Content Standards

AI.3.0: Students solve equations and inequalities involving absolute values.

Inequalities Involving Absolute Values

AI.4.0: Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x - 5) + 4(x - 2) = 12.

Modeling and Solving Two-Step Equations

Solving Equations By Graphing Each Side

Solving Two-Step Equations

AI.5.0: Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Modeling and Solving Two-Step Equations

Solving Equations By Graphing Each Side

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

Solving Two-Step Equations

AI.6.0: Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

Defining a Line with Two Points

Inequalities Involving Absolute Values

Linear Inequalities in Two Variables - Activity A

Linear Programming - Activity A

Point-Slope Form of a Line - Activity A

Slope-Intercept Form of a Line - Activity A

Standard Form of a Line

Systems of Linear Inequalities (Slope-intercept form) - Activity A

AI.7.0: Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Defining a Line with Two Points

Point-Slope Form of a Line - Activity A

AI.8.0: Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

AI.9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Linear Programming - Activity A

Modeling Linear Systems - Activity A

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

Solving Linear Systems by Graphing

Special Types of Solutions to Linear Systems

Systems of Linear Equations - Activity A

Systems of Linear Inequalities (Slope-intercept form) - Activity A

AI.10.0: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

Addition of Polynomials - Activity A

Dividing Exponential Expressions

Dividing Polynomials Using Synthetic Division

Multiplying Exponential Expressions

Simplifying Radicals - Activity A

AI.11.0: Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Factoring Special Products

Modeling the Factorization of *x*^{2}+*bx*+*c*

AI.12.0: Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

Factoring Special Products

Modeling the Factorization of *x*^{2}+*bx*+*c*

AI.14.0: Students solve a quadratic equation by factoring or completing the square.

Factoring Special Products

Modeling the Factorization of *ax*^{2}+*bx*+*c*

Modeling the Factorization of *x*^{2}+*bx*+*c*

AI.16.0: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.

Introduction to Functions

Linear Functions

AI.17.0: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

AI.18.0: Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

Introduction to Functions

Linear Functions

AI.19.0: Students know the quadratic formula and are familiar with its proof by completing the square.

AI.20.0: Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

AI.21.0: Students graph quadratic functions and know that their roots are the x-intercepts.

Parabolas - Activity A

Polynomials and Linear Factors

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

AI.22.0: Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

Modeling the Factorization of *ax*^{2}+*bx*+*c*

Modeling the Factorization of *x*^{2}+*bx*+*c*

Roots of a Quadratic

AI.24.1: Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

Biconditional Statement

Conditional Statement

AI.25.2: Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

AI.25.3: Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

Inequalities Involving Absolute Values

Modeling and Solving Two-Step Equations

Quadratic Inequalities - Activity A

Roots of a Quadratic

Solving Equations By Graphing Each Side

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

Solving Two-Step Equations

G.1.0: Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Biconditional Statement

Conditional Statement

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

G.2.0: Students write geometric proofs, including proofs by contradiction.

Biconditional Statement

Conditional Statement

Proving Triangles Congruent

G.4.0: Students prove basic theorems involving congruence and similarity.

Congruence in Right Triangles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

G.5.0: Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.

Biconditional Statement

Conditional Statement

Congruence in Right Triangles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

G.6.0: Students know and are able to use the triangle inequality theorem.

G.7.0: Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

Investigating Angle Theorems - Activity A

Parallelogram Conditions

G.8.0: Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.

Area of Parallelograms - Activity A

Circle: Circumference and Area

Perimeter, Circumference, and Area - Activity B

Rectangle: Perimeter and Area

Surface and Lateral Area of Prisms and Cylinders

Surface and Lateral Area of Pyramids and Cones

G.9.0: Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.

Prisms and Cylinders - Activity A

Pyramids and Cones - Activity A

Surface and Lateral Area of Prisms and Cylinders

Surface and Lateral Area of Pyramids and Cones

G.10.0: Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

Area of Parallelograms - Activity A

Perimeter, Circumference, and Area - Activity B

Rectangle: Perimeter and Area

G.11.0: Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

Area of Parallelograms - Activity A

Circle: Circumference and Area

Minimize Perimeter

Perimeter, Circumference, and Area - Activity B

Prisms and Cylinders - Activity A

Pyramids and Cones - Activity A

Rectangle: Perimeter and Area

G.12.0: Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

Classifying Triangles

Isosceles and Equilateral Triangles

Triangle Angle Sum - Activity A

G.13.0: Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.

Biconditional Statement

Conditional Statement

Investigating Angle Theorems - Activity A

Triangle Angle Sum - Activity A

G.14.0: Students prove the Pythagorean theorem.

Geoboard: The Pythagorean Theorem

Pythagorean Theorem - Activity B

G.15.0: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.

Distance Formula - Activity A

Geoboard: The Pythagorean Theorem

Pythagorean Theorem - Activity A

Pythagorean Theorem - Activity B

G.16.0: Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.

Construct Parallel and Perpendicular Lines

Constructing Congruent Segments and Angles

G.17.0: Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.

Distance Formula - Activity A

Pythagorean Theorem - Activity B

G.18.0: Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x))² + (cos(x))² = 1.

Simplifying Trigonometric Expressions

Sine and Cosine Ratios - Activity A

G.19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

Sine Function

Sine and Cosine Ratios - Activity A

Tangent Function

G.21.0: Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

Biconditional Statement

Chords and Arcs

Conditional Statement

Inscribing Angles

G.22.0: Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.

Dilations

Reflections

Rotations, Reflections and Translations

Translations

AII.1.0: Students solve equations and inequalities involving absolute value.

Inequalities Involving Absolute Values

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

AII.2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

Linear Programming - Activity A

Modeling Linear Systems - Activity A

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

Special Types of Solutions to Linear Systems

Systems of Linear Equations - Activity A

Systems of Linear Inequalities (Slope-intercept form) - Activity A

AII.3.0: Students are adept at operations on polynomials, including long division.

Dividing Polynomials Using Synthetic Division

AII.4.0: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

AII.5.0: Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.

Points in the Complex Plane - Activity A

Roots of a Quadratic

AII.7.0: Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

Addition of Polynomials - Activity A

Dividing Exponential Expressions

Dividing Polynomials Using Synthetic Division

Multiplying Exponential Expressions

AII.8.0: Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.

Factoring Special Products

Modeling the Factorization of *ax*^{2}+*bx*+*c*

Modeling the Factorization of *x*^{2}+*bx*+*c*

Roots of a Quadratic

AII.9.0: Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x - b)² + c.

Parabolas - Activity A

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

Translating and Scaling Functions

AII.10.0: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Parabolas - Activity A

Polynomials and Linear Factors

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

AII.11.2: Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

Dividing Exponential Expressions

Exponents and Power Rules

Multiplying Exponential Expressions

AII.12.0: Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

Exponential Functions - Activity A

Exponential Growth and Decay - Activity B

Half-life

Simple and Compound Interest

AII.15.0: Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.

Exponential Functions - Activity A

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Simplifying Radicals - Activity A

AII.16.0: Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.

Ellipse - Activity A

Hyperbola - Activity A

Parabolas - Activity A

AII.17.0: Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.

Circles

Ellipse - Activity A

Hyperbola - Activity A

Parabolas - Activity A

AII.18.0: Students use fundamental counting principles to compute combinations and permutations.

Permutations

Permutations and Combinations

AII.19.0: Students use combinations and permutations to compute probabilities.

Binomial Probabilities

Permutations

Permutations and Combinations

AII.22.0: Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

AII.24.0: Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.

Addition and Subtraction of Polynomials

AII.25.0: Students use properties from number systems to justify steps in combining and simplifying functions.

Addition and Subtraction of Polynomials

T.2.0: Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.

Cosine Function

Sine Function

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Tangent Function

Unit Circle

T.3.1: Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).

Simplifying Trigonometric Expressions

T.3.2: Students prove other trigonometric identities and simplify others by using the identity cos²(x) + sin²(x) = 1. For example, students use this identity to prove that sec²(x) = tan²(x) + 1.

Biconditional Statement

Conditional Statement

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

T.4.0: Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions - Activity A

T.5.0: Students know the definitions of the tangent and cotangent functions and can graph them.

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

T.6.0: Students know the definitions of the secant and cosecant functions and can graph them.

Simplifying Trigonometric Expressions

T.7.0: Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.

Slope - Activity B

Tangent Function

Tangent Ratio

T.9.0: Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.

Cosine Function

Sine Function

Tangent Function

T.10.0: Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/ or simplify other trigonometric identities.

Biconditional Statement

Conditional Statement

Sum and Difference Identities for Sine and Cosine

T.11.0: Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/ or simplify other trigonometric identities.

Biconditional Statement

Conditional Statement

Sum and Difference Identities for Sine and Cosine

T.12.0: Students use trigonometry to determine unknown sides or angles in right triangles.

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

T.15.0: Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.

Complex Numbers in Polar Form

Points in Polar Coordinates

T.16.0: Students represent equations given in rectangular coordinates in terms of polar coordinates.

Complex Numbers in Polar Form

Points in Polar Coordinates

T.17.0: Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form.

T.18.0: Students know DeMoivre's theorem and can give nth roots of a complex number given in polar form.

MA.1.0: Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically.

Points in Polar Coordinates

Points in the Coordinate Plane - Activity A

Vectors

MA.2.0: Students are adept at the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre's theorem.

Biconditional Statement

Complex Numbers in Polar Form

Conditional Statement

MA.5.1: Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).

Rotations, Reflections and Translations

Translations

MA.5.2: Students can take a geometric description of a conic section - for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6 - and derive a quadratic equation representing it.

Circles

Ellipse - Activity A

Hyperbola - Activity A

Parabolas - Activity A

MA.6.0: Students find the roots and poles of a rational function and can graph the function and locate its asymptotes.

General Form of a Rational Function

Rational Functions

LA.4.0: Students perform addition on matrices and vectors.

LA.7.0: Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.

LA.8.0: Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.

Modeling Linear Systems - Activity A

Solving Linear Systems by Graphing

Special Types of Solutions to Linear Systems

Systems of Linear Equations - Activity A

LA.12.0: Students compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

PS.1.0: Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.

Compound Independent Events

Compound Independent and Dependent Events

Independent and Dependent Events

PS.3.0: Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.

Binomial Probabilities

Compound Independent Events

Compound Independent and Dependent Events

Independent and Dependent Events

PS.4.0: Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families.

PS.6.0: Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.

Describing Data Using Statistics

Line Plots

Mean, Median and Mode

PS.8.0: Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

Box-and-Whisker Plots

Correlation

Histograms

Scatter Plots - Activity A

Solving Using Trend Lines

Stem-and-Leaf Plots

APPS.1.0: Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.

Binomial Probabilities

Compound Independent Events

Compound Independent and Dependent Events

Independent and Dependent Events

APPS.3.0: Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.

APPS.5.0: Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

Describing Data Using Statistics

Line Plots

Mean, Median and Mode

APPS.7.0: Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

APPS.9.0: Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

APPS.10.0: Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.

Describing Data Using Statistics

Line Plots

Mean, Median and Mode

APPS.12.0: Students find the line of best fit to a given distribution of data by using least squares regression.

APPS.13.0: Students know what the correlation coefficient of two variables means and are familiar with the coefficient's properties.

APPS.14.0: Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

Box-and-Whisker Plots

Correlation

Histograms

Scatter Plots - Activity A

Solving Using Trend Lines

Stem-and-Leaf Plots

APPS.15.0: Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.

APPS.16.0: Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.

C.1.3: Students prove and use special limits, such as the limits of (sin(x))/x and (1 - cos(x))/x as x tends to 0.

Sine Function

Tangent Function

C.2.0: Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

Functions Involving Square Roots

C.13.0: Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.

C.16.0: Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

Correlation last revised: 11/2/2009