New+Lesson+Plan


 * Lesson Outline **


 * 1. Established Goals **

** Content Standards ** Students will notice if calculations are repeated, and notice the regularity in the way terms might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

** Activity Objectives ** The main focus of the activity is to develop an conceptual understanding of infinite series, directing the students to understand the differences between series and sequences, and divergence and convergence of series.

In addition to this, the activity intents to develop a visual understanding and mental conceptualization of the academic terms.

** Learning Outcomes ** In the long term, the activity is geared toward creating a long lasting learning experience base on a tangible practical real life exercise. The hope is that the students learn to create connections between the academic content of the lesson and other possible applications of the theory in real life.

The main ideas of this activity are centered in the concepts of sequences, series, concepts of divergence or convergence, possible development of a formula.
 * 2. Understandings **

The specific understandings that is desired is to help students to differentiate between the concepts of sequence and series, to help students understand the concepts of divergence of convergence, and to expand the knowledge towards visual representations of infinite series.

It is predictable that since this is a subject newly introduced to students that the definitions and concepts of the academic language pertinent to this chapter, the students will need refreshing of some of the concepts. For this reason, the "Additional Resource" page has been uploaded, to allow students to refresh of familiarize themselves with the concepts.

Some essential questions that will foster inquiry, understanding and transfer of learning are:
 * 3. Essential questions **


 * Does the series converges or diverges for all values of n.
 * If the series diverge for some intervals and converges for another, list those intervals.
 * What mathematical expression describes the series covered in the exercise.

The key knowledge and skills will students acquire as a result of this unit is to be able to expand their academic language regarding the concepts of sequences and series. The students should develop critical thinking process for understanding and generalizations of infinite series concepts. The students should be able to develop more applications of the theory to real life situations.
 * 4. What students will not and will be able to do **

The learning experiences and instruction will enable students to visual understand and create a mental representation of the concepts involving infinite series by using a piece of paper as a learning tool. The design will attempt to help the students know where the academic unit is going and what is expected. The lesson will also help the teacher to identify where students are at in terms of prior knowledge of the subject or the need to reinforce what is being learned. By making the class lesson into an interactive project, the purpose is to hook all students and hold their interests, equip students, help them experience the key ideas and explored the issues. By providing opportunities to rethink and revise their understandings and work through a series of questions and the help of a calculator, the teacher allows students to evaluate their work and its implications. The lesson is tailored to the different needs, interests and abilities of learners by providing additional resources of instruction in the “Additional Resources Page,” and step by step procedures to attain full understanding of concepts. The flow of the lesson is organized to maximize initial and sustained engagement as well as effective learning.
 * 5. Learning Plan **

**Materials and Resources Used**

 * Blank square piece of paper and scissors
 * Calculator
 * Slides presentation of procedure
 * Videos describing concepts
 * Widgets to calculate results
 * Chat feature to share possible applications of the theory
 * Additional website support

**Lesson Outline**

 * Shortly refresh the concepts of sequence, series, infinite series and geometric series. Ask students to refer to the “Additional Resources Page,” to receive a video overview of main concepts.
 * Shortly refresh the concept of divergence or convergence of series.
 * Ask students to write out a few terms of the geometric series ½+1/4+1/8 +…
 * Ask students to try to guess weather this series diverges or converges.
 * Ask students to calculate the value of the sum of the infinite series by using a calculator.
 * Ask students to reflect on the kind of assumptions that need to be made to calculate the answer to the sum of the geometric series.
 * Ask students to use square piece of paper to make a visual representation of the geometric series sum.
 * Use the square sheet of paper to model ½ + ½ = 1
 * Use the square sheet of paper to model ½ + ¼ + ¼ = 1
 * Use the square sheet of paper to model ½ + ¼ + 1/8 + 1/8 = 1
 * Ask students to think about the number of times the procedure can be continued
 * Ask students to infer about the assumptions that can be made about the result of the sum of ½ + ¼ + 1/8 + 1/16 + …. = ?
 * Ask students to share their ideas of the possible applications of the theory in other real life examples using the chat feature on this web site.


 * 6. Assessment Evidence **

The authentic activity of folding of the paper to derive the understandingof an infinite series in itself serves as an assessment tool to determine weather or not the students are able to follow the rational regarding the development of the concept being taught. During the activity, students are asked questions regarding expected answers or outcomes of each step. Although it was not implemented in this lesson, an interactive quiz could also serve to assess the retention of the main concepts such as understanding of divergence or convergence, and the ability to identify the concept of sequences, series, infinite series, and geometric series.