## Repeating and Growing Patterns

In this activity, students create and explore more complex patterns such as "growing patterns" which have related but different relationships to "repeating patterns". Students form generalizations, analyze, and describe growing patterns using connecting-cubes, and explore what happens when growing patterns "double" or "split."

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 fully developed Standards-based lesson, but it should not be used as a complete stand-alone lesson.

Patterns are a way for young students to recognize order and to organize their world and are important in all aspects of mathematics. Help students develop the ability to form generalizations by asking questions such as "How could you describe this pattern?" or "How are these patterns alike?" The Describing Patterns 2 activity sheet guides students to analyze and describe how growing patterns are generated, encourages them to explore different ways to interpret the patterns and translate from one form to another, and challenges students to compare growing patterns to repeating patterns.

Before students visit the Web site, introduce the excursion by holding up a series of alternating red and blue cubes.

Have students discuss the number of connecting-cubes that would be generated if this unit of 12 red and blue connecting-cubes "doubled" in size.

Discuss the number of connecting-cubes that would be generated if this unit not only "doubled" once, but also "doubled" again. Write a numerical pattern representing the growing pattern, (12 + 24 + 48).

Have students imagine how many connecting-cubes it would take to create this pattern if it kept growing and growing!

Inform students that today they will be working with interactive figure "2b," using connecting-cubes, to create a growing pattern. Ask students to predict what growing patterns might be.

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

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 manipulated the cubes.

As students work through the activity, walk from group to group, encouraging them to describe the "growing pattern" using color, letters, number sentences, and sounds. This recognition lays the foundation for the idea that two very different situations can have the same mathematical features and thus, in some important ways are the same. Challenge students to compare the "growing pattern" in figure 2b, to the "repeating patterns" in figure 1b, and 1c.

When students have finished the Describing Patterns 2 activity sheet, the class should meet to debrief the lesson and learning objectives.

- Describing Patterns 2 Activity Sheet
- 6 red and 6 blue connecting-cubes
- Creating, Describing, and Analyzing Patterns Website

**Assessments**

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**

It would be useful for students to explore related activities using physical manipulatives. Partners can create a variety of growing patterns. Have students write number sentences to match their patterns. Students can also create charts and tables for recording and organizing how many times a pattern "grows" before it reaches 100 as discussed in the Algebra Standard.

**Questions for Students**

- Describe how the interactive tool created the growing pattern.
- Describe the growing pattern using colors.
- Describe the growing pattern using a clap/snap rhythm.
- Describe the growing pattern using A and B.
- Describe the growing pattern with a number sentence.
- How many times does the growing pattern in figure in 2b, need to "grow" in order to be larger than 10 connecting cubes? Larger than 20 connecting cubes?
- Describe how this pattern in Figure 2b is different from a growing pattern that "doubles" in size each time it grows.
- Describe how a repeating pattern is different from a growing pattern.
- Demonstrate how a repeating pattern sounds differently from a growing pattern.
- How many times does the growing pattern in figure 2b, need to "grow" in order to be larger than 25 connecting cubes? How can we prove it?
- Draw a table or chart showing the number of connecting cubes we would have after this pattern has grown 10 times.

### Two-Square Repeating Patterns

### Multiple-Square Repeating Patterns

### Different Representations

### Learning Objectives

Students will:

- Analyze and create growing patterns using red and blue connecting-cubes and compare growing patterns to repeating patterns
- Recognize, describe, and extend growing patterns such as sequences of sounds and shapes and translate from one representation to another
- Analyze how growing patterns are generated