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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. The 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. In the second part, Describing Patterns, examples of various ways students might interpret the same sequence of cubes are given. These differences illustrate the importance of discussing and analyzing patterns in the classroom. This 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.
When students create and describe their own patterns, teachers are better able to learn what students are thinking. Using pattern blocks, students might create a pattern like that in figure 4.3.
Fig. 4.3 In describing figure 4.3, students might describe several patterns, such as a color pattern (green-yellow-green-yellow ) or a size pattern (small-big-small-big) or a shape pattern (triangle-hexagon-triangle-hexagon). Students can create different patterns and challenge classmates to determine what pattern they had in mind. Activities like these help students think flexibly. Along with explorations with concrete manipulatives, students can work with virtual manipulatives, where a "unit" of more than one block, such as red-green-red, can be created and then iterated as easily as a single block can be iterated physically. Students create patterns with complex units-of-units consisting of many blocks. For example, a student could use the Glue Tool in a shapes program to make a new shape out of four pattern blocks (see fig. 4.4a). Using the tool to make these units-of-units and calling them "units" should be a topic for class discussion.
Fig. 4.4a Students can copy this new unit-of-units, slide it into place, and repeatedly use the Pattern Tool in shapes programs to fill a row (see fig. 4.4b).
Fig. 4.4b Teachers can encourage students to use the same strategy to fill the screen with their patterns (see fig. 4.4c) and then to describe their units.
Fig. 4.4c Extending Pattern Understandings in the ClassroomIn 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 socan 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?
ReferenceSmith, E. "Making Functions Accessible across the K–12 Curriculum: Covariation and Functional Reasoning." Educational Studies in Mathematics, forthcoming.
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