The Common Core in Michigan

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Skills available for Michigan high school math standards

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Common Core State Standards: Algebra 1 Common Core State Standards: Algebra 1
Common Core State Standards: Algebra 2 Common Core State Standards: Algebra 2
Common Core State Standards: All High School Pathways Common Core State Standards: All High School Pathways
Common Core State Standards: Fourth Courses Common Core State Standards: Fourth Courses
Common Core State Standards: Geometry Common Core State Standards: Geometry
Common Core State Standards: Mathematical Practices Common Core State Standards: Mathematical Practices
Common Core State Standards: Mathematics I Common Core State Standards: Mathematics I
Common Core State Standards: Mathematics II Common Core State Standards: Mathematics II
Common Core State Standards: Mathematics III Common Core State Standards: Mathematics III
Common Core State Standards: Precalculus Common Core State Standards: Precalculus

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9-12.HSN Number and Quantity

9-12.HSN-CN The Complex Number System
- 9-12.HSN-CN.C Use complex numbers in polynomial identities and equations.
  - 9-12.HSN-CN.C.8 Extend polynomial identities to the complex numbers.
  - 9-12.HSN-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
    - Fundamental Theorem of Algebra (A2-L.17)

9-12.HSA Algebra

9-12.HSA-SSE Seeing Structure in Expressions
- 9-12.HSA-SSE.A Interpret the structure of expressions
  - 9-12.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
    - 9-12.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.
      - Interpret parts of quadratic expressions: word problems (A2-K.)
      - Polynomial vocabulary (A2-L.1)
    - 9-12.HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.
      - Interpret parts of quadratic expressions: word problems (A2-K.)
  - 9-12.HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
- 9-12.HSA-SSE.B Write expressions in equivalent forms to solve problems
  - 9-12.HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
    - Find the sum of a finite geometric series (A2-CC.16)
9-12.HSA-APR Arithmetic with Polynomials and Rational Expressions
- 9-12.HSA-APR.A Perform arithmetic operations on polynomials
  - 9-12.HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
    - Add and subtract polynomials (A2-L.2)
    - Multiply polynomials (A2-L.3)
- 9-12.HSA-APR.B Understand the relationship between zeros and factors of polynomials
  - 9-12.HSA-APR.B.2 Know and 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).
  - 9-12.HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
- 9-12.HSA-APR.C Use polynomial identities to solve problems
  - 9-12.HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.
  - 9-12.HSA-APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
- 9-12.HSA-APR.D Rewrite rational expressions
  - 9-12.HSA-APR.D.6 Rewrite simple 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), using inspection, long division, or, for the more complicated examples, a computer algebra system.
  - 9-12.HSA-APR.D.7 Understand 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.
    - Multiply and divide rational expressions (A2-O.5)
    - Add and subtract rational expressions (A2-O.6)
9-12.HSA-CED Creating Equations
- 9-12.HSA-CED.A Create equations that describe numbers or relationships
  - 9-12.HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems.
    - Write exponential functions: word problems (A2-T.)
    - Solve quadratic equations: word problems (PC-C.)
  - 9-12.HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - 9-12.HSA-CED.A.3 Represent 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.
  - 9-12.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
    - Solve multi-variable equations (A2-B.6)
9-12.HSA-REI Reasoning with Equations and Inequalities
- 9-12.HSA-REI.A Understand solving equations as a process of reasoning and explain the reasoning
  - 9-12.HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
    - Solve radical equations (A2-M.14)
    - Solve rational equations (A2-O.7)
- 9-12.HSA-REI.D Represent and solve equations and inequalities graphically
  - 9-12.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

9-12.HSF Functions

9-12.HSF-IF Interpreting Functions
- 9-12.HSF-IF.B Interpret functions that arise in applications in terms of the context
  - 9-12.HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  - 9-12.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
    - Domain and range of polynomials (A2-L.16)
    - Domain and range of radical functions (A2-M.12)
  - 9-12.HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
    - Average rate of change (A2-D.12)
- 9-12.HSF-IF.C Analyze functions using different representations
  - 9-12.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - 9-12.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
      - Graph an absolute value function (A2-D.13)
      - Graph piecewise-defined functions (A2-D.17)
    - 9-12.HSF-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
    - 9-12.HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - 9-12.HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - 9-12.HSF-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
    - 9-12.HSF-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions.
  - 9-12.HSF-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
9-12.HSF-BF Building Functions
- 9-12.HSF-BF.A Build a function that models a relationship between two quantities
  - 9-12.HSF-BF.A.1 Write a function that describes a relationship between two quantities.
    - 9-12.HSF-BF.A.1b Combine standard function types using arithmetic operations.
- 9-12.HSF-BF.B Build new functions from existing functions
  - 9-12.HSF-BF.B.3 Identify the effect on the graph of 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); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
  - 9-12.HSF-BF.B.4 Find inverse functions.
    - 9-12.HSF-BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
9-12.HSF-LE Linear, Quadratic, and Exponential Models
- 9-12.HSF-LE.A Construct and compare linear, quadratic, and exponential models and solve problems
  - 9-12.HSF-LE.A.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
9-12.HSF-TF Trigonometric Functions
- 9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle
  - 9-12.HSF-TF.A.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)
  - 9-12.HSF-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
- 9-12.HSF-TF.B Model periodic phenomena with trigonometric functions
  - 9-12.HSF-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

9-12.HSG Geometry

9-12.HSG-SRT Similarity, Right Triangles, and Trigonometry
- 9-12.HSG-SRT.D Apply trigonometry to general triangles
  - 9-12.HSG-SRT.D.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
    - Area of a triangle: sine formula (A2-Z.22)
  - 9-12.HSG-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
  - 9-12.HSG-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
    - Solve a triangle (A2-Z.21)
9-12.HSG-GMD Geometric Measurement and Dimension
- 9-12.HSG-GMD.B Visualize relationships between two-dimensional and three-dimensional objects
  - 9-12.HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
    - Cross sections of three-dimensional figures (G-H.4)
    - Solids of revolution (G-H.5)
9-12.HSG-MG Modeling with Geometry
- 9-12.HSG-MG.A Apply geometric concepts in modeling situations
  - 9-12.HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
    - Pythagorean theorem (G-Q.1)
  - 9-12.HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
    - Calculate density, mass, and volume (G-W.9)
  - 9-12.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

9-12.HSS Statistics and Probability

9-12.HSS-ID Interpreting Categorical and Quantitative Data
- 9-12.HSS-ID.A Summarize, represent, and interpret data on a single count or measurement variable
  - 9-12.HSS-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
9-12.HSS-IC Making Inferences and Justifying Conclusions
- 9-12.HSS-IC.A Understand and evaluate random processes underlying statistical experiments
  - 9-12.HSS-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
    - Identify biased samples (A2-FF.1)
  - 9-12.HSS-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
- 9-12.HSS-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies
  - 9-12.HSS-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
    - Experiment design (A2-FF.12)
  - 9-12.HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
    - Find confidence intervals for population means (A2-FF.9)
    - Find confidence intervals for population proportions (A2-FF.10)
  - 9-12.HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
    - Analyze the results of an experiment using simulations (A2-FF.13)
  - 9-12.HSS-IC.B.6 Evaluate reports based on data.
    - Interpret confidence intervals for population means (A2-FF.11)
    - Analyze the results of an experiment using simulations (A2-FF.13)
9-12.HSS-MD Using Probability to Make Decisions
- 9-12.HSS-MD.B Use probability to evaluate outcomes of decisions
  - 9-12.HSS-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  - 9-12.HSS-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
    - Choose the better bet (A2-EE.9)

The Common Core in Michigan

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9-12.HSN Number and Quantity

9-12.HSN-CN The Complex Number System

9-12.HSN-CN.C Use complex numbers in polynomial identities and equations.

9-12.HSN-CN.C.8 Extend polynomial identities to the complex numbers.

9-12.HSN-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

9-12.HSA Algebra

9-12.HSA-SSE Seeing Structure in Expressions

9-12.HSA-SSE.A Interpret the structure of expressions

9-12.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

9-12.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients.

9-12.HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity.

9-12.HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.

9-12.HSA-SSE.B Write expressions in equivalent forms to solve problems

9-12.HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

9-12.HSA-APR Arithmetic with Polynomials and Rational Expressions

9-12.HSA-APR.A Perform arithmetic operations on polynomials

9-12.HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

9-12.HSA-APR.B Understand the relationship between zeros and factors of polynomials

9-12.HSA-APR.B.2 Know and 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).

9-12.HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

9-12.HSA-APR.C Use polynomial identities to solve problems

9-12.HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

9-12.HSA-APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

9-12.HSA-APR.D Rewrite rational expressions

9-12.HSA-APR.D.7 Understand 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.

9-12.HSA-CED Creating Equations

9-12.HSA-CED.A Create equations that describe numbers or relationships

9-12.HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems.

9-12.HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

9-12.HSA-CED.A.3 Represent 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.

9-12.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

9-12.HSA-REI Reasoning with Equations and Inequalities

9-12.HSA-REI.A Understand solving equations as a process of reasoning and explain the reasoning

9-12.HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

9-12.HSA-REI.D Represent and solve equations and inequalities graphically

9-12.HSF Functions

9-12.HSF-IF Interpreting Functions

9-12.HSF-IF.B Interpret functions that arise in applications in terms of the context

9-12.HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

9-12.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

9-12.HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

9-12.HSF-IF.C Analyze functions using different representations

9-12.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

9-12.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

9-12.HSF-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

9-12.HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

9-12.HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

9-12.HSF-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

9-12.HSF-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions.

9-12.HSF-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

9-12.HSF-BF Building Functions

9-12.HSF-BF.A Build a function that models a relationship between two quantities

9-12.HSF-BF.A.1 Write a function that describes a relationship between two quantities.

9-12.HSF-BF.A.1b Combine standard function types using arithmetic operations.

9-12.HSF-BF.B Build new functions from existing functions

9-12.HSF-BF.B.3 Identify the effect on the graph of 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); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

9-12.HSF-BF.B.4 Find inverse functions.

9-12.HSF-BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

9-12.HSF-LE Linear, Quadratic, and Exponential Models

9-12.HSF-LE.A Construct and compare linear, quadratic, and exponential models and solve problems

9-12.HSF-LE.A.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

9-12.HSF-TF Trigonometric Functions

9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle

9-12.HSF-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

9-12.HSF-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

9-12.HSF-TF.B Model periodic phenomena with trigonometric functions

9-12.HSF-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

9-12.HSG Geometry

9-12.HSG-SRT Similarity, Right Triangles, and Trigonometry

9-12.HSG-SRT.D Apply trigonometry to general triangles

9-12.HSG-SRT.D.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

9-12.HSG-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.

9-12.HSG-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

9-12.HSG-GMD Geometric Measurement and Dimension

9-12.HSG-GMD.B Visualize relationships between two-dimensional and three-dimensional objects

9-12.HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

9-12.HSG-MG Modeling with Geometry