Academic Standards

1.1: The complex number system includes real numbers and imaginary numbers

1.1.a: Students can: Extend the properties of exponents to rational exponents.

1.1.a.i: 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.

1.1.c: Students can: Perform arithmetic operations with complex numbers.

1.1.c.i: Define the complex number i such that i² = –1, and show that every complex number has the form a + bi where a and b are real numbers.

Points in the Complex Plane

Roots of a Quadratic

1.1.c.ii: Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

1.1.d: Students can: Use complex numbers in polynomial identities and equations.

1.1.d.i: Solve quadratic equations with real coefficients that have complex solutions.

Points in the Complex Plane

Roots of a Quadratic

2.1: Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables

2.1.a: Students can: Formulate the concept of a function and use function notation.

2.1.a.i: Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

Absolute Value with Linear Functions

Exponential Functions

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Introduction to Exponential Functions

Introduction to Functions

Linear Functions

Logarithmic Functions

Parabolas

Point-Slope Form of a Line

Points, Lines, and Equations

Quadratics in Factored Form

Quadratics in Polynomial Form

Quadratics in Vertex Form

Radical Functions

Standard Form of a Line

2.1.a.ii: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Absolute Value with Linear Functions

Translating and Scaling Functions

2.1.a.iii: Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Arithmetic Sequences

Geometric Sequences

2.1.b: Students can: Interpret functions that arise in applications in terms of the context

2.1.b.i: 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.

Absolute Value with Linear Functions

Exponential Functions

Function Machines 3 (Functions and Problem Solving)

General Form of a Rational Function

Graphs of Polynomial Functions

Logarithmic Functions

Points, Lines, and Equations

Quadratics in Factored Form

Quadratics in Polynomial Form

Quadratics in Vertex Form

Radical Functions

2.1.b.ii: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

General Form of a Rational Function

Introduction to Functions

Logarithmic Functions

Radical Functions

Rational Functions

2.1.b.iii: Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

Cat and Mouse (Modeling with Linear Systems)

Slope

2.1.c: Students can: Analyze functions using different representations

2.1.c.i: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Absolute Value with Linear Functions

Exponential Functions

General Form of a Rational Function

Graphs of Polynomial Functions

Introduction to Exponential Functions

Logarithmic Functions

Point-Slope Form of a Line

Quadratics in Factored Form

Quadratics in Polynomial Form

Quadratics in Vertex Form

Radical Functions

Standard Form of a Line

2.1.c.ii: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Absolute Value with Linear Functions

Cat and Mouse (Modeling with Linear Systems)

Exponential Functions

Graphs of Polynomial Functions

Linear Functions

Point-Slope Form of a Line

Points, Lines, and Equations

Polynomials and Linear Factors

Quadratics in Factored Form

Quadratics in Polynomial Form

Quadratics in Vertex Form

Roots of a Quadratic

Slope-Intercept Form of a Line

Standard Form of a Line

Zap It! Game

2.1.c.iii: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Absolute Value with Linear Functions

Radical Functions

Translating and Scaling Functions

2.1.c.iv: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graphs of Polynomial Functions

Polynomials and Linear Factors

Quadratics in Factored Form

Roots of a Quadratic

Zap It! Game

2.1.c.v: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Cosine Function

Exponential Functions

Introduction to Exponential Functions

Logarithmic Functions

Logarithmic Functions: Translating and Scaling

Sine Function

Tangent Function

Translating and Scaling Sine and Cosine Functions

2.1.c.vi: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

2.1.c.vi.1: 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.

Factoring Special Products

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

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

Quadratics in Factored Form

Quadratics in Vertex Form

Roots of a Quadratic

2.1.c.vi.2: Use the properties of exponents to interpret expressions for exponential functions.

Compound Interest

Exponential Functions

2.1.c.vi.3: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

General Form of a Rational Function

Graphs of Polynomial Functions

Linear Functions

Logarithmic Functions

Quadratics in Polynomial Form

Quadratics in Vertex Form

2.1.d: Students can: Build a function that models a relationship between two quantities

2.1.d.i: Write a function that describes a relationship between two quantities.

2.1.d.i.1: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

2.1.d.i.2: Combine standard function types using arithmetic operations.

Addition and Subtraction of Functions

2.1.d.ii: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

2.1.e: Students can: Build new functions from existing functions

2.1.e.i: 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, and find the value of k given the graphs.

Absolute Value with Linear Functions

Exponential Functions

Introduction to Exponential Functions

Logarithmic Functions

Logarithmic Functions: Translating and Scaling

Quadratics in Vertex Form

Radical Functions

Rational Functions

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions

Translations

Zap It! Game

2.1.e.ii: Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Absolute Value with Linear Functions

Exponential Functions

Introduction to Exponential Functions

Logarithmic Functions

Logarithmic Functions: Translating and Scaling

Quadratics in Vertex Form

Radical Functions

Rational Functions

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions

Translations

Zap It! Game

2.1.e.iii: Find inverse functions.

2.1.f: Students can: Extend the domain of trigonometric functions using the unit circle

2.1.f.i: Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Sine Function

Tangent Function

2.1.f.ii: 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.

Cosine Function

Sine Function

Tangent Function

2.2: Quantitative relationships in the real world can be modeled and solved using functions

2.2.a: Students can: Construct and compare linear, quadratic, and exponential models and solve problems

2.2.a.i: Distinguish between situations that can be modeled with linear functions and with exponential functions.

2.2.a.i.1: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Compound Interest

Direct and Inverse Variation

Exponential Functions

Exponential Growth and Decay

Introduction to Exponential Functions

Linear Functions

Slope-Intercept Form of a Line

2.2.a.i.2: Identify situations in which one quantity changes at a constant rate per unit interval relative to another.

Arithmetic Sequences

Compound Interest

Direct and Inverse Variation

Linear Functions

Slope-Intercept Form of a Line

2.2.a.i.3: Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Compound Interest

Exponential Growth and Decay

2.2.a.ii: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

Absolute Value with Linear Functions

Arithmetic Sequences

Arithmetic and Geometric Sequences

Compound Interest

Exponential Functions

Function Machines 1 (Functions and Tables)

Function Machines 2 (Functions, Tables, and Graphs)

Function Machines 3 (Functions and Problem Solving)

Geometric Sequences

Introduction to Exponential Functions

Linear Functions

Logarithmic Functions

Point-Slope Form of a Line

Points, Lines, and Equations

Slope-Intercept Form of a Line

Standard Form of a Line

2.2.a.iii: Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Compound Interest

Introduction to Exponential Functions

2.2.a.iv: For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Compound Interest

Logarithmic Functions

2.2.b: Students can: Interpret expressions for functions in terms of the situation they model

2.2.b.i: Interpret the parameters in a linear or exponential function in terms of a context.

Arithmetic Sequences

Compound Interest

Exponential Growth and Decay

Introduction to Exponential Functions

2.2.c: Students can: Model periodic phenomena with trigonometric functions.

2.2.c.i: Choose the trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions

2.3: Expressions can be represented in multiple, equivalent forms

2.3.a: Students can: Interpret the structure of expressions

2.3.a.i: Interpret expressions that represent a quantity in terms of its context.

2.3.a.i.1: Interpret parts of an expression, such as terms, factors, and coefficients.

Compound Interest

Operations with Radical Expressions

Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions II

2.3.a.i.2: Interpret complicated expressions by viewing one or more of their parts as a single entity.

Compound Interest

Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions II

Translating and Scaling Functions

Using Algebraic Expressions

2.3.a.ii: Use the structure of an expression to identify ways to rewrite it.

Dividing Exponential Expressions

Equivalent Algebraic Expressions I

Equivalent Algebraic Expressions II

Exponents and Power Rules

Factoring Special Products

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

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

Multiplying Exponential Expressions

Simplifying Algebraic Expressions I

Simplifying Algebraic Expressions II

Simplifying Trigonometric Expressions

Solving Algebraic Equations II

Using Algebraic Expressions

2.3.b: Students can: Write expressions in equivalent forms to solve problems

2.3.b.i: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

2.3.b.i.1: Factor a quadratic expression to reveal the zeros of the function it defines.

Factoring Special Products

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

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

Quadratics in Factored Form

2.3.b.i.2: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

2.3.b.i.3: Use the properties of exponents to transform expressions for exponential functions.

Dividing Exponential Expressions

Exponents and Power Rules

2.3.c: Students can: Perform arithmetic operations on polynomials

2.3.c.i: Explain 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.

Addition and Subtraction of Functions

Addition of Polynomials

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

2.3.d: Students can: Understand the relationship between zeros and factors of polynomials

2.3.d.i: State and apply the Remainder Theorem.

Dividing Polynomials Using Synthetic Division

Polynomials and Linear Factors

2.3.d.ii: 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.

Graphs of Polynomial Functions

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

Polynomials and Linear Factors

Quadratics in Factored Form

Quadratics in Vertex Form

2.3.e: Students can: Use polynomial identities to solve problems

2.3.e.i: Prove polynomial identities and use them to describe numerical relationships.

2.4: Solutions to equations, inequalities and systems of equations are found using a variety of tools

2.4.a: Students can: Create equations that describe numbers or relationships

2.4.a.i: Create equations and inequalities in one variable and use them to solve problems.

Absolute Value Equations and Inequalities

Arithmetic Sequences

Compound Interest

Exploring Linear Inequalities in One Variable

Geometric Sequences

Linear Inequalities in Two Variables

Modeling One-Step Equations

Modeling and Solving Two-Step Equations

Quadratic Inequalities

Solving Equations on the Number Line

Solving Linear Inequalities in One Variable

Solving Two-Step Equations

Using Algebraic Equations

2.4.a.ii: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Absolute Value Equations and Inequalities

Circles

Compound Interest

Linear Functions

Point-Slope Form of a Line

Points, Lines, and Equations

Quadratics in Polynomial Form

Quadratics in Vertex Form

Slope-Intercept Form of a Line

Solving Equations on the Number Line

Standard Form of a Line

Using Algebraic Equations

2.4.a.iii: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

Linear Functions

Linear Inequalities in Two Variables

Linear Programming

Solving Equations by Graphing Each Side

Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Standard Form)

Systems of Linear Inequalities (Slope-intercept form)

2.4.a.iv: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Area of Triangles

Solving Formulas for any Variable

2.4.b: Students can: Understand solving equations as a process of reasoning and explain the reasoning

2.4.b.i: 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.

Modeling One-Step Equations

Modeling and Solving Two-Step Equations

Solving Algebraic Equations II

Solving Equations on the Number Line

Solving Formulas for any Variable

Solving Two-Step Equations

2.4.b.ii: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

2.4.c: Students can: Solve equations and inequalities in one variable

2.4.c.i: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Area of Triangles

Compound Inequalities

Exploring Linear Inequalities in One Variable

Linear Inequalities in Two Variables

Modeling One-Step Equations

Modeling and Solving Two-Step Equations

Solving Algebraic Equations I

Solving Algebraic Equations II

Solving Equations on the Number Line

Solving Formulas for any Variable

Solving Linear Inequalities in One Variable

Solving Two-Step Equations

2.4.c.ii: Solve quadratic equations in one variable.

2.4.c.ii.1: 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.

2.4.c.ii.2: Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.

Factoring Special Products

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

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

Points in the Complex Plane

Roots of a Quadratic

2.4.c.ii.3: Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Factoring Special Products

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

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

Points in the Complex Plane

Roots of a Quadratic

2.4.d: Students can: Solve systems of equations

2.4.d.i: 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.

Solving Equations by Graphing Each Side

Solving Linear Systems (Slope-Intercept Form)

Solving Linear Systems (Standard Form)

2.4.d.ii: Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.

Cat and Mouse (Modeling with Linear Systems)

Solving Equations by Graphing Each Side

Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Slope-Intercept Form)

Solving Linear Systems (Standard Form)

2.4.e: Students can: Represent and solve equations and inequalities graphically

2.4.e.i: Explain 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.

Absolute Value Equations and Inequalities

Circles

Ellipses

Hyperbolas

Parabolas

Point-Slope Form of a Line

Points, Lines, and Equations

Standard Form of a Line

2.4.e.ii: 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.

Cat and Mouse (Modeling with Linear Systems)

Point-Slope Form of a Line

Solving Equations by Graphing Each Side

Solving Linear Systems (Matrices and Special Solutions)

Solving Linear Systems (Slope-Intercept Form)

Standard Form of a Line

2.4.e.iii: 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.

Linear Inequalities in Two Variables

Linear Programming

Systems of Linear Inequalities (Slope-intercept form)

3.1: Visual displays and summary statistics condense the information in data sets into usable knowledge

3.1.a: Students can: Summarize, represent, and interpret data on a single count or measurement variable

3.1.a.i: Represent data with plots on the real number line (dot plots, histograms, and box plots).

Box-and-Whisker Plots

Histograms

Mean, Median, and Mode

Reaction Time 1 (Graphs and Statistics)

3.1.a.ii: 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.

Box-and-Whisker Plots

Describing Data Using Statistics

Mean, Median, and Mode

Polling: City

Populations and Samples

Reaction Time 1 (Graphs and Statistics)

Real-Time Histogram

Sight vs. Sound Reactions

3.1.a.iii: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Box-and-Whisker Plots

Describing Data Using Statistics

Least-Squares Best Fit Lines

Mean, Median, and Mode

Populations and Samples

Reaction Time 1 (Graphs and Statistics)

Reaction Time 2 (Graphs and Statistics)

Real-Time Histogram

Stem-and-Leaf Plots

3.1.a.iv: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate.

Polling: City

Populations and Samples

Real-Time Histogram

3.1.a.v: Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Polling: City

Populations and Samples

Real-Time Histogram

3.1.b: Students can: Summarize, represent, and interpret data on two categorical and quantitative variables

3.1.b.i: 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.

3.1.b.ii: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

3.1.b.ii.1: 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.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

Trends in Scatter Plots

Zap It! Game

3.1.b.ii.2: Informally assess the fit of a function by plotting and analyzing residuals.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

Trends in Scatter Plots

3.1.b.ii.3: Fit a linear function for a scatter plot that suggests a linear association.

Correlation

Least-Squares Best Fit Lines

Solving Using Trend Lines

Trends in Scatter Plots

3.1.c: Students can: Interpret linear models

3.1.c.i: Interpret the slope and the intercept of a linear model in the context of the data.

Cat and Mouse (Modeling with Linear Systems)

Correlation

Solving Using Trend Lines

Trends in Scatter Plots

3.1.c.ii: Using technology, compute and interpret the correlation coefficient of a linear fit.

3.1.c.iii: Distinguish between correlation and causation.

3.2: Statistical methods take variability into account supporting informed decisions making through quantitative studies designed to answer specific questions

3.2.a: Students can: Understand and evaluate random processes underlying statistical experiments

3.2.a.i: Describe statistics as a process for making inferences about population parameters based on a random sample from that population.

Polling: City

Polling: Neighborhood

Populations and Samples

3.2.a.ii: Decide if a specified model is consistent with results from a given data-generating process.

Polling: City

Polling: Neighborhood

Populations and Samples

3.2.b: Students can: Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3.2.b.i: Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Polling: City

Polling: Neighborhood

3.2.b.ii: Use data from a sample survey to estimate a population mean or proportion.

Polling: City

Polling: Neighborhood

3.2.b.iii: Develop a margin of error through the use of simulation models for random sampling.

Polling: City

Polling: Neighborhood

3.2.b.iv: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Polling: City

Polling: Neighborhood

3.2.b.v: Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).

3.2.b.vi: Evaluate reports based on data.

Describing Data Using Statistics

Polling: City

Polling: Neighborhood

Real-Time Histogram

3.3: Probability models outcomes for situations in which there is inherent randomness

3.3.a: Students can: Understand independence and conditional probability and use them to interpret data

3.3.a.i: Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.

Independent and Dependent Events

Probability Simulations

Theoretical and Experimental Probability

3.3.a.ii: Explain 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.

Independent and Dependent Events

3.3.a.iii: Using the conditional probability of A given B as P(A and B)/P(B), interpret the 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.

Independent and Dependent Events

3.3.a.iv: 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.

3.3.a.v: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Independent and Dependent Events

3.3.b: Students can: Use the rules of probability to compute probabilities of compound events in a uniform probability model

3.3.b.i: 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.

Independent and Dependent Events

4.1: Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically

4.1.a: Students can: Experiment with transformations in the plane

4.1.a.i: State 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.

Circles

Constructing Congruent Segments and Angles

Constructing Parallel and Perpendicular Lines

Inscribed Angles

Parallel, Intersecting, and Skew Lines

4.1.a.ii: Represent transformations in the plane using appropriate tools.

Dilations

Reflections

Rotations, Reflections, and Translations

Translations

4.1.a.iii: Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

Dilations

Reflections

Rotations, Reflections, and Translations

Translations

4.1.a.iv: Compare transformations that preserve distance and angle to those that do not.

Dilations

Introduction to Exponential Functions

Reflections

Rotations, Reflections, and Translations

Translations

4.1.a.v: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Dilations

Reflections

Rotations, Reflections, and Translations

Similar Figures

4.1.a.vi: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Circles

Dilations

Reflections

Rotations, Reflections, and Translations

Similar Figures

Translations

4.1.a.vii: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools.

Dilations

Reflections

Rotations, Reflections, and Translations

Similar Figures

Translations

4.1.a.viii: Specify a sequence of transformations that will carry a given figure onto another.

Dilations

Reflections

Rotations, Reflections, and Translations

Similar Figures

Translations

4.1.b: Students can: Understand congruence in terms of rigid motions

4.1.b.i: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.

Absolute Value with Linear Functions

Circles

Dilations

Holiday Snowflake Designer

Proving Triangles Congruent

Reflections

Rotations, Reflections, and Translations

Similar Figures

Translations

4.1.b.ii: Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Absolute Value with Linear Functions

Circles

Dilations

Holiday Snowflake Designer

Proving Triangles Congruent

Reflections

Rotations, Reflections, and Translations

Similar Figures

Translations

4.1.b.iv: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

4.1.c: Students can: Prove geometric theorems

4.1.c.i: Prove theorems about lines and angles.

4.1.c.ii: Prove theorems about triangles.

Isosceles and Equilateral Triangles

Proving Triangles Congruent

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

Triangle Angle Sum

Triangle Inequalities

4.1.c.iii: Prove theorems about parallelograms.

Parallelogram Conditions

Special Parallelograms

4.1.d: Students can: Make geometric constructions

4.1.d.i: Make formal geometric constructions with a variety of tools and methods.

Constructing Congruent Segments and Angles

Constructing Parallel and Perpendicular Lines

Segment and Angle Bisectors

4.1.d.ii: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Concurrent Lines, Medians, and Altitudes

Inscribed Angles

4.2: Concepts of similarity are foundational to geometry and its applications

4.2.a: Students can: Understand similarity in terms of similarity transformations

4.2.a.i: Verify experimentally the properties of dilations given by a center and a scale factor:

4.2.a.i.1: Show that 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.

4.2.a.i.2: Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

4.2.a.ii: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.

Circles

Dilations

Similar Figures

Similarity in Right Triangles

4.2.a.iii: 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.

Circles

Dilations

Similar Figures

Similarity in Right Triangles

4.2.a.iv: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

4.2.b: Students can: Prove theorems involving similarity

4.2.b.i: Prove theorems about triangles.

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

Similar Figures

4.2.b.iii: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs

Congruence in Right Triangles

Constructing Congruent Segments and Angles

Dilations

Perimeters and Areas of Similar Figures

Proving Triangles Congruent

Similar Figures

Similarity in Right Triangles

4.2.c: Students can: Define trigonometric ratios and solve problems involving right triangles

4.2.c.i: Explain 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.

Sine, Cosine, and Tangent Ratios

4.2.c.iii: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Cosine Function

Distance Formula

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

Sine Function

Sine, Cosine, and Tangent Ratios

Tangent Function

4.2.d: Students can: Prove and apply trigonometric identities

4.2.d.i: Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1.

Simplifying Trigonometric Expressions

Sine, Cosine, and Tangent Ratios

4.2.d.ii: Use the Pythagorean identity to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.

Simplifying Trigonometric Expressions

Sine, Cosine, and Tangent Ratios

4.2.e: Students can: Understand and apply theorems about circles.

4.2.e.i: Identify and describe relationships among inscribed angles, radii, and chords.

Chords and Arcs

Circumference and Area of Circles

Inscribed Angles

4.2.e.ii: Construct the inscribed and circumscribed circles of a triangle.

Concurrent Lines, Medians, and Altitudes

Inscribed Angles

4.2.e.iii: Prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes

Inscribed Angles

4.2.f: Students can: Find arc lengths and areas of sectors of circles.

4.2.f.i: 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.

4.2.f.ii: Derive the formula for the area of a sector.

4.3: Objects in the plane can be described and analyzed algebraically

4.3.a: Students can: Express Geometric Properties with Equations.

4.3.a.i: Translate between the geometric description and the equation for a conic section

4.3.a.i.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

Circles

Distance Formula

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

4.3.a.i.2: Complete the square to find the center and radius of a circle given by an equation.

Circles

Distance Formula

Pythagorean Theorem

Pythagorean Theorem with a Geoboard

4.3.a.i.3: Derive the equation of a parabola given a focus and directrix.

4.3.a.ii: Use coordinates to prove simple geometric theorems algebraically

4.3.a.ii.4: Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.

4.4: Attributes of two- and three-dimensional objects are measurable and can be quantified

4.4.a: Students can: Explain volume formulas and use them to solve problems

4.4.a.i: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Circumference and Area of Circles

Prisms and Cylinders

Pyramids and Cones

4.4.a.ii: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders

Pyramids and Cones

Correlation last revised: 9/22/2020

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.