## Multiple-Square Repeating Patterns

- Lesson

In this activity, students create and analyze repeating patterns using pattern units of three, four, and five squares. They predict how patterns with different numbers of squares will appear when repeated in a grid, and check their predictions. Students investigate similarities and differences between the rows, columns, and diagonal patterns created with each pattern unit.

This Internet Mathematics Excursion is a brief mathematics activity. To maximize student learning, certain prerequisites are necessary to use this activity. Therefore, it would be appropriate to include this activity as part of a more fully developed Standards-based lesson, but it should not be used as a complete stand-alone lesson.

Making Patterns 2 Activity Sheet

Patterns are a way for students to recognize order and are important in all aspects of mathematics. Creating pattern units with the interactive applet can be beneficial for students who are not yet successful in creating their own patterns with physical manipulatives. These students simply make strings of objects, without order or repetition, rather than creating units that are repeated. Using the interactive math applet, students will create and study pattern units of three, four and five squares. The interactive applet is designed so students may place entire units on the grid one at a time, or have the computer fill the entire grid. The Making Patterns 2 activity sheet encourages students to investigate row, column and diagonal patterns generated when units of three, four, and five squares are repeated. Throughout the activity, students predict, explore, and analyze similarities and differences between the repeating pattern units. Students also explore factors of ten.

Before students visit the Web site, introduce the excursion by drawing a 10 × 2 grid onto the board.

Inform the students that you are going to color in a 5-unit pattern in this grid. Fill in the pattern using 5 different colors. Define these colored squares as a "unit."

Ask the students to think about how many times this unit pattern could be repeated in the 10 × 2 grid? Will it fit exactly? How do they know?

Inform students that they will be using the computer to investigate patterns created when three, four, and five-unit patterns are repeated again and again to fill up 100 squares in a 10 × 10 grid. They will discover similarities and differences in the row, column and diagonal patterns created by the repeating unit. They will investigate which units fit exactly in one row of the grid.

Place students into teams of two and distribute a Making Patterns 2 activity sheet to each group. They should visit the following Web site: Creating, Describing, and Analyzing Patterns.

Working together, partners share the responsibility of "Mouse Driver" and "Reader/Recorder". The "Reader/Recorder" will read the directions from the activity sheet and record observations while guiding the activity. The "Mouse Driver" controls the action of the mouse and movement on the computer screen. Partners should switch roles until all have moved the mouse.

Students should click on the stand-alone applet to begin. As students work through the activity, walk from group to group, encouraging students to focus on the number of squares in their pattern unit and how this unit will look when repeated in a ten-by-ten grid. Encourage students to analyze and compare row, column, and diagonal patterns created by the various units. As they work through the activity sheet encourage students to think about factors of ten. Once students have completed exploring the applet, lead a class discussion using the following questions.

When students have finished the Making Patterns 2
activity sheet, the class should meet and discuss the learning process.
Write "Pattern Discoveries" on a piece of butcher paper. Guide students
to summarize the lesson by answering the *Questions for Students* as you record their responses.

- Making Patterns 2 Activity Sheet
- Chart paper and markers
- Creating, Describing, and Analyzing Patterns Interactive

Assessment Options

1. The *Questions for Students* will provide an opportunity for you
and the students to assess what they have learned and what they still
want or need to understand. This will
give you ideas for further instruction.

Extensions

1. It would be useful for students to explore related activities using physical manipulatives. Students can compare their grid patterns to linear patterns by using connecting cubes to build and repeat their grid pattern units in a linear fashion. For example, they can predict what color every fifth cube will be or what color the sixteenth cube will be, etc. Students may extend thinking further by trying these challenges:

- Make different grid patterns with varying size units that will all result in vertical stripes when displayed on the grid.
- Make a pattern that will have red squares appearing diagonally on the grid.
- Create a pattern that will result in the eighteenth square being green.

**Questions for Students**

- How many squares are in your pattern unit?
- How many times can you repeat your pattern unit in one row?
- Does your pattern unit fit in one row exactly? Why?
- When looking at the color of the 20th square in the grid, what color do you think the 40th square will be? Why?
- Describe row patterns created.
- Describe column patterns created.
- Describe diagonal patterns created.
- How did the patterns change when you added one square to your unit?
- What patterns appeared when a three-unit pattern is repeated in a 10 × 10 grid?
- How did the patterns change when you went to four squares?
- What was similar between three and four-unit patterns?
- When you made your four-unit pattern one square longer how did the pattern change?
- Did your five-unit patterns fit exactly in one row of the grid? Why?
- What will always be true about three, four, and five unit patterns?
- If the fifth square in our unit is blue what squares on the grid can we predict will be colored blue? How do we know?
- What does your team think will always be true about five-unit patterns when they are repeated on a 10 × 10 grid?

### Two-Square Repeating Patterns

### Different Representations

### Repeating and Growing Patterns

### Learning Objectives

Students will:

- Create pattern units of squares, predict how patterns with different numbers of squares will appear when repeated in a grid, and check their predictions
- Analyze the similarities and differences between the rows, columns, and diagonal patterns created with each pattern unit
- Analyze how repeating patterns are generated