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Rationale What is the definition of a geometric series

By Julie Nichols,2014-04-08 21:14
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Rationale What is the definition of a geometric series

     1

    Instructor’s Name:

    Course Title: Pre Calculus or Calculus

    Unit: Discrete Math, Patterns

    Topic: Geometric Series

    Grade Level: High School, 11-12

    Rationale: What is the definition of a geometric series? Understanding limits is a fundamental concept in calculus. How can a limit applied to a geometric series? Geometric series can be used to understand real world situations.

    NJ State Standards:

    ; 5.1.12. A.1; Refine interrelationships among concepts and patterns of evidence

    found in difference central scientific explanations. (2009)

    ; 5.1.12. B.2; Build, refine, and represent evidence-based models using

    mathematical, physical, and computational tools. (2009)

    ; 4.3. A.1; Use models and algebraic formulas to represent and analyze sequences

    and series. (former standards, HS-level)

    ; 4.3.A.2; Develop an informal notion of limit (former standards, HS-level)

    ; 4.3.A.3 ; Use inductive reasoning to form generalizations (former standards, HS-

    level)

    Instructional Goals: Students will restate the definition of a geometric series and give an example of an everyday application of series. Students will be asked to construct an informal definition of the concept of limit and an infinite geometric series. By the end of the lesson, students will evaluate a finite sum of a geometric series.

    Performance Objective: Students will distinguish the difference between a finite sum and an infinite sum via a combination of written instruction, note-taking, and Calculator technology. Students will evaluate a finite sum of a geometric series via a handout and TI calculator. Tasks to be done on the TI calculator include modeling data. Evaluating a finite sum for a large number of terms will lead to the concept of a limit.

    Lesson Content:

    ; Derivation of finite sum for geometric series through written instruction

    ; Analyze a finite sum for a geometric series for a very large number of terms using

    Calculator technology. Students should conclude the sum converges to a limiting

    value.

    ; Evaluate an application of geometric series, tracking medicine levels in the human

    body, using TI calculator and handout containing questions.

     2

    Instructor’s Name:

    Course Title: Pre Calculus or Calculus

    Unit: Discrete Math, Patterns

    Topic: Geometric Series

    Grade Level: High School, 11-12

Lesson Time Frame:

    Time Frame of Activity Activity to be Performed

    0-5 minutes Focusing Activity

    5-10 minutes Bridge/Connections

    10-20 minutes Hands-on Activity with Calculators

    Assessment Activity/Interdisciplinary 20-45 minutes Connection

    45-50 minutes Closure Activity

Instructional procedures:

    A. Focusing event:

    i. What is the definition of geometric series? Can you think of real-life examples of

    series?

    a. Fractal geometrysnowflakes

    b.Bouncing ball’s changing height from the first drop

    c. Radioactive decay in rocks and metals

    d.Illegal pyramid schemes

    B. Bridge/ Connections - How does this build and/or connect to students’ prior knowledge?

    i. Derive sum of finite geometric series as written exercise, building off definition

    of geometric series. (See Power Point Presentation)

    C. Teaching procedures Teaching methods you will use.

    i. Students will derive a formula for the sum of a converging infinite geometric

    series. Before this first activity, make sure the calculator is set to calculate up to at

    least 4 places behind the decimal point. Let r = 0.7 (or any value between -1 and n1). The students use the Home screen of the calculator to see how r changes

    when n=0, 10, 100, 1000, and 10000. Students can either store calculator

    variables ―r‖ and ―n‖ or just plug the values for these variables directly into the

    command line. Ideally, the presenter should define the variables on the TI-nspire. nThe students should observe that for a large or infinite value of n, r will

    approximately be zero. The point of this exercise is to show that when the limit is napplied to S = a(1-r) / (1-r), and the zero is plugged in, then the sum will be n1

    equal to S =a/ (1-r). n1

    ii. Students will apply infinite geometric series to drug dosages in the human body.

    This second activity includes a handout, labeled medicine_inf_series.doc. The

    presenter will lead the activity with the TI-nspire connected to the computer and

    monitor.

     3

    Instructor’s Name:

    Course Title: Pre Calculus or Calculus

    Unit: Discrete Math, Patterns

    Topic: Geometric Series

    Grade Level: High School, 11-12

D. Formative check

    i. The TI-nspire .tns file has a Q&A application throughout the lesson. The

    presenter will use the Question-Answer feature of the TI-nspire to confirm a

    student answer.

    a. Through student participation, members of class will answer questions

    during the infinite geometric series activity.

    b.The handout activity about geometric series used to estimate medicine

    levels in the human body also contains questions to answer.

    c. Answers are provided in the .tns file, but hidden from the students until

    the instructor chooses to reveal them to the class.

    E. Student participation- how will you get the students to participate?

    i. What will students do to demonstrate their learning?

    a. Whole group instruction using the TI-nspire notebook Q&A application

    b.Independent work through answering the questions and creating data

    graphs on TI calculators

    F. Interdisciplinary connections Biological sciences is another real-life application of

    geometric series. The assessment activity applies geometric series to medicine levels in

    the human body. This integration is a topic students can relate to on a first or third

    person basis. It also raises drug and substance abuse awareness. G. Closure- Ask students why is understanding geometric series important when taking

    prescribed medicine? Using our example, did the sum of our series have a limit? Can

    you describe what a limit is in your own words?

    Evaluation Procedure:

    ; Continuous monitoring of formative assessment activity. Handout will be graded at the

    end of class.

    ; Extensions

    o Construct applied example of geometric series

    o Practice problems from textbook

    o Graph data for medicine levels of the patient who takes the medicine every 2 hours

    Materials and Aids:

    The teacher or presenter will connect the calculator to a PC with TI-connect software.

    The PC is connected to the projector or SMART boardthis way, the students will be able to

    follow along on the monitor. Students should be supplied with TI-83 Plus calculators or TI-

    nspire calculators.

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