### AI.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.

AI.NQ.2: Students will… Rewrite expressions involving radicals and rational exponents using the properties of exponents.

AI.NQ.3: Students will… Define the imaginary number i such that i² = –1.

### AI.AF: Algebra and Functions

#### AI.AF.1: Algebra

2.1.1: 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.

AI.AF.1.4: Students will… Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

AI.AF.1.5: Students will… Use the structure of an expression to identify ways to rewrite it.

AI.AF.1.6: Students will… Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

AI.AF.1.6.a: Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

AI.AF.1.6.b: Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

AI.AF.1.6.c: Use the properties of exponents to transform expressions for exponential functions.

AI.AF.1.7: Students will… Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.

2.1.3: 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.

AI.AF.1.9: Students will… Select an appropriate method to solve a quadratic equation in one variable.

AI.AF.1.9.a: Use 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. Explain how the quadratic formula is derived from this form.

AI.AF.1.9.b: Solve quadratic equations by inspection (such as x² = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

AI.AF.1.10: Students will… Select an appropriate method to solve a system of two linear equations in two variables.

AI.AF.1.10.a: Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

AI.AF.1.10.b: Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

2.1.4: 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.

AI.AF.1.11: Students will… Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately.

AI.AF.1.12: Students will… Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions.

AI.AF.1.13: Students will… Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context.

#### AI.AF.2: Connecting Algebra to Functions

2.2.1: Functions shift the emphasis from a point-by-point relationship between two variables (input/output) to considering an entire set of ordered pairs (where each first element is paired with exactly one second element) as an entity with its own features and characteristics.

AI.AF.2.14: Students will… Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane.

AI.AF.2.15: Students will… Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

AI.AF.2.15.a: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

AI.AF.2.15.b: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

AI.AF.2.16: Students will… Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).

AI.AF.2.17: Students will… Combine different types of standard functions to write, evaluate, and interpret functions in context.

AI.AF.2.17.a: Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

2.2.2: 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).

AI.AF.2.18: Students will… Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.

AI.AF.2.19: 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).

AI.AF.2.19.a: Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.

AI.AF.2.20: Students will… Graph 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, using technology where appropriate.

#### AI.AF.3: Functions

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

AI.AF.3.21: Students will… Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

AI.AF.3.22: Students will… Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.

AI.AF.3.22.a: Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions.

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

AI.AF.3.23: 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 explain the effects on the graph, using technology as appropriate.

AI.AF.3.24: Students will… Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

AI.AF.3.24.a: Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

AI.AF.3.24.b: Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

AI.AF.3.24.c: Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

AI.AF.3.25: Students will… Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

AI.AF.3.26: Students will… Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

AI.AF.3.27: Students will… Interpret the parameters of functions in terms of a context.

2.3.3: 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.

AI.AF.3.28: 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.

AI.AF.3.29: 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.

AI.AF.3.30: 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.

AI.AF.3.30.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

AI.AF.3.30.b: Graph piecewise-defined functions, including step functions and absolute value functions.

AI.AF.3.30.c: Graph exponential functions, showing intercepts and end behavior.

2.3.4: 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.

AI.AF.3.31: Students will… Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.

### AI.DSP: Data Analysis, Statistics, and Probability

#### AI.DSP.1: Quantitative Literacy

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

AI.DSP.1.32: Students will… Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk.

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

AI.DSP.1.33: Students will… Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation.

#### AI.DSP.4: Probability

3.4.1: Two events are independent if the occurrence of one event does not affect the probability of the other event. Determining whether two events are independent can be used for finding and understanding probabilities.

AI.DSP.4.37: Students will… 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').

AI.DSP.4.38: Students will… Explain whether two events, A and B, are independent, using two-way tables or tree diagrams.

3.4.2: Conditional probabilities – that is, those probabilities that are “conditioned” by some known information – can be computed from data organized in contingency tables. Conditions or assumptions may affect the computation of a probability.

AI.DSP.4.39: Students will… Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.

AI.DSP.4.40: Students will… Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language.

AI.DSP.4.41: Students will… Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context.

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.