E-Example 4.1.1: Creating, Describing and Analyzing Patterns

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 patterns. 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. In the second part, Describing Patterns, examples of various ways students might interpret the same sequence of cubes are given. This illustrates 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:

Making Predictions from Patterns in the Classroom

Fostering the ability to create and analyze simple patterns and make predictions about them is a major learning goal 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, responding to the same questions for their computer patterns.

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:

Through the pattern activity students are also exploring the divisibility of ten by two, three, four, and five. 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 because they simply make strings of objects without order or repetition rather than creating units that are repeated. This computer environment provides a structure for success and reflection on the idea of a repeating unit.

Take Time to Reflect

What realistic applications could teachers suggest to provide students with additional opportunities to create and repeat pattern units?
Why are conversations about the pattern units students create important in helping students 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

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

Task

Making Predictions from Patterns in the Classroom

Reference