A: Algebra, Functions, And Graphs

A.1: Students will understand algebraic concepts and applications.

A.1.1: Represent and analyze mathematical situations and structures using algebraic symbols.

A.1.1.3: Evaluate the numerical value of expressions of one or more variables that are:

A.1.1.3.b: rational

Dividing Exponential Expressions

A.1.1.3.c: radical

Simplifying Radicals - Activity A

A.1.1.4: Simplify algebraic monomial expressions raised to a power (e.g., [5xy²]³) and algebraic binomial (e.g., [5x² + y]²) expressions raised to a power.

Dividing Exponential Expressions
Multiplying Exponential Expressions

A.1.1.6: Represent and analyze relationships using written and verbal expressions, tables, equations, and graphs, and describe the connections among those representations:

A.1.1.6.a: translate from verbal expression to algebraic formulae (e.g., "Set up the equations that represent the data in the following equation: John's father is 23 years older than John. John is 4 years older than his sister Jane. John's mother is 3 years younger than John's father. John's mother is 9 times as old as Jane. How old are John, Jane, John's mother, and John's father?")

Using Algebraic Equations
Using Algebraic Expressions

A.1.1.6.b: given data in a table, construct a function that represents these data (linear only)

Introduction to Functions
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Algebraic Equations
Using Tables, Rules and Graphs

A.1.1.6.c: given a graph, construct a function that represents the graph (linear only)

Introduction to Functions
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Algebraic Equations
Using Tables, Rules and Graphs

A.1.1.7: Know, explain, and use equivalent representations for the same real number including:

A.1.1.7.d: ratios

Estimating Population Size
Part:Part and Part:Whole Ratios
Polling: Neighborhood

A.1.1.7.f: numbers with integer exponents

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

A.1.1.8: Simplify algebraic expressions using the distributive property.

Solving Equations By Graphing Each Side

A.1.1.11: Simplify square roots and cube roots with monomial radicands that are perfect squares or perfect cubes (e.g., 9a²x to the 4th power).

Simplifying Radicals - Activity A
Square Roots

A.1.1.13: Solve:

A.1.1.13.a: formulas for specified variables

Solving Formulas for any Variable

A.1.1.14: Factor polynomials, difference of squares and perfect square trinomials, and the sum and difference of cubes.

Factoring Special Products
Modeling the Factorization of x²+bx+c

A.1.1.15: 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 x²+bx+c

A.1.1.16: Manipulate simple expressions with + and - exponents.

Dividing Exponential Expressions
Exponents and Power Rules

A.1.1.17: Use the four basic operations (+, -, x, ÷) with:

A.1.1.17.b: polynomial expressions

Addition of Polynomials - Activity A

A.1.2: Understand patterns, relations, functions, and graphs.

A.1.2.1: Distinguish between the concept of a relation and a function.

Introduction to Functions
Linear Functions

A.1.2.2: Determine whether a relation defined by a graph, a set of ordered pairs, a table of values, an equation, or a rule is a function.

Exponential Functions - Activity A
Introduction to Functions
Linear Functions
Logarithmic Functions: Translating and Scaling
Slope-Intercept Form of a Line - Activity A
Using Algebraic Equations
Using Tables, Rules and Graphs

A.1.2.4: Translate among tabular, symbolic, and graphical representations of functions.

Introduction to Functions
Linear Functions
Using Algebraic Equations
Using Tables, Rules and Graphs

A.1.2.6: 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.

Functions Involving Square Roots
Introduction to Functions

A.1.2.8: Describe the concept of a graph of an equation.

Defining a Line with Two Points
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Standard Form of a Line

A.1.2.9: Understand symmetry of graphs.

Cosine Function
Cubic Function Activity
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Fourth-Degree Polynomials - Activity A
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Sine Function
Tangent Function

A.1.2.10: Analyze and describe middle and end (asymptotic) behavior of linear, quadratic, and exponential functions, and sketch the graphs of functions.

Exponential Functions - Activity A
Linear Functions
Point-Slope Form of a Line - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic
Slope-Intercept Form of a Line - Activity A

A.1.2.11: Work with composition of functions (e.g., find f of g when f(x) = 2x - 3 and g(x) = 3x - 2), and find the domain, range, intercepts, zeros, and local maxima or minima of the final function.

Parabolas - Activity A
Polynomials and Linear Factors
Roots of a Quadratic

A.1.2.12: Use the quadratic formula and factoring techniques to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Polynomials and Linear Factors
Roots of a Quadratic

A.1.3: Use mathematical models to represent and understand quantitative relationships.

A.1.3.1: Model real-world phenomena using linear and quadratic equations and linear inequalities (e.g., apply algebraic techniques to solve rate problems, work problems, and percent mixture problems; solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest; apply quadratic equations to model throwing a baseball in the air).

Percent of Change
Simple and Compound Interest
Solving Linear Inequalities using Addition and Subtraction
Solving Linear Inequalities using Multiplication and Division

A.1.3.3: Express the relationship between two variables using a table with a finite set of values and graph the relationship.

Cosine Function
Cubic Function Activity
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Exponential Functions - Activity A
Fourth-Degree Polynomials - Activity A
General Form of a Rational Function
Introduction to Functions
Linear Functions
Logarithmic Functions - Activity A
Logarithmic Functions: Translating and Scaling
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Radical Functions
Rational Functions
Sine Function
Slope-Intercept Form of a Line - Activity A
Tangent Function
Using Tables, Rules and Graphs

A.1.3.4: Express the relationship between two variables using an equation and a graph:

A.1.3.4.a: graph a linear equation and linear inequality in two variables

Defining a Line with Two Points
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
Solving Linear Inequalities using Addition and Subtraction
Solving Linear Inequalities using Multiplication and Division
Standard Form of a Line

A.1.3.4.b: solve linear inequalities and equations in one variable

Modeling One-Step Equations - Activity A
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

A.1.3.4.c: solve systems of linear equations in two variables and graph the solutions

Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Solving Linear Systems by Graphing
Special Types of Solutions to Linear Systems
Systems of Linear Equations - Activity A

A.1.3.4.d: use the graph of a system of equations in two variables to help determine the solution

Modeling Linear Systems - Activity A
Solving Linear Systems by Graphing
Special Types of Solutions to Linear Systems
Systems of Linear Equations - Activity A

A.1.3.5: Solve applications involving systems of equations.

Solving Linear Systems by Graphing
Special Types of Solutions to Linear Systems
Systems of Linear Equations - Activity A

A.1.3.6: Evaluate numerical and algebraic absolute value expressions.

Inequalities Involving Absolute Values
Quadratic and Absolute Value Functions

A.1.3.7: Create a linear equation from a table of values containing co-linear data.

Exponential Functions - Activity A
Introduction to Functions
Linear Functions
Logarithmic Functions: Translating and Scaling
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

A.1.3.8: Determine the solution to a system of equations in two variables from a given graph.

Modeling Linear Systems - Activity A
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Solving Linear Systems by Graphing
Special Types of Solutions to Linear Systems
Systems of Linear Equations - Activity A

A.1.3.10: Write an equation of the line that passes through two given points.

Defining a Line with Two Points
Point-Slope Form of a Line - Activity A
Standard Form of a Line

A.1.3.11: Understand and use:

A.1.3.11.b: the rules of exponents

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

A.1.3.12: Verify that a point lies on a line, given an equation of the line, and be 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

A.1.4: Analyze changes in various contexts.

A.1.4.1: Analyze the effects of parameter changes on these functions:

A.1.4.1.a: linear (e.g., changes in slope or coefficients)

Linear Functions
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

A.1.4.1.b: quadratic (e.g., f[x-a] changes coefficients and constants)

Parabolas - Activity A
Quadratic and Absolute Value Functions
Quadratics in Factored Form
Quadratics in Polynomial Form - Activity A
Roots of a Quadratic

A.1.4.1.c: exponential (e.g., changes caused by increasing x[x + c] or [a to the x power])

Exponential Functions - Activity A
Exponential Growth and Decay - Activity B

A.1.4.1.d: polynomial (e.g., changes caused by positive or negative values of a, or in a constant c)

Cubic Function Activity
Fourth-Degree Polynomials - Activity A

A.1.4.2: Solve routine two- and three-step problems relating to change using concepts such as:

A.1.4.2.a: exponents

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

A.1.4.2.c: ratio

Similar Figures - Activity A
Similar Polygons

A.1.4.2.d: proportion

Polling: Neighborhood
Similar Figures - Activity A
Similar Polygons

A.1.4.2.e: average

Distance-Time Graphs
Distance-Time and Velocity-Time Graphs

A.1.4.3: Calculate the percentage of increase and decrease of a quantity.

Percent of Change

A.1.4.5: Estimate the rate of change of a function or equation by finding the slope between two points on the graph.

Slope - Activity B

A.1.4.6: Evaluate the estimated rate of change in the context of the problem.

Direct Variation
Direct and Inverse Variation
Distance-Time Graphs
Distance-Time and Velocity-Time Graphs

A.1.4.7: Know Pascal's triangle and use it to expand binomial expressions that are raised to positive integer powers.

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

B: Geometry and Trigonometry

B.1: Students will understand geometric concepts and applications.

B.1.1: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

B.1.1.1: Interpret and draw two-dimensional objects and find the area and perimeter of basic figures (e.g., rectangles, circles, triangles, other polygons [e.g., rhombi, parallelograms, trapezoids]).

Area of Parallelograms - Activity A
Perimeter, Circumference, and Area - Activity B
Rectangle: Perimeter and Area

B.1.1.2: Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles with just edges in common.

Special Quadrilaterals

B.1.1.3: Find and use measures of sides and interior and exterior angles of triangles and polygons to classify figures (e.g., scalene, isosceles, and equilateral triangles; rectangles [square and nonsquare]; other convex polygons).

Classifying Triangles
Isosceles and Equilateral Triangles
Triangle Angle Sum - Activity A

B.1.1.4: Interpret and draw three-dimensional objects and find the surface area and volume of basic figures (e.g., spheres, rectangular solids, prisms, polygonal cones), and calculate the surface areas and volumes of these figures as well as figures constructed from unions of rectangular solids and prisms with faces in common, given the formulas for these figures.

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

B.1.1.5: Demonstrate an understanding of simple aspects of a logical argument:

B.1.1.5.a: identify the hypothesis and conclusion in logical deduction

Biconditional Statement
Conditional Statement

B.1.1.6: Demonstrate an understanding of inductive and deductive reasoning, explain the difference between inductive and deductive reasoning, and identify and provide examples of each:

B.1.1.6.b: for deductive reasoning, prove simple theorems

Biconditional Statement
Conditional Statement
Simplifying Trigonometric Expressions
Sum and Difference Identities for Sine and Cosine

B.1.1.7: Write geometric proofs (including proofs by contradiction), including:

B.1.1.7.a: theorems involving the properties of parallel lines cut by a transversal line and the properties of quadrilaterals

Biconditional Statement
Chords and Arcs
Conditional Statement
Investigating Angle Theorems - Activity A
Parallelogram Conditions
Simplifying Trigonometric Expressions
Special Quadrilaterals
Sum and Difference Identities for Sine and Cosine
Triangle Angle Sum - Activity A

B.1.1.7.b: theorems involving complementary, supplementary, and congruent angles

Biconditional Statement
Conditional Statement
Investigating Angle Theorems - Activity A
Simplifying Trigonometric Expressions
Sum and Difference Identities for Sine and Cosine

B.1.1.7.c: theorems involving congruence and similarity

Biconditional Statement
Conditional Statement
Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures - Activity A
Similar Polygons

B.1.1.7.d: the Pythagorean theorem (tangram proof)

Biconditional Statement
Conditional Statement
Distance Formula - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Pythagorean Theorem - Activity B

B.1.2: Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

B.1.2.2: Determine the midpoint and distance between two points within a coordinate system and relate these ideas to geometric figures in the plane (e.g., find the center of a circle given two endpoints of a diameter of the circle).

Classifying Quadrilaterals - Activity B

B.1.2.3: Given two linear equations, determine whether the lines are parallel, perpendicular, or coincide.

Construct Parallel and Perpendicular Lines

B.1.2.4: Use basic geometric ideas (e.g., the Pythagorean Theorem, area, and perimeter of objects) in the context of the Euclidean Plane, calculate the perimeter of a rectangle with integer coordinates and sides parallel to the coordinate axes and with sides not parallel.

Perimeter, Circumference, and Area - Activity B
Rectangle: Perimeter and Area
Special Quadrilaterals

B.1.3: Apply transformations and use symmetry to analyze mathematical situations.

B.1.3.1: Describe the effect of rigid motions on figures in the coordinate plane and space that include rotations, translations, and reflections:

B.1.3.1.a: determine whether a given pair of figures on a coordinate plane represents the effect of a translation, reflection, rotation, and/or dilation

Dilations
Reflections
Rotations, Reflections and Translations
Translations

B.1.3.1.b: sketch the planar figure that is the result of a given transformation of this type

Dilations
Reflections
Rotations, Reflections and Translations

B.1.3.2: Deduce properties of figures using transformations that include translations, rotations, reflections, and dilations in a coordinate system:

B.1.3.2.a: identify congruency and similarity in terms of transformations

Constructing Congruent Segments and Angles
Dilations
Perimeters and Areas of Similar Figures
Reflections
Rotations, Reflections and Translations
Similar Figures - Activity A
Similar Polygons

B.1.3.2.b: determine the effects of the above transformations on linear and area measurements of the original planar figure

Area of Parallelograms - Activity A
Dilations
Minimize Perimeter
Perimeter, Circumference, and Area - Activity B
Prisms and Cylinders - Activity A
Reflections
Rotations, Reflections and Translations

B.1.4: Use visualization, spatial reasoning, and geometric modeling to solve problems.

B.1.4.1: Solve real-world problems using congruence and similarity relationships of triangles (e.g., find the height of a pole given the length of its shadow).

Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures - Activity A
Similar Polygons

B.1.4.2: Solve problems involving complementary, supplementary, and congruent angles.

Investigating Angle Theorems - Activity A

B.1.4.3: Solve problems involving the perimeter, circumference, area, volume, and surface area of common geometric figures (e.g., "Determine the surface area of a can of height h and radius r. How does the surface area change when the height is changed to 3h? How does the surface area change when the radius is changed to 3r? How does the surface area change when both h and r are doubled?").

Circle: Circumference and Area
Perimeter, Circumference, and Area - Activity B
Rectangle: Perimeter and Area

B.1.4.4: Solve problems using the Pythagorean Theorem (e.g., "Given the length of a ladder and the distance of the base of the ladder from a wall, determine the distance up the wall to the top of the ladder").

Distance Formula - Activity A
Geoboard: The Pythagorean Theorem
Pythagorean Theorem - Activity A
Pythagorean Theorem - Activity B

B.1.4.5: Understand and use elementary relationships of basic trigonometric functions defined by the angles of a right triangle (e.g., "What is the radius of a circle with an inscribed regular octagon with the length of each side being exactly 2 feet?").

Cosine Function
Sine Function
Tangent Function
Unit Circle

B.1.4.6: Use trigonometric functions to solve for the length of the second leg of a right triangle given the angles and the length of the first leg. (e.g., "A surveyor determines that the angle subtended by a two-foot stick at right angles to his transit is exactly one degree. What is the distance from the transit to the base of the measuring stick?").

Cosine Function
Sine Function
Tangent Function
Triangle Angle Sum - Activity A
Unit Circle

B.1.4.7: Know and use angle and side relationships in problems with special right triangles (e.g., 30-, 45-, 60-, and 90-degree triangles).

Triangle Angle Sum - Activity A

C: Data Analysis and Probability

C.1: Students will understand how to formulate questions, analyze data, and determine probabilities.

C.1.1: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

C.1.1.2: Know the characteristics of a well-designed and well-conducted survey:

C.1.1.2.b: differentiate between a biased and an unbiased sample

Polling: Neighborhood

C.1.1.4: Understand the role of randomization in well-designed surveys and experiments.

Polling: Neighborhood

C.1.2: Select and use appropriate statistical methods to analyze data.

C.1.2.2: Understand the meaning of "univariate" (i.e., one variable) and "bivariate" (i.e., two variable) data.

Scatter Plots - Activity A

C.1.2.3: For univariate data, be able to display the distribution and describe its shape using appropriate summary statistics, and understand the distinction between a statistic and a parameter:

C.1.2.3.a: construct and interpret frequency tables, histograms, stem and leaf plots, and box and whisker plots

Box-and-Whisker Plots
Histograms
Populations and Samples
Stem-and-Leaf Plots

C.1.2.3.b: calculate and apply measures of central tendency (mean, median, and mode) and measures of variability (range, quartiles, standard deviation)

Box-and-Whisker Plots
Describing Data Using Statistics
Line Plots
Mean, Median and Mode

C.1.2.3.c: compare distributions of univariate data using back-to-back stem and leaf plots and parallel box and whisker plots

Box-and-Whisker Plots
Stem-and-Leaf Plots

C.1.2.4: For bivariate data, be able to display a scatter plot and describe its shape:

C.1.2.4.b: describe and interpret the relationship/correlation between two variables using technological tools

Correlation
Solving Using Trend Lines

C.1.3: Develop and evaluate inferences and predictions that are based on data.

C.1.3.1: Compare and draw conclusions between two or more sets of univariate data using basic data analysis techniques and summary statistics.

Populations and Samples

C.1.3.2: Draw conclusions concerning the relationships among bivariate data:

C.1.3.2.a: make predictions from a linear pattern in data

Arithmetic and Geometric Sequences
Linear Functions

C.1.3.2.b: determine the strength of the relationship between two sets of data by examining the correlation

Correlation
Solving Using Trend Lines

C.1.3.2.c: understand that correlation does not imply a cause-and-effect relationship

Correlation
Solving Using Trend Lines

C.1.3.3: Use simulations to explore the variability of sample statistics from a known population and construct sampling distributions.

Populations and Samples
Probability Simulations

C.1.3.4: Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.

Polling: City
Populations and Samples

C.1.4: Understand and apply basic concepts of probability.

C.1.4.2: Understand the concept of probability as relative frequency.

Binomial Probabilities
Geometric Probability - Activity A

C.1.4.3: Use simulations to compute the expected value and probabilities of random variables in simple cases.

Binomial Probabilities
Probability Simulations

C.1.4.4: Distinguish between independent and dependent events.

Compound Independent Events
Compound Independent and Dependent Events
Independent and Dependent Events

C.1.4.5: Understand how to compute the probability of an event using the basic rules of probability:

C.1.4.5.c: multiplication rule (independent events)

Binomial Probabilities
Compound Independent Events
Compound Independent and Dependent Events
Geometric Probability - Activity A
Independent and Dependent Events