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HSA-REI
Algebra - Reasoning with Equations and Inequalities

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

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.

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

HSA-REI.B
Solve equations and inequalities in one variable.

HSA-REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

HSA-REI.B.4
Solve quadratic equations in one variable.

HSA-REI.B.4.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. Derive the quadratic formula from this form.

HSA-REI.B.4.b
Solve quadratic equations by inspection (e.g., for \(x^2 = 49\)), 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 \(a \pm bi\) for real numbers \(a\) and \(b\).

HSA-REI.C
Solve systems of equations.

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.

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.

HSA-REI.C.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line \(y = -3x\) and the circle \(x^2 + y^2 = 3\).

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

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).

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.

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.

HSF-IF
Functions - Interpreting Functions

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

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 \(y = f(x)\).

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.

HSF-IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by \(f(0) = f(1) = 1\), \(f(n+1) = f(n) + f(n-1)\) for \(n \ge 1\).

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

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function \(h(n)\) gives the number of person-hours it takes to assemble \(n\) engines in a factory, then the positive integers would be an appropriate domain for the function.

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.

HSF-IF.C
Analyze functions using different representations.

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.

HSF-IF.C.7.a
Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

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

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

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.

HSG-CO
Geometry - Congruence

HSG-CO.A
Experiment with transformations in the plane

HSG-CO.A.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

HSG-CO.A.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

HSG-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

HSG-CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

HSG-CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HSG-CO.B
Understand congruence in terms of rigid motions

HSG-CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

HSG-CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

HSG-CO.B.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

HSG-CO.C
Prove geometric theorems

HSG-CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to \(180^\circ\); base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

HSG-CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

HSG-CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

HSG-CO.D
Make geometric constructions

HSG-CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

HSG-CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

HSS-CP
Statistics and Probability - Conditional Probability and the Rules of Probability

HSS-CP.A
Understand independence and conditional probability and use them to interpret data

HSS-CP.A.1
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”).

HSS-CP.A.2
Understand that two events \(A\) and \(B\) are independent if the probability of \(A\) and \(B\) occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

HSS-CP.A.3
Understand the conditional probability of \(A\) given \(B\) as \$, and interpret independence of \(A\) and \(B\) as saying that the conditional probability of \(A\) given \(B\) is the same as the probability of \(A\), and the conditional probability of \(B\) given \(A\) is the same as the probability of \(B\).

HSS-CP.A.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

HSS-CP.A.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

HSS-CP.B
Use the rules of probability to compute probabilities of compound events in a uniform probability model

HSS-CP.B.6
Find the conditional probability of \(A\) given \(B\) as the fraction of \(B\)'s outcomes that also belong to \(A\), and interpret the answer in terms of the model.

HSS-CP.B.7
Apply the Addition Rule, \$, and interpret the answer in terms of the model.

HSS-ID
Statistics and Probability - Interpreting Categorical and Quantitative Data

HSS-ID.A
Summarize, represent, and interpret data on a single count or measurement variable

HSS-ID.A.1
Represent data with plots on the real number line (dot plots, histograms, and box plots).

HSS-ID.A.2
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HSS-ID.A.3
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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.

HSS-ID.B
Summarize, represent, and interpret data on two categorical and quantitative variables

HSS-ID.B.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

HSS-ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

HSS-ID.B.6.a
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

HSS-ID.B.6.b
Informally assess the fit of a function by plotting and analyzing residuals.

HSS-ID.B.6.c
Fit a linear function for a scatter plot that suggests a linear association.

HSS-ID.C
Interpret linear models

HSS-ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

HSS-ID.C.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.

HSS-ID.C.9
Distinguish between correlation and causation.