Table of Contents Appendix E-examples Resources

Creating, Describing, and Analyzing Patterns to Recognize Relationships and Make Predictions: Making Patterns Making Patterns Describing Patterns Extending Pattern Understandings

This three-part example highlights different aspects of students' understanding and use of patterns as they analyze relationships and make predictions, as discussed in the Algebra Standard. This first part, Making Patterns, includes an interactive figure for creating, comparing, and viewing multiple repetitions of pattern units. The interactive figure illustrates how students can create pattern units of squares, then predict how patterns with different numbers of squares will appear when repeated in a grid and check their predictions. The second part, Describing Patterns, gives examples of various ways students might interpret the same sequence of cubes. These differences illustrate the importance of discussing and analyzing patterns in the classroom. The third part, Extending Pattern Understandings, demonstrates ways in which students begin to create a "unit of units," or a grouping that can be repeated, and begin to relate two patterns in a functional relationship.

Task

Create pattern units of two to five squares and display them on the grid. Can you visualize how the grid will look when your pattern is repeated? Try these challenges:

• Make different patterns with units of varying size. The units you create should all result in a pattern with vertical stripes when displayed on the grid.
• Make a unit that will create a pattern with red squares appearing diagonally on the grid.
• Create a pattern whose eighteenth square is green.

[How to Use the Interactive Figure]

[Stand-alone applet]

Making Predictions from Patterns in the Classroom

The ability to create and analyze simple patterns and make predictions about them is a major learning goal for students in the primary grades. Using cubes and a grid or the interactive computer applet, students can create and study different pattern units. With physical manipulatives, they can repeat their pattern units in a linear fashion, predicting what the next cube will be or what color the sixteenth cube will be. The interactive applet is designed so that students can place squares one at a time as they extend their patterns, place entire units on the grid one at a time, or have the computer fill the entire grid. Students can complete the task using either method.

This example encourages students to explore what new designs their pattern units will generate when repeated on the grid. Teachers should help students focus on the number of squares in students' pattern units and how these units will look when repeated in a ten-by-ten grid. Questions such as these are helpful:

• 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 one row exactly?
• What happens if you make your pattern unit one square longer (or shorter)?

Through the pattern activity students can also explore the divisibility of 10 by 2, 3, 4, and 5. A similar activity, "Mr. Stripes Paper Company," appears in Burton et al. (1992).

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. With physical objects, students may simply make strings of objects without order or repetition instead of creating units that are repeated. The computer environment provides a structure for success and for reflection on the idea of a repeating unit.

Take Time to Reflect

What realistic applications could teachers suggest to give students additional opportunities to create and repeat pattern units? Why are conversations about the pattern units students create important in helping them learn to analyze patterns made by others?

Reference

Burton, Grace, Douglas Clements, Terrence Coburn, John Del Grande, John Firkins, Jeane Joyner, Miriam A. Leiva, Mary M. Lindquist, and Lorna Morrow. Third-Grade Book. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades K–6, edited by Miriam A. Leiva. Reston, Va.: National Council of Teachers of Mathematics, 1992.

Making Patterns

Describing Patterns

Extending Pattern Understandings

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