__Enduring Understandings__1. At a basic level, the student will become conversant in the language and notation of math. The student will be able to translate mathematical concepts into “normal” language and vice versa.

2. The student will be able to reason logically through a problem.

3. The student will begin to see the underlying structure of the mathematical system so as to recognize the assumptions, flaws, and logical conclusions. The student may begin to see alternatives to the system. Ideally, the student will recognize this systemic approach in other fields and apply the same analysis to them.

4. The student will hopefully go beyond the utilitarian mode of learning (“When are we going to use this?” or “Is this going to be on the test?”) and begin to learn new things for the sake of learning them. Ultimately, the student might begin to appreciate the power and the intrinsic beauty people find in the material.

5. The student will begin to develop a meta-cognitive approach to learning. That is, the student will begin to think about his or her thinking process and how that affects his or her learning.

6. The student will develop a confidence and tenacity when approaching lengthy and intricate math problems. By breaking down a problem into its component parts, analyzing and resolving each part, and then reassembling the whole, the student will develop a sense that no problem is beyond his or her grasp.

7. The student will develop the ability to manage large amounts of information by learning to prioritize time, tasks, and goals.

__Essential Questions--Overall__1. What does this “math stuff” mean?

2. Where should this problem start and where does it end? How can I connect those?

3. How do all the problems relate to one another?

4. Why do some people find this interesting? Can it interest me?

5. What are learning preferences? What are my learning preferences? How might other approaches (that I do not prefer) help me learn?

6. How can I make this problem more manageable? What are the pieces of a particular problem and how do they fit together?

7. What information or techniques require details? When can I just know generalities? When should I skip a problem and move on?

__Unit Expectations__

__Unit I. Derivatives__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· The meaning of a derivative in context of graphing, motion, and rate of change

· How to calculate the derivative for various kinds of functions

· How a derivative helps to estimate function values

**Unit Essential Questions**

1. What is a derivative and how is this different in different contexts?

2. What are derivatives used for?

3. Why are there so many different rules for differentiation?

__Unit II Anti-Derivatives__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What an Anti-Derivative is and how it is calculated

· The graphic, algebraic and numerical representations of a differential equation are and how they are related

· What a differential equation is and how to solve one

· What a Slope Field is and how to use is to envision its solution curve

· What the solution function to a slope field might look like

**Unit Essential Questions**

1. What is an ant-derivative and how is it calculated?

2. What is a differential equation and how is it solved?

3. What is a slope field and what is its relationship to particular solution curves?

4. How does Euler’s Method relate to Separation of Variables and Slope Fields?

__Unit III Limits__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a limit is and how it is calculated

· The difference between a one-sided limit and a two-sided limit

· The difference between an infinite limit and a limit at infinity, and how to calculate each

· The abstract nature of the underpinnings of the Calculus and how they relate to the concept of transfinite and indeterminate numbers.

· The concepts of continuity, differentiability, and smoothness

**Unit Essential Questions**

1. What are the different kinds of limits and how are they calculated?

2. What are the various kinds of discontinuity?

3. What do the various kinds of limits have to do with continuity and differentiability?

4. What is differentiability and how is it proven?

5. What is L’Hopital’s Rule and how and when do I use it?

6. What are improper integrals and how are they calculated?

7. How can an infinite region have a finite area?

Unit IV Applications of the DerivativeUnit IV Applications of the Derivative

**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a derivative tells about extremes

· The relationship between a function and its graph and how the derivative helps define that relationship

· What optimization is and how to find it

· The vector nature of position and its derivatives in 2D motion

· What concavity and inflection are

· How the graphs of a function, its derivative and its second derivative are related to one another

**Unit Essential Questions**

1. What does optimization mean?

2. What do derivative graphs tell about and unknown “original” function’s graph? How are the traits of one related to the traits of the others?

3. How are position, velocity, speed, and acceleration related?

__Unit V Integrals__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· What a Definite Integral is and how it is calculated

· The relationship between a definite integral and the are under a curve

· The interpretation of the definite integral in context of graphing and motion

· The difference between net change and total change in context of graphing and motion

· The idea of the definite integral as an accumulator.

· How trapezoids and Riemann Rectangles estimate the definite integral

· How the increasing or decreasing nature of a function relates to where an approximation is an under- or over-estimate.

**Unit Essential Questions**

1. What is a definite integral and how do I calculate it?

2. How is the definite integral set up in different contexts?

3. What are Riemann Rectangles and how are they used? How do I evaluate an integral when the equation for the integrand is known?

4. What is average value and how do I find it?

5. How do I tell the difference between the theorems (Average Value, Mean Value, Intermediate Value, etc) that all sound alike?

6. What does it mean for an integral to be “an accumulator?”

__Unit VI Applications of the Integral__**Unit Enduring Understandings**

By the end of this unit, the student will understand:

· How to calculate volumes of solids formed by rotation or cross-section

· How to calculate volumes of solids formed by cross-sections arising from a base region

· How to ignore the distracter of visualizing three dimensional shapes in order to see the appropriate set-up of the problem

· How the Riemann Rectangles relate to the Volume of a solid

**Unit Essential Questions**

1. How do I calculate area for an irregular region?

2. How do I calculate volume of a solid formed by rotating irregular region?

3. How do I calculate volume of a solid formed by polygonal cross-sections rising from an irregular region?

4. How does the sample Riemann Rectangle determine which volume formula to use?

5. How do I see through the “white-noise” of a complicated 3D problem?

6. How do I calculate arc-length of an irregular region?