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Skills available for New York high school math standards

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Next Generation Learning Standards: Algebra I Next Generation Learning Standards: Algebra I
Next Generation Learning Standards: Algebra II Next Generation Learning Standards: Algebra II
Next Generation Learning Standards: Geometry Next Generation Learning Standards: Geometry
Next Generation Learning Standards: The Plus Standards Next Generation Learning Standards: The Plus Standards
New York State P-12 Common Core Learning Standards New York State P-12 Common Core Learning Standards

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AII-N Number and Quantity

AII-N.RN The Real Number System
- Extend the properties of exponents to rational exponents.
  - AII-N.RN.1 Explore how the meaning of rational exponents follows from extending the properties of integer exponents.
    - Evaluate rational exponents (A2-N.1)
  - AII-N.RN.2 Convert between radical expressions and expressions with rational exponents using the properties of exponents.
AII-N.CN The Complex Number System
- Perform arithmetic operations with complex numbers.
  - AII-N.CN.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
    - Introduction to complex numbers (A2-I.1)
  - AII-N.CN.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

AII-A Algebra

AII-A.SSE Seeing Structure in Expressions
- Interpret the structure of expressions.
  - AII-A.SSE.2 Recognize and use the structure of an expression to identify ways to rewrite it.
- Write expressions in equivalent forms to reveal their characteristics.
  - 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.
    - AII-A.SSE.3a Factor quadratic expressions including leading coefficients other than 1 to reveal the zeros of the function it defines.
    - AII-A.SSE.3c Use the properties of exponents to rewrite exponential expressions.
AII-A.APR Arithmetic with Polynomials and Rational Expressions
- Understand the relationship between zeros and factors of polynomials.
  - AII-A.APR.2 Apply 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.3 Identify zeros of polynomial functions when suitable factorizations are available.
- Rewrite rational expressions.
  - AII-A.APR.6 Rewrite 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).
    - Divide polynomials using long division (A2-L.4)
    - Simplify rational expressions (A2-O.4)
AII-A.CED Creating Equations
- Create equations that describe numbers or relationships.
  - AII-A.CED.1 Create equations and inequalities in one variable to represent a real-world context.
AII-A.REI Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning.
  - AII-A.REI.1b Explain 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.2 Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise.
    - Solve radical equations (A2-M.14)
    - Solve rational equations (A2-O.7)
- Solve equations and inequalities in one variable.
  - AII-A.REI.4 Solve quadratic equations in one variable.
    - AII-A.REI.4b Solve quadratic equations by: inspection, taking square roots, factoring, completing the square, the quadratic formula, and graphing. Write complex solutions in a + bi form.
- Solve systems of equations.
  - AII-A.REI.7b Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
    - Solve a system of linear and quadratic equations by graphing: parabolas (A2-E.15)
    - Solve a system of linear and quadratic equations: parabolas (A2-E.16)
- Represent and solve equations and inequalities graphically.
  - AII-A.REI.11 Given the equations y = f(x) and y = g(x):
    - AII-A.REI.11.i recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x);
      - Find solutions using a table (A2-D.6)
    - AII-A.REI.11.ii find the solutions approximately using technology to graph the functions or make tables of values;
    - AII-A.REI.11.iii find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and
      - Solve systems of linear inequalities by graphing (A2-F.2)
      - Solve systems of linear and absolute value inequalities by graphing (A2-F.3)
    - AII-A.REI.11.iv interpret the solution in context.
      - Solve a system of equations by graphing: word problems (A2-E.3)
      - Find the vertices of a solution set (A2-F.4)

AII-F Functions

AII-F.IF Interpreting Functions
- Understand the concept of a function and use function notation.
  - AII-F.IF.3 Recognize that a sequence is a function whose domain is a subset of the integers.
- Interpret functions that arise in applications in terms of the context.
  - AII-F.IF.4 For a function that models a relationship between two quantities:
    - AII-F.IF.4.i interpret key features of graphs and tables in terms of the quantities; and
    - AII-F.IF.4.ii sketch graphs showing key features given a verbal description of the relationship.
  - AII-F.IF.6 Calculate and interpret the average rate of change of a function over a specified interval.
    - Find the slope of a linear function (A2-D.8)
    - Average rate of change (A2-D.12)
- Analyze functions using different representations.
  - AII-F.IF.7 Graph functions and show key features of the graph by hand and using technology when appropriate.
    - AII-F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
    - AII-F.IF.7e Graph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.
- Analyze functions using different representations.
  - AII-F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function.
    - AII-F.IF.8b Use the properties of exponents to interpret exponential functions, and classify them as representing exponential growth or decay.
  - AII-F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
AII-F.BF Building Functions
- Build a function that models a relationship between two quantities.
  - AII-F.BF.1 Write a function that describes a relationship between two quantities.
    - AII-F.BF.1a Determine a function from context. Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - AII-F.BF.1b Combine standard function types using arithmetic operations.
  - AII-F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- Build new functions from existing functions.
  - AII-F.BF.3b Using f(x) + k, k f(x), f(kx), and f(x + k):
    - AII-F.BF.3b.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);
      - Function transformation rules (A2-Q.1)
      - Describe function transformations (A2-Q.6)
    - AII-F.BF.3b.ii find the value of k given the graphs;
    - AII-F.BF.3b.iii write a new function using the value of k; and
    - AII-F.BF.3b.iv use technology to experiment with cases and explore the effects on the graph.
  - AII-F.BF.4a Find the inverse of a one-to-one function both algebraically and graphically.
  - AII-F.BF.5a Understand inverse relationships between exponents and logarithms algebraically and graphically.
  - AII-F.BF.6 Represent and evaluate the sum of a finite arithmetic or finite geometric series, using summation (sigma) notation.
  - AII-F.BF.7 Explore the derivation of the formulas for finite arithmetic and finite geometric series. Use the formulas to solve problems.
    - Find the sum of an arithmetic series (A2-CC.15)
    - Find the sum of a finite geometric series (A2-CC.16)
AII-F.LE Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems.
  - AII-F.LE.2 Construct a linear or exponential function symbolically given:
    - AII-F.LE.2.i a graph;
      - Slope-intercept form: write an equation from a graph (A2)
      - Match exponential functions and graphs (A2-T.4)
    - AII-F.LE.2.ii a description of the relationship; and
    - AII-F.LE.2.iii two input-output pairs (include reading these from a table).
      - Write linear, quadratic, and exponential functions (A2)
  - AII-F.LE.4 Use logarithms to solve exponential equations, such as ab to the ct power = d (where a, b, c, and d are real numbers and b > 0) and evaluate the logarithm using technology.
- Interpret expressions for functions in terms of the situation they model.
  - AII-F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
AII-F.TF Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle.
  - AII-F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
    - Convert between radians and degrees (A2-Y.1)
    - Radians and arc length (A2-Y.2)
  - AII-F.TF.2 Apply 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.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
    - Symmetry and periodicity of trigonometric functions (A2-BB.2)
- Model periodic phenomena with trigonometric functions.
  - AII-F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, and midline.
- Prove and apply trigonometric identities.
  - AII-F.TF.8 Prove 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.
    - Trigonometric identities I (A2-BB.3)
    - Trigonometric identities II (A2-BB.4)

AII-S Statistics and Probability

AII-S.ID Interpreting Categorical and Quantitative Data
- Summarize, represent, and interpret data on a single count or measurement variable.
  - AII-S.ID.4a Recognize whether or not a normal curve is appropriate for a given data set.
  - AII-S.ID.4b If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.
    - Find probabilities using the normal distribution I (A2-EE.11)
    - Find probabilities using the normal distribution II (A2-EE.12)
- Summarize, represent, and interpret data on two categorical and quantitative variables.
  - AII-S.ID.6 Represent bivariate data on a scatter plot, and describe how the variables' values are related.
    - AII-S.ID.6a Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.
AII-S.IC Making Inferences and Justifying Conclusions
- Understand and evaluate random processes underlying statistical experiments.
  - AII-S.IC.2 Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation.
- Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
  - AII-S.IC.3 Recognize the purposes of and differences among surveys, experiments, and observational studies. Explain how randomization relates to each.
    - Experiment design (A2-FF.12)
  - AII-S.IC.4 Given 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.
    - Find confidence intervals for population means (A2-FF.9)
    - Find confidence intervals for population proportions (A2-FF.10)
  - AII-S.IC.6a Use the tools of statistics to draw conclusions from numerical summaries.
    - Interpret confidence intervals for population means (A2-FF.11)
    - Analyze the results of an experiment using simulations (A2-FF.13)
  - AII-S.IC.6b Use the language of statistics to critique claims from informational texts. For example, causation vs correlation, bias, measures of center and spread.
    - Correlation and causation (A2)
    - Identify biased samples (A2-FF.1)
AII-S.CP Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data.
  - AII-S.CP.1 Describe 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").
    - Introduction to probability (A2-DD.1)
    - Calculate probabilities of events (A2-DD.2)
  - AII-S.CP.4 Interpret 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.
    - Find probabilities using two-way frequency tables (A2-DD.6)
    - Find conditional probabilities using two-way frequency tables (A2-DD.11)
- Use the rules of probability to compute probabilities of compound events in a uniform probability model.
  - AII-S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
    - Find probabilities using the addition rule (A2-DD.12)

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AII-N Number and Quantity

AII-N.RN The Real Number System

Extend the properties of exponents to rational exponents.

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

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

AII-N.CN The Complex Number System

Perform arithmetic operations with complex numbers.

AII-N.CN.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

AII-N.CN.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

AII-A Algebra

AII-A.SSE Seeing Structure in Expressions

Interpret the structure of expressions.

AII-A.SSE.2 Recognize and use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to reveal their characteristics.

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.

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

AII-A.SSE.3c Use the properties of exponents to rewrite exponential expressions.

AII-A.APR Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

AII-A.APR.2 Apply 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.3 Identify zeros of polynomial functions when suitable factorizations are available.

Rewrite rational expressions.

AII-A.APR.6 Rewrite 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 Creating Equations

Create equations that describe numbers or relationships.

AII-A.CED.1 Create equations and inequalities in one variable to represent a real-world context.

AII-A.REI Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning.

AII-A.REI.1b Explain 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.2 Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise.

Solve equations and inequalities in one variable.

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

AII-A.REI.4b Solve quadratic equations by: inspection, taking square roots, factoring, completing the square, the quadratic formula, and graphing. Write complex solutions in a + bi form.

Solve systems of equations.

AII-A.REI.7b Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Represent and solve equations and inequalities graphically.

AII-A.REI.11 Given the equations y = f(x) and y = g(x):

AII-A.REI.11.i recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x);

AII-A.REI.11.ii find the solutions approximately using technology to graph the functions or make tables of values;

AII-A.REI.11.iii find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and

AII-A.REI.11.iv interpret the solution in context.

AII-F Functions

AII-F.IF Interpreting Functions

Understand the concept of a function and use function notation.

AII-F.IF.3 Recognize that a sequence is a function whose domain is a subset of the integers.

Interpret functions that arise in applications in terms of the context.

AII-F.IF.4 For a function that models a relationship between two quantities:

AII-F.IF.4.i interpret key features of graphs and tables in terms of the quantities; and

AII-F.IF.4.ii sketch graphs showing key features given a verbal description of the relationship.

AII-F.IF.6 Calculate and interpret the average rate of change of a function over a specified interval.

Analyze functions using different representations.

AII-F.IF.7 Graph functions and show key features of the graph by hand and using technology when appropriate.

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

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

Analyze functions using different representations.

AII-F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function.

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

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

AII-F.BF Building Functions

Build a function that models a relationship between two quantities.

AII-F.BF.1 Write a function that describes a relationship between two quantities.

AII-F.BF.1a Determine a function from context. Determine an explicit expression, a recursive process, or steps for calculation from a context.

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

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

Build new functions from existing functions.

AII-F.BF.3b Using f(x) + k, k f(x), f(kx), and f(x + k):

AII-F.BF.3b.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);

AII-F.BF.3b.ii find the value of k given the graphs;

AII-F.BF.3b.iii write a new function using the value of k; and

AII-F.BF.3b.iv use technology to experiment with cases and explore the effects on the graph.

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

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

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

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

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

Construct and compare linear, quadratic, and exponential models and solve problems.

AII-F.LE.2 Construct a linear or exponential function symbolically given: