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Skills available for Alabama Algebra 2 standards

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Alabama Course of Study (adopted in 2019): Algebra II with Statistics Alabama Course of Study (adopted in 2019): Algebra II with Statistics
Alabama Course of Study (Common Core): Algebra II Alabama Course of Study (Common Core): Algebra II
Alabama Course of Study (Common Core): Algebra II With Trigonometry Alabama Course of Study (Common Core): Algebra II With Trigonometry
Alabama Course of Study (Common Core): Algebraic Connections Alabama Course of Study (Common Core): Algebraic Connections

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AII.NQ Number and Quantity

Together, irrational numbers and rational numbers complete the real number system, representing all points on the number line, while there exist numbers beyond the real numbers called complex numbers.
- 1 Identify numbers written in the form a + bi, where a and b are real numbers and i² = –1, as complex numbers.
  - Introduction to complex numbers (A2-I.1)
  - 1.a Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties.
Matrices are a useful way to represent information.
- 2 Use matrices to represent and manipulate data.
  - Matrix vocabulary (A2-G.1)
- 3 Multiply matrices by scalars to produce new matrices.
  - Multiply a matrix by a scalar (A2-G.4)
- 4 Add, subtract, and multiply matrices of appropriate dimensions.
- 5 Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.
  - 5.a Find the additive and multiplicative inverses of square matrices, using technology as appropriate.
    - Inverse of a matrix (A2-G.12)
    - Identify inverse matrices (A2-G.13)
  - 5.b Explain the role of the determinant in determining if a square matrix has a multiplicative inverse.
    - Determinant of a matrix (A2-G.10)
    - Is a matrix invertible? (A2-G.11)

AII.AF Algebra and Functions

Focus 1: Algebra
- Expressions can be rewritten in equivalent forms by using algebraic properties, including properties of addition, multiplication, and exponentiation, to make different characteristics or features visible.
  - 6 Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.
  - 7 Prove polynomial identities and use them to describe numerical relationships.
- Finding solutions to an equation, inequality, or system of equations or inequalities requires the checking of candidate solutions, whether generated analytically or graphically, to ensure that solutions are found and that those found are not extraneous.
  - 8 Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. Extend to radical equations.
    - Solve radical equations (A2-M.14)
- The structure of an equation or inequality (including, but not limited to, one-variable linear and quadratic equations, inequalities, and systems of linear equations in two variables) can be purposefully analyzed (with and without technology) to determine an efficient strategy to find a solution, if one exists, and then to justify the solution.
  - 9 For exponential models, express as a logarithm the solution to ab^ct = d, where a, c, and d are real numbers and the base b is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.
- Expressions, equations, and inequalities can be used to analyze and make predictions, both within mathematics and as mathematics is applied in different contexts—in particular, contexts that arise in relation to linear, quadratic, and exponential situations.
  - 10 Create equations and inequalities in one variable and use them to solve problems. Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
  - 11 Solve quadratic equations with real coefficients that have complex solutions.
    - Solve a quadratic equation using square roots (A2-K.6)
    - Using the discriminant (A2-K.12)
  - 12 Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.
  - 13 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
Focus 2: Connecting Algebra to Functions
- Graphs can be used to obtain exact or approximate solutions of equations, inequalities, and systems of equations and inequalities—including systems of linear equations in two variables and systems of linear and quadratic equations (given or obtained by using technology).
  - 14 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).
    - 14.a Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Extend to cases where f(x) and/or g(x) are polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
Focus 3: Functions
- Functions can be described by using a variety of representations: mapping diagrams, function notation (e.g., f(x) = x²), recursive definitions, tables, and graphs.
  - 15 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend to polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
- Functions that are members of the same family have distinguishing attributes (structure) common to all functions within that family.
  - 16 Identify the effect on the graph of replacing f(x) by f(x) + k, k ⋅ f(x), f(k ⋅ x), 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. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
- Functions can be represented graphically, and key features of the graphs, including zeros, intercepts, and, when relevant, rate of change and maximum/minimum values, can be associated with and interpreted in terms of the equivalent symbolic representation.
  - 17 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. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
  - 18 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
  - 19 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. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
    - Find the slope of a linear function (A2-D.8)
    - Average rate of change (A2-D.12)
  - 20 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
    - 20.a Graph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior.
    - 20.b Graph sine and cosine functions expressed symbolically, showing period, midline, and amplitude.
    - 20.c Graph logarithmic functions expressed symbolically, showing intercepts and end behavior.
      - Graph logarithmic functions (A2-T.)
    - 20.d Graph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes.
    - 20.e Graph square root and cube root functions expressed symbolically.
      - Graph square root functions (A2-M.13)
    - 20.f Compare the graphs of inverse functions and the relationships between their key features, including but not limited to quadratic, square root, exponential, and logarithmic functions.
  - 21 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, building on work with non-right triangle trigonometry.
- Functions model a wide variety of real situations and can help students understand the processes of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
  - 22 Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution's feasibility.

AII.DSP Data Analysis, Statistics, and Probability

Focus 1: Quantitative Literacy
- Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.
  - 23 Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.
- Making and defending informed data-based decisions is a characteristic of a quantitatively literate person.
  - 24 Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.
    - Find probabilities using the normal distribution II (A2-EE.12)
Focus 2: Visualizing and Summarizing Data
- Distributions of quantitative data (continuous or discrete) in one variable should be described in the context of the data with respect to what is typical (the shape, with appropriate measures of center and variability, including standard deviation) and what is not (outliers), and these characteristics can be used to compare two or more subgroups with respect to a variable.
  - 25 From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.
    - 25.a Use technology to determine if a given set of data is normal by applying the empirical rule.
    - 25.b Estimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.
Focus 3: Statistical Inference
- Study designs are of three main types: sample survey, experiment, and observational study.
  - 26 Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
    - Experiment design (A2-FF.12)
- The role of randomization is different in randomly selecting samples and in randomly assigning subjects to experimental treatment groups.
  - 27 Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.
  - 28 Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.
    - Analyze the results of an experiment using simulations (A2-FF.13)
- The scope and validity of statistical inferences are dependent on the role of randomization in the study design.
  - 29 Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.
- Bias, such as sampling, response, or nonresponse bias, may occur in surveys, yielding results that are not representative of the population of interest.
  - 30 Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.
    - Identify biased samples (A2-FF.1)
- The larger the sample size, the less the expected variability in the sampling distribution of a sample statistic.
  - 31 Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.
    - 31.a Simulate a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.
    - 31.b Demonstrate that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.
      - Distributions of sample means (A2-EE.15)
- The sampling distribution of a sample statistic formed from repeated samples for a given sample size drawn from a population can be used to identify typical behavior for that statistic. Examining several such sampling distributions leads to estimating a set of plausible values for the population parameter, using the margin of error as a measure that describes the sampling variability.
  - 32 Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.
    - 32.a Verify that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.
    - 32.b Verify that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.
    - 32.c Create and interpret a 95% confidence interval based on an observed mean from a sampling distribution.
      - Find confidence intervals for population means (A2-FF.9)
      - Interpret confidence intervals for population means (A2-FF.11)
  - 33 Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.
    - Analyze the results of an experiment using simulations (A2-FF.13)

AII.GM Geometry and Measurement

Focus 1: Measurement
- When an object is the image of a known object under a similarity transformation, a length, area, or volume on the image can be computed by using proportional relationships.
  - 34 Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.
    - Convert between radians and degrees (A2-Y.1)
    - Radians and arc length (A2-Y.2)
Focus 4: Solving Applied Problems and Modeling in Geometry
- Recognizing congruence, similarity, symmetry, measurement opportunities, and other geometric ideas, including right triangle trigonometry in real-world contexts, provides a means of building understanding of these concepts and is a powerful tool for solving problems related to the physical world in which we live.
  - 35 Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.
  - 36 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.
    - Trigonometric identities I (A2-BB.3)
    - Trigonometric identities II (A2-BB.4)
  - 37 Derive and apply the formula A = ½ ⋅ ab ⋅ sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.
    - Area of a triangle: sine formula (A2-Z.22)
  - 38 Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Extend the domain of sine and cosine to include right and obtuse angles.

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AII.NQ Number and Quantity

Together, irrational numbers and rational numbers complete the real number system, representing all points on the number line, while there exist numbers beyond the real numbers called complex numbers.

1 Identify numbers written in the form a + bi, where a and b are real numbers and i2 = –1, as complex numbers.

1.a Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties.

Matrices are a useful way to represent information.

2 Use matrices to represent and manipulate data.

3 Multiply matrices by scalars to produce new matrices.

4 Add, subtract, and multiply matrices of appropriate dimensions.

5 Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.

5.a Find the additive and multiplicative inverses of square matrices, using technology as appropriate.

5.b Explain the role of the determinant in determining if a square matrix has a multiplicative inverse.

AII.AF Algebra and Functions

Focus 1: Algebra

Expressions can be rewritten in equivalent forms by using algebraic properties, including properties of addition, multiplication, and exponentiation, to make different characteristics or features visible.

6 Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.

7 Prove polynomial identities and use them to describe numerical relationships.

Finding solutions to an equation, inequality, or system of equations or inequalities requires the checking of candidate solutions, whether generated analytically or graphically, to ensure that solutions are found and that those found are not extraneous.

8 Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. Extend to radical equations.

9 For exponential models, express as a logarithm the solution to abct = d, where a, c, and d are real numbers and the base b is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.

Expressions, equations, and inequalities can be used to analyze and make predictions, both within mathematics and as mathematics is applied in different contexts—in particular, contexts that arise in relation to linear, quadratic, and exponential situations.

10 Create equations and inequalities in one variable and use them to solve problems. Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.

11 Solve quadratic equations with real coefficients that have complex solutions.

12 Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.

Focus 2: Connecting Algebra to Functions

Graphs can be used to obtain exact or approximate solutions of equations, inequalities, and systems of equations and inequalities—including systems of linear equations in two variables and systems of linear and quadratic equations (given or obtained by using technology).

14 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).

Focus 3: Functions

Functions can be described by using a variety of representations: mapping diagrams, function notation (e.g., f(x) = x2), recursive definitions, tables, and graphs.

15 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend to polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.

Functions that are members of the same family have distinguishing attributes (structure) common to all functions within that family.

Functions can be represented graphically, and key features of the graphs, including zeros, intercepts, and, when relevant, rate of change and maximum/minimum values, can be associated with and interpreted in terms of the equivalent symbolic representation.

18 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

19 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. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

20 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

20.a Graph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior.

20.b Graph sine and cosine functions expressed symbolically, showing period, midline, and amplitude.

20.c Graph logarithmic functions expressed symbolically, showing intercepts and end behavior.

20.d Graph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes.

20.e Graph square root and cube root functions expressed symbolically.

20.f Compare the graphs of inverse functions and the relationships between their key features, including but not limited to quadratic, square root, exponential, and logarithmic functions.

21 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, building on work with non-right triangle trigonometry.

Functions model a wide variety of real situations and can help students understand the processes of making and changing assumptions, assigning variables, and finding solutions to contextual problems.

AII.DSP Data Analysis, Statistics, and Probability

Focus 1: Quantitative Literacy

Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.

23 Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.

Making and defending informed data-based decisions is a characteristic of a quantitatively literate person.

24 Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.

Focus 2: Visualizing and Summarizing Data

25 From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.

25.a Use technology to determine if a given set of data is normal by applying the empirical rule.

25.b Estimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.

Focus 3: Statistical Inference

Study designs are of three main types: sample survey, experiment, and observational study.

26 Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

The role of randomization is different in randomly selecting samples and in randomly assigning subjects to experimental treatment groups.

27 Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.

28 Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.

The scope and validity of statistical inferences are dependent on the role of randomization in the study design.

29 Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.

Bias, such as sampling, response, or nonresponse bias, may occur in surveys, yielding results that are not representative of the population of interest.

30 Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.

The larger the sample size, the less the expected variability in the sampling distribution of a sample statistic.

31 Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.

31.a Simulate a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.

31.b Demonstrate that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.

32 Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.

32.a Verify that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.

32.b Verify that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.

32.c Create and interpret a 95% confidence interval based on an observed mean from a sampling distribution.

AII.GM Geometry and Measurement

Focus 1: Measurement

When an object is the image of a known object under a similarity transformation, a length, area, or volume on the image can be computed by using proportional relationships.

34 Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.

Focus 4: Solving Applied Problems and Modeling in Geometry

35 Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.

36 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios.

1 Identify numbers written in the form a + bi, where a and b are real numbers and i² = –1, as complex numbers.

9 For exponential models, express as a logarithm the solution to ab^ct = d, where a, c, and d are real numbers and the base b is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.

Functions can be described by using a variety of representations: mapping diagrams, function notation (e.g., f(x) = x²), recursive definitions, tables, and graphs.

36 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.