The purpose of including these sequences is to encourage students to notice and make use of structure (MP7). The last two sequences may present a challenge since the growth factor is less than 1.

They might first propose less formal or precise language, and then restate their observation with more precise language in order to communicate more clearly. When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). (Students likely encountered it in an earlier course when they studied exponential functions.) Students notice and describe that each sequence is characterized by the same type of relationship between consecutive terms. The purpose of this task is to re-introduce growth factor. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task accessible to all students with these two approachable questions. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to write down things they notice and things they wonder. This is the first Notice and Wonder activity in the course.

Geometric sequences activity