## 4.1.3 Extending Pattern Understanding

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.

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### Extending Pattern Understandings in the Classroom

In situations such as this example, the computer software's capability to "glue" shapes together can facilitate students' understanding of the idea of a unit-of-units. The connecting cubes example shows a variety of ways to think about the unit of a pattern, and the pattern-block example above illustrates how computers can support young students' learning about units-of-units, that is, "composite" units that contain other units. This is an important idea in the development of place-value concepts. "Ten," for instance, can be seen as "one more than nine," but in our number system, it also plays an important role as a unit-of-units. Students who think about both "one ten" and "ten ones" are poised to understand place value.

Teachers can use pattern activities such as these to assess whether students have a basic understanding of how an arrangement might be generated. Such questions as, How would you tell someone else to build this pattern? or Is there another way you can make this pattern? assess students' abilities to identify different units in a given arrangement and to articulate their ideas.

The examples above included two- and three-dimensional patterns that required students to think of various units-of-units and how doing so can change one's view and description of a pattern. Not as clear, perhaps, is how patterns such as these connect to ideas of function and algebra. One way to illustrate the connection is to pair the counting sequence with the units of a pattern, creating two repeatable patterns. This is a function (Smith, forthcoming).

New kinds of questions lead students to search for relationships: What shape goes with 2? To continue the pattern, what shape is next? What number? Can you predict what shape will go with 12?

### Take Time to Reflect

- How does a strong recognition of patterns benefit students in learning about our place-value system?
- When students begin to organize sets of information into tables, why is it important for teachers to help them focus on both horizontal and vertical patterns and relationships?

### Reference

Smith, E. "Making Functions Accessible across the K–12 Curriculum: Covariation and Functional Reasoning."

Educational Studies in Mathematics, forthcoming.