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### 9-12.HSN Number and Quantity

- 9-12.HSN-RN The Real Number System
- 9-12.HSN-RN.A Extend the properties of exponents to rational exponents.
- 9-12.HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
- 9-12.HSN-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

- 9-12.HSN-RN.B Use properties of rational and irrational numbers.
- 9-12.HSN-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

- Checkpoint opportunity

- 9-12.HSN-RN.A Extend the properties of exponents to rational exponents.
- 9-12.HSN-Q Quantities
- 9-12.HSN-Q.A Reason quantitatively and use units to solve problems.
- 9-12.HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- 9-12.HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- 9-12.HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

- Checkpoint opportunity

- 9-12.HSN-Q.A Reason quantitatively and use units to solve problems.

### 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.A.1 Interpret expressions that represent a quantity in terms of its context.
- 9-12.HSA-SSE.B Write expressions in equivalent forms to solve problems
- 9-12.HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- 9-12.HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.
- 9-12.HSA-SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
- 9-12.HSA-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions.

- 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-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- Checkpoint opportunity

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

- Checkpoint opportunity

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

- Checkpoint opportunity

- 9-12.HSA-CED.A Create equations that describe numbers or relationships
- 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.1 Explain each step in solving a simple equation 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.

- 9-12.HSA-REI.B Solve equations and inequalities in one variable
- 9-12.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- 9-12.HSA-REI.B.4 Solve quadratic equations in one variable.
- 9-12.HSA-REI.B.4a 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. Derive the quadratic formula from this form.
- 9-12.HSA-REI.B.4b Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

- Checkpoint opportunity
- 9-12.HSA-REI.C Solve systems of equations
- 9-12.HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- 9-12.HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- 9-12.HSA-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
- 9-12.HSA-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.
- 9-12.HSA-REI.C.9 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

- 9-12.HSA-REI.D Represent and solve equations and inequalities graphically
- 9-12.HSA-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
- 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.HSA-REI.D.12 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.

- Checkpoint opportunity

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

### 9-12.HSF Functions

- 9-12.HSF-IF Interpreting Functions
- 9-12.HSF-IF.A Understand the concept of a function and use function notation
- 9-12.HSF-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- 9-12.HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- 9-12.HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

- 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.

- Checkpoint opportunity
- 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.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.
- 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.7e Graph exponential and

- 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.A Understand the concept of a function and use function notation