: Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data.
A.S.6. Organization and Display of Data: Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box and- whisker plot.
A.S.7. Organization and Display of Data: Create a scatter plot of bivariate data.
A.S.8. Organization and Display of Data: Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line.
A.S.9. Analysis of Data: Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot.
A.S.10. Analysis of Data: Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions.
A.S.11. Analysis of Data: Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles.
A.S.12. Analysis of Data: Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none).
A.S.13. Analysis of Data: Understand the difference between correlation and causation.
A.S.14. Analysis of Data: Identify variables that might have a correlation but not a causal relationship.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
A.S.15. Predictions from Data: Identify and describe sources of bias and its effect, drawing conclusions from data.
A.S.16. Predictions from Data: Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range.
A.S.17. Predictions from Data: Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
A.S.18. Probability: Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
A.S.19. Probability: Determine the number of elements in a sample space and the number of favorable events.
A.S.20. Probability: Calculate the probability of an event and its complement.
A.S.21. Probability: Determine empirical probabilities based on specific sample data.
A.S.22. Probability: Determine, based on calculated probability of a set of events, if some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen.
A.S.23. Probability: Calculate the probability of a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive.
G.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
G.PS.2. Observe and explain patterns to formulate generalizations and conjectures.
G.PS.3. Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations).
G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.
G.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
G.PS.6. Use a variety of strategies to extend solution methods to other problems.
G.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
G.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
G.PS.9. Interpret solutions within the given constraints of a problem.
G.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
G.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
G.RP.2. Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies.
G.RP.3. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
G.RP.4. Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments.
G.RP.5. Present correct mathematical arguments in a variety of forms.
G.RP.6. Evaluate written arguments for validity.
G.RP.7. Construct a proof using a variety of methods (e.g., deductive, analytic, transformational).
G.RP.8. Devise ways to verify results or use counterexamples to refute incorrect statements.
G.RP.9. Apply inductive reasoning in making and supporting mathematical conjectures.
G.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
G.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
G.CM.4. Explain relationships among different representations of a problem.
G.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
G.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
G.CM.8. Reflect on strategies of others in relation to one's own strategy.
G.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
G.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
G.CM.11. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
G.CM.12. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
G.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
G.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
G.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
G.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
G.CN.5. Understand how quantitative models connect to various physical models and representations.
G.CN.6. Recognize and apply mathematics to situations in the outside world.
G.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
G.CN.8. Develop an appreciation for the historical development of mathematics.
G.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
G.R.2. Recognize, compare, and use an array of representational forms.
G.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
G.R.4. Select appropriate representations to solve problem situations.
G.R.5. Investigate relationships between different representations and their impact on a given problem.
G.R.6. Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank).
G.R.7. Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation).
G.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent).
G.G.1. Geometric Relationships: Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
G.G.2. Geometric Relationships: Know and apply that through a given point there passes one and only one plane perpendicular to a given line.
G.G.3. Geometric Relationships: Know and apply that through a given point there passes one and only one line perpendicular to a given plane.
G.G.4. Geometric Relationships: Know and apply that two lines perpendicular to the same plane are coplanar.
G.G.5. Geometric Relationships: Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
G.G.6. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
G.G.7. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
G.G.8. Geometric Relationships: Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines.
G.G.9. Geometric Relationships: Know and apply that if two planes are perpendicular to the same line, they are parallel.
G.G.10. Geometric Relationships: Know and apply that the lateral edges of a prism are congruent and parallel.
G.G.11. Geometric Relationships: Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
G.G.12. Geometric Relationships: Know and apply that the volume of a prism is the product of the area of the base and the altitude.
G.G.13. Geometric Relationships: Apply the properties of a regular pyramid, including lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude.
G.G.14. Geometric Relationships: Apply the properties of a cylinder, including bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base.
G.G.15. Geometric Relationships: Apply the properties of a right circular cone, including lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude.
G.G.16. Geometric Relationships: Apply the properties of a sphere, including the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; surface area is 4 pi r to the power of 2; volume is 4/3 pi r to the power of 3.
G.G.17. Constructions: Construct a bisector of a given angle, using a straightedge and compass, and justify the construction.
G.G.18. Constructions: Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction.
G.G.19. Constructions: Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction.
G.G.20. Constructions: Construct an equilateral triangle, using a straightedge and compass, and justify the construction.
G.G.21. Locus: Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles.
G.G.22. Locus: Solve problems using compound loci.
G.G.23. Locus: Graph and solve compound loci in the coordinate plane.
G.G.24. Informal and Formal Proofs: Determine the negation of a statement and establish its truth value.
G.G.25. Informal and Formal Proofs: Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true.
G.G.26. Informal and Formal Proofs: Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences.
G.G.27. Informal and Formal Proofs: Write a proof arguing from a given hypothesis to a given conclusion.
G.G.28. Informal and Formal Proofs: Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles.
G.G.29. Informal and Formal Proofs: Identify corresponding parts of congruent triangles.
G.G.30. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle.
G.G.31. Informal and Formal Proofs: Investigate, justify, and apply the isosceles triangle theorem and its converse.
G.G.32. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem.
G.G.33. Informal and Formal Proofs: Investigate, justify, and apply the triangle inequality theorem.
G.G.34. Informal and Formal Proofs: Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle.
G.G.35. Informal and Formal Proofs: Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines.
G.G.36. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons.
G.G.37. Informal and Formal Proofs: Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons.
G.G.38. Informal and Formal Proofs: Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals.
G.G.39. Informal and Formal Proofs: Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals.
G.G.40. Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals.
G.G.41. Informal and Formal Proofs: Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids.
G.G.42. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle.
G.G.43. Informal and Formal Proofs: Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1.
G.G.44. Informal and Formal Proofs: Establish similarity of triangles, using the following theorems: AA, SAS, and SSS.
G.G.45. Informal and Formal Proofs: Investigate, justify, and apply theorems about similar triangles.
G.G.46. Informal and Formal Proofs: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.
G.G.47. Informal and Formal Proofs: Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg.
G.G.48. Informal and Formal Proofs: Investigate, justify, and apply the Pythagorean theorem and its converse.
G.G.49. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle.
G.G.50. Informal and Formal Proofs: Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles.
G.G.51. Informal and Formal Proofs: Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two secants, or tangent and secant).
G.G.52. Informal and Formal Proofs: Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines.
G.G.53. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two intersecting chords of a given circle.
G.G.54. Transformational Geometry: Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
G.G.55. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections.
G.G.56. Transformational Geometry: Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism.
G.G.57. Transformational Geometry: Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections).
G.G.58. Transformational Geometry: Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries).
G.G.59. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under similarities.
G.G.60. Transformational Geometry: Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism.
G.G.61. Transformational Geometry: Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin.
G.G.62. Coordinate Geometry: Find the slope of a perpendicular line, given the equation of a line.
G.G.63. Coordinate Geometry: Determine whether two lines are parallel, perpendicular, or neither, given their equations.
G.G.64. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
G.G.65. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line.
G.G.66. Coordinate Geometry: Find the midpoint of a line segment, given its endpoints.
G.G.67. Coordinate Geometry: Find the length of a line segment, given its endpoints.
G.G.68. Coordinate Geometry: Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment.
G.G.69. Coordinate Geometry: Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas.
G.G.70. Coordinate Geometry: Solve systems of equations involving one linear equation and one quadratic equation graphically.
G.G.71. Coordinate Geometry: Write the equation of a circle, given its center and radius or given the endpoints of a diameter.
G.G.72. Coordinate Geometry: Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
G.G.73. Coordinate Geometry: Find the center and radius of a circle, given the equation of the circle in center-radius form.
G.G.74. Coordinate Geometry: Graph circles of the form (x - h) to the power of 2 + (j - k) to the power of 2 = r to the power of 2.
A2.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A2.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
A2.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
A2.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A2.PS.6. Use a variety of strategies to extend solution methods to other problems.
A2.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
A2.PS.8. Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A2.PS.9. Interpret solutions within the given constraints of a problem.
A2.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
A2.RP.1. Support mathematical ideas using a variety of strategies.
A2.RP.2. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
A2.RP.3. Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer.
A2.RP.4. Recognize when an approximation is more appropriate than an exact answer.
A2.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A2.RP.6. Construct logical arguments that verify claims or counterexamples that refute claims.
A2.RP.7. Present correct mathematical arguments in a variety of forms.
A2.RP.8. Evaluate written arguments for validity.
A2.RP.9. Support an argument by using a systematic approach to test more than one case.
A2.RP.10. Devise ways to verify results, using counterexamples and informal indirect proof.
A2.RP.11. Extend specific results to more general cases.
A2.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
A2.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A2.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A2.CM.4. Explain relationships among different representations of a problem.
A2.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A2.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
A2.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A2.CM.8. Reflect on strategies of others in relation to one's own strategy.
A2.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
A2.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A2.CM.11. Represent word problems using standard mathematical notation.
A2.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A2.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
A2.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A2.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
A2.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A2.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A2.CN.5. Understand how quantitative models connect to various physical models and representations.
A2.CN.6. Recognize and apply mathematics to situations in the outside world.
A2.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A2.CN.8. Develop an appreciation for the historical development of mathematics.
A2.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A2.R.2. Recognize, compare, and use an array of representational forms.
A2.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.4. Select appropriate representations to solve problem situations.
A2.R.5. Investigate relationships among different representations and their impact on a given problem.
A2.R.6. Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions).
A2.R.7. Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll).
A2.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss).
A2.N.1. Operations: Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers).
A2.N.2. Operations: Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form.
A2.N.3. Operations: Perform arithmetic operations with polynomial expressions containing rational coefficients.
A2.N.4. Operations: Perform arithmetic operations on irrational expressions.
A2.N.5. Operations: Rationalize a denominator containing a radical expression.
A2.N.6. Operations: Write square roots of negative numbers in terms of i.
A2.N.7. Operations: Simplify powers of i.
A2.N.8. Operations: Determine the conjugate of a complex number.
A2.N.9. Operations: Perform arithmetic operations on complex numbers and write the answer in the form a + bi. Note: This includes simplifying expressions with complex denominators.
A2.N.10. Operations: Know and apply sigma notation.
A2.A.1. Equations and Inequalities: Solve absolute value equations and inequalities involving linear expressions in one variable.
A2.A.2. Equations and Inequalities: Use the discriminant to determine the nature of the roots of a quadratic equation.
A2.A.3. Equations and Inequalities: Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots.
A2.A.4. Equations and Inequalities: Solve quadratic inequalities in one and two variables, algebraically and graphically.
A2.A.5. Equations and Inequalities: Use direct and inverse variation to solve for unknown values.
A2.A.6. Equations and Inequalities: Solve an application which results in an exponential function.
A2.A.7. Variables and Expressions: Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials.
A2.A.8. Variables and Expressions: Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents.
A2.A.9. Variables and Expressions: Rewrite algebraic expressions that contain negative exponents using only positive exponents.
A2.A.10. Variables and Expressions: Rewrite algebraic expressions with fractional exponents as radical expressions.
A2.A.11. Variables and Expressions: Rewrite algebraic expressions in radical form as expressions with fractional exponents.
A2.A.12. Variables and Expressions: Evaluate exponential expressions, including those with base e.
A2.A.13. Variables and Expressions: Simplify radical expressions.
A2.A.14. Variables and Expressions: Perform addition, subtraction, multiplication and division of radical expressions.
A2.A.15. Variables and Expressions: Rationalize denominators involving algebraic radical expressions.
A2.A.16. Variables and Expressions: Perform arithmetic operations with rational expressions and rename to lowest terms.
A2.A.17. Variables and Expressions: Simplify complex fractional expressions.
A2.A.18. Variables and Expressions: Evaluate logarithmic expressions in any base.
A2.A.19. Variables and Expressions: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A.20. Equations and Inequalities: Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
A2.A.21. Equations and Inequalities: Determine the quadratic equation, given the sum and product of its roots.
A2.A.22. Equations and Inequalities: Solve radical equations.
A2.A.23. Equations and Inequalities: Solve rational equations and inequalities.
A2.A.24. Equations and Inequalities: Know and apply the technique of completing the square.
A2.A.25. Equations and Inequalities: Solve quadratic equations, using the quadratic formula.
A2.A.26. Equations and Inequalities: Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula.
A2.A.27. Equations and Inequalities: Solve exponential equations with and without common bases.
A2.A.28. Equations and Inequalities: Solve a logarithmic equation by rewriting as an exponential equation.
A2.A.29. Patterns, Relations, and Functions: Identify an arithmetic or geometric sequence and find the formula for its nth term.
A2.A.30. Patterns, Relations, and Functions: Determine the common difference in an arithmetic sequence.
A2.A.31. Patterns, Relations, and Functions: Determine the common ratio in a geometric sequence.
A2.A.32. Patterns, Relations, and Functions: Determine a specified term of an arithmetic or geometric sequence.
A2.A.33. Patterns, Relations, and Functions: Specify terms of a sequence, given its recursive definition.
A2.A.34. Patterns, Relations, and Functions: Represent the sum of a series, using sigma notation.
A2.A.35. Patterns, Relations, and Functions: Determine the sum of the first n terms of an arithmetic or geometric series.
A2.A.36. Patterns, Relations, and Functions: Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion.
A2.A.37. Patterns, Relations, and Functions: Define a relation and function.
A2.A.38. Patterns, Relations, and Functions: Determine when a relation is a function.
A2.A.39. Patterns, Relations, and Functions: Determine the domain and range of a function from its equation.
A2.A.40. Patterns, Relations, and Functions: Write functions in functional notation.
A2.A.41. Patterns, Relations, and Functions: Use functional notation to evaluate functions for given values in the domain.
A2.A.42. Patterns, Relations, and Functions: Find the composition of functions.
A2.A.43. Patterns, Relations, and Functions: Determine if a function is one-to-one, onto, or both.
A2.A.44. Patterns, Relations, and Functions: Define the inverse of a function.
A2.A.45. Patterns, Relations, and Functions: Determine the inverse of a function and use composition to justify the result.
A2.A.46. Patterns, Relations, and Functions: Perform transformations with functions and relations: f(x + a), f(x) + a, f(-x), -f(x), af(x).
A2.A.47. Coordinate Geometry: Determine the center-radius form for the equation of circle in standard form.
A2.A.48. Coordinate Geometry: Write the equation of a circle, given its center and a point on the circle.
A2.A.49. Coordinate Geometry: Write the equation of a circle from its graph.
A2.A.50. Coordinate Geometry: Approximate the solution to polynomial equations of higher degree by inspecting the graph.
A2.A.51. Coordinate Geometry: Determine the domain and range of a function from its graph.
A2.A.52. Coordinate Geometry: Identify relations and functions, using graphs.
A2.A.53. Coordinate Geometry: Graph exponential functions of the form y = b to the power of x for positive values of b, including b = e.
A2.A.54. Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.55. Trigonometric Functions: Express and apply the six trigonometric functions as ratios of the sides of a right triangle.
A2.A.56. Trigonometric Functions: Know the exact and approximate values of the sine, cosine, and tangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.57. Trigonometric Functions: Sketch and use the reference angle for angles in standard position.
A2.A.58. Trigonometric Functions: Know and apply the co-function and reciprocal relationships between trigonometric ratios.
A2.A.59. Trigonometric Functions: Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.60. Trigonometric Functions: Sketch the unit circle and represent angles in standard position.
A2.A.61. Trigonometric Functions: Determine the length of an arc of a circle, given its radius and the measure of its central angle.
A2.A.62. Trigonometric Functions: Find the value of trigonometric functions, if given a point on the terminal side of angle theta.
A2.A.63. Trigonometric Functions: Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function.
A2.A.64. Trigonometric Functions: Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent.
A2.A.65. Trigonometric Functions: Sketch the graph of the inverses of the sine, cosine, and tangent functions.
A2.A.66. Trigonometric Functions: Determine the trigonometric functions of any angle, using technology.
A2.A.67. Trigonometric Functions: Justify the Pythagorean identities.
A2.A.68. Trigonometric Functions: Solve trigonometric equations for all values of the variable from 0 degrees to 360 degrees.
A2.A.69. Trigonometric Functions: Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function.
A2.A.70. Trigonometric Functions: Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx.
A2.A.71. Trigonometric Functions: Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x).
A2.A.72. Trigonometric Functions: Write the trigonometric function that is represented by a given periodic graph.
A2.A.73. Trigonometric Functions: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines.
A2.A.74. Trigonometric Functions: Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle.
A2.A.75. Trigonometric Functions: Determine the solution(s) from the SSA situation (ambiguous case).
A2.A.76. Trigonometric Functions: Apply the angle sum and difference formulas for trigonometric functions.
A2.A.77. Trigonometric Functions: Apply the double-angle and half-angle formulas for trigonometric functions.
A2.M.1. Units of Measurement: Define radian measure.
A2.M.2. Units of Measurement: Convert between radian and degree measures.
A2.S.1. Collection of Data: Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment).
A2.S.2. Collection of Data: Determine factors which may affect the outcome of a survey.
A2.S.3. Organization and Display of Data: Calculate measures of central tendency with group frequency distributions.
A2.S.4. Organization and Display of Data: Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations.
A2.S.5. Organization and Display of Data: Know and apply the characteristics of the normal distribution.
A2.S.6. Predictions from Data: Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate.
A2.S.7. Predictions from Data: Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.
A2.S.8. Predictions from Data: Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship.
A2.S.9. Probability: Differentiate between situations requiring permutations and those requiring combinations.
A2.S.10. Probability: Calculate the number of possible permutations (nPr) of n items taken r at a time.
A2.S.11. Probability: Calculate the number of possible combinations (nCr) of n items taken r at a time.
A2.S.12. Probability: Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event).
A2.S.13. Probability: Calculate theoretical probabilities, including geometric applications.
A2.S.14. Probability: Calculate empirical probabilities.
A2.S.15. Probability: Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most.
A2.S.16. Probability: Use the normal distribution as an approximation for binomial probabilities.