Note that the perimeter includes the lengths of the edges of the "holes" that are left by removing the "middle square in each group of 9." For example, the perimeter of the figure in the third step is The figures students are generating at each step are the figures whose limit is called "Sierpinski's carpet." This is a fractal whose area is 0 and perimeter is infinite! A very challenging extension is to ask students to find the perimeter of each figure in the task. Another solution approach students might take is to subtract the "holes" left by removing squares from 1, and it would be instructive for students who approach this task in different ways to compare their solutions to verify they give the same results. Given the complexity of the task, this would be a good activity for students to work on in groups. Students might not realize that the last figure is the same size as the others, just zoomed in to better see the details.
Because students are most likely to find an answer to part (d) by noting that at each step all squares are replaced by 8 squares that are $\frac19$ as big, this task naturally engages students in MP8, Look for and express regularity in repeated reasoning. If they haven't, part (d) will be very challenging. Even so, students might not think to use exponential notation to represent the areas.
The task assumes that students have seen and evaluated numeric expressions that involve whole-number exponents. The purpose of this task is to help motivate the usefulness of exponential notation in a geometric context and to give students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression, which is an important facet of MP7, Look for and make use of structure.
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