Illuminations: Patterns That Grow

Patterns That Grow

Unit Overview

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Growing Patterns

Students use numbers to make growing patterns. They create, analyze, and describe growing patterns and then record them. They also analyze a special growing pattern called Pascal’s triangle.

Learning Objectives

Students will:

create growing patterns
describe growing patterns
analyze how growing patterns are created
find number patterns in Pascal’s triangle

Materials

"The House that Jack Built" by Mother Goose
Exploring Pascal's Triangle

Instructional Plan

To begin the lesson, recite "The House that Jack Built" by Mother Goose. Then ask the students to tell what happened in the story.

Next explain that students will be exploring patterns that grow according to a rule. Display the “bowling pin” pattern (which is a “counting-on” pattern):

•
•   •
•   •   •
•   •   •   •

Then ask, “What will come next in this pattern?” (Students may find this question easier to answer if they use cubes on a blank grid or copy the pattern onto paper.) When students give the correct answer [a row of five dots], ask them to explain how they got that answer. Repeat with several more rows. Then ask the students to state the rule that they would use to add more figures to the pattern. Encourage alternate expressions of the rule. [This activity requires higher-order thinking and involves solving numerical problems.]

Next display the pattern below, and ask students what they might call the pattern [a T pattern.]

• • • • •

•

• • • •

• •

• • •

Then repeat the steps used in the counting-on pattern above with the new pattern below. [The next 3 figures will contain 13, 17, and 21 dots.] Ask several students to state the rule that they would use to add more figures to the pattern. [Each time, two dots are added to each part of the previous figure.] Then ask them to add more rows to the table below.

Figure	Dots
1	1
2	5
3	9

Next introduce the students to Pascal’s triangle by opening Exploring Pascal's Triangle. Ask for volunteers to name any patterns that they see. (You may wish to print out copies of the triangle and provide calculators for the students to use.) Partners should take turns answering questions at the site.

To conclude the lesson, have the students make patterns that grow and exchange them with a friend to extend. At the end of the class, ask for volunteers to share their growing patterns and their rules.

Questions for Students

How many dots are in the first figure of the T growing pattern? How many dots will be in the fourth figure? In the fifth? How do you know? What is the rule? How would you demonstrate that you have found the rule?

[1; 17; each figure contains four dots more than the previous figure.]

How long could we continue this pattern?

[Forever.]

Make a "corner pattern" by making a right angle with three dots, then one with five dots, then one with seven dots. What will come next in this growing pattern? Then what? How do you know? Have you seen this pattern before?

[9; 11; It is a list of the odd numbers.]

What was the rule for one of the patterns that you made? Did anyone else make a different pattern with that rule?

[Student responses may vary.]

Here is a row from Pascal’s triangle (1 4 6 4 1). What would be the row after that? How do you know? What is the sum of this row? What is the sum of the first four rows? Do you notice a pattern in the sum of the rows?

[1 5 10 10 5 1; 16; 1, 2, 4, 8; Each row's sum is double the previous sum.]

Do you notice any patterns in the triangle?

[Student responses may vary.]

Assessment Options

At this stage of the unit, it is important for students to know how to:
- create growing patterns;
- describe growing patterns;
- analyze growing patterns;
- find number patterns in Pascal’s triangle.

Teacher Reflection

Which students were able to analyze and describe growing patterns? What extension activities would be appropriate for these students?
Which students had difficulty analyzing and describing growing patterns? What instructional experiences do they need next?
Which students met all the objectives of this lesson? What extension activities are appropriate for these students?

NCTM Standards and Expectations

Algebra 3-5

Describe, extend, and make generalizations about geometric and numeric patterns.
Represent and analyze patterns and functions, using words, tables, and graphs.

This lesson prepared by Grace M. Burton.


		1 period
		NCTM Resources Principles and Standards for School Mathematics Web Sites Pascal’s Triangle