Standards for Teaching and Learning

AI.N.3: Calculate and apply ratios, proportions, rates, and percentages to solve a range of consumer and practical problems.

Beam to Moon (Ratios and Proportions)

Estimating Population Size

Part:Part and Part:Whole Ratios

Percent of Change

Polling: Neighborhood

Simple and Compound Interest

AI.N.4: Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers, including approximate error in measurement and the approximate value of square roots. (Reminder: This is without the use of calculators.)

AI.P.1: Recognize, describe, and extend patterns governed by a linear, quadratic, or exponential functional relationship or by a simple iterative process (e.g., the Fibonacci sequence).

Arithmetic Sequences

Arithmetic and Geometric Sequences

Exponential Functions - Activity A

Finding Patterns

Geometric Sequences

Linear Functions

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

AI.P.3: Demonstrate an understanding of relations and functions. Identify the domain, range, and dependent and independent variables of functions.

Cosine Function

Functions Involving Square Roots

General Form of a Rational Function

Introduction to Functions

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Radical Functions

Rational Functions

Sine Function

Tangent Function

AI.P.4: Translate between different representations of functions and relations: graphs, equations, sets of ordered pairs (scatter plots), verbal, and tabular.

Introduction to Functions

Linear Functions

Polynomials and Linear Factors

Scatter Plots - Activity A

Using Algebraic Equations

Using Algebraic Expressions

AI.P.5: Demonstrate an understanding of the relationship between various representations of a line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that represents the line.

Defining a Line with Two Points

Modeling Linear Systems - Activity A

Point-Slope Form of a Line - Activity A

Slope - Activity B

Slope-Intercept Form of a Line - Activity A

Solving Equations By Graphing Each Side

Standard Form of a Line

AI.P.6: Find a linear function describing a line from a graph or a geometric description of the line (e.g., by using the point-slope or slope y-intercept formulas). Explain the significance of a positive, negative, zero, or undefined slope.

Defining a Line with Two Points

Linear Functions

Modeling Linear Systems - Activity A

Point-Slope Form of a Line - Activity A

Slope - Activity B

Slope-Intercept Form of a Line - Activity A

Using Tables, Rules and Graphs

AI.P.7: Find linear functions that represent lines either perpendicular or parallel to a given line and through a point (e.g., by using the point-slope form of the equation).

Linear Functions

Point-Slope Form of a Line - Activity A

Slope-Intercept Form of a Line - Activity A

Using Tables, Rules and Graphs

AI.P.8: Add, subtract, and multiply polynomials with emphasis on 1st- and 2nd-degree polynomials.

Addition of Polynomials - Activity A

AI.P.9: Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms, factoring [e.g., a² – b² = (a + b)(a – b), x² + 10x + 21 = (x + 3) (x + 7), 5x the the 4th power + 10x³ – 5x² = 5x² (x² + 2x – 1)], identifying and canceling common factors in rational expressions, and applying the properties of positive integer exponents.

Dividing Exponential Expressions

Exponents and Power Rules

Factoring Special Products

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

AI.P.10: Divide polynomials by monomials with emphasis on 1st- and 2nd-degree polynomials.

Dividing Exponential Expressions

Dividing Polynomials Using Synthetic Division

AI.P.11: Perform basic arithmetic operations with rational expressions and functions.

General Form of a Rational Function

Rational Functions

AI.P.12: Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods.

Factoring Special Products

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

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

Roots of a Quadratic

AI.P.13: Solve equations and inequalities, including those involving absolute value of linear expressions (e.g., |x – 2| > 5), and apply to the solution of problems.

Inequalities Involving Absolute Values

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

AI.P.14: Solve everyday problems (e.g., compound interest and direct and inverse variation problems) that can be modeled using linear or quadratic functions. Apply appropriate graphical or symbolic methods to the solution.

Direct Variation

Direct and Inverse Variation

Linear Functions

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

Simple and Compound Interest

AI.P.15: Solve everyday problems (e.g., mixture, rate, and work problems) that can be modeled using systems of linear equations or inequalities. Apply algebraic and graphical methods to the solution.

Linear Programming - Activity A

Modeling Linear Systems - Activity A

Special Types of Solutions to Linear Systems

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

AI.D.1: Select, create, and interpret an appropriate graphical representation (e.g., scatter plot, table, stem-and-leaf plots, circle graph, line graph, and line plot) for a set of data, and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.

Box-and-Whisker Plots

Describing Data Using Statistics

Line Plots

Mean, Median and Mode

Populations and Samples

Scatter Plots - Activity A

Stem-and-Leaf Plots

AII.P.1: Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Fibonacci Numbers and Pascal’s Triangle.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Finding Patterns

Geometric Sequences

AII.P.2: Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

AII.P.3: Understand functional notation, evaluate a function at a specified point in its domain, and perform operations on functions with emphasis on the domain and range.

Addition and Subtraction of Polynomials

Logarithmic Functions: Translating and Scaling

AII.P.4: Understand exponential and logarithmic functions and their basic arithmetic properties, including change of base and formulas for exponential of a sum and logarithm of a product.

Exponential Functions - Activity A

Exponential Growth and Decay - Activity B

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

AII.P.5: Given algebraic, numeric, and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential, and describe their behavior.

Cubic Function Activity

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Rational Functions

Using Algebraic Equations

AII.P.6: Find solutions to radical equations; find solutions to quadratic equations (with real coefficients and real or complex roots) graphically, by factoring, by completing the square, or by using the quadratic formula.

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

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

Roots of a Quadratic

AII.P.7: Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula. Include polynomial, exponential, and logarithmic functions, expressions involving the absolute values, and simple rational expressions.

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

General Form of a Rational Function

Inequalities Involving Absolute Values

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Quadratic Inequalities - Activity A

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Rational Functions

Roots of a Quadratic

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

AII.P.9: Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Describe the relationships among the methods.

Modeling Linear Systems - Activity A

Special Types of Solutions to Linear Systems

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

AII.P.10: Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, and step functions; absolute values; and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include compound interest, exponential growth and decay, and direct and inverse variation problems.

Determining a Spring Constant

Direct Variation

Direct and Inverse Variation

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

Functions Involving Square Roots

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Quadratic and Absolute Value Functions

Radical Functions

Rational Functions

Simple and Compound Interest

AII.P.11: Recognize translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af(b(x + c/b)) + d. In particular, describe qualitatively the effect of such changes on polynomial, rational, exponential, and logarithmic functions.

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Rational Functions

AII.P.12: Simplify rational expressions. Solve rational equations and inequalities.

Dividing Exponential Expressions

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

AII.G.1: Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

Cosine Function

Sine Function

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

Unit Circle

AII.G.2: Explain the identity sin²q + cos²q = 1. Relate the identity to the Pythagorean theorem.

Distance Formula - Activity A

Geoboard: The Pythagorean Theorem

Pythagorean Theorem - Activity A

Pythagorean Theorem - Activity B

Simplifying Trigonometric Expressions

AII.G.3: Relate geometric and algebraic representations of lines and simple curves.

Cosine Function

Ellipse - Activity A

Hyperbola - Activity A

Point-Slope Form of a Line - Activity A

Sine Function

Slope-Intercept Form of a Line - Activity A

Tangent Function

AII.D.1: Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data, including box plots.

Box-and-Whisker Plots

Histograms

Line Plots

Scatter Plots - Activity A

Stem-and-Leaf Plots

AII.D.2: Use combinatorics (e.g., fundamental counting principle, permutations, and combinations) to solve problems, including computing geometric probabilities and probabilities of compound events.

Binomial Probabilities

Compound Independent Events

Compound Independent and Dependent Events

Geometric Probability - Activity A

Independent and Dependent Events

Permutations

Permutations and Combinations

G.G.1: Know correct geometric notation, including the notation for line segment (AB) and angle (

Classifying Quadrilaterals - Activity B

G.G.2: Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses).

G.G.3: Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, midsegments); determine interior angles for regular polygons.

Classifying Triangles

Isosceles and Equilateral Triangles

G.G.5: Detect symmetries of geometric figures.

G.G.6: Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and to solve problems.

G.G.7: Use properties and theorems about congruent and similar figures and about perpendicular and parallel lines to solve problems.

Congruence in Right Triangles

Constructing Congruent Segments and Angles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

G.G.8: Write simple proofs of theorems in geometric situations, such as theorems about triangles, congruent and similar figures, and perpendicular and parallel lines (e.g., the longest side is opposite the greatest angle, two lines parallel to a third are parallel to each other; perpendicular bisectors of line segments are the set of all points equidistant from the two end points).

Congruence in Right Triangles

Constructing Congruent Segments and Angles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

Triangle Angle Sum - Activity A

G.G.9: Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive.

Biconditional Statement

Conditional Statement

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

G.G.11: Draw congruent and similar figures using a compass, straightedge, or protractor. Justify the constructions by logical argument.

Congruence in Right Triangles

Constructing Congruent Segments and Angles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

G.G.12: Apply congruence and similarity correspondences (e.g., DABC @ DXYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.

Classifying Quadrilaterals - Activity B

Congruence in Right Triangles

Constructing Congruent Segments and Angles

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures - Activity A

Similar Polygons

G.G.13: Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

G.G.14: Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean theorem; study and understand more than one proof of this theorem.

Biconditional Statement

Conditional Statement

Geoboard: The Pythagorean Theorem

Investigating Angle Theorems - Activity A

Pythagorean Theorem - Activity B

Triangle Angle Sum - Activity A

G.G.15: Use the properties of special triangles (e.g., isosceles, equilateral, 30º-60º-90º, 45º-45º-90º) to solve problems.

Isosceles and Equilateral Triangles

Triangle Angle Sum - Activity A

G.G.16: Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

Cosine Function

Sine Function

Sine and Cosine Ratios - Activity A

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

Unit Circle

G.G.17: Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line (e.g., by using the point-slope or slope y-intercept formulas). Explain the significance of a positive, negative, zero, or undefined slope.

Defining a Line with Two Points

Linear Functions

Point-Slope Form of a Line - Activity A

Slope - Activity B

Slope-Intercept Form of a Line - Activity A

Solving Equations By Graphing Each Side

Standard Form of a Line

Using Tables, Rules and Graphs

G.G.18: Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.

Distance Formula - Activity A

Geoboard: The Pythagorean Theorem

Pythagorean Theorem - Activity A

Slope - Activity B

G.G.19: Find linear equations that represent lines either perpendicular or parallel to a given line and through a point (e.g., by using the point-slope form of the equation).

Construct Parallel and Perpendicular Lines

Linear Functions

Point-Slope Form of a Line - Activity A

G.G.20: Draw the results and interpret transformations on figures in the coordinate plane such as translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems.

Dilations

Reflections

Rotations, Reflections and Translations

Translations

G.G.21: Demonstrate the ability to visualize solid objects and recognize their projections, cross sections, and graph points in 3-D.

3D and Orthographic Views - Activity A

G.G.22: Find and use measures of perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles.

Circle: Circumference and Area

Parallelogram Conditions

Perimeter, Circumference, and Area - Activity B

G.G.23: Find and use measures of lateral areas, surface areas, and volumes of prisms, pyramids, spheres, cylinders, and cones, and relate these measures to each other using formulas.

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.G.24: Relate changes in the measurement (including units) of one attribute of an object to changes in other attributes.

Prisms and Cylinders - Activity A

PCT.N.2: Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi = r (cos theta + i sin theta).

PCT.P.1: Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients, including use of completing the square.

Polynomials and Linear Factors

Roots of a Quadratic

PCT.P.2: Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.

Cosine Function

Sine Function

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

Unit Circle

PCT.P.4: Given algebraic, numeric, and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential.

Cubic Function Activity

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Rational Functions

Using Algebraic Equations

PCT.P.5: Combine functions by composition, as well as by addition, subtraction, multiplication, and division.

Addition and Subtraction of Polynomials

PCT.P.6: Identify whether a function has an inverse and when functions are inverses of each other; explain why the graph of a function and its inverse are reflections of one another over the line y = x.

Absolute Value with Linear Functions - Activity B

PCT.P.7: Identify maximum and minimum values of functions. Apply to the solution of problems.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Parabolas - Activity A

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

PCT.P.8: Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = a f(b (x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, and logarithmic functions.

Exponential Functions - Activity A

Fourth-Degree Polynomials - Activity A

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Rational Functions

PCT.P.9: Derive and apply basic trigonometric identities i.e., sin²theta + cos²theta = 1, tan²theta + 1 = sec²theta and the laws of sines and cosines.

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

PCT.P.10: Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre's theorem and use them to prove other trigonometric identities. Apply to the solution of problems.

Simplifying Trigonometric Expressions

Sum and Difference Identities for Sine and Cosine

PCT.P.11: Understand, predict, and interpret the effects of the parameters a, w, b, and c on the graph of y = asin (infinity (x — b)) + c; do the same for the cosine and tangent. Use to model periodic processes.

Translating and Scaling Sine and Cosine Functions - Activity A

PCT.P.16: Identify maximum and minimum values of functions in simple situations. Apply to the solution of problems.

Cubic Function Activity

Fourth-Degree Polynomials - Activity A

Parabolas - Activity A

Quadratic and Absolute Value Functions

Quadratics in Factored Form

Quadratics in Polynomial Form - Activity A

Roots of a Quadratic

PCT.G.2: Use vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results.

PCT.G.3: Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

PCT.D.3: Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities.

Geometric Probability - Activity A

Probability Simulations

Compound Independent Events

Compound Independent and Dependent Events

Independent and Dependent Events

Compound Independent Events

Compound Independent and Dependent Events

Independent and Dependent Events

Describing Data Using Statistics

Line Plots

Mean, Median and Mode

Describing Data Using Statistics

Box-and-Whisker Plots

Correlation

Histograms

Scatter Plots - Activity A

Solving Using Trend Lines

Stem-and-Leaf Plots

Correlation

Lines of Best Fit Using Least Squares - Activity A

Solving Using Trend Lines

Correlation last revised: 12/2/2009