Algebra, Geometry, & Algebra 2

Table of Contents

Core Connections Geometry

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

7.2.1 Special Quadrilaterals and Proof 

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: Solids and Constructions

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

Core Connections
Algebra

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

Chapter 8: Quadratic Functions

Section 8.1

8.1.1 Introduction to Factoring Quadratics 

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 

8.2.4 Completing the Quadratic Web 

8.2.5 Completing the Square 

Chapter Closure

Chapter 9: Solving Quadratics and Inequalities

Section 9.1

9.1.1 Solving Quadratic Equations 

9.1.2 Introduction to the Quadratic Formula 

9.1.3 More Solving Quadratic Equations 

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

Core Connections
Algebra 2

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 

7.1.5 Defining a Radian 

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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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

This series contains three different courses, taken in either order. The courses are designed for schools and teachers with a minimum of one year of experience teaching with CPM curriculum materials. Teachers will develop further understanding of strategies and tools for instructional practices and assessment.

Building on Equity

In this course, participants will learn how to include equitable practices in their  classroom and support traditionally underserved students in becoming leaders of their own learning. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for implementing Chapter 1 that creates an equitable classroom culture and curate strategies for supporting all students in becoming leaders of their own learning. Follow-up during the school year will support ongoing implementation of equitable classroom practices.

Building on Assessment

In this course, participants will apply assessment research to develop methods to provide feedback to students and to inform equitable assessment decisions. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

This professional learning builds upon the Foundations for Implementation Series by improving teachers’ 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 rigorous, team-worthy tasks with all elements of the Effective Mathematics Teaching Practices.