An Introduction to Arithmetic and Geometric Sequences

Abstract

This lesson is designed to introduce students to the arithmetic and geometric sequences.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to sequences
  • understand the terminology used with sequences
  • understand how to vary a sequence by changing the starting number, multiplier, and add-on values used to produce the sequence
  • be able to determine the starting values that should be used to produce a desired sequence.

Standards Addressed

Grade 6

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 7

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 8

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Fourth Grade

  • Operations and Algebraic Thinking

    • Use the four operations with whole numbers to solve problems.

Fifth Grade

  • Operations and Algebraic Thinking

    • Analyze patterns and relationships.

Functions

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems

Grades 3-5

  • Algebra

    • Understand patterns, relations, and functions

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

  • Numbers and Operations

    • Understand meanings of operations and how they relate to one another

Algebra I

  • Algebra

    • Competency Goal 4: The learner will use relations and functions to solve problems.

Advanced Functions and Modeling

  • Algebra

    • Competency Goal 2: The learner will use functions to solve problems.

Discrete Mathematics

  • Algebra

    • Competency Goal 3: The learner will describe and use recursively-defined relationships to solve problems.

6th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities.

7th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of proportional relationships.

Intermediate Algebra

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of sequences and series.

Geometry

  • Geometry

    • Standard G-2: The student will demonstrate through the mathematical processes an understanding of the properties of basic geometric figures and the relationships between and among them.

Grade 3

  • Patterns, Relationships, and Algebraic Thinking

    • 6. The student uses patterns to solve problems.
    • 7. The student uses lists, tables, and charts to express patterns and relationships.

7th Grade

  • Patterns, Functions, and Algebra

    • 7.19 The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.
    • 7.20 The student will write verbal expressions as algebraic expressions and sentences as equations.

Secondary

  • Algebra II

    • AII.01 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices.
    • AII.02 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.
    • AII.03a The student will add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents.
    • AII.03b The student will write radical expressions as expressions containing rational exponents and vice versa.
    • AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ and an.
    • AII.1
    • AII.2
    • AII.3.a
    • AII.3.b
    • AII.16

Textbooks Aligned

Grade Seven

  • Variables and Patterns

    • Investigation Four: Patterns and Rules

Book 3

  • Module 4 - Patterns and Discoveries

    • Section 1: Sequences

6th

  • Module 1 - Patterns and Problem Solving

    • Section 2: Patterns and Sequences

7th

  • Module 2 - Bright Ideas

    • Section 3: Sequences and Equivalent Equations
      • Reason for Alignment: The Introduction to Sequences lesson accompanies the Sequencer activity. It should provide good background and the necessary steps to use the activity in investigating some arithmetic sequences.

8th

  • Module 8 - MATH-Thematical Mix

    • Section 1: Patterns and Sequences
      • Reason for Alignment: This is at the introductory level for both types of sequences containing discussions, along with inclusion of vocabulary terms.

Book 1

  • Patterns in Numbers and Shapes

    • Lesson 8: Points, Plots, and Patterns

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 for experimenting with the activities

Teacher Preparation

Key Terms

iteration

Repeating a set of rules or steps over and over. One step is called an iterate

recursion

Given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information

sequence

An ordered set whose elements are usually determined based on some function of the counting numbers

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 "1" 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 Outline

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.