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Below are all of the objectives students are required to learn in Math II, written with the original Common Core descriptions
Note:
(*) Indicates a modeling standard linking mathematics to everyday life, work, and decision-making.
(+) Indicates additional mathematics to prepare students for advanced courses.
(CA) Indicates only for California.
I. Geometry [G.]
[G.CO.] Congruence
Understand congruence in terms of rigid motions. [Build on rigid motions as a familiar starting point for development of concept of geometric proof.]
[G.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.
[G.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.
[G.CO.8] Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs.]
[G.CO.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.
[G.CO.10] Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.
[G.CO.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.
Make geometric constructions. [Formalize and explain processes.]
[G.CO.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.
[G.CO.13] Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
[G.SRT.] Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
[G.SRT.1] Verify experimentally the properties of dilations given by a center and a scale factor:
[G.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.
[G.SRT.1.b] The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
[G.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.
[G.SRT.3] Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
Prove theorems involving similarity.
[G.SRT.4] Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
[G.SRT.5] Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles.
[G.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.
[G.SRT.7] Explain and use the relationship between the sine and cosine of complementary angles.
[G.SRT.8] (*) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
[G.SRT.8.1] Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°). (CA)
Apply trigonometry to general triangles.
[G.SRT.9] (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
[G.SRT.10] (+) Prove the Laws of Sines and Cosines and use them to solve problems.
[G.SRT.11] (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
[G.C.] Circles
Understand and apply theorems about circles.
[G.C.1] Prove that all circles are similar.
[G.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.
Find arc lengths and areas of sectors of circles. [Radian introduced only as unit of measure]
[G.C.5] Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Convert between degrees and radians. (CA)
[G.GPE.] Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section.
[G.GPE.1] Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
[G.GPE.2] Derive the equation of a parabola given a focus and directrix.
Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean Theorem.]
[G.GPE.4] Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
[G.GPE.5] Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
[G.GMD.] Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
[G.GMD.1] Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
[G.GMD.2] (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
[G.GMD.3] (*) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects.
[G.GMD.5] Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k2, and k3, respectively; determine length, area and volume measures using scale factors. (CA)
[G.GMD.6] Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems. (CA)
[G.MG.] Modeling With Geometry
Apply geometric concepts in modeling situations.
[G.MG.1] (*) Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).'
[G.MG.2] (*) Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
[G.MG.3] Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
II. Statistics and Probability [S.]
[S.ID.] Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on on two categorical and quantitative variables.
[S.ID.6] Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
[S.ID.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.
III. Number and Quantity [N.]
[N.Q.] Quantities
Reason quantitatively and use units to solve problems.
[N.Q.2] Define appropriate quantities for the purpose of descriptive modeling.
[N.RN.] The Real Number System
Extend the properties of exponents to rational exponents.
[N.RN.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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
[N.RN.2] Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
[N.RN.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.
[N.CN.] The Complex Number System
Perform arithmetic measures with complex numbers.
[N.CN.1] Know that there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
[N.CN.2] Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
[N.CN.3] (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients.]
[N.CN.7] Solve quadratic equations with real coefficients that have complex solutions.
[N.CN.8] (+) Extend polynomial identities to complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).
[N.CN.9] (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
IV. Algebra [A.]
[A.SSE.] Seeing Structure In Expressions
Interpret the structure of expressions. [Polynomial and rational.]
[A.SSE.1] (*) Interpret expressions that represent a quantity in terms of its context.
[A.SSE.1a] (*) Interpret parts of an expression, such as terms, factors, and coefficients.
[A.SSE.1b] (*) Interpret complication expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
[A.SSE.2] Use the structure of an expression to identify ways to rewrite it.
[A.APR.] Seeing Structure In Expressions
Perform arithmetic operations on polynomials. [Beyond quadratic.]
[A.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.
[A.CED.] Creating Equations
Create equations that describe numbers or relationships. [Equations using all available types of expresions, including simple root functions.]
[A.CED.1] (*) Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA
[A.CED.2] (*) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
[A.CED.4] (*) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
[A.REI.] Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning. [Simple radical and rational.]
[A.REI.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.
Solve equations and inequalities in one variable.
[A.REI.4] Solve quadratic equations in one variable.
[A.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.
[A.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.
[A.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.
V. Functions [F.]
[F.IF.] Interpreting Functions
Interpret functions that arise in applications in terms of the context. [Emphasize selection of appropriate models.]
[F.IF.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 a function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
[F.IF.5] (*) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
[F.IF.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.
Analyze functions using different representations. [Focus on using key features to guide selection of appropriate type of model function.]
[F.IF.7] (*) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
[F.IF.7b] (*) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
[F.IF.7c] (*) Graph polynomials functions, identifying zeros when suitable factorizations are available, and showing end behavior.
[F.IF.7e] (*) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
[F.IF.8] Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
[F.IF.8a] Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
[F.IF.9] Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
[F.BF.] Building Functions
Build a function that models a relationship between two quantities. [Include all types of functions studied.]
[F.BF.1] (*) Write a function that describes a relationship between two quantities.
[F.BF.1a] Determine an explicit expression, a recursive process, or steps for calculation from a context.
[F.BF.1b] (*) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Build new functions from existing functions. [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.]
[F.BF.3] Identify the effect on the graph by replacing f(x) by f(x) + k, kf(x), f(kx), 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
[F.LE.] Linear, Quadratic, and Exponential Models
Construct and compare linear, quadratic, and exponential models and solve problems
[F.LE.3] Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
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