### AII.NQ: Number and Quantity

#### 1.1: 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.

AII.NQ.1: Students will… Identify numbers written in the form a + bi, where a and b are real numbers and i² = –1, as complex numbers.

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

#### 1.2: Matrices are a useful way to represent information.

AII.NQ.2: Students will… Use matrices to represent and manipulate data.

AII.NQ.3: Students will… Multiply matrices by scalars to produce new matrices.

AII.NQ.4: Students will… Add, subtract, and multiply matrices of appropriate dimensions.

AII.NQ.5: Students will… 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.

AII.NQ.5.a: Students will… Find the additive and multiplicative inverses of square matrices, using technology as appropriate.

AII.NQ.5.b: Students will… Explain the role of the determinant in determining if a square matrix has a multiplicative inverse.

#### AII.NQ.1: Algebra

1.3.1: 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.

AII.NQ.1.6: Students will… Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.

AII.NQ.1.7: Students will… Prove polynomial identities and use them to describe numerical relationships.

1.3.3: 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.

AII.NQ.1.9: Students will… 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.

1.3.4: 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.

AII.NQ.1.10: Students will… Create equations and inequalities in one variable and use them to solve problems.

AII.NQ.1.11: Students will… Solve quadratic equations with real coefficients that have complex solutions.

AII.NQ.1.12: Students will… Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.

AII.NQ.1.13: Students will… 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.

#### AII.NQ.2: Connecting Algebra to Functions

1.4.1: 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).

AII.NQ.2.14: Students will… 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).

AII.NQ.2.14.a: Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.

#### AII.NQ.3: Functions

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

AII.NQ.3.15: Students will… Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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

AII.NQ.3.16: Students will… 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.

1.5.3: 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.

AII.NQ.3.17: Students will… 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.

AII.NQ.3.18: Students will… Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

AII.NQ.3.19: Students will… 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.

AII.NQ.3.20: Students will… Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

AII.NQ.3.20.a: Graph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior.

AII.NQ.3.20.b: Graph sine and cosine functions expressed symbolically, showing period, midline, and amplitude.

AII.NQ.3.20.c: Graph logarithmic functions expressed symbolically, showing intercepts and end behavior.

AII.NQ.3.20.d: Graph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes.

AII.NQ.3.20.e: Graph square root and cube root functions expressed symbolically.

AII.NQ.3.20.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.

AII.NQ.3.21: Students will… 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.

1.5.4: 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.NQ.3.22: Students will… 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

#### AII.DSP.1: Quantitative Literacy

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

AII.DSP.1.23: Students will… Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.

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

AII.DSP.1.24: Students will… Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.

#### AII.DSP.2: Visualizing and Summarizing Data

2.2.1: 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.

AII.DSP.2.25: Students will… From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.

AII.DSP.2.25.a: Use technology to determine if a given set of data is normal by applying the empirical rule.

#### AII.DSP.3: Statistical Inference

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

AII.DSP.3.26: Students will… Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

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

AII.DSP.3.27: Students will… 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.

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

AII.DSP.3.30: Students will… Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.

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

AII.DSP.3.31: Students will… Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.

AII.DSP.3.31.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.

AII.DSP.3.31.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.

2.3.6: 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.

AII.DSP.3.32: Students will… 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.

AII.DSP.3.32.a: Verify that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.

AII.DSP.3.32.b: Verify that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.

AII.DSP.3.32.c: Create and interpret a 95% confidence interval based on an observed mean from a sampling distribution.

### AII.GM: Geometry and Measurement

#### AII.GM.1: Measurement

3.1.1: 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.

AII.GM.1.34: Students will… 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.

#### AII.GM.4: Solving Applied Problems and Modeling in Geometry

3.4.1: 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.

AII.GM.4.35: Students will… Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.

AII.GM.4.36: Students will… Prove the Pythagorean identity sin² (theta) + cos² (theta) = 1 and use it to calculate trigonometric ratios.

Correlation last revised: 3/2/2020

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.