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## New York Math Standards - 9th Grade

MathScore aligns to the New York Math Standards for 9th Grade. The standards appear below along with the MathScore topics that match. If you click on a topic name, you will see sample problems at varying degrees of difficulty that MathScore generated. When students use our program, the difficulty of the problems will automatically adapt based on individual performance, resulting in not only true differentiated instruction, but a challenging game-like experience.

View the New York Math Standards at other levels.

## 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
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)
A.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) (Age Problems )
A.PS.6 Use a variety of strategies to extend solution methods to other problems (Area And Volume Proportions )
A.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to 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

## Reasoning and Proof

A.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies
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
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
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 (Function Tables , Function Tables 2 )
A.RP.11 Use a Venn diagram to support a logical argument
A.RP.12 Apply inductive reasoning in making and supporting mathematical conjectures

## 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
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
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 conjecturesof others
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 (Mixture Word Problems , Work Word Problems , Integer Word Problems )
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

## Connections

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
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 (Triangle Angles 2 )
A.CN.5 Understand how quantitative models connect to various physical models and representations
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

## Representation

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
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
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 = x2 and y = - x2 )

## Number Sense and Operations

A.N.1 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.
A.N.3 Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form (Adding and Subtracting Radical Expressions , Multiplying and Dividing Radical Expressions )
A.N.4 Understand and use scientific notation to compute products and quotients of numbers greater than 100% (Scientific Notation 2 , Scientific Notation )
A.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation (Percentage Change , Purchases At Stores , Restaurant Bills , Commissions , Solving For Angles , Proportions 2 )
A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) (Absolute Value 1 , Absolute Value 2 , Absolute Value Equations )
A.N.7 Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting
A.N.8 Determine the number of possible arrangements (permutations) of a list of items

## Algebra

A.A.1 Translate a quantitative verbal phrase into an algebraic expression (Phrases to Algebraic Expressions , Algebraic Sentences 2 , Algebraic Sentences )
A.A.2 Write verbal expressions that match given mathematical expressions
A.A.3 Distinguish the difference between an algebraic expression and an algebraic equation (Algebraic Terms )
A.A.4 Translate verbal sentences into mathematical equations or inequalities (Algebraic Sentences 2 , Algebraic Sentences )
A.A.5 Write algebraic equations or inequalities that represent a situation (Algebraic Sentences 2 , Algebraic Sentences )
A.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable (Algebraic Word Problems )
A.A.7 Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables (Age Problems , Mixture Word Problems , Work Word Problems )
A.A.8 Analyze and solve verbal problems that involve quadratic equations (Area And Volume Proportions , Integer Word Problems )
A.A.9 Analyze and solve verbal problems that involve exponential growth and decay (Continuous Compound Interest )
A.A.10 Solve systems of two linear equations in two variables algebraically (See A.G.7) (System of Equations Substitution , System of Equations Addition )
A.A.11 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.
A.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only. (Multiplying and Dividing Exponent Expressions , Exponent Rules For Fractions , Roots Of Exponential Expressions , Estimating Square Roots )
A.A.13 Add, subtract, and multiply monomials and polynomials (Simplifying Algebraic Expressions , Simplifying Algebraic Expressions 2 , Foil Method , Binomial Fraction Simplification )
A.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainder (Polynomial Fraction Simplification )
A.A.15 Find values of a variable for which an algebraic fraction is undefined. (Domain and Range )
A.A.16 Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms (Polynomial Fraction Simplification )
A.A.17 Add or subtract fractional expressions with monomial or like binomial denominators
A.A.18 Multiply and divide algebraic fractions and express the product or quotient in simplest form
A.A.19 Identify and factor the difference of two perfect squares (Trinomial Factoring , Quadratic Zero Equations )
A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF) (Trinomial Factoring )
A.A.21 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 Solve all types of linear equations in one variable (Single Variable Equations , Single Variable Equations 2 , Single Variable Equations 3 )
A.A.23 Solve literal equations for a given variable (Two Variable Equations )
A.A.24 Solve linear inequalities in one variable (Single Variable Inequalities )
A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations
A.A.27 Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots (Quadratic Zero Equations , Quadratic Formula , Quadratic X-Intercepts )
A.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression (Quadratic X-Intercepts , Quadratic Zero Equations , Quadratic Formula )
A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form
A.A.30 Find the complement of a subset of a given set, within a given universe
A.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets) (Intersection and Union )
A.A.32 Explain slope as a rate of change between dependent and independent variables
A.A.33 Determine the slope of a line, given the coordinates of two points on the line (Determining Slope )
A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line (Applied Linear Equations 1 )
A.A.35 Write the equation of a line, given the coordinates of two points on the line (Applied Linear Equations 1 )
A.A.36 Write the equation of a line parallel to the x- or y-axis (Applied Linear Equations 2 )
A.A.37 Determine the slope of a line, given its equation in any form
A.A.38 Determine if two lines are parallel, given their equations in any form (Applied Linear Equations 2 )
A.A.39 Determine whether a given point is on a line, given the equation of the line
A.A.40 Determine whether a given point is in the solution set of a system of linear inequalities
A.A.41 Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 )
A.A.42 Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides
A.A.43 Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle
A.A.44 Find the measure of a side of a right triangle, given an acute angle and the length of another side
A.A.45 Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides (Pythagorean Theorem )

## Geometry

A.G.1 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). (Perimeter and Area of Composite Figures )
A.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders (Rectangular Solids , Rectangular Solids 2 , Cylinders )
A.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations
A.G.4 Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions
A.G.5 Investigate and generalize how changing the coefficients of a function affects its graph
A.G.6 Graph linear inequalities
A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10)
A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9 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 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.

## Measurement

A.M.1 Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail)
A.M.2 Solve problems involving conversions within measurement systems, given the relationship between the units (Distance Conversion , Time Conversion , Volume Conversion , Weight Conversion , Temperature Conversion , Area and Volume Conversions )
A.M.3 Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure

## Statistics and Probability

A.S.1 Categorize data as qualitative or quantitative
A.S.2 Determine whether the data to be analyzed is univariate or bivariate
A.S.3 Determine when collected data or display of data may be biased
A.S.4 Compare and contrast the appropriateness of different measures of central tendency for a given data set
A.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data
A.S.6 Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plot
A.S.7 Create a scatter plot of bivariate data
A.S.8 Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line
A.S.9 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 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 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 Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none) (Independent and Dependent Variables )
A.S.13 Understand the difference between correlation and causation
A.S.14 Identify variables that might have a correlation but not a causal relationship
A.S.15 Identify and describe sources of bias and its effect, drawing conclusions from data
A.S.16 Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range
A.S.17 Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation
A.S.18 Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces (Object Picking Probability )
A.S.19 Determine the number of elements in a sample space and the number of favorable events
A.S.20 Calculate the probability of an event and its complement (Probability , Probability 2 )
A.S.21 Determine empirical probabilities based on specific sample data
A.S.22 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 Calculate the probability
• a series of independent events
• two mutually exclusive events
• two events that are not mutually exclusive
(Probability 2 , Object Picking Probability )