TITLE:  AREA AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS
TASK DEVELOPER:  MATHEW ACHERIL
CONTENT AREA AND GRADE: MATHEMATICS (CALCULUS)  GRADE 12
TARGET TEACHING DATE:  MAY 7TH,  8TH  ,9TH  & 10TH  2007
SCHOOL: John F. Kennedy High School, Paterson, NJ 07522 

MATHEMATICAL PROCESSES - K-12

STANDARD 4.5 MATHEMATICAL PROCESSES: All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.

Strand A. Problem Solving: At each grade level, with respect to content appropriate for that grade level, students will:

1. Learn mathematics through problem solving, inquiry, and discovery.
2. Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.
3. Pose problems of various types and levels of difficulty.
4. Monitor their progress and reflect on the process of their problem solving activity.

Strand D. Reasoning: At each grade level, with respect to content appropriate for that grade level, students will:

1. Recognize that mathematical facts, procedures, and claims must be justified.

Strand E. Representations: At each grade level, with respect to content appropriate for that grade level, students will:

1. Create and use representations to organize, record, and communicate mathematical ideas.

• Symbolic representations (e.g., a formula)

Strand F. Technology: Students will

1. Use graphing calculators and computer software to investigate properties of functions and their graphs.
2. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

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1. The students will write formulas and steps to be followed to evaluate an integral from the board
2. Students will evaluate definite integrals by sketching the graph and then use a geometric formula
3. Students will use Fundamental Theorem Of Integral Calculus to evaluate the integrals
4. Students will verify their answers by graphing calculator 

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Real World Setting: (1)  Medical breakthroughs

You are a medical research analyst. You are faced with a problem of finding the average velocity of blood flow along the radius of an artery.

LET US SEE A PROBLEM:

The velocity v of blood at a distance r from the center of the artery of radius R is given by

v = k ( R= -- r= ) where k is a constant. You are required to find the average velocity along a radius of the artery.

You must analyze the velocity function and decide what should be done in order to reach a solution.  . Once you have completed your analysis, and by integrating the velocity function with lower limit 0 and upper limit R you will have your result

Real World Setting: (2)   Marketing

You are a company manager. The company purchased a new machine. You are faced with a problem of selling the machine after 3 years. Based on a depreciation function you are required to find the total loss of the value of the machine after 3 years.  

For example if the depreciation function is given by  dV/dT = 10,000(t -- 6)  ; 0= t = 5 where V is the value of the machine after t years.

You must analyze the problem first. Set up and evaluate the definite integral that yields the total loss of the value of the machine after 3 years.

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Level I: Acquiring Data - Data students will acquire in this standards-based task:

• Vocabulary: Definite integral and indefinite integral,  upper limit, lower limit,
  
• Concepts: Distinguishing definite and indefinite integrals
  
• Processes: Sketch the graph of a linear function

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task:

• Arranging: Step by step procedure to reach the solution. First sketch the region, identify the area to be calculated and then apply a geometrical formula to find the area.
  
• Storing: Understand the fact that definite integral is the area of the region bounded by the function and some constraints. Also store the properties of definite integrals.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

• Making decisions: Students decide to use fundamental theorem of integral Calculus to obtain the area of a region bounded by a given functions and its limits.
• Solving problems: Students will be able to solve new problems related to depreciation, compound interest, medicine, revenue or biology and so on that uses definite integrals.

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Student Involvement - The students will complete the task:

• Individually:   Students work independently on first set of problems.
  
• Cooperative group:  Students work in groups on second set of problems.

Instruction - Activities will be organized and delivered by differentiating the complexity of the products and performances expected of students in a student note book during class time. Problem sets will vary in the complexity of their solutions.

Use of Resources - The school will provide text books and graphic calculator

Customer for Student Work - The students will present their work as evidence of task completion to teachers

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: The student's teacher, Peers

Assessment of Student Work - The following forms of assessment will be used to determine progress and results:

• Performance assessment: A group of 10 questions ( Class test) regarding definite integral from easy to hard level will be given to students during class time.

Reporting Results - The assessment results will be reported as a letter grade

Timeline - The estimated time needed to plan, teach, and score this task is three  to four class periods

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Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson and assigns a problem that was not finished

• Step 1: Ask students about graphing techniques of linear, quadratic functions and identify the shapes of regions under the graph of a function with given limits of the variable
• Step 2: Teacher presents the concept of the relationship of definite integrals of a function and the area under the graph bounded by the limits of the variable. Students understand that "area under the graph which is above X axis and definite integral are the same".

Students will solve some problems with definite integrals by finding the areas of the regions under the graph and above the X axis  with certain limits and by using some geometrical formulas.

For example: Students  find   ? 2x dx  with limits x=0 and  x=3

First, students sketch the graph of  y = 2x and identify the region under the graph from x=0 to x=3.

Students see that a right triangle having base 3 units and height 6 units. They could find its area using the formula    A =O bh

Students will find areas by using geometrical formula . Then they check the results from evaluating with a graphing calculator.

Home work will be assigned to students based on activity 1. Page 353 (1 - 4 all) & (8 - 14 evens) from the text "Calculus- An Applied Approach" by "Larson & Edwards" 6th Edition

Step 3: Check the homework on yesterday's lesson.

Technology:   Students use graphing calculator to see the graph of the functions

Materials:  Notebook, Text book

Student product or performance: Students produce their homework and check their results.

Links or connections between different parts of the lesson:  The ability to find derivatives of functions will help them to find the anti derivatives

Scoring:   Teacher records homework assignment.

Activity 2: The students present solution methods they have developed and the teacher summarizes

• Step 1:

  The teacher presents on chalk board the properties of definite integrals. Students practice on a problem set.

  For example :  Evaluate  ? [ f(x) - 5g(x)]dx  with limits x=0 & x=5

  if given  ? f(x) = 18  and ? g(x)= 3  with same limits.

• Step 2:  Teacher presents "Fundamental Theorem Of Calculus"

Students understand that this theorem can be used to evaluate a definite integral without sketching graphs and without using a geometrical formula to evaluate the area.

For example:  Let the students evaluate   ? (2x 1)dx  with lower limit x= 1 & upper limit x=3

Students first sketch the graph of y = 2x   1, then use a geometrical formula to find the area of the region.

Then students use Fundamental Theorem of Calculus to evaluate the integral and they find that answers are the same.  

• Step 3:  Students do more problems from their text book on evaluating definite integrals.

Home work from text book will be assigned to students based on activity 2 (Page 353 6a,6b,6c,6d &

16 - 28 evens from the text "Calculus- An Applied Approach" by "Larson & Edwards

(6th Edition)

Technology: graphing calculator to verify their answers
Materials:   Text book and note book
Student product or performance:  The students will do class work and their assigned homework.
Links or connections between different parts of the lesson: Ability to apply correct property will help the students to evaluate the problems from the text book. Also algebraic skills help them to evaluate the integrals correctly.

Activity 3: The teacher presents the task for the day and asks the students to work on it independently (Task is to invent a problem for classmates to solve.)

• Step 1 :  Students try to analyze and interpret word problems. The word problems are problems they face when they are in real world settings. Real problems could be in the field of science, medicine, accounting, or any other discipline.
• Step 2: Students work from the text book pages 354 and 355 which has many word problems
• Step 3: Students understand about even and odd functions and their integrals and work on even and odd functions (Page 354 65 - 68) from the above mentioned text

Home work assignment will be given to students based on activity 3 (Page 355 # 86, 88, 90) from the above mentioned text.

Technology:  Graphing calculator
Materials:      Text book and note book
Student product or performance:  Class work and home work

Activity 4: The teacher suggests that students continue their work in small groups. Leaders of groups share their problems with the teacher, who makes them public, e.g., writes them on the board. Students copy the problems and begin working on them.

(Estimated time: 20 minutes)

• Step 1: Clarify and troubleshoot problems.
• Step 2: Students  teach each other and develop new problems based on the lesson.
  

Technology: Graphing Calculator
Materials:     Text book and notebook
Student product or performance:   Class work, peer tutoring, developing new problems

Activity 5: The teacher highlights a good method for solving these problems.

Source: Adapted from:

1. The 1996 Third International Mathematics and Science Study (TIMSS)
   
2. Stigler, J. & Hiebert, J. (1999) The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: The Free Press.

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Student Performance One:

Students can create their own examples (about 5 Questions)  of definite integrals and they themselves evaluate their integrals. These examples should be based on the activities 1 through 5

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Cognitive Information: I will collect the following information by asking questions about the lesson and summarize their verbal responses.

1. Describe what skills you needed to complete this task.
   
2. Explain how you solved the goal, problem, or issue in this task.
   

Attitude Information: I will collect the following information

1. Do you feel that you are good in evaluating integrals?
   
2. Did you find this task to be difficult? Which part was difficult?
   
3. Did you see the usefulness of what you were asked to do in real life?
   
4. Did you enjoy the task?

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Reflect: I noticed that students enjoyed group activities and some of them really like to solve problems on the black board and explain to their peers. I noticed that a small percent of the class still struggle to solve more difficult level problems but they are being helped by some other students and by the teacher to make them more comfortable with the topic in the lesson.

Summarize: I assessed and scored the content standard and discovered that 38% of my students performed at or above the distinguished  level on my scoring rubric. 39% of the students were placed in the proficient level 15% of the students fell in apprentice level and the remaining 8% were in the novice level. The above results were determined by assessing answers to questions presented during the lesson, observing class work, active participation, observing student's board work, the written test and assessing homework.

Improvement: Next time I would pay more attention to under achievers and make them pair with distinguished level students. I would do more examples on board that range from an easy level to a more difficult level. Novice learners can make use of the extra help option given during my tutoring period. 

My  next standard based task will focus on:

Title: The Area of a Region Bounded by two Graphs

CALCULUS - An Applied approach : by Larson and Edwards 6th Edition

SEC 5.5

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