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.
As students are creating patterns using connecting cubes, teachers can ask them to describe different patterns that they find in different sequences of cubes. Consider what might happen when a class is presented with the arrangement of cubes shown in figure 1a and asked to describe the pattern.
Many students explain the pattern by saying, "It's a red cube, then a blue cube, and it keeps going like that." Some students might describe it as an "ABAB" pattern. Most students see the pattern being formed as a sequence of single cubes of alternating colors, as shown in the animation in figure 1b.
The teacher might ask the students to find other ways in which this pattern can be created. For example, students might respond, "You could connect two blocks together a red and a blue. Make a pile of them like that. Then just connect them all together." They see the pattern being formed, as shown in the animation in figure 1c.
The class might discuss how the same pattern could be generated in different ways. Some students may see it as a pattern of three blocks (red, blue, red) being connected to form part Aand three other blocks (blue, red, blue) being connected to form part B, making a pattern that is like the ABAB pattern in figure 1a.
Teachers might offer more-complex patterns for students to explore, such as the growing pattern in figure 2a. The relationship among the elements of the pattern, although related in some ways, is different from the relationship among the elements of the previous pattern.
In figure 2a, the first red cube is followed by one blue cube, the second single red cube is followed by two blue cubes, and so on, as demonstrated in the animation in figure 2b. Other growing patterns might build on students' intuitions of "doubling" or "splitting" (Confrey 1995). For example, each additional unit might double the number of cubes or shapes. Students can be asked to describe this pattern and talk about how it is different from the patterns in figures 1a, 1b, and 1c.
Take Time to Reflect
Confrey, Jere. "Student Voice in Examining "splitting" as an Approach to Ratio, Proportions, and Fractions." PME-19, Recife, Brazil, 1995.
There are four buttons to help use each animation (from left to right):
See the interactive for exploration options.
Confrey, Jere. "Student Voice in Examining 'splitting' as an Approach to Ratio, Proportions, and Fractions." PME-19, Recife, Brazil, 1995.
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