## Different Representations

In this activity, students create patterns using connecting cubes and describe various patterns they find in different sequences of cubes. Students explore and describe connecting patterns, and extend their patterns using a sequence of sounds and shapes. Students investigate various ways to interpret the same sequence of cubes by exploring ways to describe patterns translating from one representation to another.

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.

Even before formal schooling, children develop beginning concepts related to patterns, functions, and algebra. They learn repetitive songs, rhythmic chants, and predictable poems that are based on repeating patterns. In this activity, students use the interactive math applet to create and study red and blue connecting-cube patterns. The interactive tool is designed so students can create the entire pattern one connecting-cube at a time, or create the pattern, two connecting- cubes at a time. The Describing Patterns 1 activity sheet, guides students to describe the different patterns, encourages them to explore different ways to interpret the patterns, and challenges students to translate the patterns generated, from one representation to another.

Before students visit the Web site, introduce the excursion by holding up a series of 12 red and blue connecting cubes.

Ask students to describe this "connecting-cube pattern" using the colors they see. Discuss with students why this pattern could be named "ABABABAB." Inform students that they will be using the computer to explore similar "ABABAB" patterns, and investigating different ways to create the same pattern. They will also analyze different ways to describe an "ABABAB" pattern. Many students explain the pattern by saying, "It's a red cube then a blue cube and it keeps going like that." Some students might describe it as an "ABAB" pattern. Most students see the pattern being formed as a sequence of single cubes of alternating colors.

Place students into teams of two and distribute a Describing Patterns 1 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 both have manipulated the cubes.

As students work through the activity, walk from group to group, encouraging them to describe the connecting-cube pattern using color, letters and sounds. Challenge students to explore several ways to describe the pattern. The teacher’s role during his activity is to help students draw connections between what is happening to the patterns while moving the cubes. Suggestions for guiding questions will help facilitate this understanding.

When students have finished the Describing Patterns 1 activity sheet, the class should meet to debrief the lesson and learning objectives.

- Describing Patterns 1 Activity Sheet
- Connecting cubes (6 red, 6 blue)
- Creating, Describing, and Analyzing Patterns Website

**Assessments**

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**

- It would be useful for students to explore related activities using physical manipulatives. Partners can create a variety of connecting-cube patterns; for example, "ABC" patterns or "AABB" patterns, and investigate how to describe each pattern in words, sounds and letters. This recognition lays the foundation for the idea that two very different situations can have the same mathematical features. Students can further extend their thinking by using charts and tables for recording and organizing how many times a pattern repeats before it reaches a certain number of connecting- cubes, predicting along the way what color various cubes will be. The fourth and final i-ME in this series provides opportunities for extending the concepts discussed in this activity.

**Questions for Students**

- Describe how the interactive tool created the connecting cube pattern.
- Describe the connecting-cube pattern using colors.
- Describe the connecting cube pattern using a clap/snap rhythm.
- Describe the connecting-cube pattern using letters.
- When looking at the color of the 12
^{th }connecting-cube, what color do you think the 18^{th}connecting-cube would be if this pattern were extended? How do you know? - When looking at the color of the 12
^{th }connecting-cube, what color do you think the 19^{th}connecting cube would be, if this pattern were extended? How do you know? - Describe other ways you discovered to create an "ABABAB" pattern.
- Demonstrate how an "ABABAB" pattern sounds differently from an "AB, AB, AB," pattern.

### Two-Square Repeating Patterns

### Multiple-Square Repeating Patterns

### Repeating and Growing Patterns

### Learning Objectives

- Analyze a series of twelve red and blue connecting cubes, and describe different patterns they find in different sequences of cubes
- Recognize, describe, and extend patterns; such as, sequences of sounds and shapes and translate from one representation to another
- Analyze how repeating patterns are generated