To download a pdf of the lesson plan, please click here: Lesson Plan Please note: the PDF uses the backward design template (Wiggins & McTighe, 2005)

I. Goals

To understand the patterns in infinite series in order to approximate the sum of an infinite series. (NCTM Algebra Standard: "Generalize patterns using explicitly defined and recursively defined functions") (NCTM, 2012)

To develop the ability to use symbolic algebra to represent the sum of infinite series and situations which require the use of infinite series. (NCTM Algebra Standard: "Use symbolic algebra to represent and explain mathematical relationships") (NCTM, 2012)

II. Objectives

Students will create their own understanding of infinity and infinite series through modeling and experimentation.

Students will use modeling to visualize and comprehend the abstract idea of infinite series.

Students will discuss how an infinite series can equal a finite number preparing them for the concept of limits.

III. Materials and Resources

For each student:

Blank sheet of paper (construction paper or printer paper)

Scissors

Notebook or a few sheets of lined paper to write down thoughts or notes

IV. Motivation

Ask students how they would solve the series: 1/2 + 1/4 + 1/8 + 1/16 + ...

After a few suggestions from students, inform them that they will be using a model to find a solution.

V. Lesson Procedure

Ask students to fold the blank sheet of paper in half and cut it into two rectangles. Remind students that the sum they have in front of them is "1/2 + 1/2 = 1."

Ask students how they can represent 1/2 + 1/4 using the pieces in front of them. If students do not figure it out on their own after a couple of minutes, instruct them to put aside one of the half rectangles and to take the other half, fold that one in half, and cut it into two squares, thus creating a piece that is one-fourth of the original blank sheet.

Before students continue the pattern, ask students to hypothesize what they think will happen if they continue this pattern and to write down their ideas in their notebooks.

After students finish their hypothesis, ask them to cut one of the fourth pieces in half and then one of the eighth pieces in half.

Raise the question, "What fraction do these two pieces represent?" holding or pointing to the one-fourth piece and one-eight piece.

Have students label each piece with the appropriate fraction, then ask them to write down the mathematical equation they have created with all of their pieces (ie. 1/2 + 1/4 + 1/8 + 1/8 = 1).

Discuss the hypothesis that students have previously written down and if they still believe that their hypothesis is correct.

Draw a 2-D diagram of the process students have completed by cutting the blank sheet. Then, ask students to continue the diagram to the smallest size possible. Remind students that they can use the model in front of them to help them continue the pattern.

Raise the question, "From your model and diagram, what will 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + .... equal?" Students should be able to assume that they will get really close to one.

With the help of student diagrams and models, explain to students how one is the correct answer.

Ask students to write this series as an equation, which can be used to figure out each term of the series.

Introduce students to the mathematical symbol "∑" and how it can be used to express an infinite series.

VI. Closure

Raise the question, "What real life scenario can you think of where you will need to solve an infinite series?"

Discuss different occasions when professionals use infinite series.

VII. Extension

Ask students to briefly write in their notebooks a strategy they could use to solve a different infinite series, such as 1/3 + 1/9 + 1/27 + 1/81.

Discuss students' strategies and ideas on how to approach a different infinite series. Finally, ask students to solve this infinite series as well as two other infinite series for homework and to explain in at least one paragraph the strategies they used to reach their solution. Remind students to use appropriate mathematical notation.

VIII. Assessment

Students' understanding is assessed through informal observations of their entries in their notebooks and comments during class discussion. Additionally, the extension will assess students' understanding of infinite series and how well they can use their understanding to develop an appropriate strategy to solve such problems.

## Table of Contents

Please note: the PDF uses the backward design template (Wiggins & McTighe, 2005)

## I. Goals

## II. Objectives

## III. Materials and Resources

For each student:## IV. Motivation

## V. Lesson Procedure

## VI. Closure

- Raise the question, "What real life scenario can you think of where you will need to solve an infinite series?"
- Discuss different occasions when professionals use infinite series.

VII. Extension## VIII. Assessment

Sample Virtual Lesson:

An Example of Limits. (2008, April 28). Retrieved January 20, 2012, from http://www.youtube.com/watch?v=o7GLWMXq7jo&feature=youtu.be

Brahier, D. J. (2009).

Teaching Secondary and Middle School Mathematics. Boston, MA: Pearson Education, Inc.Honner, P. (2010, October 30). Proofs without words. Retrieved January 20, 2012, from http://mrhonner.com/2010/10/30/proofs-without-words/

National Council of Teachers of Mathematics (NCTM). (2012). Algebra Standard for Grades 9-12. Retrieved January 20, 2012, from http://nctm.org/standards/content.aspx?id=26831

Wiggins, G. & McTighe, J. (2005).

Understanding by Design(2nd ed.). New Jersey: Merrill/Prentice Hall. (p. 22).