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, HSlevel)
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
0-5 minutes
5-10 minutes
10-20 minutes
20-45 minutes
45-50 minutes
Activity to be Performed
Focusing Activity
Bridge/Connections
Hands-on Activity with Calculators
Assessment Activity/Interdisciplinary
Connection
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 geometry—snowflakes
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
1). The students use the Home screen of the calculator to see how rn 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.
The students should observe that for a large or infinite value of n, rn will
approximately be zero. The point of this exercise is to show that when the limit is
applied to Sn = a1(1-rn) / (1-r), and the zero is plugged in, then the sum will be
equal to Sn =a1/ (1-r).
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 board—this way, the students will be able to
follow along on the monitor. Students should be supplied with TI-83 Plus calculators or TInspire calculators.