New York Math Standards - Next Gen

755 standards, Kindergarten through AP.

New York's Next Generation Mathematics Learning Standards, adopted in 2017, build on Common Core foundations with state specific modifications and additions. The result is a New York centered take on strengthening the original Common Core structure.

The Plus (+) Standards

(+)-A

(+)-A.APR Algebra - Arithmetic with Polynomial and Rational Expressions

(+)-A.APR.4Prove polynomial identities and use them to describe numerical relationships.

(+)-A.APR.5Use the Binomial Theorem for the expansion of (x + y)^n for a positive integer n.

(+)-A.APR.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

(+)-A.REI Algebra - Reasoning with Equations and Inequalities

(+)-A.REI.6bSolve systems of linear equations in three variables.

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

(+)-A.REI.9Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

(+)-F

(+)-F.BF Functions - Building Functions

(+)-F.BF.1cCompose functions and state resulting domain.

(+)-F.BF.3cDetermine algebraically whether or not a function is even or odd.

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

(+)-F.BF.4cGiven the graph or table of an invertible function, determine coordinates of its inverse.

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

(+)-F.BF.5bUse inverse relationships to solve problems involving logarithms and exponents.

(+)-F.BF.5cApply the properties of logarithms to rewrite logarithmic expressions in equivalent forms and solve logarithmic equations.

(+)-F.IF Functions - Interpreting Functions

(+)-F.IF.7dGraph rational functions, identifying zeros and asymptotes when suitable factorizations are available.

(+)-F.TF Functions - Trigonometric Functions

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

(+)-F.TF.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

(+)-F.TF.7Solve trigonometric equations: analytically with inverse functions and; graphically with technology and interpret solutions in terms of the context.

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

(+)-G

(+)-G.C Geometry – Circles

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

(+)-G.GMD Geometry - Geometric Measurement and Dimension

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

(+)-G.GPE Geometry - Expressing Geometric Properties with Equations

(+)-G.GPE.2Explore the relationship among the parabola, focus, and directrix and use the equation to model a real-life situation, using technology as appropriate.

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

(+)-G.GPE.3bUse these equations to model real life situations.

(+)-G.SRT Geometry - Similarity, Right Triangles, and Trigonometry

(+)-G.SRT.10Prove the Law of Sines and the Law of Cosines and apply in all cases, including the ambiguous case.

(+)-G.SRT.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

(+)-N

(+)-N.CN Number and Quantity - The Complex Number System

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

(+)-N.CN.4aRepresent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and convert between rectangular and polar forms of a given complex number.

(+)-N.CN.4bDetermine whether rectangular or polar form is more efficient given the context.

(+)-N.CN.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

(+)-N.CN.6aCalculate the distance between two points in the complex plane.

(+)-N.CN.6bFind the midpoint of the segment whose endpoints are in the complex plane.

(+)-N.CN.8Extend polynomial identities to the complex numbers.

(+)-N.CN.9State the Fundamental Theorem of Algebra and use it to find roots of polynomials.

(+)-N.VM Number and Quantity - Vector and Matrix Quantities

(+)-N.VM.1Represent a vector analytically and geometrically.

(+)-N.VM.11Use matrices to perform linear transformations in the plane.

(+)-N.VM.12Calculate and interpret the determinant of a matrix.

(+)-N.VM.2Find the magnitude and direction of a given vector.

(+)-N.VM.3Solve problems using vectors analytically and geometrically.

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

(+)-N.VM.5Multiply a vector by a scalar analytically and geometrically.

(+)-N.VM.6Use matrices to represent and model real world situations.

(+)-N.VM.7Multiply matrices by scalars.

(+)-N.VM.8Add, subtract, and multiply matrices.

(+)-N.VM.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

(+)-S

(+)-S.CP Statistics and Probability - Conditional Probability and the Rules of Probability

(+)-S.CP.9Solve problems using permutations and combinations to compute probabilities of compound events.

(+)-S.ID Statistics and Probability - Interpreting Categorical and Quantitative Data

(+)-S.ID.6bInformally assess the fit of a function by plotting and analyzing residuals.

(+)-S.MD Statistics and Probability - Using Probability to Make Decisions

(+)-S.MD.1aDefine a random variable for a quantity of interest.

(+)-S.MD.1bGraph a probability distribution for a discrete random variable based on either empirical or theoretical probabilities.

(+)-S.MD.2Calculate and interpret the expected value of a random variable.

(+)-S.MD.5Use expected values from probability distributions to evaluate and compare the outcomes of decisions.

(+)-S.MD.6Use probabilities to make fair decisions.

(+)-S.MD.7Using probability concepts, evaluate decisions and strategies.

Algebra I

AI-A

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

AI-A.APR.1Add, subtract, and multiply polynomials and recognize that the result of the operation is also a polynomial. This forms a system analogous to the integers.

AI-A.APR.3Identify zeros of polynomial functions when suitable factorizations are available. (Shared standard with Algebra II)

AI-A.CED Algebra - Creating Equations

AI-A.CED.1Create equations and inequalities in one variable to represent a real-world context. (Shared standard with Algebra II)

AI-A.CED.2Create equations and linear inequalities in two variables to represent a real-world context.

AI-A.CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

AI-A.CED.4Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

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

AI-A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

AI-A.REI.11Given the equations y = f(x) and y = g(x): i) recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x); ii) find the solutions approximately using technology to graph the functions or make tables of values; and iii) interpret the solution in context. (Shared standard with Algebra II)

AI-A.REI.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

AI-A.REI.1aExplain each step when solving a linear or quadratic equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

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

AI-A.REI.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Understand that the quadratic formula is a derivative of this process.

AI-A.REI.4.bSolve quadratic equations by: i) inspection, ii) taking square roots, iii) factoring, iv) completing the square, v) the quadratic formula, and vi) graphing. Recognize when the process yields no real solutions. (Shared standard with Algebra II)

AI-A.REI.6aSolve systems of linear equations in two variables both algebraically and graphically.

AI-A.REI.7aSolve a system, with rational solutions, consisting of a linear equation and a quadratic equation (parabolas only) in two variables algebraically and graphically. (Shared standard with Algebra II)

AI-A.SSE Algebra - Seeing Structure in Expressions

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

AI-A.SSE.1.aWrite the standard form of a given polynomial and identify the terms, coefficients, degree, leading coefficient, and constant term.

AI-A.SSE.1.bInterpret expressions by viewing one or more of their parts as a single entity.

AI-A.SSE.2Recognize and use the structure of an expression to identify ways to rewrite it. (Shared standard with Algebra II)

AI-A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Shared standard with Algebra II)

AI-A.SSE.3.cUse the properties of exponents to rewrite exponential expressions. (Shared standard with Algebra II)

AI-F

AI-F.BF Functions - Building Functions

AI-F.BF.1 Write a function that describes a relationship between two quantities. (Shared standard with Algebra II)

AI-F.BF.1.aDetermine a function from context. Define a sequence explicitly or steps for calculation from a context. (Shared standard with Algebra II)

AI-F.BF.3aUsing f(x) + k, k f(x), and f(x + k): i) identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), and f(x + k) for specific values of k (both positive and negative); ii) find the value of k given the graphs; iii) write a new function using the value of k; and iv) use technology to experiment with cases and explore the effects on the graph. (Shared standard with Algebra II)

AI-F.IF Functions - Interpreting Functions

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

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

AI-F.IF.3Recognize that a sequence is a function whose domain is a subset of the integers. (Shared standard with Algebra II)

AI-F.IF.4For a function that models a relationship between two quantities: i) interpret key features of graphs and tables in terms of the quantities; and ii) sketch graphs showing key features given a verbal description of the relationship. (Shared standard with Algebra II)

AI-F.IF.5Determine the domain of a function from its graph and, where applicable, identify the appropriate domain for a function in context.

AI-F.IF.6Calculate and interpret the average rate of change of a function over a specified interval. (Shared standard with Algebra II)

AI-F.IF.7 Graph functions and show key features of the graph by hand and by using technology where appropriate. (Shared standard with Algebra II)

AI-F.IF.7.aGraph linear, quadratic, and exponential functions and show key features.

AI-F.IF.7.bGraph square root, and piecewise-defined functions, including step functions and absolute value functions and show key features.

AI-F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function. (Shared standard with Algebra II)

AI-F.IF.8.aFor a quadratic function, use an algebraic process to find zeros, maxima, minima, and symmetry of the graph, and interpret these in terms of context.

AI-F.IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Shared standard with Algebra II)

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

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

AI-F.LE.1.aJustify that a function is linear because it grows by equal differences over equal intervals, and that a function is exponential because it grows by equal factors over equal intervals.

AI-F.LE.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another, and therefore can be modeled linearly.

AI-F.LE.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another, and therefore can be modeled exponentially.

AI-F.LE.2Construct a linear or exponential function symbolically given: i) a graph; ii) a description of the relationship; iii) two input-output pairs (include reading these from a table). (Shared standard with Algebra II)

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

AI-F.LE.5Interpret the parameters in a linear or exponential function in terms of a context. (Shared standard with Algebra II)

AI-N

AI-N.Q Number and Quantity – Quantities

AI-N.Q.1Select quantities and use units as a way to: i) interpret and guide the solution of multi-step problems; ii) choose and interpret units consistently in formulas; and iii) choose and interpret the scale and the origin in graphs and data displays.

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

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

AI-N.RN.3 Use properties and operations to understand the different forms of rational and irrational numbers.

AI-N.RN.3.aPerform all four arithmetic operations and apply properties to generate equivalent forms of rational numbers and square roots.

AI-N.RN.3.bCategorize the sum or product of rational or irrational numbers: The sum and product of two rational numbers is rational; The sum of a rational number and an irrational number is irrational; The product of a nonzero rational number and an irrational number is irrational; The sum and product of two irrational numbers could be either rational or irrational.

AI-S

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

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

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

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

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

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

AI-S.ID.6.aFit a function to real-world data; use functions fitted to data to solve problems in the context of the data. (Shared standard with Algebra II)

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

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

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

AII-A

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

AII-A.APR.2Apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

AII-A.APR.3Identify zeros of polynomial functions when suitable factorizations are available. (Shared standard with Algebra I)

AII-A.APR.6Rewrite rational expressions in different forms: Write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x).

AII-A.CED Algebra - Creating Equations

AII-A.CED.1Create equations and inequalities in one variable to represent a real-world context. (Shared standard with Algebra I)

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

AII-A.REI.11Given the equations y = f(x) and y = g(x): i) recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x); ii) find the solutions approximately using technology to graph the functions or make tables of values; iii) find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and iv) interpret the solution in context. (Shared standard with Algebra I)

AII-A.REI.1bExplain each step when solving rational or radical equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

AII-A.REI.2Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise.

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

AII-A.REI.4.bSolve quadratic equations by: i) inspection, ii) taking square roots, iii) factoring, iv) completing the square, v) the quadratic formula, and vi) graphing. Write complex solutions in a + bi form. (Shared standard with Algebra I)

AII-A.REI.7bSolve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (Shared standard with Algebra I)

AII-A.SSE Algebra - Seeing Structure in Expressions

AII-A.SSE.2Recognize and use the structure of an expression to identify ways to rewrite it. (Shared standard with Algebra I)

AII-A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Shared standard with Algebra I)

AII-A.SSE.3.aFactor quadratic expressions including leading coefficients other than 1 to reveal the zeros of the function it defines.

AII-A.SSE.3.cUse the properties of exponents to rewrite exponential expressions. (Shared standard with Algebra I)

AII-F

AII-F.BF Functions - Building Functions

AII-F.BF.1 Write a function that describes a relationship between two quantities. (Shared standard with Algebra I)

AII-F.BF.1.aDetermine a function from context. Determine an explicit expression, a recursive process, or steps for calculation from a context. (Shared standard with Algebra I)

AII-F.BF.1.bCombine standard function types using arithmetic operations.

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

AII-F.BF.3bUsing f(x) + k, k f(x), f(kx), and f(x + k): i) identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); ii) find the value of k given the graphs; iii) write a new function using the value of k; and iv) use technology to experiment with cases and explore the effects on the graph. Include recognizing even and odd functions from their graphs. (Shared standard with Algebra I)

AII-F.BF.4aFind the inverse of a one-to-one function both algebraically and graphically.

AII-F.BF.5aUnderstand inverse relationships between exponents and logarithms algebraically and graphically.

AII-F.BF.6Represent and evaluate the sum of a finite arithmetic or finite geometric series, using summation (sigma) notation.

AII-F.BF.7Explore the derivation of the formulas for finite arithmetic and finite geometric series. Use the formulas to solve problems.

AII-F.IF Functions - Interpreting Functions

AII-F.IF.3Recognize that a sequence is a function whose domain is a subset of the integers. (Shared standard with Algebra I)

AII-F.IF.4For a function that models a relationship between two quantities: i) interpret key features of graphs and tables in terms of the quantities; and ii) sketch graphs showing key features given a verbal description of the relationship. (Shared standard with Algebra I)

AII-F.IF.6Calculate and interpret the average rate of change of a function over a specified interval. (Shared standard with Algebra I)

AII-F.IF.7 Graph functions and show key features of the graph by hand and using technology when appropriate. (Shared standard with Algebra I)

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

AII-F.IF.7.eGraph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.

AII-F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function. (Shared standard with Algebra I)

AII-F.IF.8.bUse the properties of exponents to interpret exponential functions, and classify them as representing exponential growth or decay.

AII-F.IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Shared standard with Algebra I)

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

AII-F.LE.2Construct a linear or exponential function symbolically given: i) a graph; ii) a description of the relationship; and iii) two input-output pairs (include reading these from a table). (Shared standard with Algebra I)

AII-F.LE.4Use logarithms to solve exponential equations, such as abct = d (where a, b, c, and d are real numbers and b > 0) and evaluate the logarithm using technology.

AII-F.LE.5Interpret the parameters in a linear or exponential function in terms of a context. (Shared standard with Algebra I)

AII-F.TF Functions - Trigonometric Functions

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

AII-F.TF.2Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions given angles in radian measure.

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

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

AII-F.TF.8Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. Find the value of any of the six trigonometric functions given any other trigonometric function value and when necessary find the quadrant of the angle.

AII-N

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

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

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

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

AII-N.RN.1Explore how the meaning of rational exponents follows from extending the properties of integer exponents.

AII-N.RN.2Convert between radical expressions and expressions with rational exponents using the properties of exponents.

AII-S

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

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

AII-S.CP.4Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and calculate conditional probabilities.

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

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

AII-S.IC.2Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation.

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

AII-S.IC.4Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/- two standard deviations) and determine if a suggested parameter is plausible.

AII-S.IC.6aUse the tools of statistics to draw conclusions from numerical summaries.

AII-S.IC.6bUse the language of statistics to critique claims from informational texts. For example, causation vs correlation, bias, measures of center and spread.

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

AII-S.ID.4aRecognize whether or not a normal curve is appropriate for a given data set.

AII-S.ID.4bIf appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.

AII-S.ID.6 Represent bivariate data on a scatter plot, and describe how the variables’ values are related.

AII-S.ID.6.aFit a function to real-world data; use functions fitted to data to solve problems in the context of the data. (Shared standard with Algebra I)

GEO-G

GEO-G.C Geometry – Circles

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

GEO-G.C.2aIdentify, describe and apply relationships between the angles and their intercepted arcs of a circle.

GEO-G.C.2bIdentify, describe and apply relationships among radii, chords, tangents, and secants of a circle.

GEO-G.C.5Using proportionality, find one of the following given two others; the central angle, arc length, radius or area of sector.

GEO-G.CO Geometry – Congruence

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

GEO-G.CO.10Prove and apply theorems about triangles.

GEO-G.CO.11Prove and apply theorems about parallelograms.

GEO-G.CO.12Make, justify, and apply formal geometric constructions.

GEO-G.CO.13Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle.

GEO-G.CO.2Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

GEO-G.CO.3Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that carry the polygon onto itself.

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

GEO-G.CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.

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

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

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

GEO-G.CO.9Prove and apply theorems about lines and angles.

GEO-G.GMD Geometry - Geometric Measurement and Dimension

GEO-G.GMD.1Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

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

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

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

GEO-G.GPE.1aDerive the equation of a circle of given center and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the equation of the circle.

GEO-G.GPE.4On the coordinate plane, algebraically prove geometric theorems and properties.

GEO-G.GPE.5 On the coordinate plane:

GEO-G.GPE.5.aExplore the proof for the relationship between slopes of parallel and perpendicular lines;

GEO-G.GPE.5.bDetermine if lines are parallel, perpendicular, or neither, based on their slopes; and

GEO-G.GPE.5.cApply properties of parallel and perpendicular lines to solve geometric problems.

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

GEO-G.GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

GEO-G.MG Geometry - Modeling with Geometry

GEO-G.MG.1Use geometric shapes, their measures, and their properties to describe objects.

GEO-G.MG.2Apply concepts of density based on area and volume of geometric figures in modeling situations.

GEO-G.MG.3Apply geometric methods to solve design problems.

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

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

GEO-G.SRT.1.aVerify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

GEO-G.SRT.1.bVerify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

GEO-G.SRT.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

GEO-G.SRT.3Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.

GEO-G.SRT.4Prove and apply similarity theorems about triangles.

GEO-G.SRT.5 Use congruence and similarity criteria for triangles to:

GEO-G.SRT.5.aSolve problems algebraically and geometrically.

GEO-G.SRT.5.bProve relationships in geometric figures.

GEO-G.SRT.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent ratios for acute angles.

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

GEO-G.SRT.8Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems.

GEO-G.SRT.9Justify and apply the formula A=(1/2)ab sin (C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Grade 1

NY-1.G Geometry

NY-1.G.1Distinguish between defining attributes versus non-defining attributes for a wide variety of shapes. Build and/or draw shapes to possess defining attributes.

NY-1.G.2Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

NY-1.G.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

NY-1.MD Measurement and Data

NY-1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.

NY-1.MD.2Measure the length of an object using same-size “length units” placed end to end with no gaps or overlaps. Express the length of an object as a whole number of “length units.”

NY-1.MD.3aTell and write time in hours and half-hours using analog and digital clocks. Develop an understanding of common terms, such as, but not limited to, o’clock and half past.

NY-1.MD.3bRecognize and identify coins (penny, nickel, dime, and quarter) and their value and use the cent symbol (¢) appropriately.

NY-1.MD.3cCount a mixed collection of dimes and pennies and determine the cent value (total not to exceed 100 cents).

NY-1.MD.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

NY-1.NBT Number and Operations in Base Ten

NY-1.NBT.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

NY-1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones.

NY-1.NBT.2.aUnderstand 10 can be thought of as a bundle of ten ones, called a "ten".

NY-1.NBT.2.bUnderstand the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

NY-1.NBT.2.cUnderstand the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

NY-1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

NY-1.NBT.4Add within 100, including: a two-digit number and a one-digit number, a two-digit number and a multiple of 10. Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones, and sometimes it is necessary to compose a ten. Relate the strategy to a written representation and explain the reasoning used.

NY-1.NBT.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

NY-1.NBT.6Subtract multiples of 10 from multiples of 10 in the range 10-90 using: concrete models or drawings, and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy used to a written representation and explain the reasoning.

NY-1.OA Operations and Algebraic Thinking

NY-1.OA.1Use addition and subtraction within 20 to solve one step word problems involving situations of adding to, taking from, putting together, taking apart, and/or comparing, with unknowns in all positions.

NY-1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.

NY-1.OA.3Apply properties of operations as strategies to add and subtract.

NY-1.OA.4Understand subtraction as an unknown-addend problem within 20.

NY-1.OA.5Relate counting to addition and subtraction.

NY-1.OA.6aAdd and subtract within 20. Use strategies such as: • counting on; • making ten; • decomposing a number leading to a ten; • using the relationship between addition and subtraction; and • creating equivalent but easier or known sums.

NY-1.OA.6bFluently add and subtract within 10.

NY-1.OA.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

NY-1.OA.8Determine the unknown whole number in an addition or subtraction equation with the unknown in all positions.

Grade 2

NY-2.G Geometry

NY-2.G.1Classify two-dimensional figures as polygons or non-polygons.

NY-2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

NY-2.G.3Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the words halves, thirds, half of, a third of, etc. Describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

NY-2.MD Measurement and Data

NY-2.MD.1Measure the length of an object to the nearest whole by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

NY-2.MD.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a picture graph or a bar graph.

NY-2.MD.2Measure the length of an object twice, using different “length units” for the two measurements; describe how the two measurements relate to the size of the unit chosen.

NY-2.MD.3Estimate lengths using units of inches, feet, centimeters, and meters.

NY-2.MD.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard “length unit.”

NY-2.MD.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.

NY-2.MD.6Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line.

NY-2.MD.7Tell and write time from analog and digital clocks in five minute increments, using a.m. and p.m. Develop an understanding of common terms, such as, but not limited to, quarter past, half past, and quarter to.

NY-2.MD.8aCount a mixed collection of coins whose sum is less than or equal to one dollar.

NY-2.MD.8bSolve real world and mathematical problems within one dollar involving quarters, dimes, nickels, and pennies, using the ¢ (cent) symbol appropriately.

NY-2.MD.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Present the measurement data in a line plot, where the horizontal scale is marked off in whole-number units.

NY-2.NBT Number and Operations in Base Ten

NY-2.NBT.1 Understand that the digits of a three-digit number represent amounts of hundreds, tens, and ones.

NY-2.NBT.1.aUnderstand 100 can be thought of as a bundle of ten tens, called a “hundred.”

NY-2.NBT.1.bUnderstand the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

NY-2.NBT.2Count within 1000; skip-count by 5s, 10s, and 100s.

NY-2.NBT.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

NY-2.NBT.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

NY-2.NBT.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

NY-2.NBT.6Add up to four two-digit numbers using strategies based on place value and properties of operations.

NY-2.NBT.7aAdd and subtract within 1000, using concrete models or drawings, and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written representation.

NY-2.NBT.7bUnderstand that in adding or subtracting up to three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and sometimes it is necessary to compose or decompose tens or hundreds.

NY-2.NBT.8Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

NY-2.NBT.9Explain why addition and subtraction strategies work, using place value and the properties of operations.

NY-2.OA Operations and Algebraic Thinking

NY-2.OA.1aUse addition and subtraction within 100 to solve one-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

NY-2.OA.1bUse addition and subtraction within 100 to develop an understanding of solving two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

NY-2.OA.2aFluently add and subtract within 20 using mental strategies. Strategies could include: counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums.

NY-2.OA.2bKnow from memory all sums within 20 of two one-digit numbers.

NY-2.OA.3aDetermine whether a group of objects (up to 20) has an odd or even number of members.

NY-2.OA.3bWrite an equation to express an even number as a sum of two equal addends.

NY-2.OA.4Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as a sum of equal addends.

Grade 3

NY-3.G Geometry

NY-3.G.1Recognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons, and hexagons). Identify shapes that do not belong to one of the given subcategories.

NY-3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

NY-3.MD Measurement and Data

NY-3.MD.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve one-step word problems involving addition and subtraction of time intervals in minutes.

NY-3.MD.2aMeasure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (l).

NY-3.MD.2bAdd, subtract, multiply, or divide to solve one-step word problems involving masses or liquid volumes that are given in the same units.

NY-3.MD.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in a scaled picture graph or a scaled bar graph.

NY-3.MD.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

NY-3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.

NY-3.MD.5.aRecognize a square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.

NY-3.MD.5.bRecognize a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

NY-3.MD.6Measure areas by counting unit squares.

NY-3.MD.7 Relate area to the operations of multiplication and addition.

NY-3.MD.7.aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

NY-3.MD.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

NY-3.MD.7.cUse tiling to show in a concrete case that the area of a rectangle with whole-number side length a and side length b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

NY-3.MD.7.dRecognize area as additive. Find areas of figures composed of non-overlapping rectangles, and apply this technique to solve real world problems.

NY-3.MD.8aSolve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths or finding one unknown side length given the perimeter and other side lengths.

NY-3.MD.8bIdentify rectangles with the same perimeter and different areas or with the same area and different perimeters.

NY-3.NBT Number and Operations in Base Ten

NY-3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.

NY-3.NBT.2Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

NY-3.NBT.3Multiply one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties of operations.

NY-3.NBT.4aUnderstand that the digits of a four-digit number represent amounts of thousands, hundreds, tens, and ones.

NY-3.NBT.4bRead and write four-digit numbers using base-ten numerals, number names, and expanded form.

NY-3.NF Number and Operations – Fractions

NY-3.NF.1Understand a unit fraction, 1/b, is the quantity formed by 1 part when a whole is partitioned into b equal parts. Understand a fraction a/b as the quantity formed by a parts of size 1/b.

NY-3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line.

NY-3.NF.2.aRepresent a fraction 1/b on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part starting at 0 locates the number 1/b on the number line.

NY-3.NF.2.bRepresent a fraction a/b on a number line by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

NY-3.NF.3 Explain equivalence of fractions and compare fractions by reasoning about their size.

NY-3.NF.3.aUnderstand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

NY-3.NF.3.bRecognize and generate equivalent fractions. Explain why the fractions are equivalent.

NY-3.NF.3.cExpress whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

NY-3.NF.3.dCompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons rely on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

NY-3.OA Operations and Algebraic Thinking

NY-3.OA.1Interpret products of whole numbers.

NY-3.OA.2Interpret whole-number quotients of whole numbers.

NY-3.OA.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.

NY-3.OA.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

NY-3.OA.5Apply properties of operations as strategies to multiply and divide.

NY-3.OA.6Understand division as an unknown-factor problem.

NY-3.OA.7aFluently solve single-digit multiplication and related divisions, using strategies such as the relationship between multiplication and division or properties of operations.

NY-3.OA.7bKnow from memory all products of two one-digit numbers.

NY-3.OA.8 Solve two-step word problems posed with whole numbers and having whole-number answers using the four operations.

NY-3.OA.8.aRepresent these problems using equations or expressions with a letter standing for the unknown quantity.

NY-3.OA.8.bAssess the reasonableness of answers using mental computation and estimation strategies including rounding.

NY-3.OA.9Identify and extend arithmetic patterns (including patterns in the addition table or multiplication table).

Grade 4

NY-4.G Geometry

NY-4.G.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

NY-4.G.2aIdentify and name triangles based on angle size (right, obtuse, acute).

NY-4.G.2bIdentify and name all quadrilaterals with 2 pairs of parallel sides as parallelograms.

NY-4.G.2cIdentify and name all quadrilaterals with four right angles as rectangles.

NY-4.G.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

NY-4.MD Measurement and Data

NY-4.MD.1Know relative sizes of measurement units: ft., in.; km, m, cm. Know the conversion factor and use it to convert measurements in a larger unit in terms of a smaller unit: ft., in.; km, m, cm; hr., min., sec. Given the conversion factor, convert all other measurements within a single system of measurement from a larger unit to a smaller unit. Record measurement equivalents in a two-column table.

NY-4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money.

NY-4.MD.2.aSolve problems involving fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

NY-4.MD.2.bRepresent measurement quantities using diagrams that feature a measurement scale, such as number lines.

NY-4.MD.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

NY-4.MD.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.

NY-4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.

NY-4.MD.5.aRecognize an angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

NY-4.MD.5.bRecognize an angle that turns through n one-degree angles is said to have an angle measure of n degrees.

NY-4.MD.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

NY-4.MD.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.

NY-4.NBT Number and Operations in Base Ten

NY-4.NBT.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

NY-4.NBT.2aRead and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.

NY-4.NBT.2bCompare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

NY-4.NBT.3Use place value understanding to round multi-digit whole numbers to any place.

NY-4.NBT.4Fluently add and subtract multi-digit whole numbers using a standard algorithm.

NY-4.NBT.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

NY-4.NBT.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

NY-4.NF Number and Operations—Fractions

NY-4.NF.1Explain why a fraction a/b is equivalent to a fraction (a × n)/(b × n) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

NY-4.NF.2Compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions.

NY-4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

NY-4.NF.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.

NY-4.NF.3.bDecompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions.

NY-4.NF.3.cAdd and subtract mixed numbers with like denominators.

NY-4.NF.3.dSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

NY-4.NF.4 Apply and extend previous understandings of multiplication to multiply a whole number by a fraction.

NY-4.NF.4.aUnderstand a fraction a/b as a multiple of 1/b.

NY-4.NF.4.bUnderstand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a whole number by a fraction.

NY-4.NF.4.cSolve word problems involving multiplication of a whole number by a fraction.

NY-4.NF.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.

NY-4.NF.6Use decimal notation for fractions with denominators 10 or 100.

NY-4.NF.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

NY-4.OA Operations and Algebraic Thinking

NY-4.OA.1Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations.

NY-4.OA.2Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison. Use drawings and equations with a symbol for the unknown number to represent the problem.

NY-4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.

NY-4.OA.3.aRepresent these problems using equations or expressions with a letter standing for the unknown quantity.

NY-4.OA.3.bAssess the reasonableness of answers using mental computation and estimation strategies including rounding.

NY-4.OA.4Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

NY-4.OA.5Generate a number or shape pattern that follows a given rule. Identify and informally explain apparent features of the pattern that were not explicit in the rule itself.

Grade 5

NY-5.G Geometry

NY-5.G.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond.

NY-5.G.2Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

NY-5.G.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

NY-5.G.4Classify two-dimensional figures in a hierarchy based on properties.

NY-5.MD Measurement and Data

NY-5.MD.1Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

NY-5.MD.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

NY-5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

NY-5.MD.3.aRecognize that a cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

NY-5.MD.3.bRecognize that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

NY-5.MD.4Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.

NY-5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

NY-5.MD.5.aFind the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.

NY-5.MD.5.bApply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

NY-5.MD.5.cRecognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

NY-5.NBT Number and Operations in Base Ten

NY-5.NBT.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 110 of what it represents in the place to its left.

NY-5.NBT.2Use whole-number exponents to denote powers of 10. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

NY-5.NBT.3 Read, write, and compare decimals to thousandths.

NY-5.NBT.3.aRead and write decimals to thousandths using base-ten numerals, number names, and expanded form.

NY-5.NBT.3.bCompare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

NY-5.NBT.4Use place value understanding to round decimals to any place.

NY-5.NBT.5Fluently multiply multi-digit whole numbers using a standard algorithm.

NY-5.NBT.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

NY-5.NBT.7Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations: add and subtract decimals to hundredths; multiply and divide decimals to hundredths. Relate the strategy to a written method and explain the reasoning used.

NY-5.NF Number and Operations—Fractions

NY-5.NF.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

NY-5.NF.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

NY-5.NF.3Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

NY-5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction.

NY-5.NF.4.aInterpret the product a/b × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

NY-5.NF.4.bFind the area of a rectangle with fractional side lengths by tiling it with rectangles of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

NY-5.NF.5 Interpret multiplication as scaling (resizing).

NY-5.NF.5.aCompare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

NY-5.NF.5.bExplain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case). Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. Relate the principle of fraction equivalence a/b = a/b × n/n to the effect of multiplying a/b by 1.

NY-5.NF.6Solve real world problems involving multiplication of fractions and mixed numbers.

NY-5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

NY-5.NF.7.aInterpret division of a unit fraction by a non-zero whole number, and compute such quotients.

NY-5.NF.7.bInterpret division of a whole number by a unit fraction, and compute such quotients.

NY-5.NF.7.cSolve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

NY-5.OA Operations and Algebraic Thinking

NY-5.OA.1Apply the order of operations to evaluate numerical expressions.

NY-5.OA.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

NY-5.OA.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Grade 6

NY-6.EE Expressions, Equations, and Inequalities

NY-6.EE.1Write and evaluate numerical expressions involving whole-number exponents.

NY-6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

NY-6.EE.2.aWrite expressions that record operations with numbers and with letters standing for numbers.

NY-6.EE.2.bIdentify parts of an expression using mathematical terms (term, coefficient, sum, difference, product, factor, and quotient); view one or more parts of an expression as a single entity.

NY-6.EE.2.cEvaluate expressions given specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order (Order of Operations).

NY-6.EE.3Apply the properties of operations to generate equivalent expressions.

NY-6.EE.4Identify when two expressions are equivalent.

NY-6.EE.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

NY-6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. Understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

NY-6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q; x – p = q; px = q; and x/p = q for cases in which p, q, and x are all nonnegative rational numbers.

NY-6.EE.8Write an inequality of the form x > c, x ≥ c, x ≤ c, or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on a number line.

NY-6.EE.9Use variables to represent two quantities in a real-world problem that change in relationship to one another. Given a verbal context and an equation, identify the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

NY-6.G Geometry

NY-6.G.1Find area of triangles, trapezoids, and other polygons by composing into rectangles or decomposing into triangles and quadrilaterals. Apply these techniques in the context of solving real-world and mathematical problems.

NY-6.G.2Find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

NY-6.G.3Draw polygons in the coordinate plane given coordinates for the vertices. Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

NY-6.G.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

NY-6.G.5Use area and volume models to explain perfect squares and perfect cubes.

NY-6.NS The Number System

NY-6.NS.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.

NY-6.NS.2Fluently divide multi-digit numbers using a standard algorithm.

NY-6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.

NY-6.NS.4Find the greatest common factor of two whole numbers less than or equal to 100. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor other than 1. Find the least common multiple of two whole numbers less than or equal to 12.

NY-6.NS.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

NY-6.NS.6 Understand a rational number as a point on the number line. Use number lines and coordinate axes to represent points on a number line and in the coordinate plane with negative number coordinates.

NY-6.NS.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself, and that 0 is its own opposite.

NY-6.NS.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane. Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

NY-6.NS.6.cFind and position integers and other rational numbers on a horizontal or vertical number line. Find and position pairs of integers and other rational numbers on a coordinate plane.

NY-6.NS.7 Understand ordering and absolute value of rational numbers.

NY-6.NS.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line.

NY-6.NS.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.

NY-6.NS.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line. Interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

NY-6.NS.7.dDistinguish comparisons of absolute value from statements about order.

NY-6.NS.8Solve real-world and mathematical problems by graphing points on a coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

NY-6.RP Ratios and Proportional Relationships

NY-6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

NY-6.RP.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship.

NY-6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems.

NY-6.RP.3.aMake tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

NY-6.RP.3.bSolve unit rate problems.

NY-6.RP.3.cFind a percent of a quantity as a rate per 100. Solve problems that involve finding the whole given a part and the percent, and finding a part of a whole given the percent.

NY-6.RP.3.dUse ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

NY-6.SP Statistics and Probability

NY-6.SP.1aRecognize that a statistical question is one that anticipates variability in the data related to the question and accounts for it in the answers.

NY-6.SP.1bUnderstand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population.

NY-6.SP.1cUnderstand that the method and sample size used to collect data for a particular question is intended to reduce the difference between a population and a sample taken from the population so valid inferences can be drawn about the population. Generate multiple samples (or simulated samples) of the same size to recognize the variation in estimates or predictions.

NY-6.SP.2Understand that a set of quantitative data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

NY-6.SP.3Recognize that a measure of center for a quantitative data set summarizes all of its values with a single number while a measure of variation describes how its values vary with a single number.

NY-6.SP.4Display quantitative data in plots on a number line, including dot plots, and histograms.

NY-6.SP.5 Summarize quantitative data sets in relation to their context.

NY-6.SP.5.aReport the number of observations.

NY-6.SP.5.bDescribe the nature of the attribute under investigation, including how it was measured and its units of measurement.

NY-6.SP.5.cCalculate range and measures of center, as well as describe any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

NY-6.SP.5.dRelate the range and the choice of measures of center to the shape of the data distribution and the context in which the data were gathered.

NY-6.SP.6Understand that the probability of a chance event is a number between 0 and 1 inclusive, that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

NY-6.SP.7Approximate the probability of a simple event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

NY-6.SP.8 Develop a probability model and use it to find probabilities of simple events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

NY-6.SP.8.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of simple events.

NY-6.SP.8.bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Grade 7

NY-7.EE Expressions, Equations, and Inequalities

NY-7.EE.1Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations.

NY-7.EE.2Understand that rewriting an expression in different forms in real-world and mathematical problems can reveal and explain how the quantities are related.

NY-7.EE.3Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies.

NY-7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

NY-7.EE.4.aSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

NY-7.EE.4.bSolve word problems leading to inequalities of the form px + q > r, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem.

NY-7.G Geometry

NY-7.G.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

NY-7.G.2Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

NY-7.G.3Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the base.

NY-7.G.4Apply the formulas for the area and circumference of a circle to solve problems.

NY-7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

NY-7.G.6Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids. Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids. Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms.

NY-7.NS The Number System

NY-7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. Represent addition and subtraction on a horizontal or vertical number line.

NY-7.NS.1.aDescribe situations in which opposite quantities combine to make 0.

NY-7.NS.1.bUnderstand addition of rational numbers; p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

NY-7.NS.1.cUnderstand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

NY-7.NS.1.dApply properties of operations as strategies to add and subtract rational numbers.

NY-7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

NY-7.NS.2.aUnderstand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

NY-7.NS.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = −p/q = p/-q. Interpret quotients of rational numbers by describing real-world contexts.

NY-7.NS.2.cApply properties of operations as strategies to multiply and divide rational numbers.

NY-7.NS.2.dConvert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

NY-7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.

NY-7.RP Ratios and Proportional Relationships

NY-7.RP.1Compute unit rates associated with ratios of fractions.

NY-7.RP.2 Recognize and represent proportional relationships between quantities.

NY-7.RP.2.aDecide whether two quantities are in a proportional relationship.

NY-7.RP.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

NY-7.RP.2.cRepresent a proportional relationship using an equation.

NY-7.RP.2.dExplain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

NY-7.RP.3Use proportional relationships to solve multistep ratio and percent problems.

NY-7.SP Statistics and Probability

NY-7.SP.1Construct and interpret box-plots, find the interquartile range, and determine if a data point is an outlier.

NY-7.SP.3Informally assess the degree of visual overlap of two quantitative data distributions.

NY-7.SP.4Use measures of center and measures of variability for quantitative data from random samples or populations to draw informal comparative inferences about the populations.

NY-7.SP.8 Find probabilities of compound events using organized lists, sample space tables, tree diagrams, and simulation.

NY-7.SP.8.aUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

NY-7.SP.8.bRepresent sample spaces for compound events using methods such as organized lists, sample space tables, and tree diagrams. For an event described in everyday language, identify the outcomes in the sample space which compose the event.

NY-7.SP.8.cDesign and use a simulation to generate frequencies for compound events.

Grade 8

NY-8.EE Expressions, Equations, and Inequalities

NY-8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.

NY-8.EE.2Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational.

NY-8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

NY-8.EE.4Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.

NY-8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

NY-8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

NY-8.EE.7 Solve linear equations in one variable.

NY-8.EE.7.aRecognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms.

NY-8.EE.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms.

NY-8.EE.8 Analyze and solve pairs of simultaneous linear equations.

NY-8.EE.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Recognize when the system has one solution, no solution, or infinitely many solutions.

NY-8.EE.8.bSolve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve simple cases by inspection.

NY-8.EE.8.cSolve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.

NY-8.F Functions

NY-8.F.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

NY-8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

NY-8.F.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of functions that are linear and non-linear.

NY-8.F.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

NY-8.F.5Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described in a real-world context.

NY-8.G Geometry

NY-8.G.1 Verify experimentally the properties of rotations, reflections, and translations.

NY-8.G.1.aVerify experimentally lines are mapped to lines, and line segments to line segments of the same length.

NY-8.G.1.bVerify experimentally angles are mapped to angles of the same measure.

NY-8.G.1.cVerify experimentally parallel lines are mapped to parallel lines.

NY-8.G.2Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane.

NY-8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

NY-8.G.4Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.

NY-8.G.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

NY-8.G.6Understand a proof of the Pythagorean Theorem and its converse.

NY-8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

NY-8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

NY-8.G.9Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real-world problems.

NY-8.NS The Number System

NY-8.NS.1Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational.

NY-8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.

NY-8.SP Statistics and Probability

NY-8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

NY-8.SP.2Understand that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

NY-8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Kindergarten

NY-K.CC Counting and Cardinality

NY-K.CC.1Count to 100 by ones and by tens.

NY-K.CC.2Count to 100 by ones beginning from any given number (instead of beginning at 1).

NY-K.CC.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

NY-K.CC.4 Understand the relationship between numbers and quantities up to 20; connect counting to cardinality.

NY-K.CC.4.aWhen counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. (1:1 correspondence)

NY-K.CC.4.bUnderstand that the last number name said tells the number of objects counted, (cardinality). The number of objects is the same regardless of their arrangement or the order in which they were counted.

NY-K.CC.4.cUnderstand the concept that each successive number name refers to a quantity that is one larger.

NY-K.CC.4.dUnderstand the concept of ordinal numbers (first through tenth) to describe the relative position and magnitude of whole numbers.

NY-K.CC.5aAnswer counting questions using as many as 20 objects arranged in a line, a rectangular array, and a circle. Answer counting questions using as many as 10 objects in a scattered configuration.

NY-K.CC.5bGiven a number from 1–20, count out that many objects.

NY-K.CC.6Identify whether the number of objects in one group is greater than (more than), less than (fewer than), or equal to (the same as) the number of objects in another group.

NY-K.CC.7Compare two numbers between 1 and 10 presented as written numerals.

NY-K.G Geometry

NY-K.G.1Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

NY-K.G.2Name shapes regardless of their orientation or overall size.

NY-K.G.3Understand the difference between two-dimensional (lying in a plane, “flat”) and three-dimensional (“solid”) shapes.

NY-K.G.4Analyze, compare, and sort two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts, and other attributes.

NY-K.G.5Model objects in their environment by building and/or drawing shapes.

NY-K.G.6Compose larger shapes from simple shapes.

NY-K.MD Measurement and Data

NY-K.MD.1Describe measurable attributes of an object(s), such as length or weight, using appropriate vocabulary.

NY-K.MD.2Directly compare two objects with a common measurable attribute and describe the difference.

NY-K.MD.3Classify objects into given categories; count the objects in each category and sort the categories by count.

NY-K.MD.4Explore coins (pennies, nickels, dimes, and quarters) and begin identifying pennies and dimes.

NY-K.NBT Number and Operations in Base Ten

NY-K.NBT.1Compose and decompose the numbers from 11 to 19 into ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

NY-K.OA Operations and Algebraic Thinking

NY-K.OA.1Represent addition and subtraction using objects, fingers, pennies, drawings, sounds, acting out situations, verbal explanations, expressions, equations, or other strategies.

NY-K.OA.2aAdd and subtract within 10.

NY-K.OA.2bSolve addition and subtraction word problems within 10.

NY-K.OA.3Decompose numbers less than or equal to 10 into pairs in more than one way.

NY-K.OA.4Find the number that makes 10 when given a number from 1 to 9.

NY-K.OA.5Fluently add and subtract within 5.

NY-K.OA.6Duplicate, extend, and create simple patterns using concrete objects.

AP Calculus Learning Objectives

CB.Calc.LIM Limits

CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

CB.Calc.LIM-1.ARepresent limits analytically using correct notation.

CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.

CB.Calc.LIM-1.CEstimate limits of functions.

CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.

CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.

CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.

CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.

CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.

CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.

CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.

CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.

CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.

CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.

CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.

CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.

CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.

CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.

CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.

CB.Calc.LIM-7.ADetermine whether a series converges or diverges.

CB.Calc.LIM-7.BApproximate the sum of a series.

CB.Calc.LIM-8 Power series and Taylor series can represent functions.

CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.

CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.

CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.

CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.

CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.

CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.

CB.Calc.LIM-8.GRepresent a given function as a power series.

CB.Calc.CHA Change

CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.

CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.

CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.

CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.

CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.

CB.Calc.CHA-2.DEstimate derivatives.

CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.

CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.

CB.Calc.CHA-3.BCalculate rates of change in applied contexts.

CB.Calc.CHA-3.CInterpret rates of change in applied contexts.

CB.Calc.CHA-3.DCalculate related rates in applied contexts.

CB.Calc.CHA-3.EInterpret related rates in applied contexts.

CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.

CB.Calc.CHA-3.GCalculate derivatives of parametric functions.

CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.

CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.

CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.

CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.

CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.

CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.

CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.

CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.

CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.

CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.

CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.

CB.Calc.CHA-6 Arc length can be calculated using definite integrals.

CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.

CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.

CB.Calc.FUN Analysis of Functions

CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.

CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.

CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.

CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.

CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.

CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.

CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.

CB.Calc.FUN-3.ACalculate derivatives of familiar functions.

CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.

CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.

CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.

CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.

CB.Calc.FUN-3.FDetermine higher order derivatives of a function.

CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.

CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.

CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.

CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.

CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.

CB.Calc.FUN-4.DDetermine critical points of implicit relations.

CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.

CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.

CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.

CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.

CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.

CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.

CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.

CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.

CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.

CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.

CB.Calc.FUN-7.BVerify solutions to differential equations.

CB.Calc.FUN-7.CEstimate solutions to differential equations.

CB.Calc.FUN-7.DDetermine general solutions to differential equations.

CB.Calc.FUN-7.EDetermine particular solutions to differential equations.

CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.

CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.

CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.

CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.

CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.

CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution

CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.

CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.

CB.Stats.VAR-1.BIdentify variables in a set of data.

CB.Stats.VAR-1.CClassify types of variables.

CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.

CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.

CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.

CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.

CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.

CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.

CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.

CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.

CB.Stats.VAR-2 Normal distributions and comparing data.

CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.

CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.

CB.Stats.VAR-2.CCompare measures of relative position in data sets.

CB.Stats.VAR-3 Designing and analyzing experiments.

CB.Stats.VAR-3.AIdentify the components of an experiment.

CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.

CB.Stats.VAR-3.CCompare experimental designs and methods.

CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.

CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.

CB.Stats.VAR-4 Probability rules and calculations.

CB.Stats.VAR-4.ACalculate probabilities for events and their complements.

CB.Stats.VAR-4.BInterpret probabilities for events.

CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.

CB.Stats.VAR-4.DCalculate conditional probabilities.

CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.

CB.Stats.VAR-5 Discrete random variables and probability distributions.

CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.

CB.Stats.VAR-5.BInterpret a probability distribution.

CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.

CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.

CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.

CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.

CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.

CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.

CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.

CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.

CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.

CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.

CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.

CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.

CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.

CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.

CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.

CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.

CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.

CB.Stats.VAR-7.ADescribe t-distributions.

CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.

CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.

CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.

CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.

CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.

CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.

CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.

CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.

CB.Stats.VAR-8 Chi-square distributions and tests.

CB.Stats.VAR-8.ADescribe chi-square distributions.

CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.

CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.

CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.

CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.

CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.

CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.

CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.

CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.

CB.Stats.UNC Uncertainty and Inference

CB.Stats.UNC-1 Representing and describing categorical and quantitative data.

CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.

CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.

CB.Stats.UNC-1.CRepresent categorical data graphically.

CB.Stats.UNC-1.DDescribe categorical data represented graphically.

CB.Stats.UNC-1.ECompare multiple sets of categorical data.

CB.Stats.UNC-1.FClassify types of quantitative variables.

CB.Stats.UNC-1.GRepresent quantitative data graphically.

CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.

CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.

CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.

CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.

CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.

CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.

CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.

CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.

CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.

CB.Stats.UNC-1.QCalculate statistics for two categorical variables.

CB.Stats.UNC-1.RCompare statistics for two categorical variables.

CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.

CB.Stats.UNC-2 Probability and simulation.

CB.Stats.UNC-2.AEstimate probabilities using simulation.

CB.Stats.UNC-3 Random variables and sampling distributions.

CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.

CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.

CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.

CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.

CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.

CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.

CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.

CB.Stats.UNC-3.HEstimate sampling distributions using simulation.

CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.

CB.Stats.UNC-3.JCalculate estimates for a population parameter.

CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.

CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.

CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.

CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.

CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.

CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.

CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.

CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.

CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.

CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.

CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.

CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.

CB.Stats.UNC-4 Confidence intervals for proportions and means.

CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.

CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.

CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.

CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.

CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.

CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.

CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.

CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.

CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.

CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.

CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.

CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.

CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.

CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.

CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.

CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.

CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.

CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.

CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.

CB.Stats.UNC-5 Errors in statistical inference.

CB.Stats.UNC-5.AIdentify Type I and Type II errors.

CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.

CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.

CB.Stats.UNC-5.DInterpret Type I and Type II errors.

CB.Stats.DAT Data Analysis

CB.Stats.DAT-1 Bivariate data analysis and linear regression.

CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.

CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.

CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.

CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.

CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.

CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.

CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.

CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.

CB.Stats.DAT-1.IIdentify influential points in regression.

CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.

CB.Stats.DAT-2 Studies, sampling, and sources of bias.

CB.Stats.DAT-2.AIdentify the type of a study.

CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.

CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.

CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.

CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.

CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.

CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.

CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.

CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.

CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.

CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.

CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.

CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.

CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.

CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.

CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.

CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.

CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.

CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

Standards on this page are sourced from published state frameworks. Goblins aligns tutoring to these standards.