Arithmetic With Polynomials And Rational Expressions 
HSA.APR: Perform Arithmetic Operations On Polynomials 
HSA.APR.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. 
Circles 
HSG.C: Understand And Apply Theorems About Circles 
HSG.C.1  Prove that all circles are similar. 

HSG.C.2  Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 

HSG.C.3  Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 

HSG.C.4  (+) Construct a tangent line from a point outside a given circle to the circle. 
Congruence 
HSG.CO: Understand Congruence In Terms Of Rigid Motions 
HSG.CO.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.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.8  Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 
Functions 
8.F: Define, Evaluate, And Compare Functions. 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 

8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 

8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 
Operations And Algebraic Thinking 
1.OA: Represent And Solve Problems Involving Addition And Subtraction. 
1.OA.1  Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.2 

1.OA.2  Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 
2.OA: Represent And Solve Problems Involving Addition And Subtraction. 
2.OA.1  Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
3.OA: Represent And Solve Problems Involving Multiplication And Division. 
3.OA.1  Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 

3.OA.2  Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 

3.OA.3  Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 

3.OA.4  Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?. 
4.OA: Use The Four Operations With Whole Numbers To Solve Problems. 
4.OA.1  Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 

4.OA.2  Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1 

4.OA.3  Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 
5.OA: Write And Interpret Numerical Expressions. 
5.OA.1  Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 

5.OA.2  Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. 
Reasoning With Equations And Inequalities 
HSA.REI: Solve Equations And Inequalities In One Variable 
HSA.REI.3  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

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

HSA.REI.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.4.b  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. 
Solve Systems Of Equations 
HSA.REI.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.6  Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 

HSA.REI.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 x2 + y2 = 3. 

HSA.REI.8  (+) Represent a system of linear equations as a single matrix equation in a vector variable. 

HSA.REI.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). 
Represent And Solve Equations And Inequalities Graphically 
HSA.REI.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.11  Explain why the xcoordinates 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.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 halfplanes. 
Seeing Structure In Expressions 
HSA.SSE: Write Expressions In Equivalent Forms To Solve Problems 
HSA.SSE.3  Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ 

HSA.SSE.3.a  Factor a quadratic expression to reveal the zeros of the function it defines. 

HSA.SSE.3.b  Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 

HSA.SSE.3.c  Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 

HSA.SSE.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. For example, calculate mortgage payments.★ 
Similarity, Right Triangles, And Trigonometry 
HSG.SRT: Understand Similarity In Terms Of Similarity Transformations 
HSG.SRT.1  Verify experimentally the properties of dilations given by a center and a scale factor: 

HSG.SRT.1.a  A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. 

HSG.SRT.1.b  The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 

HSG.SRT.2  Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 

HSG.SRT.3  Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 
Define Trigonometric Ratios And Solve Problems Involving Right Triangles 
HSG.SRT.6  Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 

HSG.SRT.7  Explain and use the relationship between the sine and cosine of complementary angles. 

HSG.SRT.8  Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ 
The Number System 
6.NS: Compute Fluently With MultiDigit Numbers And Find Common Factors And Multiples. 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm. 

6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. 

6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). 
Vector And Matrix Quantities 
HSN.VM: Perform Operations On Vectors. 
HSN.VM.4  (+) Add and subtract vectors. 

HSN.VM.4.a  Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 

HSN.VM.4.b  Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 

HSN.VM.4.c  Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 

HSN.VM.5  (+) Multiply a vector by a scalar. 

HSN.VM.5.a  Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). 

HSN.VM.5.b  Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 