Integrated I,II and III
Table of Contents

Integrated I

Chapter 1: Functions

Section 1.1

1.1.1 Solving Puzzles in Teams 

1.1.2 Investigating the Growth of Patterns 

1.1.3 Multiple Representations of Functions 

Section 1.2

1.2.1 Function Machines

1.2.3 Domain and Range 

Section 1.3

1.3.1 Rewriting Expressions with Exponents 

1.3.2 Zero and Negative Exponents 

Chapter Closure

Chapter 2: Linear Functions

Section 2.1

2.1.1 Seeing Growth in Linear Functions

2.1.2 Comparing △y and △x 57

2.1.3 Slope 

2.1.4 y = mx + b and More on Slope 

Section 2.2

2.2.1 Modeling Linear Functions 

2.2.2 Rate of Change 

2.2.3 Equations of Lines in a Situation 

2.2.4 Dimensional Analysis 

Section 2.3

2.3.1 Writing the Equation of a Line Given the Slope and a Point 

2.3.2 Writing the Equation of a Line Through Two Points 

2.3.3 Writing y = mx + b from Graphs and Tables 

Chapter Closure

Chapter 3:Transformations and Solving

Section 3.1

3.1.1 Spatial Visualization and Reflections 

3.1.2 Rotations and Translations 

3.1.3 Slopes of Parallel and Perpendicular Lines 

3.1.4 Defining Rigid Transformations 

3.1.5 Using Transformations to Create Polygons 

3.1.6 Symmetry 

Section 3.2

3.2.1 Modeling Area and Perimeter with Algebra Tiles 

3.2.2 Exploring an Area Model 

3.2.3 Multiplying Polynomials and the Distributive Property 

Section 3.3

3.3.1 Multiple Methods for Solving Equations 

3.3.2 Fraction Busters 

3.3.3 Solving Exponential and Complex Equations 

Chapter Closure

Chapter 4: Modeling Two-Variable Data

Section 4.1

4.1.1 Line of Best Fit 

4.1.2 Residuals 

4.1.3 Upper and Lower Bounds 

4.1.4 Least Squares Regression Line 

Section 4.2

4.2.1 Residual Plots 

4.2.2 Correlation 

4.2.3 Association is Not Causation 

4.2.4 Interpreting Correlation in Context 

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 Comparing Growth in Tables and Graphs 

5.3.2 Using Multipliers to Solve Problems 

5.3.3 Comparing Sequences to Functions 

Chapter Closure

Chapter 6: Systems of Equations

Section 6.1

6.1.1 Working with Multi-Variable Equations 

6.1.2 Summary of Solving Equations 

6.1.3 Solving Word Problems by Using Different Representations 

6.1.4 Solving Word Problems by Writing Equations 

Section 6.2

6.2.1 Solving Systems of Equations Using the Equal Values Method 

6.2.2 Solving Systems of Equations Using Substitution 

6.2.3 Making Connections: Systems and Multiple Representations 

Section 6.3

6.3.1 Solving Systems Using Elimination 

6.3.2 More Elimination 

6.3.3 Making Connections: Systems, Solutions, and Graphs 

Section 6.4

6.4.1 Choosing a Strategy for Solving a System 

6.4.2 Pulling it all Together 

Chapter Closure

Chapter 7: Congruence and Coordinate Geometry

Section 7.1

7.1.1 Defining Congruence 

7.1.2 Conditions for Triangle Congruence 

7.1.3 Creating a Flowchart 

7.1.4 Justifying Triangle Congruence Using Flowcharts 

7.1.5 More Conditions for Triangle Congruence

7.1.6 Congruence of Triangles Through Rigid Transformations 

7.1.7 More Congruence Flowcharts 

Section 7.2

7.2.1 Studying Quadrilaterals on a Coordinate Grid 

7.2.2 Coordinate Geometry and Midpoints 

7.2.3 Identifying Quadrilaterals on a Coordinate Grid 

Chapter Closure

Chapter 8:Exponential Functions

Section 8.1

8.1.1 Investigating 

8.1.2 Multiple Representations of Exponential Functions

8.1.3 More Applications of Exponential Functions 

8.1.4 Exponential Decay 

8.1.5 Graph → Equation 

8.1.6 Completing the Multiple Representations Web 

Section 8.2

8.2.1 Curve Fitting 

8.2.2 Curved Best-Fit Models 

8.2.3 Solving a System of Exponential Functions Graphically 

Chapter Closure

Chapter 9: Inequalities

Section 9.1

9.1.1 Solving Linear, One-Variable Inequalities 

9.1.2 More Solving Inequalities 

9.1.3 Solving Absolute Value Equations and Inequalities 

Section 9.2

9.2.1 Graphing Two-Variable Inequalities 

9.2.2 Graphing Linear and Nonlinear Inequalities 

Section 9.3

9.3.1 Systems of Inequalities 

9.3.2 More Systems of Inequalities 

9.3.3 Applying Inequalities to Solve Problems 

Chapter Closure

Chapter 10: Functions and Data

Section 10.1

10.1.1 Association in Two-Way Tables 

10.1.2 Investigating Data Representations 

10.1.3 Comparing Data 

10.1.4 Standard Deviation 

Section 10.2

10.2.1 Transforming Functions 

10.2.2 Arithmetic Operations with Functions 

10.2.3 Proving Linear and Exponential Growth Patterns 

Chapter Closure

Chapter 11: Constructions and Closure

Section 11.1

11.1.1 Introduction to Constructions 

11.1.2 Constructing Bisectors 

11.1.3 More Explorations with Constructions 

Section 11.2

11.2.1 Solving Work and Mixing Problems 

11.2.2 Solving Equations and Systems Graphically 

11.2.3 Using a Best-Fit Line to Make a Prediction 

11.2.4 Treasure Hunt 

11.2.5 Using Coordinate Geometry and Constructions to Explore Shapes 

11.2.6 Modeling with Exponential Functions and Linear Inequalities 

Chapter Closure

Appendix: Solving Equations

Section A.1

A.1.1 Exploring Variables and Expressions 

A.1.2 Using Zero to Simplify Algebraic Expressions 

A.1.3 Using Algebra Tiles to Compare Expressions 

A.1.4 Justifying and Recording Work 

A.1.5 Using Algebra Tiles to Solve for x 

A.1.6 More Solving Equations 

A.1.7 Checking Solutions 

A.1.8 Determining the Number of Solutions 

A.1.9 Using Equations to Solve Problems 

Appendix Closure

Checkpoint Materials

Checkpoint 1: Solving Linear Equations, Part 1 (Integer Coefficients) 

Checkpoint 2: Evaluating Expressions and the Order of Operations 

Checkpoint 3: Operations with Rational Numbers 

Checkpoint 4: Laws of Exponents and Scientific Notation 

Checkpoint 5: Writing the Equation of a Line 

Checkpoint 6A: Solving Linear Equations, Part 2 (Fractional Coefficients) 

Checkpoint 6B: Multiplying Binomials and Solving Equations with Parentheses 

Checkpoint 7: Interpreting Associations 

Checkpoint 8A: Rewriting Equations with More Than One Variable 

Checkpoint 8B: Solving Problems by Writing Equations 

Checkpoint 9: Solving Linear Systems of Equations 

Checkpoint 10: Determining Congruent Triangles 

Checkpoint 11: The Exponential Web

Integrated II

Chapter 1: Exploring Algebraic and Geometric Relationships

Section 1.1

1.1.1 Attributes of Polygons 

1.1.2 More Attributes of Polygons 

Section 1.2

1.2.1 Making Predictions and Investigating Results 

1.2.2 Perimeters and Areas of Enlarging Patterns 

1.2.3 Area as a Product and a Sum 

1.2.4 Describing a Graph 

Section 1.3

1.3.1 Angle Pair Relationships 

1.3.2 Angles Formed by Transversals 

1.3.3 More Angles Formed by Transversals 

1.3.4 Angles and Sides of a Triangle 

Chapter Closure

Chapter 2: Justification and Similarity

Section 2.1

2.1.1 Triangle Congruence Theorems 

2.1.2 Flowcharts for Congruence 

2.1.3 Converses 

2.1.4 Proof by Contradiction 

Section 2.2

2.2.1 Dilations 

2.2.2 Similarity 

Section 2.3

2.3.1 Conditions for Triangle Similarity 

2.3.2 Determining Similar Triangles 

2.3.3 Applying Similarity 

2.3.4 Similar Triangle Proofs 

Chapter Closure

Chapter 3: Probability and Trigonometry

Section 3.1

3.1.1 Using an Area Model 

3.1.2 Using a Tree Diagram 

3.1.3 Probability Models 

3.1.4 Unions, Intersections, and Complements 

3.1.5 Expected Value 

Section 3.2

3.2.1 Constant Ratios in Right Triangles 

3.2.2 Connecting Slope Ratios to Specific Angles 

3.2.3 Expanding the Trig Table 

3.2.4 The Tangent Ratio 

3.2.5 Applying the Tangent Ratio 

Chapter Closure

Chapter 4: Factoring and More Trigonometry

Section 4.1

4.1.1 Introduction to Factoring Expressions 

4.1.2 Factoring with Area Models 

4.1.3 Factoring More Quadratics 

4.1.4 Factoring Completely 

4.1.5 Factoring Special Cases 

Section 4.2

4.2.1 Sine and Cosine Ratios 

4.2.2 Selecting a Trig Tool 

4.2.3 Inverse Trigonometry 

4.2.4 Trigonometric Applications 

Chapter Closure

Chapter 5: Quadratic Functions

Section 5.1

5.1.1 Investigating the Graphs of Quadratic Functions 

5.1.2 Multiple Representations of Quadratic Functions 

5.1.3 Zero Product Property 

5.1.4 Writing Equations for Quadratic Functions 

5.1.5 Completing the Quadratic Web 

Section 5.2

5.2.1 Perfect Square Equations 

5.2.2 Completing the Square 

5.2.3 More Completing the Square 

5.2.4 Introduction to the Quadratic Formula 

5.2.5 Solving and Applying Quadratic Equations 

5.2.6 Introducing Complex Numbers

Chapter Closure

Chapter 6: More Right Triangles

Section 6.1

6.1.1 Special Right Triangles 

6.1.2 Pythagorean Triples 

6.1.3 Special Right Triangles and Trigonometry 

6.1.4 Radicals and Fractional Exponents 

Section 6.2

6.2.1 At Your Service 

6.2.2 Angles on a Pool Table 

6.2.3 Shortest Distance Problems 345

6.2.4 The Number System and Deriving the Quadratic Formula 

6.2.5 Using Algebra to Find a Maximum 

6.2.6 Analyzing a Game 

Chapter Closure

Chapter 7: Proof and Conditional Probability

Section 7.1

7.1.1 Explore-Conjecture-Prove 

7.1.2 Properties of Rhombi 

7.1.3 Two Column Proofs 

7.1.4 More Geometric Proofs 

7.1.5 Using Similar Triangles to Prove Theorems 

Section 7.2

7.2.1 Conditional Probability and Independence 

7.2.2 More Conditional Probability 

7.2.3 Applications of Probability 

Chapter Closure

Chapter 8: Polygons and Circles

Section 8.1

8.1.1 Constructing Triangle Centers 

Section 8.2

8.2.1 Angles of Polygons 

8.2.2 Areas of Regular Polygons 

Section 8.3

8.3.1 Area Ratios of Similar Figures 

8.3.2 Ratios of Similarity 

Section 8.4

8.4.1 A Special Ratio 458

8.4.2 Arcs and Sectors 463

8.4.3 Circles in Context 469

Chapter Closure

Chapter 9: Modeling with Functions

Section 9.1

9.1.1 Modeling Nonlinear Data 

9.1.2 Parabola Investigation 

9.1.3 Graphing Form of a Quadratic Function 

9.1.4 Transforming the Absolute Value Function 

Section 9.2

9.2.1 Quadratic Applications with Inequalities 

9.2.2 Solving Systems of Equations 

Section 9.3

9.3.1 Average Rate of Change and Projectile Motion 

9.3.2 Comparing the Growth of Functions 

9.3.3 Piecewise-Defined Functions 

9.3.4 Combining Functions 

Section 9.4

9.4.1 Inverse Functions 

Chapter Closure

Chapter 10: Circles and More

Section 10.1

10.1.1 The Equation of a Circle 

10.1.2 Completing the Square for Equations of Circles 

10.1.3 The Geometric Definition of a Parabola 

Section 10.2

10.2.1 Introduction to Chords 

10.2.2 Angles and Arcs

10.2.3 Chords and Angles

10.2.4 Tangents 

10.2.5 Tangents and Arcs 

Chapter Closure

Chapter 11: Solids

Section 11.1

11.1.1 Prisms and Cylinders 

11.1.2 Volumes of Similar Solids 

11.1.3 Ratios of Similarity 

Section 11.2

11.2.1 Volume of a Pyramid 

11.2.2 Surface Area and Volume of a Cone 

11.2.3 Surface Area and Volume of a Sphere 

Chapter Closure

Chapter 12: Counting and Closure

Section 12.1

12.1.1 The Fundamental Counting Principle 

12.1.2 Permutations 

12.1.3 Combinations 

12.1.4 Categorizing Counting Problems 

Section 12.2

12.2.1 Using Geometry to Calculate Probabilities 

12.2.2 Choosing a Model 

12.2.3 The Golden Ratio 

12.2.4 Some Challenging Probability Problems 

Chapter Closure

Checkpoint Materials

Checkpoint 1: Solving Problems with Linear and Exponential Relationships

Checkpoint 2: Calculating Areas and Perimeters of Complex Shapes 

Checkpoint 3: Angle Relationships in Geometric Figures 

Checkpoint 4: Solving Proportions and Similar Figures 

Checkpoint 5: Calculating Probabilities 

Checkpoint 7: Factoring Quadratic Expressions 

Checkpoint 8: Applying Trigonometric Ratios and the Pythagorean Theorem 

Checkpoint 9: The Quadratic Web 

Checkpoint 10: Solving Quadratic Equations 

Checkpoint 11: Angle Measures and Areas of Regular Polygons 

Checkpoint 12: Circles, Arcs, Sectors, Chords, and Tangents

Integrated III

Chapter 1: Investigations and Functions

Section 1.1

1.1.1 Solving a Function Puzzle in Teams 

1.1.2 Using a Graphing Calculator to Explore a Function 

1.1.3 Function Investigation 

1.1.4 Combining Linear Functions 

Section 1.2

1.2.1 Representing Points of Intersection 

1.2.2 Modeling a Geometric Relationship 

1.2.3 Describing Data 

Chapter Closure

Chapter 2:Transformations of Parent Graphs

Section 2.1

2.1.1 Transforming Quadratic Functions 

2.1.2 Modeling with Parabolas 

Section 2.2

2.2.1 Transforming Other Parent Graphs 

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

2.2.3 Transformations of Functions 

2.2.4 Transforming Non-Functions 

2.2.5 Developing a Mathematical Model 

Section 2.3

2.3.1 Completing the Square 

Chapter Closure

Chapter 3: Solving and Inequalities

Section 3.1

3.1.1 Strategies for Solving Equations 

3.1.2 Solving Equations Graphically 

3.1.3 Multiple Solutions to Systems of Equations 

3.1.4 Using Systems of Equations to Solve Problems 

Section 3.2

3.2.1 Solving Inequalities with One or Two Variables 

3.2.2 Using Systems to Solve a Problem 

3.2.3 Applications of Systems of Inequalities 

3.2.4 Using Graphs to Determine Solutions 

Chapter Closure

Chapter 4: Normal Distributions and Geometric Modeling

Section 4.1

4.1.1 Survey Design 

4.1.2 Samples and the Role of Randomness 

4.1.3 Bias in Convenience Samples 

Section 4.2

4.2.1 Testing Cause and Effect with Experiments 

4.2.2 Conclusions from Studies 

Section 4.3

4.3.1 Relative Frequency Histograms 

4.3.2 The Normal Probability Density Function 

4.3.3 Percentiles 

Section 4.4

4.4.1 Cross-Sections and Solids of a Revolution 

4.4.2 Modeling with Geometric Solids 

4.4.3 Designing to Meet Constraints 

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 More Inverse Functions 

Section 5.2

5.2.1 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 

Chapter Closure

Chapter 6: Simulating Sampling Variability

Section 6.1

6.1.1 Simulations of Probability 

6.1.2 More Simulations of Probability 

6.1.3 Simulating Sampling Variability 

Section 6.2

6.2.1 Statistical Test Using Sampling Variability 

6.2.2 Variability in Experimental Results 

6.2.3 Quality Control 

6.2.4 Statistical Process Control 

Section 6.3

6.3.1 Analyzing Decisions and Strategies 

Chapter Closure

Chapter 7: Logarithms and Triangles

Section 7.1

7.1.1 Using Logarithms to Solve Exponential Equations 

7.1.2 Investigating the Properties of Logarithms 

7.1.3 Writing Equations of Exponential Functions 

7.1.4 An Application of Logarithms 

Section 7.2

7.2.1 Determining Missing Parts of Triangles 

7.2.2 Law of Sines 

7.2.3 Law of Cosines 

7.2.4 The Ambiguous Case 

7.2.5 Choosing a Tool

Chapter Closure

Chapter 8: Polynomials

Section 8.1

8.1.1 Sketching Graphs of Polynomial Functions 

8.1.2 More Graphs of Polynomial Functions 

8.1.3 Stretch Factors for Polynomial Functions 

Section 8.2

8.2.1 Writing Equations Using Complex Roots 

8.2.2 More Real and Complex Roots 

Section 8.3

8.3.1 Polynomial Division 

8.3.2 Factors and Rational Zeros 

8.3.3 An Application of Polynomials 

8.3.4 Special Cases of Factoring417

Chapter Closure

Chapter 9: Trigonometric Functions

Section 9.1

9.1.1 Introductions to Periodic Models 

9.1.2 Graphing the Sine Functions 

9.1.3 Unit Circle ↔ Graph 

9.1.4 Graphing and Interpreting the Cosine Function 

9.1.5 Defining a Radian 

9.1.6 Building a Unit Circle 

9.1.7 The Tangent Function 

Section 9.2

9.2.1 Transformations of y = sin(x) 

9.2.2 One More Parameter for a Periodic Function 

9.2.3 Period of a Trigonometric Function 

9.2.4 Graph ↔ Equation 

Chapter Closure

Chapter 10: Series

Section 10.1

10.1.1 Introduction to Arithmetic Series 489

10.1.2 More Arithmetic Series 497

10.1.3 General Arithmetic Series 501

10.1.4 Summation Notation and Combinations of Series 506

10.1.5 Mathematical Induction 510

Section 10.2

10.2.1 Geometric Series 518

10.2.2 Infinite Series 528

Section 10.3

10.3.1 Using a Binomial Probability Model 535

10.3.2 Pascal’s Triangle and the Binomial Theorem 541

10.3.3 The Number e 549

Chapter Closure

Chapter 11: Rational Expressions and Three-Variable Systems

Section 11.1

11.1.1 Simplifying Rational Expressions 

11.1.2 Multiplying and Dividing Rational Expressions 

11.1.3 Adding and Subtracting Rational Expressions 

11.1.4 Operations with Rational Expressions 

Section 11.2

11.2.1 Creating a Three-Dimensional Model 

11.2.2 Graphing Equations in Three Dimensions

11.2.3 Solving Systems of Three Equations with Three Variables 

11.2.4 Using Systems of Three Equations for Curve Fitting 

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

Checkpoint Materials

Checkpoint 2: Solving Quadratic Equations 

Checkpoint 3: Function Notation and Describing a Function 

Checkpoint 4: Expressions with Integer and Rational Exponents 

Checkpoint 5: Transformations of Functions 

Checkpoint 6: Solving Complicated Equations and Systems 663

Checkpoint 7: Solving and Graphing Inequalities 667

Checkpoint 8: Determining the Equation for the Inverse of a Function 670

Checkpoint 9A: Solving Equations with Exponents 673

Checkpoint 9B: Rewriting Expressions and Solving Equations with Logarithms 675

Checkpoint 10: Solving Triangles 678

Checkpoint 11: Roots and Graphs of Polynomial Functions 682

Checkpoint 12: Periodic Functions

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Algebra Tiles Blue Icon

Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

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.