Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to nctm.org/illuminations. Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on nctm.org.

## Two-Square Repeating Patterns

Pre-K-2
1

In this activity, students analyze how repeating patterns are generated. Using the interactive computer applet, students create, compare, and contrast pattern units of two squares and predict how patterns with different colors will appear when repeated in a grid and check their predictions.

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.

Fostering the ability to create and analyze simple patterns and make predictions about them is a major learning goal in the primary grades. Using the interactive math applet, students can create and study different pattern units. The interactive applet is designed so students will place unit patterns one at a time on the grid as they extend their patterns, or have the computer fill in the entire grid. This example encourages students to explore what new designs their pattern units will generate when repeated on the grid.

The Making Patterns 1 Activity Sheet encourages students to predict, explore, and analyze repeating patterns. Throughout the activity, students also explore the divisibility of ten by two.

To introduce the excursion, tape one blue square and one red square onto the board. Define these two colored squares as a "unit pattern" that can be repeated again and again. Ask a volunteer to tape the remaining two squares on the board and demonstrate a repeating pattern. Inform students that they will be using the computer to explore similar two-unit patterns, which repeat to fill up 100 squares on a grid.

Place students into teams of two and distribute a Making Patterns 1 Activity Sheet to each group. They should visit the website listed 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 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 them 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. Also, encourage students as they work through the activity sheet to think about the relationship of dividing 10 by 2. Ask questions like the following as you monitor and facilitate the group work.

When students have finished the Making Patterns 1 Activity Sheet, lead a class discussion about what they have learned. Write "Pattern Discoveries" on a piece of butcher paper and ask questions like those listed in the "Questions for Students" section.

Assessment Option

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.

Extension

Move on to the next lesson, Multi-Square Repeating Patterns

Questions for Students

1. How many squares are in your pattern unit?
2. How many times can you repeat your pattern unit in one row?
3. Does your pattern unit fit in one row exactly? Why?
4. Describe the row patterns your two-unit pattern created.
5. Describe the column pattern your two-unit pattern created.
6. Did you discover any diagonal patterns? Describe any that you found.
7. What discovery did we make about rows of a two-unit pattern when we placed the pattern in a 10x10 grid?
8. What did we discover about the columns?
9. How could we describe the diagonal pattern?
10. Once the colors changed in the two-unit pattern, how did the pattern change?
11. What was similar between your two patterns?
12. What do we think will always be true about two-unit patterns when we place them in a 10 × 10 grid?
13. Will two-unit patterns always fit exactly on a 10 × 10 grid? Why?
14. If we changed our two-unit pattern to a three-unit pattern, what changes could we predict would occur when we placed it in the 10 × 10 grid? Why?

### Multiple-Square Repeating Patterns

Pre-K-2
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.

### Different Representations

Pre-K-2
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.

### Repeating and Growing Patterns

Pre-K-2
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."

### 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 how repeating patterns are generated.

### NCTM Standards and Expectations

• Sort, classify, and order objects by size, number, and other properties.
• 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.