Khan Common Core Exercises for High School Maths

[chris macrae]

Curriculum specs High School: Number and Quantity High School: Algebra High School: Functions High School: Geometry High School: Statistics & Probability

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source https://www.khanacademy.org/commoncore/map

Maths choices rom the learning menu top left at www.khanacademy.org

Arithmetic and pre-algebra

• Probability and statistics
• Algebra
• Differential equations
• Geometry
• Linear algebra
• Trigonometry and precalculus
• Applied math
• Calculus
• Recreational mathematics

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As at 21 Aug 2013, Khan Academy maths dashboard offers 508 exercises- provided you use one login name you can track your progress on each - grey not started, dark blue accomplished , pale blue studying

exercises line 4 of khan dashboard line 4 comparing linear functions comparing linear function applications solve y intercept slope intercept form point slope form converting slope intercept and standard form converting point slope & slope intercept finding equation of a line midpoint formula pythagorean theorem distance formula equations of parallel and perpendicular lines distance between point and line graph of inequalities graphing linear inequalities graphing and solving linear inequalities slope and triangle similarity graphing systems of equations

..some end line 3 entries relevant to precalculus graphing points graphing points and naming quadrants points on the coordinate plane coordinate plane word problems reflecting points equations from tables ordered pair solutions to linear equations identifying linear relationships linear and nonlinear functions equations from tables solving for the x intercept linear function intercepts interpreting and finding intercepts of linear functions rates & proportional relationships analyzing and identifying proportional relationships graphing proportional relationships rate problems 1 constructing and comparing proportional relationship identifying slope of a line line graph intuition average rate of change graphing linear equations comparing linear functions interpreting features of linear functions comparing linear function applications solve y intercept slope intercept form

systems of equations with simple elimination

\systems of equations with eliminations

systems of equationn with substitution

systems of equations

systems of equations word problems

solutions to systems of equations

graphical solutions to systems

constructing consistent and inconsistent systems

graphing systems of inequalities

graphing and solving systems of inequalities

systems of nonlinear equations

factoring differences of squares 1

factoring of differences of squares 2

factoring differences of squares 3

multiplying expressions 0.5

multiplying expressions 1

factoring linear binomials

factoring polynomials 1

factoring polynomials 2

factoring polynomials by grouping

factoring polynomials with 2 variables

adding and subtracting polynomials

multiplying polynomials

solving quadratics by factoring

solving quadratics by factoring 2

solving quadratics by taking the square root

completing the square 1

completing the square 2

quadratic formula

quadratic formula with complex solutions

solutions to quadratic equations

vertex of a parabola

graphing parabolas in standard form

graphing parabolas in vertex form

graphing parabolas in all forms

parabola intuition 3

simplifying rational expression with exponent properties

simplifying expressions with exponents

radical equations

understanding function notation

recognizing functions

recognizing functions2

domain of a function

range of a function

domain and range from graph

direct and inverse variation

views of a function

evaluating expressions with function notation

evaluating composite functions

inverses of functions

new operator definitions 1

new operator definitions 2

writing proportions

proportions 1

proportions 2

dividing polynomials by binomials 1

 

line 5

line 5

dividing polynomials by binomials 2

simplifying rational expressions 2

dividing polynomials by binomials 3

simplifying rational expressions 3

simplifying rational experssions 4

adding and subtracting rational expressions 0,5

adding and subtracting rational expressions 1

adding and subtracting rational expressions 1,5

adding and subtracting rational expressions 2

adding and subtracting rational expressions 3

multiplying and dividing rational expressions 1

multiplying and dividing rational expressions 2

multiplying and dividing rational expressions 3

multiplying and dividing rational expressions 4

multiplying and dividing rational expressions 5

partial fraction expansion

solving rational equations 1

solving rational equations 2

extraneous solutions

evaluating logarithms

evaluating logarithms 2

operations with logarithms

recognizing conic sections

graphing circles

equation of a circle in factored form

equation of a circle in non-factored form

graphing circles 2

equation of an ellipse

parabola intuition 1

parabola intuition 2

equation of a hyperbola

matrix dimensions

scalar matrix multiplication

matrix addition and subtraction

matrix transpose

multiplying a matrix by a vector

multiplying a matrix by a matrix

defined and undefined matrix operations

determinant of 2X2 matrix

inverse of 2X2 matrix

determinant of 3X3 matrix

inverse of 3X3 matrix

imaginary unit powers

the complex plane

adding and subtracting complex numbers

multiplying complex numbers

dividing complex numbers

absolute value of complex numbers

recognizing rays lines and line segments

points lines and planes

measuring segments

congruent segments

segment addition

midpoint of a segment

measuring angles

angle types

exploring angle pairs

vertical angles

vertical angles 2

congruent angles

parallel lines 1

parallel lines 2

corresponding angles 2

same side exterior angles 2

same side interior angles 2

alternate interior angles 2

alternate exterior angles 2

inverses of functions

angles 2

angles of a polygon

complementary and supplementary angles

angle addition postulate

congruent postulates

congruent triangle 1

 

line 6

line 6

congruent triangles 2

triangle types

triangle angles 1

basic triangle proofs

fill-in-the-blank triangle proofs

wrong statements in triangle profs

perieter1

finding dimensions given perimeter

area 1

finding dimensions given area

area of triangles

shaded areas

triangle inequality theorem

heron's formula

radius diameter and circumference

area of a circle

area of parallelograms

area of trapezoids, rhombi and kites

solid geometry

similar triangles 1

similar triangles 2

solving similar triangles 1

solving similar triangles 2

special right angles

angle bisector theorem

compass constructions

quadrilateral types

quadrilateral angles

circles and arcs

areas of circles and sectors

inscribed angles 1

inscribed angles 2

inscribed angles 3

conditional statements

logical arguments and deductive reasoning

conditional states and truth value

axis of symmetry

translations of polygon

rotation of polygons

positive and negative parts of functions

even and odd functions

shifting and reflecting functions

recognizing features of functions 2

interpreting features of functions 2

comparing features of functions 2

trigonometry 0.5

trigonometry 1

reciprocal trig functions

trigonometry 1.5

trigonometry 2

degrees to radians

radians to degrees

radians and degrees

Unit circle

graphs of sine and cosine

inverse trig functions

pythagorean identities

addition and subtraction trig identities

law of cosines

arithmetic sequences 1

arithmetic sequences 2

geometric sequences 1

geometric sequences 2

recursive and explicit functions

arithmetic series

probability 1

independent probability

dependent probability

counting 2

permutations

combinations

permutations and combinations

probability with permutations and combinations

complex number polar form intuition

 

line 7

 

line 7

complex plane operations

multiplying and dividing complex numbers in polar form

powers of complex numbers

limits 1

limits 2

one-sided limits from graphs

slope of secant lines

tangent slope in limiting value of secant slope

derivatives 1

the formal and alternate form of the derivative

recognizing slope of curves

derivative intuition

visualizing derivatives

power rule

special derivatives

chain rule 1

product rule

quotient rule

implicit differentiation

recognizing concavity

L'Hopital rule

Probability space

Basic set notation

Expected value

Mean median and mode

exploring mean and median

Average word problems

Creating box & whisker plots

Variance

Exploring standard deviation 1

standard deviation

Estimating line of best fit

Empirical rule

Z scores 1

Z scores 2

Z scores 3

scaling vectors

adding vectors

[chris macrae]

Common Core Mapping for High School: Algebra

Seeing Structure in Expressions

Code	Standard	Exercises
HSA-SSE.A.1	Interpret expressions that represent a quantity in terms of its context.★	Structure in expressions 1
HSA-SSE.A.1a	Interpret parts of an expression, such as terms, factors, and coefficients.	Structure in expressions 1
HSA-SSE.A.1b	Interpret complicated 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.	Structure in expressions 1
HSA-SSE.A.2	Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).	Radical equations
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.★	Completing the square 1 Factoring polynomials 1 Solving quadratics by factoring Solving quadratics by factoring 2 Vertex of a parabola
HSA-SSE.B.3a	Factor a quadratic expression to reveal the zeros of the function it defines.	Factoring polynomials 1 Solving quadratics by factoring Solving quadratics by factoring 2
HSA-SSE.B.3b	Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.	Completing the square 1 Vertex of a parabola
HSA-SSE.B.3c	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.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. For example, calculate mortgage payments.★

Arithmetic with Polynomials and Rational Expressions

Code	Standard	Exercises
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.	Adding and subtracting polynomials Multiplying polynomials
HSA-APR.B.2	Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
HSA-APR.B.3	Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
HSA-APR.C.4	Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
HSA-APR.C.5	(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1
HSA-APR.D.6	Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
HSA-APR.D.7	(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Creating Equations*

Code	Standard	Exercises
HSA-CED.A.1	Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
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.
HSA-CED.A.3	Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
HSA-CED.A.4	Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Reasoning with Equations and Inequalities

Code	Standard	Exercises
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.	Graphing and solving linear inequalities Graphing and solving systems of inequalities Graphing linear inequalities Graphing systems of inequalities
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.3	Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.	Equations with variables on both sides Linear equation word problems Multi-step linear inequalities Multistep equations with distribution Solving rational equations 1
HSA-REI.B.4	Solve quadratic equations in one variable.	Completing the square 1 Quadratic formula Quadratic formula with complex solutions Solving quadratics by factoring Solving quadratics by factoring 2 Solving quadratics by taking the square root
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.	Completing the square 1
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.	Quadratic formula Quadratic formula with complex solutions Solving quadratics by factoring Solving quadratics by factoring 2 Solving quadratics by taking the square root
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.	Systems of equations
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 x2 + y2 = 3.	Systems of nonlinear equations
HSA-REI.C.8	(+) Represent a system of linear equations as a single matrix equation in a vector variable.
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).
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.	Graphing and solving linear inequalities Graphing and solving systems of inequalities Graphing linear inequalities Graphing systems of inequalities

[chris macrae]

Common Core Mapping for High School: Functions

Interpreting Functions

Code	Standard	Exercises
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).	Domain and range from graph Recognizing functions 2
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 ≥ 1.	Recursive and explicit functions
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.★	Interpreting features of functions 2 Recognizing features of functions 2
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.★	Domain and range from graph
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.★	Average rate of change
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.7a	Graph linear and quadratic functions and show intercepts, maxima, and minima.
HSF-IF.C.7b	Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
HSF-IF.C.7c	Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
HSF-IF.C.7d	(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
HSF-IF.C.7e	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.
HSF-IF.C.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.
HSF-IF.C.8b	Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
HSF-IF.C.9	Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.	Comparing features of functions 2

Building Functions

Code	Standard	Exercises
HSF-BF.A.1	Write a function that describes a relationship between two quantities.★	Recursive and explicit functions
HSF-BF.A.1a	Determine an explicit expression, a recursive process, or steps for calculation from a context.	Recursive and explicit functions
HSF-BF.A.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.
HSF-BF.A.1c	(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
HSF-BF.A.2	Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★	Recursive and explicit functions
HSF-BF.B.3	Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.	Even and odd functions Shifting and reflecting functions
HSF-BF.B.4	Find inverse functions.	Inverses of functions
HSF-BF.B.4a	Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.	Inverses of functions
HSF-BF.B.4b	(+) Verify by composition that one function is the inverse of another.
HSF-BF.B.4c	(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
HSF-BF.B.4d	(+) Produce an invertible function from a non-invertible function by restricting the domain.
HSF-BF.B.5	(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Linear, Quadratic, and Exponential Models

Code	Standard	Exercises
HSF-LE.A.1	Distinguish between situations that can be modeled with linear functions and with exponential functions.
HSF-LE.A.1a	Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
HSF-LE.A.1b	Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
HSF-LE.A.1c	Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
HSF-LE.A.2	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).
HSF-LE.A.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.
HSF-LE.A.4	For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
HSF-LE.B.5	Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions

Code	Standard	Exercises
HSF-TF.A.1	Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HSF-TF.A.2	Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF-TF.A.3	(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
HSF-TF.A.4	(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
HSF-TF.B.5	Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
HSF-TF.B.6	(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
HSF-TF.B.7	(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★
HSF-TF.C.8	Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find  sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the  quadrant of the angle.
HSF-TF.C.9	(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

[chris macrae]

Common Core Mapping for High School: Geometry

Congruence

Code	Standard	Exercises
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.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.	Congruent triangles 1 Congruent triangles 2
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.	Congruency postulates
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.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.C.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.
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.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.

Similarity, Right Triangles, and Trigonometry

Code	Standard	Exercises
HSG-SRT.A.1	Verify experimentally the properties of dilations given by a center and a scale factor:
HSG-SRT.A.1a	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.A.1b	The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
HSG-SRT.A.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.A.3	Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.	Similar triangles 1 Similar triangles 2 Solving similar triangles 1
HSG-SRT.B.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.
HSG-SRT.B.5	Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.	Basic triangle proofs Fill-in-the-blank triangle proofs Solving similar triangles 2 Wrong statements in triangle proofs
HSG-SRT.C.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.C.7	Explain and use the relationship between the sine and cosine of complementary angles.
HSG-SRT.C.8	Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
HSG-SRT.D.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.
HSG-SRT.D.10	(+) Prove the Laws of Sines and Cosines and use them to solve problems.
HSG-SRT.D.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).

Circles

Code	Standard	Exercises
HSG-C.A.1	Prove that all circles are similar.
HSG-C.A.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.A.3	Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
HSG-C.A.4	(+) Construct a tangent line from a point outside a given circle to the circle.
HSG-C.B.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.

Expressing Geometric Properties with Equations

Code	Standard	Exercises
HSG-GPE.A.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.
HSG-GPE.A.2	Derive the equation of a parabola given a focus and directrix.
HSG-GPE.A.3	(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
HSG-GPE.B.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).
HSG-GPE.B.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).
HSG-GPE.B.6	Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
HSG-GPE.B.7	Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

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Geometric Measurement and Dimension

Code	Standard	Exercises
HSG-GMD.A.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.
HSG-GMD.A.2	(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
HSG-GMD.A.3	Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
HSG-GMD.B.4	Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Modeling with Geometry

Code	Standard	Exercises
HSG-MG.A.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).★
HSG-MG.A.2	Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
HSG-MG.A.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).★

[chris macrae]

Common Core Mapping for High School: Statistics & Probability

Interpreting Categorical and Quantitative Data

Code	Standard	Exercises
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.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.6a	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.6b	Informally assess the fit of a function by plotting and analyzing residuals.
HSS-ID.B.6c	Fit a linear function for a scatter plot that suggests a linear association.
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.

Making Inferences and Justifying Conclusions

Code	Standard	Exercises
HSS-IC.A.1	Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
HSS-IC.A.2	Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
HSS-IC.B.3	Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
HSS-IC.B.4	Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
HSS-IC.B.5	Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
HSS-IC.B.6	Evaluate reports based on data.

Conditional Probability & the Rules of Probability

Code	Standard	Exercises
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 P(A and B)/P(B), 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.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, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
HSS-CP.B.8	(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
HSS-CP.B.9	(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Using Probability to Make Decisions

Code	Standard	Exercises
HSS-MD.A.1	(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
HSS-MD.A.2	(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
HSS-MD.A.3	(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
HSS-MD.A.4	(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
HSS-MD.B.5	(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
HSS-MD.B.5a	Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
HSS-MD.B.5b	Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

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HSS-MD.B.6	(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
HSS-MD.B.7	(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

[chris macrae]

Common Core Mapping for High School: Number and Quantity

The Real Number System

Code	Standard	Exercises
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. 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.
HSN-RN.A.2	Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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.

Quantities

Code	Standard	Exercises
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.
HSN-Q.A.2	Define appropriate quantities for the purpose of descriptive modeling.
HSN-Q.A.3	Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

The Complex Number System

Code	Standard	Exercises
HSN-CN.A.1	Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
HSN-CN.A.2	Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN-CN.A.3	(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
HSN-CN.B.4	(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
HSN-CN.B.6	(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
HSN-CN.B.5	(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
HSN-CN.C.7	Solve quadratic equations with real coefficients that have complex solutions.
HSN-CN.C.8	(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
HSN-CN.C.9	(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Vector and Matrix Quantities

Code	Standard	Exercises
HSN-VM.A.1	(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
HSN-VM.A.2	(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
HSN-VM.A.3	(+) Solve problems involving velocity and other quantities that can be represented by vectors.
HSN-VM.B.4	(+) Add and subtract vectors.
HSN-VM.B.4a	Add vectors end-to-end, component-wise, 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.B.4b	Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
HSN-VM.B.4c	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 component-wise.
HSN-VM.B.5	(+) Multiply a vector by a scalar.
HSN-VM.B.5a	Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
HSN-VM.B.5b	Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
HSN-VM.C.6	(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
HSN-VM.C.7	(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
HSN-VM.C.8	(+) Add, subtract, and multiply matrices of appropriate dimensions.
HSN-VM.C.9	(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
HSN-VM.C.10	(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
HSN-VM.C.11	(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
HSN-VM.C.12	(+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.