There are several aspects of the interactive activity to focus on.

Different representations bring different aspects of the problem into focus. Tearing the paper clearly models the sums, but the graph makes the limit of 1 more obvious. The summation notation is abstract, but it would provide a basis for a formal proof, which the paper tearing in itself does not provide. One way to discuss this variation is to speak of how "transparent" a representation is in the sense of how much prior learning it requires for accurate interpretation. Effective teaching and learning require moving across the range of transparency with ease, always keeping in mind that any one representation does not exhaust or completely capture the underlying mathematical phenomena. Different levels of transparency are inherent in mathematics.

Individuals have different levels of comfort, skill, and talent with respect to different forms of representation. This is manifested as you work through the halves activity, as you learn, and also as you teach. The goal of the standard is to ensure skill and fluency among varied representations. Differences in initial preferences of representation are not inherently negative and can serve as launching points for rich discussions of the variety of representations and the connections among them. In this way, students can expand their comfort, familiarity, and appreciation for different forms. The benefit is having a wider repertoire of tools for thinking about and communicating mathematical ideas.

As we help students learn, we should attend to how our own skill level and approach to representation influence our students. A "simple" representation for the teacher may not be so for a student. Likewise, a teacher may be surprised if a student intuitively offers a geometric insight to an algebra problem that the teacher, with a long-established algebraic solution, may struggle to visualize. In such circumstances, being open to alternate representations and skilled in communicating about them will improve not only students' learning, but the teacher's as well.

You just had an opportunity to consider a variety of representations. We started with a physical representation, then made visual representations of the physical experience, and used symbols to show a data table, an equation, and a graph that depicted the situation. This allowed you to notice that each type of representation highlighted some mathematical features and downplayed others. Such activities help us learn as we create, critique, and interpret various forms of representation in mathematics work.

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