The unit was supported through group tasks as the topics moved along. Arithmetic sequences and series were linked to linear functions through the toothpick problem. Students were asked to arrange toothpicks into boxes and record how many toothpicks it took to make 'x' number of boxes. Their results were extrapolated and tied to variables from the linear functions notation. From there, I introduced the new terms of "common difference" and "term one" instead of slope and y-intercept. The arithmetic portion usually goes smoother than its geometric cousin for two reasons:
1) The members of the sequence are equally spaced in an arithmetic sequence
2) The arithmetic situation avoids large and ugly fractions
I introduced the concept of geometric sequences with a video clip from "Pay it Forward". The clip, found here, used a familiar situation to show the new relationship between terms. Students commented on how quickly the results grew in this new case. One student asked if all the answers were going to be so huge. Before I could respond, another student retorted that very large numbers are better than very small ones. With an inward smile, I seized my chance to begin the shaded square.
I handed out a simple sheet to every student. On it were 2 squares--one on each side--and room for a few calculations below. I put three steps on the board, modeled the process for one iteration, and stepped back to let them answer the question.
1) Cut the unshaded area in half.
2) Shade in one side of your bisection.
How much area is shaded with each iteration?
What pattern do you notice between each successive number?
From here, I had to work quickly. Students' pages are only large enough to give 6-7 cuts of the square before they run out of feasible space. I circulate to make sure that the cuts are being made as logically as possible. Most students develop a picture much like this: