# Algebra, Geometry, & Algebra 2

### Chapter 1: Functions

Section 1.1

1.1.1 Solving Puzzles in Teams

1.1.2 Investigating the Growth of Patterns

1.1.3 Investigating the Graphs of Quadratic Functions

Section 1.2

1.2.1 Describing a Graph

1.2.2 Cube Root and Absolute Value Functions

1.2.3 Function Machines

1.2.4 Functions

1.2.5 Domain and Range

Chapter Closure

### Chapter 2: Linear Relationships

Section 2.1

2.1.1 Seeing Growth in Linear Representations

2.1.2 Slope

2.1.3 Comparing Δy and Δx

2.1.4 y = mx + b and More on Slope

Section 2.2

2.2.1 Slope as Motion

2.2.2 Rate of Change

2.2.3 Equations of Lines in Situations

Section 2.3

2.3.1 Finding an Equation Given a Slope and a Point

2.3.2 Finding the Equation of a Line Through Two Points

Extension Activity Finding y = mx + b from Graphs and Tables

Chapter Closure

### Chapter 3: Simplifying and Solving

Section 3.1

3.1.1 Simplifying Exponential Expressions

3.1.2 Zero and Negative Exponents

Section 3.2

3.2.1 Equations Algebra Tiles

3.2.2 Exploring an Area Model

3.2.3 Multiplying Binomials and the Distributive Property

3.2.4 Using Generic Rectangles to Multiply

Section 3.3

3.3.1 Solving Equations With Multiplication and Absolute Value

3.3.2 Working With Multi-Variable Equations

3.3.3 Summary of Solving Equations

Chapter Closure

### Chapter 4: Systems of Equations

Section 4.1

4.1.1 Solving Word Problems by Writing Equations

4.1.2 One Equation or Two?

Section 4.2

4.2.1 Solving Systems of Equations Using Substitution

4.2.2 Making Connections: Systems, Solutions, and Graphs

4.2.3 Solving Systems Using Elimination

4.2.4 More Elimination

4.2.5 Choosing a Strategy for Solving Systems

Section 4.3

4.3.1 Pulling it all Together

Chapter Closure

### Chapter 5: Sequences

Section 5.1

5.1.1 Representing Exponential Growth

5.1.2 Rebound Ratios

5.1.3 The Bouncing Ball and Exponential Decay

Section 5.2

5.2.1 Generating and Investigating Sequences

5.2.2 Generalizing Arithmetic Sequences

5.2.3 Recursive Sequences

Section 5.3

5.3.1 Patterns of Growth in Tables and Graphs

5.3.2 Using Multipliers to Solve Problems

5.3.3 Comparing Sequences to Functions

Chapter Closure

### Chapter 6: Modeling Two-Variable Data

Section 6.1

6.1.1 Line of Best Fit

6.1.2 Residuals

6.1.3 Upper and Lower Bounds

6.1.4 Least Squares Regression Line

Section 6.2

6.2.1 Residual Plots

6.2.2 Correlation

6.2.3 Association is Not Causation

6.2.4 Interpreting Correlation in Context

6.2.5 Curved Best-Fit Models

Chapter Closure

### Chapter 7: Exponential Functions

Section 7.1

7.1.1 Investigating y = b

7.1.2 Multiple Representations of Exponential Functions

7.1.3 More Applications of Exponential Growth

7.1.4 Exponential Decay

7.1.5 Graph → Equation

7.1.6 Completing the Multiple Representations Web

Section 7.2

7.2.1 Curve Fitting and Fractional Exponents

7.2.2 More Curve Fitting

7.2.3 Solving a System of Exponential Functions Graphically

Chapter Closure

Section 8.1

8.1.2 Factoring with Generic Rectangles

8.1.3 Factoring with Special Cases

8.1.4 Factoring Completely

8.1.5 Factoring Shortcuts

Section 8.2

8.2.1 Multiple Representations for Quadratic Functions

8.2.2 Zero Product Property

8.2.3 More Ways To Find the x-Intercepts

8.2.5 Completing the Square

Chapter Closure

### Chapter 9: Solving Quadratics and Inequalities

Section 9.1

9.1.2 Introduction to the Quadratic Formula

9.1.4 Choosing a Strategy

Section 9.2

9.2.1 Solving Linear, One-Variable Inequalities

9.2.2 More Solving Inequalities

Section 9.3

9.3.1 Graphing Two-Variable Inequalities

9.3.2 Graphing Linear and Non-Linear Inequalities

Section 9.4

9.4.1 Systems of Inequalities

9.4.2 More Systems of Inequalities

9.4.3 Applying Inequalities to Solve Problems

Chapter Closure

### Chapter 10: Solving Complex Equations

Section 10.1

10.1.1 Association in Two-Way Tables

Section 10.2

10.2.1 Solving by Rewriting

10.2.2 Fraction Busters

10.2.3 Multiple Methods for Solving Equations

10.2.4 Determining the Number of Solutions

10.2.5 Deriving the Quadratic Formula and the Number System

10.2.6 More Solving and an Application

Section 10.3

10.3.1 Intersection of Two Functions

10.3.2 Number of Parabola Intersections

10.3.3 Solving Quadratic and Absolute Value Inequalities

Chapter Closure

### Chapter 11: Functions and Data

Section 11.1

11.1.1 Transforming Functions

11.1.2 Inverse Functions

Section 11.2

11.2.1 Investigating Data Representations

11.2.2 Comparing Data

11.2.3 Standard Deviation

Section 11.3

11.3.1 Using a Best-Fit Line to Make a Prediction

11.3.2 Relation Treasure Hunt

11.3.3 Investigating a Complex Function

11.3.4 Using Algebra to Find a Maximum

11.3.5 Exponential Functions and Linear Inequalities

Chapter Closure

### Appendix: Representing Expressions

Section A.1

A.1.1 Exploring Variables and Combining Like Terms

A.1.2 Simplifying Expressions by Combining Like Terms

A.1.3 Writing Algebraic Expressions

A.1.4 Using Zero to Simplify Algebraic Expressions

A.1.5 Using Algebra Tiles to Simplify Algebraic Expressions

A.1.6 Using Algebra Tiles to Compare Expressions

A.1.7 Simplifying and Recording Work

A.1.8 Using Algebra Tiles to Solve for x

A.1.9 More Solving Equations

Chapter Closure

### Chapter 1: Shapes and Transformations

Section 1.1

1.1.1 Creating Quilt Using Symmetry

1.1.2 Making Predictions and Investigating Results

1.1.3 Perimeters and Areas of Enlarging Tile Patterns

1.1.4 Logical Arguments

1.1.5 Building a Kaleidoscope

Section 1.2

1.2.1 Spatial Visualization and Reflection

1.2.2 Rigid Transformations: Rotation and Translations

1.2.3 Slope of Parallel and Perpendicular Lines

1.2.4 Defining Transformations

1.2.5 Using Transformations to Create Shapes

1.2.6 Symmetry

Section 1.3

1.3.1 Attributes and Characteristics of Shapes

1.3.2 More Characteristics of Shapes

Chapter Closure

### Chapter 2:Angles and Measurement

Section 2.1

2.1.1 Complementary, Supplementary, and Vertical Angles

2.1.2 Angles Formed by Transversals 5

2.1.3 More Angles Formed by Transversals

2.1.4 Angles in a Triangle

2.1.5 Applying Angle Relationships

Section 2.2

2.2.1 Units of Measure

2.2.2 Areas of Triangles and Composite Shapes

2.2.3 Areas of Parallelograms and Trapezoids

2.2.4 Heights and Areas

Section 2.3

2.3.1 Triangle Inequality

2.3.2 The Pythagorean Theorem

Chapter Closure

### Chapter 3: Justification and Similarity

Section 3.1

3.1.1 Dilations

3.1.2 Similarity

3.1.3 Using Ratios of Similarity

3.1.4 Applications and Notation

Section 3.2

3.2.1 Conditions for Triangle Similarity

3.2.2 Creating a Flowchart

3.2.3 Triangle Similarity and Congruence

3.2.4 More Conditions for Triangle Similarity

3.2.5 Determining Similarity

3.2.6 Applying Similarity

Chapter Closure

### Chapter 4: Trigonometry and Probability

Section 4.1

4.1.1 Constant Ratios in Right Triangles

4.1.2 Connecting Slope Ratios to Specific Angles

4.1.3 Expanding the Trig Table

4.1.4 The Tangent Ratio

4.1.5 Applying the Tangent Ratio

Section 4.2

4.2.1 Using an Area Model

4.2.2 Using a Tree Diagram

4.2.3 Probability Models

4.2.4 Unions, Intersections, and Complements

4.2.5 Expected Value

Chapter Closure

### Chapter 5: Completing the Triangle Toolkit

Section 5.1

5.1.1 Sine and Cosine Ratios

5.1.2 Selecting a Trig Tool

5.1.3 Inverse Trigonometry

5.1.4 Applications

Section 5.2

5.2.1 Special Right Triangles

5.2.2 Pythagorean Triples

Section 5.3

5.3.1 Finding Missing Parts of Triangles

5.3.2 Law of Sines

5.3.3 Law of Cosines

5.3.4 Ambiguous Triangles (Optional)

5.3.5 Choosing a Tool

Chapter Closure

### Chapter 6: Congruent Triangles

Section 6.1

6.1.1 Congruent Triangles

6.1.2 Conditions for Triangle Congruence

6.1.3 Congruence of Triangles Through Rigid Transformations

6.1.4 Flowcharts for Congruence

6.1.5 Converses

Section 6.2

6.2.1 Angles on a Pool Table

6.2.2 Investigating a Triangle

6.2.3 Creating a Mathematical Model

6.2.4 Analyzing a Game

6.2.5 Using Transformations and Symmetry to Design Snowflakes

Chapter Closure

### Chapter 7: Proof and Quadrilaterals

Section 7.1

7.1.1 Properties of a Circle

7.1.2 Building a Tetrahedron

7.1.3 Shortest Distance Problems

7.1.4 Using Symmetry to Study Polygons

Section 7.2

7.2.2 Properties of Rhombi

7.2.3 More Proofs with Congruent Triangles

7.2.4 More Properties of Quadrilaterals 7.2.5 Two-Column Proofs

7.2.6 Explore-Conjecture-Prove

Section 7.3

7.3.1 Studying Quadrilaterals on a Coordinate Grid

7.3.2 Coordinate Geometry and Midpoints

7.3.3 Identifying Quadrilaterals on a Coordinate Grid

Chapter Closure

### Chapter 8: Polygons and Circles

Section 8.1

8.1.1 Pinwheels and Polygons

8.1.2 Interior Angles of Polygons

8.1.3 Angles of Regular Polygons

8.1.4 Regular Polygon Angle Connections

8.1.5 Finding Areas of Regular Polygons

Section 8.2

8.2.1 Area Ratios of Similar Figures

8.2.2 Ratios of Similarity

Section 8.3

8.3.1 A Special Ratio

8.3.2 Area and Circumference of a Circle

8.3.3 Circles in Context

Chapter Closure

### Chapter 9: Solids and Constructions

Section 9.1

9.1.1 Three-Dimensional Solids

9.1.2 Volumes and Surface Areas of Prisms

9.1.3 Prisms and Cylinders

9.1.4 Volumes of Similar Solids

9.1.5 Ratios of Similarity

Section 9.2

9.2.1 Introduction to Constructions

9.2.2 Constructing Bisectors

9.2.3 More Explorations with Constructions

9.2.4 Other Constructions

Chapter Closure

### Chapter 10: Circles and Conditional Probability

Section 10.1

10.1.1 Introduction to Chords

10.1.2 Angles and Arcs

10.1.3 Chords and Angles

10.1.4 Tangents and Secants

10.1.5 Problem Solving with Circles

Section 10.2

10.2.1 Conditional Probability and Independence

10.2.2 Two-Way Tables

10.2.3 Applications of Probability

Section 10.3

10.3.1 The Fundamental Principle of Counting

10.3.2 Permutations

10.3.3 Combinations

10.3.4 Categorizing Counting Problems

10.3.5 Some Challenging Probability Problems

Chapter Closure

### Chapter 11: Solids and Circles

Section 11.1

11.1.1 Platonic Solids

11.1.2 Pyramids

11.1.3 Volume of a Pyramid

11.1.4 Surface Area and Volume of a Cone

11.1.5 Surface Area and Volume of a Sphere

Section 11.2

11.2.1 Coordinates on a Sphere

11.2.2 Tangents and Arcs

11.2.3 Secant and Tangent Relationships

Chapter Closure

### Chapter 12: Conics and Closure

Section 12.1

12.1.1 The Equation of a Circle

12.1.2 Completing the Square for Equations of Circles

12.1.3 Introduction to Conic Sections

12.1.4 Graphing a Parabola Using the Focus and Directrix

Section 12.2

12.2.1 Using Coordinate Geometry and Constructions to Explore Shapes

12.2.2 Euler’s Formula for Polyhedra

12.2.3 The Golden Ratio

12.2.4 Using Geometry to find Probabilities

Chapter Closure

### Chapter 1: Investigations and Functions

Section 1.1

1.1.1 Solving Puzzles in Teams

1.1.2 Using a Graphing Calculator to Explore a Function

1.1.3 Domain and Range

1.1.4 Points of Intersection in Multiple Representations

Section 1.2

1.2.1 Modeling a Geometric Relationship

1.2.2 Function Investigation

1.2.3 The Family of Linear Functions

1.2.4 Function Investigation Challenge

Chapter Closure

### Chapter 2: Transformations of Parent Graphs

Section 2.1

2.1.1 Modeling Non-Linear Data 55

2.1.2 Parabola Investigation 60

2.1.3 Graphing a Parabola Without a Table 66

2.1.4 Rewriting in Graphing Form 71

2.1.5 Mathematical Modeling with Parabolas 79

Section 2.2

2.2.1 Transforming Other Parent Graphs 83

2.2.2 Describing (h, k) for Each Family of Functions 91

2.2.3 Transformations of Functions 96

2.2.4 Transforming Non-Functions 100

2.2.5 Transforming Piecewise-Defined Functions 105

Chapter Closure

### Chapter 3: Equivalent Forms

Section 3.1

3.1.1 Equivalent Expressions

3.1.2 Rewriting Expressions and Determining Equivalence

3.1.3 Solving by Rewriting

Section 3.2

3.2.1 Investigating Rational Functions

3.2.2 Simplifying Rational Expressions

3.2.3 Multiplying and Dividing Rational Expressions

3.2.4 Adding and Subtracting Rational Expressions

3.2.5 Creating New Functions

Chapter Closure

### Chapter 4: Solving and Intersections

Section 4.1

4.1.1 Strategies for Solving Equations

4.1.2 Solving Equations and Systems Graphically

4.1.3 Finding Multiple Solutions to Systems of Equations

4.1.4 Using Systems of Equations to Solve Problems

Section 4.2

4.2.1 Solving Inequalities with One or Two Variables

4.2.2 Using Systems to Solve a Problem

4.2.3 Application of Systems of Linear Inequalities

4.2.4 Using Graphs to Find Solutions

Chapter Closure

### Chapter 5: Inverses and Logarithms

Section 5.1

5.1.1 “Undo” Equations

5.1.2 Using a Graph to Find an Inverse

5.1.3 Finding Inverses and Justifying Algebraically

Section 5.2

5.2.1 Finding the Inverse of an Exponential Function

5.2.2 Defining the Inverse of an Exponential Function

5.2.3 Investigating the Family of Logarithmic Functions

5.2.4 Transformations of Logarithmic Functions

5.2.5 Investigating Compositions of Functions

Chapter Closure

### Chapter 6: 3-D Graphing and Logarithms

Section 6.1

6.1.1 Creating a Three-Dimensional Model

6.1.2 Graphing Equations in Three Dimensions

6.1.3 Systems of Three-Variable Equations

6.1.4 Solving Systems of Three Equations with Three Unknowns

6.1.5 Using Systems of Three Equations for Curve Fitting

Section 6.2

6.2.1 Using Logarithms to Solve Exponential Equations

6.2.2 Investigating the Properties of Logarithms

6.2.3 Writing Equations of Exponential Functions

6.2.4 An Application of Logarithms

Chapter Closure

### Chapter 7: Trigonometric Functions

Section 7.1

7.1.1 Introduction to Cyclic Models

7.1.2 Graphing the Sine Function

7.1.3 Unit Circle Graph

7.1.4 Graphing and Interpreting the Cosine Function

7.1.6 Building a Unit Circle

7.1.7 The Tangent Function

Section 7.2

7.2.1 Transformations of y = sin x

7.2.2 One More Parameter for a Cyclic Function

7.2.3 Period of a Cyclic Function

7.2.4 Graph Equation

Chapter Closure

### Chapter 8: Polynomials

Section 8.1

8.1.1 Sketching Graphs of Polynomial Functions

8.1.2 More Graphs of Polynomials

8.1.3 Stretch Factors for Polynomial Functions

Section 8.2

8.2.1 Introducing Imaginary Numbers

8.2.2 Complex Roots

8.2.3 More Complex Numbers and Equations

Section 8.3

8.3.1 Polynomial Division

8.3.2 Factors and Integral Roots

8.3.3 An Application of Polynomials

Chapter Closure

### Chapter 9: Randomization and Normal Distributions

Section 9.1

9.1.1 Survey Design

9.1.2 Samples and the Role of Randomness

9.1.3 Bias in Convenience Samples

Section 9.2

9.2.1 Testing Cause and Effect with Experiments

9.2.2 Conclusions From Studies

Section 9.3

9.3.1 Relative Frequency Histograms

9.3.2 The Normal Probability Density Function

9.3.3 Percentiles

Chapter Closure

### Chapter 10: Series

Section 10.1

10.1.1 Introduction to Arithmetic Series

10.1.2 More Arithmetic Series

10.1.3 General Arithmetic Series

10.1.4 Summation Notation and Combinations of Series

Section 10.2

10.2.1 Geometric Series

10.2.2 Infinite Series

Section 10.3

10.3.1 Pascal’s Triangle and the Binomial Theorem

10.3.2 The Number e

Chapter Closure

### Chapter 11: Simulating Sampling Variability

Section 11.1

11.1.1 Simulations of Probability

11.1.2 More Simulations of Probability

11.1.3 Simulating Sampling Variability

Section 11.2

11.2.1 Statistical Test Using Sampling Variability

11.2.2 Variability in Experimental Results

11.2.3 Quality Control

11.2.4 Statistical Process Control

Section 11.3

11.3.1 Analyzing Decisions and Strategies

Chapter Closure

### Chapter 12: Analytic Trigonometry

Section 12.1

12.1.1 Analyzing Trigonometric Equations

12.1.2 Solutions to Trigonometric Equations

12.1.3 Inverses of Trigonometric Functions

12.1.4 Reciprocal Trigonometric Functions

Section 12.2

12.2.1 Trigonometric Identities

12.2.2 Proving Trigonometric Identities

12.2.3 Angle Sum and Difference Identities

Chapter Closure

### Appendix A: Sequences

Section A.1

A.1.1 Representing Exponential Growth

A.1.2 Rebound Ratios

A.1.3 The Bouncing Ball and Exponential Decay

Section A.2

A.2.1 Generating and Investigating Sequences

A.2.2 Generalizing Arithmetic Sequences

A.2.3 Recursive Sequences

Section A.3

A.3.1 Patterns of Growth in Tables and Graphs

A.3.2 Using Multipliers to Solve Problems

A.3.3 Comparing Sequences to Functions

Appendix Closure

### Appendix B: Exponential Functions

Section B.1

B.1.1 Investigation y = b

B.1.2 Multiple Representations of Exponential Functions

B.1.3 More Applications of Exponential Growth

B.1.4 Exponential Decay

B.1.5 Graph → Equation

B.1.6 Completing the Multiple Representations Web

Section B.2

B.2.1 Curve Fitting and Fractional Exponents

B.2.2 More Curve Fitting

B.2.3 Solving a System of Exponential Functions Graphically

Appendix Closure

### Appendix C: Comparing Single-Variable Data

Section C.1

C.1.1 Investigating Data Representations

C.1.2 Comparing Data

C.1.3 Standard Deviation

Appendix Closure

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# Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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# Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

# Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

# Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

# Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.