Use the following interactive activity to investigate different ways to represent the sum of consecutive powers of 1/2; that is, 1/2 + 1/4 + 1/8 + 1/16 + ..., etc. Think of yourself as a mathematical tourist, hiking through a range of possible representations.
After you have explored the activity, please consider these questions in light of how you worked with representations in the activity.
How do the wording of the problem and the initial physical activity support the subsequent development of understanding of representations, including abstract forms?
What aspects of the problem are readily seen in each representation, and what information must be interpreted based on prior learning about the particular type of representation?
How does this problem lead you to connect various forms of representation, including physical and abstract representations?
Different representations may have required different skills and different insights. Which approach did you find easiest to understand? Which was most challenging? What might account for these differences in your thinking? What about in your students' thinking?
Did you think of representations or connections that were not in the activity? If so, what were they?