Patterns and Sequences

Objectives

• Given the first few elements of a pattern, extend the pattern.
• Given the elements in a pattern, identify and distinguish between arithmetic and geometric sequences.
• Given an arithmetic sequence, write an expression for finding the nth term of the sequence.
• Given a geometric sequence, write an expression for finding the nth term of the sequence.

Teaching Plan (3 Days)

DAY 1 (50 Minutes)

Patterns

The purplse of this lesson is to help students practice pattern recognition and understand sequences.

This lesson could also be taught as a presentation with the whole class working together.

Using a computer projector, demonstrate how to go to the eNLVM website, go to your school and class, and login. (5 minutes)
(5 activities) Students are presented with several patterns of colors, shapes, and numbers. They are asked to extend and desribe each pattern.
(1 activity) Students are presented with the definition of arithmetic and geometric sequences. They are asked to identify number sequences as arithmetic, geometric, or neither.
(1 activity) Students create arithmetic and geometric sequences.
(1 activity) Students are introduced to the Fibonacci Sequence. They are asked to extend the sequence .

DAY 2 (50 Minutes)

Arithmetic Sequences

The purpose of this lesson is to help students write expressions of arithmetic sequences and find the nth term.

This lesson could also be taught as a presentation with the whole class working together.

(1 activity) As a motivator for the activities that follow, students are given a complex problem to solve using methods for finding the nth term of an arithmetic sequence. They are asked to identify the sequence as arithmetic or geometric.
(1 activities) Students are presented a sequence with the expression n + 6.  They are asked to find n = (large numbers) using the expression. They are also asked to formulate other expressions of similar nature.
(1 activities) Students are presented a sequence with the expression 2n.  They are asked to find n = (large numbers) using the expression. They are also asked to formulate other expressions of similar nature.
(1 activities) Students are presented a sequence with the expression 3n+2.  They are asked to find n = (large numbers) using the expression. They are also asked to formulate other expressions of similar nature.
(1 activity) Students are reintroduced to the beginning problem. They are asked to write an expression for the sequence and solve it for n = (large numbers).
(1 activity) Students are introduced to an algorithm for finding expressions of arithmetic sequences.
(1 activity) Students use the algorithm to find an expression for an arithmetic sequence.
(2 activities) Students are guided through the process of solving a problem involving an arithmetic sequence.

DAY 3 (50 Minutes)

Geometric Sequences

The purpose of this lesson is to help students write expressions of geometric sequences and find the nth term.

This lesson could also be taught as a presentation with the whole class working together.

(1 activity) As a motivator for the activities that follow, students are given a complex problem to solve using methods for finding the nth term of a geometric sequence. They are asked to identify the sequence as arithmetic or geometric.
(1 activities) Students are presented a sequence with the expression 4^n.  They are asked to find n = (large numbers) using the expression. They are also asked to formulate other expressions of similar nature.
(1 activities) Students are presented a sequence with the expression 2(3)^n.  They are asked to find n = (large numbers) using the expression. They are also asked to formulate other expressions of similar nature.
(1 activity) Students are reintroduced to the beginning problem. They are asked to write an expression for the sequence and solve it for n = (large numbers).
(1 activity) Students are introduced to an algorithm for finding expressions of geometric sequences.
(1 activity) Students use the algorithm to find an expression for a geometric sequence.
(1 activity) Students are guided through the process of solving a problem involving a geometric sequence.

Credits

Correlation to Standards

Correlation to NCTM Standards

Algebra - Standard 1. Understand patterns, relations, and functions.

• Grades PreK-2
  • Recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another.
  • Analyze how both repeating and growing patterns are generated.
• Grades 3-5
  • Describe, extend, and make generalizations about geometric and numeric patterns.
  • Represent and analyze patterns and functions, using words, tables, and graphs.
• Grades 6-8
• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.  

Correlation to Utah Standards

• Kindergarten - II.2.d Duplicate, extend, and create simple patterns using objects and pictorial representations.
• Math 1 - II.1.b Identify, create, and label simple patterns using manipulatives, pictures, and symbolic notation.
• Math 3 - II.1 Recognize and create patterns with given attributes.
  • Create and extend repeating and growing patterns using objects, numbers, and tables.
  • Record results of patterns created using manipulatives, pictures, and numeric representations and describe how they are extended.
• Math 4 - II.1.a Represent and analyze repeating and growing patterns using objects, pictures, numbers, and tables.
• Math 5 - II.1.b Extend patterns and describe a rule for predicting the next element.
• Math 6 - II.1 Recognize, analyze, and use multiple representations of patterns and functions and describe their attributes.
  • Analyze patterns on graphs and tables and write a generalization to predict how the patterns will continue.
  • Write an algebraic expression from a graph or a table of values.
• Math 7 - 2.1 Understand patterns, relations, and functions.
  • Describe a pattern using a mathematical rule or algebraic expression.
  • Create simple numerical and visual patterns.
  • Extend simple numerical and visual patterns.
• PreAlgebra - 2.1 Use patterns, relations, and functions to represent mathematical situations.
  • Represent a variety of relations and functions using tables, graphs, manipulatives, verbal rules, or algebraic rules.
  • Describe simple patterns using a mathematical rule or algebraic expression.
  • Create and extend simple numeric and visual patterns, including those that have a recursive nature (e.g., Fibonacci numbers, triangular and square numbers).
• Elemetary Algebra - 2.1.1 Write algebraic expressions or equations to generalize visual patterns, numerical patterns, relations, data sets, or scatter plots.
• Precalculus -2.1.5 Identify and generate arithmetic and geometric sequences and series recursively and explicitly using correct notation.