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.

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Fourth Grade

Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems.

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

Advanced Functions and Modeling

Algebra

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

Algebra 1

Algebra

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

Algebra I

Algebra

Competency Goal 4: The learner will use relations and 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.

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.

Intermediate Algebra

Algebra

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

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.

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.

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

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:

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.

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.

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.

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.

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.

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.

Suggested Follow-Up

The next lesson,
Patterns in Fractals will teach students to identify patterns in fractals.