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Computer Math

 

COURSE:  CALCULUS (A.P. /U. Conn.)

TOPIC: Functions, Graphs and Limits

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

 

NCTM Standards:                                

a)     Analyze change in various contexts.

b)     Represent and analyze mathematical situations using algebraic symbols.

c)     Use mathematical models to represent and understand quantitative relationships.

d)     Apply and adapt a variety of appropriate strategies to solve problems.

e)     Apply appropriate techniques, tools, and formulas to determine measurements.

f)      Use the language of mathematics to express mathematical ideas precisely.

g)     Understand how mathematical ideas interconnect and build on one another.

 

CT Frameworks: 

a)     Functions can be viewed as objects on which operations can be performed.

b)     Operations such as addition and subtraction can be applied to objects such as vectors and matrices that are not numbers.

c)     Measurements that are not directly determined can be approximated with some degree of precision.

Enduring Understandings

Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

 The student will understand that:

  1. Calculus, along with geometric and analytic information, can explain the observed local and global behavior of a function.
  2. Limits can be determined using algebra, graphs and/or tables of data.
  3. Some graphs demonstrate asymptotic and unbounded behavior.
  4. Close values of the domain lead to close values of the range.

 
  1. How can calculus be used to help describe the behavior of a function?
  2. What is the best method to use to find the limit of a function?
  3. How do limits approaching infinity help describe the asymptotic behavior of a function?
  4. How do limits help determine the continuity of a function?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

  1. How to describe the various behaviors of a function using calculus techniques.
  2. How to choose the most appropriate technique for determining the limit of a function.
  3. How to determine the asymptotes of a function using the limiting process.
  4. How to describe the continuity of a function at a point using limits.

 

Students will be able to…

  1. Calculate limits using algebra.
  2. Estimate limits from graphs or tables of data.
  3. Describe asymptotic behavior in terms of limits involving infinity.
  4. Compare relative magnitudes of functions and their rates of change.
  5. Determine continuity in terms of limits.
  6. Use the Intermediate and Extreme Value Theorems.

 

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

The Fabulous Calculus Roller Coaster – Students will, as amusement park designers, draw plans (poster) for a theoretical roller coaster that utilizes the ideas of limits and continuity.  This plan is to be presented at the national amusement park conference. 

 

 

In addition to tests and quizzes, one or more of the following will be used:

  1. Computer laboratory simulations with lab reports.
  2. Theoretical discovery exercises.
  3. Research paper on the underpinnings of calculus.

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

  1. Cooperative learning activities.
  2. Graphing calculator activities.
  3. Homework and review worksheets.
  4. Work stations.

 

 

COURSE:  CALCULUS (A.P. /U. Conn.)

TOPIC: Derivatives

 

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

 

NCTM Standards:                               

a)     Analyze change in various contexts.

b)     Represent and analyze mathematical situations using algebraic symbols.

c)     Use mathematical models to represent and understand quantitative relationships.

d)     Apply and adapt a variety of appropriate strategies to solve problems.

e)     Apply appropriate techniques, tools, and formulas to determine measurements.

f)      Use the language of mathematics to express mathematical ideas precisely.

g)     Understand how mathematical ideas interconnect and build on one another.

 

CT Frameworks: 

a)     A wide variety of functions can be used to model real world situations.

b)     Functions can be viewed as objects on which operations can be performed.

 

Enduring Understandings

Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

 The student will understand that:

1.     Derivatives can be used to analyze curves.

2.     Derivatives can be used to model rates of change.

3.     Derivatives can be used in optimization problems.

4.     There is a relationship between the solution curves for differential equations and slope fields.

5.     The physics concepts of position, velocity and acceleration are related mathematically by the derivative.

6.     Finding the derivative of a function may require the use of several rules, including rules for:  sums, products, quotients, powers, exponentials, logarithms, trigonometric and inverse trigonometric functions, and the chain rule.

1.     How can calculus be used to help describe the behavior of a function?

2.     When a quantity is changing with respect to time, how are associated quantities changing?

3.     How can you find the greatest (smallest) value needed to solve a particular problem?

4.     How are slope fields and differential equations related?

5.     How are position, velocity, and acceleration related?

6.     How does the composition of functions tell you how many derivative rules must be applied?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

  1. How to completely analyze a function, including intervals where the function is increasing or decreasing.
  2. How to solve related rates problems.
  3. How to solve optimization problems.
  4. How to utilize slope fields to determine the shape of an implicitly defined function.
  5. How to find the position, velocity, and acceleration of a point moving along a line.

 

Students will be able to…

1.     Present the derivative graphically, numerically, and analytically.

2.     Interpret the derivative as a rate of change.

3.     Define the derivative as the limit of the difference quotient.

4.     Find derivatives using formulas.

5.     Describe the relationship between differentiability and continuity.

6.     Find the slope of a curve at a point.

7.     Find the equation of the tangent line to a curve at a point. (Use local linearity.)

8.     Express instantaneous rates of change as the limit of the average rate of change.

9.     Approximate rates of change from graphs and tables of values.

10. Find the corresponding characteristics of f, f 'and f ' '.

11. Use the derivative to determine intervals where a function is increasing or decreasing, concave up or concave down, or points of inflection.

12. Use the Mean Value Theorem and describe its geometric consequences.

13. Translate verbal descriptions into equations involving derivatives.

14. Differentiate using the rules for power, exponential, logarithmic, trigonometric and inverse trigonometric functions.

15. Differentiate using the rules for sums, products and quotients of functions.\

16. Use the chain rule and implicit differentiation.

 

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

As members of the Tootsie Roll Pop Company’s quality assurance team, students will determine the rate of change (experimentally) of the pop as it is licked (sucked).  The rate of change of the volume when the radius is three-fourths its original size will also be determined by modeling the data with an algebraic function of time.  The report written to the company’s executives should clearly outline the techniques used to gather the data, the data, the function used and the values of dr/dt and dV/dt.

In addition to tests and quizzes, one or more of the following will be used:

1.     Computer laboratory simulations with lab reports.

2.     Theoretical discovery exercises.

 

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

  1. Cooperative learning activities.
  2. Graphing calculator activities.
  3. Homework and review worksheets.
  4. Work stations.

 

 

COURSE:  CALCULUS (A.P. /U. Conn.)

TOPIC: Integrals

 

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

 

NCTM Standards:                               

a)     Analyze change in various contexts.

b)     Represent and analyze mathematical situations using algebraic symbols.

c)     Use mathematical models to represent and understand quantitative relationships.

d)     Apply and adapt a variety of appropriate strategies to solve problems.

e)     Apply appropriate techniques, tools, and formulas to determine measurements.

f)      Use the language of mathematics to express mathematical ideas precisely.

g)     Understand how mathematical ideas interconnect and build on one another.

 

CT Frameworks: 

a)     A wide variety of functions can be used to model real world situations.

b)     Functions can be viewed as objects on which operations can be performed.

c)     Measurements that are not directly determined can be approximated with some degree of precision.

 

Enduring Understandings

Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

 The student will understand that:

1.     Integration is a summation process.

2.     The Fundamental Theorems of Calculus relate differentiation and integration as inverse functions.

3.     Antiderivatives follow directly from derivatives.

4.     Antiderivatives can be used to solve initial condition problems, including separable differential equations.

5.     There are several numerical techniques to approximate the definite integral.

1.     How is sigma notation related to integration?

2.     How are the derivative and the integral related?

3.     What does a definite integral represent?

4.     How can you use your knowledge of derivatives to find the integral of a function?

5.     How can substitution make an integral simpler to evaluate?

6.     How can you find a specific antiderivative using initial conditions?

7.     How can the area under a curve be approximated?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

  1. That integration is the limit of a summation.
  2. That the derivative and the integral are related (using the Fundamental Theorem of Calculus).
  3. How to evaluate definite integrals.
  4. How to solve initial condition problems.
  5. How to utilize slope fields to determine the shape of an implicitly defined function.
  6. How to find the position, velocity, and acceleration of a point moving along a line.

 

Students will be able to…

  1. Compute Riemann sums using left, right, and midpoint evaluation points.
  2. Write the definite integral as the  limit of a Riemann sum.
  3. Apply the properties of definite integrals (additivity and linearity).
  4. Evaluate the definite integral using the Fundamental Theorem of Calculus.
  5. Show that antiderivatives follow directly from the derivatives of basic functions.
  6. Solve initial condition problems, including separable differential equations.
  7. Use Riemann and trapezoidal sums (and Simpson’s Rule) to approximate definite integrals represented algebraically, graphically and by tables of values.

 

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

As a ghost writer for Sir Arthur Conan Doyle, students will complete (and solve) a mystery story using their knowledge of calculus. The story must incorporate, as smoothly as possible, the mathematics Holmes (and/or Watson) used to solve the crime.

In addition to tests and quizzes, one or more of the following will be used:

1.     Computer laboratory simulations with lab reports.

2.     Theoretical discovery exercises.

 

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

1.     Cooperative learning activities.

2.     Graphing calculator activities.

3.     Homework and review worksheets.

4.     Work stations.