Advanced High School Courses
Table of Contents

Third Edition

Chapter 1: Preparing for Your Journey

Section 1.1

1.1.1 Interpreting Graphs 

1.1.2 The Spring Problem 

1.1.3 Modeling with Functions 

1.1.4 Rates of Change 

1.1.5 Setting Up Word Problems 

1.1.6 Equivalent Expressions 

Section 1.2

1.2.1 Composition of Functions 

1.2.2 Inverse Functions 

1.2.3 Piecewise-Defined Functions and Continuity 

Section 1.3

1.3.1 Radians as a Unit of Measure 

1.3.2 Radian Measure in the Unit Circle 

1.3.3 Applications of Radian Measure 


Chapter 2: Functions and Trigonometry

Section 2.1

2.1.1 Characteristics of Functions 

2.1.2 Even and Odd Functions 

2.1.3 Transformations of Functions 

Section 2.2

2.2.1 Special Angles in the Unit Circle 

2.2.2 Trigonometric Ratios in the Unit Circle 

2.2.3 Graphs of Sine and Cosine 

2.2.4 Transformations of Sine and Cosine 

2.2.5 Horizontal Stretches of Sine and Cosine Graphs 

Section 2.3

2.3.1 Solving Trigonometric Equations 

2.3.2 Inverse Sine and Cosine 

2.3.3 Graphs of Tangent and Inverse Tangent 


Chapter 3: Algebra and Area Under a Curve

Section 3.1

3.1.1 Operations with Rational Expressions 

3.1.2 Rewriting Expressions and Equations 

3.1.3 Solving Nonlinear Systems of Equations 

3.1.4 Polynomial Division 

3.1.5 Solving Classic Word Problems 

Section 3.2

3.2.1 Using Sigma Notation 

3.2.2 Area Under a Curve: Part One 

3.2.3 Area Under a Curve: Part Two 

3.2.4 Area Under a Curve: Part Three 


Chapter 4:Polynomial and Rational Functions

Section 4.1

4.1.1 Graphs of Polynomial Functions in Factored Form 

4.1.2 Writing Equations of Polynomial Functions 

4.1.3 Identifying and Using Roots of Polynomials 

Section 4.2

4.2.1 Graphing Transformations of y = 1x

4.2.2 Graphing Rational Functions 

4.2.3 Graphing Reciprocal Functions 

Section 4.3

4.3.1 Polynomial and Rational Inequalities 

4.3.2 Applications of Polynomial and Rational Functions 


Chapter 5: Exponentials and Logarithms

Section 5.1

5.1.1 Applications of Exponential Functions 

5.1.2 Stretching Exponential Functions 

5.1.3 The Number e 

Section 5.2

5.2.1 Logarithms 

5.2.2 Properties of Logarithms 

5.2.3 Solving Exponential and Logarithmic Equations 

5.2.4 Graphing Logarithmic Functions 

5.2.5 Applications of Exponentials and Logarithms 


Chapter 6: Triangles and Vectors

Section 6.1

6.1.1 The Law of Sines and Area

6.1.2 The Law of Cosines 

6.1.3 The Ambiguous Case of the Law of Sines 

Section 6.2

6.2.1 An Introduction to Vectors 

6.2.2 Operations with Vectors 

6.2.3 Applications of Vectors 

6.2.4 The Dot Product 


Chapter 7: Limits and Rates

Section 7.1

7.1.1 An Introduction to Limits 

7.1.2 Working With One-Sided Limits 

7.1.3 The Definition of a Limit 

7.1.4 Limits and Continuity 

7.1.5 Special Limits 

Section 7.2

7.2.1 Rates of Change from Data 

7.2.2 Slope and Rates of Change 

7.2.3 Average Velocity and Rates of Change 

7.2.4 Moving from AROC to IROC 

7.2.5 Rate of Change Applications 


Chapter 8: Extending Periodic Functions

Section 8.1

8.1.1 Graphing y = asin(b(x – h)) + k 

8.1.2 Modeling With Periodic Functions 

8.1.3 Improving the Spring Problem 

Section 8.2

8.2.1 Graphing Reciprocal Trigonometric Functions 

8.2.2 Trigonometric Functions, Geometrically 

Section 8.3

8.3.1 Simplifying Trigonometric Expressions 

8.3.2 Proving Trigonometric Identities 

8.3.3 Angle Sum and Difference Identities 

8.3.4 Double-Angle and Half-Angle Identities 

8.3.5 Solving Complex Trigonometric Equations 


Chapter 9: Matrices

Section 9.1

9.1.1 Introduction to Matrices 

9.1.2 Matrix Multiplication 

9.1.3 Determinants and Inverse Matrices

9.1.4 Solving Systems Using Matrix Equations 

Section 9.2

9.2.1 Linear Transformations 

9.2.2 Compositions of Transformations 

9.2.3 Properties of Linear Transformations Closure

Chapter 10: Conics and Parametric Functions

Section 10.1

10.1.1 Circles and Completing the Square 

10.1.2 Ellipses 

10.1.3 Hyperbolas 

10.1.4 Parabolas 

10.1.5 Identifying and Graphing Conic Sections 

Section 10.2

10.2.1 Parametrically-Defined Functions 

10.2.2 Applications of Parametrically-Defined Functions 

10.2.3 Conic Sections in Parametric Form


Chapter 11: Polar Functions and Complex Numbers

Section 11.1

11.1.1 Plotting Polar Coordinates 

11.1.2 Graphs of Polar Functions 

11.1.3 Families of Polar Functions 

11.1.4 Converting Between Polar and Rectangular Forms 

Section 11.2

11.2.1 Using the Complex Plane 

11.2.2 Operations with Complex Numbers Geometrically 

11.2.3 Polar Form of Complex Numbers 

11.2.4 Operations with Complex Numbers in Polar Form 

11.2.5 Powers and Roots of Complex Numbers 


Chapter 12: Series and Statistics

Section 12.1

12.1.1 Arithmetic Series 

12.1.2 Geometric Series 

12.1.3 Infinite Geometric Series 12.1.4 Applications of Geometric Series 

12.1.5 The Sum of the Harmonic Series 

Section 12.2

12.2.1 The Binomial Theorem 

12.2.2 Binomial Probabilities 

Section 12.3

12.3.1 Expected Value of a Discrete Random Variable 

12.3.2 Expected Value and Decision Making 


Chapter 13: Precalculus Finale

Section 13.1

13.1.1 A Race to Infinity 

13.1.2 Limits to Infinity 

13.1.3 Evaluating Limits at a Point Algebraically 

13.1.4 Another Look at e 

Section 13.2

13.2.1 Trapping Area With Trapezoids 

13.2.2 Area as a Function 

13.2.2A Going all to Pieces: Writing an Area Program 

13.2.3 Rocket Launch 

Section 13.3

13.3.1 Velocity and Position Graphs 

13.3.2 Instantaneous Velocity 

13.3.3 Slope Functions 

13.3.4 The Definition of Derivative 

13.3.5 Slope and Area Under a Curve 


Third Edition

Chapter 1:A Beginning Look at Calculus

Section 1.1

1.1.1 Applying Rates and Distance 

Section 1.2

1.2.1 Piecewise-Defined Functions and Continuity 

1.2.2 End Behavior and Asymptotes 

1.2.3 Holes, Vertical Asymptotes, and Approach Statements 

1.2.4 Composite Functions and Inverse Functions 

1.2.5 Attributes of Even and Odd Functions 

1.2.6 Design a Flag 

Section 1.3

1.3.1 Finite Differences 

1.3.2 Slope Statements and Finite Differences of Non-Polynomials 

1.3.3 The Slope Walk 

Section 1.4

1.4.1 Distance and Velocity 

1.4.2 Average Velocity on a Position Graph 

1.4.3 Average Velocity on a Velocity Graph 

1.4.4 Acceleration 

Section 1.5

1.5.1 Area and Slope

Chapter 2: Rates, Sums, Limits, and Continuity

Section 2.1

2.1.1 Area Under a Curve Using Trapezoids 

2.1.2 Methods to Calculate Area Under a Curve 

2.1.3 Area Under a Curve as a Riemann Sum 

Section 2.2

2.2.1 Introduction to Limits as Predictions 

2.2.2 Intuitive Ideas of Continuity

2.2.3 Definition of Continuity 

2.2.4 Evaluating Limits 

Section 2.3

2.3.1 Ramp Lab 

2.3.2 Sudden Impact 

2.3.3 Local Linearity 

Section 2.4

2.4.1 Improving Approximation

Chapter 3: Slope and Curve Analysis

Section 3.1

3.1.1 The Power Rule 

3.1.2 Secants to Tangents, AROC to IROC 

Section 3.2

3.2.1 Definition of a Derivative 

3.2.2 Derivatives Using Multiple Strategies120

3.2.3 Derivatives of Sine and Cosine 

Section 3.3

3.3.1 Curve Constructor: Part One 

3.3.2 The Shape of a Curve 

3.3.3 Curve Sketching: Derivatives 

3.3.4 Ways to Describe f ′ and f ′′ 

Section 3.4

3.4.1 Conditions for Differentiability 

3.4.2 Curve Constructor: Part Two 

3.4.3 Differentiability of Specific Functions 

3.4.4 Intersection of Tangents

Chapter 4: The Fundamental Theorem of Calculus

Section 4.1

4.1.1 Definite Integrals 

4.1.2 Properties of Definite Integrals 

4.1.3 More Properties of Definite Integrals 

Section 4.2

4.2.1 Deriving “Area Functions” 

4.2.2 Indefinite and Definite Integrals 

4.2.3 The Fundamental Theorem of Calculus 

4.2.4 The Fundamental Theorem of Calculus 

4.2.5 Integrals as Accumulators 1

Section 4.3

4.3.1 Fast Times: Parts One & Two 

4.3.2 Fast Times: Parts Three & Four 

4.3.3 Fast Times: Part Five 

Section 4.4

4.4.1 Area Between Curves 197

4.4.2 More Area Between Curves 200

4.4.3 Multiple Methods for Calculating Area Between Curves 

Section 4.5

4.5.1 Newton’s Method

Chapter 5: Derivative Tools and Applications

Section 5.1

5.1.1 Distance, Velocity, and Acceleration Functions 

5.1.2 Optimization 

5.1.3 Using the Frist and Second Derivatives 

5.1.4 Applying the First and Second Derivative Tests 

Section 5.2

5.2.1 The Product Rule 

5.2.2 The Chain Rule and Application: Part One 

5.2.3 The Chain Rule and Application: Part Two 

5.2.4 The Quotient Rule 

5.2.5 More Trigonometric Derivatives 

Section 5.3

5.3.1 Optimization Problems: Part One 

5.3.2 Optimization Problems: Part Two 

5.3.3 Optimization Problems: Part Three 

Section 5.4

5.4.1 Chain Rule Extension of the Fundamental Theorem of Calculus 

Section 5.5

5.5.1 Evaluating Limits of Indeterminate Forms 

5.5.2 Using l’Hôpital’s Rule

Chapter 6: More Tools and Theorems

Section 6.1

6.1.1 Exponential Functions 

6.1.2 Derivatives of Exponential Functions 

6.1.3 Derivatives Using Multiple Tools 

6.1.4 Integrals of Exponential Functions 

Section 6.2

6.2.1 Implicit Differentiation 

6.2.2 Implicit Differentiation Practice 

Section 6.3

6.3.1 Inverse Trigonometric Derivatives 

6.3.2 Derivatives of Natural Logarithms 

6.3.3 Derivatives of Inverse Functions 

Section 6.4

6.4.1 Mean Value 

6.4.2 Mean Value Theorem 

6.4.3 Mean Value Theorem: Applications 

Section 6.5

6.5.1 Improper Integrals 

Chapter 7: Related Rates and Integration Tools

Section 7.1

7.1.1 Related Rates Introduction 

7.1.2 Related Rates Applications: The Pythagorean Theorem 

7.1.3 Related Rates Applications: Similar Triangles 323

7.1.4 Related Rates Applications: Choosing the Best Formula 

7.1.5 Related Rates Applications: Trigonometry 

Section 7.2

7.2.1 Undoing the Chain Rule 

7.2.2 Integration with u-Substitution 

7.2.3 Definite Integrals and u-Substitution 

7.2.4 Varied Integration Techniques 

Section 7.3

7.3.1 Solving Differential Equations 

7.3.2 Newton’s Law of Cooling 

7.3.3 Solving Separable Differential Equations 

7.3.4 Slope Fields with Parallel Tangents 

7.3.5 Plotting Slope Efficiently 

7.3.6 Differential Equation and Slope Field Applications 

Section 7.4

7.4.1 Euler’s Method

7.4.2 Integration By Parts 

7.4.3 Integration By Parts with Substitution 

7.4.4 Integration by Partial Fractions

Chapter 8: Volume

8.1.1 Volumes by Slicing 

8.1.2 The Disk Method 

8.1.3 The Washer Method 

8.1.4 Revolution About Horizontal and Vertical Lines 

8.1.5 Changing the Axis of Rotation 

8.1.6 Disk and Washer Problems 

Section 8.2

8.2.1 Shell Lab 

8.2.2 Comparing the Disk and Shell Methods 

8.2.3 Using an Appropriate Method to Calculate Volume 

Section 8.3

8.3.1 Cross-Sections Lab: General Case 

8.3.2 Cross-Sections Lab: Functions Given

8.3.3 Cross-Section Problems 

Section 8.4

8.4.1 Arc Length

Chapter 9: Pre-Calculus Review

Section 9.1

9.1.1 Infinite Geometric Series 

9.1.2 More Infinite Geometric Series 

9.1.3 Convergence and Divergence 

Section 9.2

9.2.1 Parametric Equations 

9.2.2 Converting Between Parametric and Rectilinear Form 

Section 9.3

9.3.1 Introduction to Vectors 

9.3.2 Vector Operations 

Section 9.4

9.4.1 Graphs of Polar Equations 

9.4.2 Converting Between Polar and Rectilinear Form 

9.4.3 Polar Families

Chapter 10: Convergence of Series

Section 10.1

10.1.1 Convergence of Series 

10.1.2 The Divergence Test 

10.1.3 The Alternating Series Test 

10.1.4 The Integral Test 

10.1.5 The p-Series Test 

10.1.6 The Comparison Test 

10.1.7 The Limit Comparison Test 

10.1.8 The Ratio Test 

Section 10.2

10.2.1 The Cootie Lab 

10.2.2 More Logistic Differential Equations 

Section 10.3

10.3.1 Power Series Convergence 

10.3.2 Using Polynomials to Approximate Curves 

Section 10.4

10.4.1 Absolute Convergence 

10.4.2 Regrouping and Rearranging Series

Chapter 11: Polar and Parametric Functions

Section 11.1

11.1.1 Area Bounded by a Polar Curve 

11.1.2 More Polar Area 

11.1.3 Area Between Polar Curves 

Section 11.2

11.2.1 Applied Calculus in Component Form 

11.2.2 Second Derivatives in Component Form 

11.2.3 Total Distance and Arc Length 

Section 11.3

11.3.1 Slopes of Polar Curves 

11.3.2 More Slopes of Polar Curves 

Section 11.4

11.4.1 Battling Robots

Chapter 12: Approximating Functions and Error

Section 12.1

12.1.1 Approximating with Polynomial Functions 

12.1.2 Taylor Polynomials About x = 0 

12.1.3 Taylor Polynomials About x = c 

12.1.4 Taylor Series 

12.1.5 Building Taylor Series Using Substitution 

Section 12.2

12.2.1 Interval of Convergence Using Technology 

12.2.2 Interval of Convergence Analytically 

Section 12.3

12.3.1 Error Bound for Alternating Taylor Polynomials 

12.3.2 Lagrange Error Bound 

Section 12.4

12.4.1 Evaluating Indeterminate Forms Using Taylor Series

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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.