sequences and series

Shading Squares

I recently finished up a unit on sequences and series with my grade 11 pre-calculus students. The unit is somewhat of an enigma because it contains relatively simple ideas bogged down in complex notation. This coupled with the overlapping definitions makes for a fortnight of rather rigorous cognitive exercise.

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:

In the Footsteps of Gauss

I like to introduce each topic with a task or activity. These do not necessarily have to be long, but should activate mathematical thinking. The idea has slowly evolved for me throughout my short career. They are the amalgamation of the ideas of a “motivational set” and discovery learning. I felt that both components are positive things to include in a math class, but both had severe implementation problems.

The motivational set is far too passive. In my college, a picture, story, or conversation could serve as a motivational set. It was essentially a transition tool that was completely void of any mathematics. Every lesson begins with the same routine whether it be a national anthem, attendance, or a short time of homework recap, but each learning experience needs to begin with an active brain. I found that the purpose of the motivational set was important, but needed a stronger method to get brains engaged in the day’s learning.