An Introduction to Arithmetic and Geometric Sequences

Abstract

This lesson is designed to introduce students to the idea of an arithmetic sequence.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to sequences,
  • have learned the terminology used with sequences,
  • have experimented with creating a sequence by varying the starting number, multiplier, and add-on values used to produce the sequence.
  • have practiced determining the starting values that should be used to produce a desired sequence.

Standards

The activities and discussions in this lesson address the following NCTM Standards:

Numbers and Operations

Understand meanings of operations and how they relate to one another

  • understand the meaning and effects of arithmetic operations with fractions, decimals, and integers
Algebra

Understand patterns, relations, and functions

  • represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
  • identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations
Analyze change in various contexts
  • use graphs to analyze the nature of changes in quantities in linear relationships
Links to other standards.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer and fractional arithmetic
  • Technological Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

Key Terms

This lesson introduces students to the following terms through the included discussions:

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Walk through the discussion on recursion.

    • Present a few elements of a sequence to students and have them determine what should come next. Ask the class, "If I listed the following numbers, what would come next: 5, 10, 15, 20... ?"

    • If a student answers "25," then have the student suggest why s/he knew that was the next number.

    • Ask the students what is being added or multiplied to get each new number. Assist the students in understanding that each number is obtained by adding 5 to the previous number.

    • Ask the students similar questions for a sequence such as 2, 4, 8, 16, 32.... Help the students understand that each number is obtained by multiplying the previous number by 2.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will be talking about sequences. These lists of numbers that we have been discussing are sequences. A sequence is a list of numbers in which each number depends on the one before it. If we add a number to get from one element to the next, we call it an arithmetic sequence. If we multiply, it is a geometric sequence.

    • We are going to use the computers to learn about sequences and to create our own sequences.

  3. Teacher Input

    In this part of the lesson you will explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

    • Open your browser to The Sequencer Activity. You may need to instruct students not to open their browsers until told to do so.

    • Show the students how to input the initial values for the starting number, multiplier, and add-on and how to obtain the new sequence. Explain to students that if they wish to see a sequence that is strictly arithmetic, they may enter "0" in the multiplier box. Similarly, if they wish to see only a geometric sequence, they may enter a "0" in the add-on box.

    • Pass out the Sequences Exploration Questions

  4. Guided Practice

    Your students may be ready to move along on their own, or they may need a little more instruction:

    • If your class seems to understand the process for doing this assignment, simply ask, "Can anyone tell me what I need to do to complete this worksheet?" or ask, "How do I run this applet?"

    • If your class seems to be having a little trouble with this process, do another example together, but let the students direct your actions.

    • You may choose to do the first problem on the worksheet together. Let the students suggest possible values for the starting number, multiplier, and add-on. If the answer is not correct, have the students talk about how to change the numbers to correct the mistake.

    • After practicing together, ask if there are any more questions before proceeding to let the class work on the worksheet individually or in groups.

  5. Independent Practice

    • Allow the students to work on their own to complete the rest of the worksheet. Monitor the room for questions and to be sure that the students are on the correct web site.

  6. Closure

    It is important to verify that all of the students made progress toward understanding the concepts presented in this lesson. You may do this in one of several ways:

    • Bring the class together and share some of the answers that the students obtained for each item on the worksheet. Students may be surprised to find that there are several ways to obtain a sequence in which all the elements end in 3, for example.

    • Let the students write a breif definition of a sequence on paper and provide an example to ensure that they have understood the lesson.

Alternate Outlines

This lesson can be rearranged in several ways.

  • You may choose not to pass out the worksheet, but rather to dictate the problems to the students and have groups working on the same problem and the same time. Students make make a note of their findings on notebook paper.

  • You may choose to allow students to design their own sequences and make a statement about what makes it special.

Suggested Follow-Ups or Extensions

As sequences and patterns are usually grouped together in the curriculum, you may consider the following lessons and activities: