The purpose of this lesson is for students to understand what makes a sequence a geometric sequence and to begin to connect that idea with their learning about exponential functions in an earlier course. The lesson also gives students opportunity to use precise language to describe the relationship between consecutive terms in a sequence (MP6). In particular, how the terms of a geometric sequence grow by the same factor from one term to the next. For example, this is a geometric sequence: 0.5, 2, 8, 32, 128, . . . Each term is 4 times the previous term. Two ways to think about how you know the sequence is geometric are:
- Each term is multiplied by a factor of 4 to get the next term.
- The ratio of each term and the previous term is 4.
We call 4 the growth factor or the common ratio. After considering some examples of geometric sequences in the warm-up and how they are similar, students then develop two different sequences from the context of continually cutting a piece of paper in half. Using tables and graphs to identify the growth factor of these sequences, geometric sequences are defined. Next, students practice calculating missing terms of geometric sequences by first identifying the growth factor.
In an earlier course, students encountered the idea of functions and studied exponential functions specifically. Some students may see that a geometric sequence is simply an exponential function whose outputs are the terms and whose inputs are the positions of the terms. Later lessons will elaborate on this idea. This lesson invites recall of those ideas with a light touch by referring to, for example, “the size of each piece as a function of the number of cuts” and by using the term growth factor rather than common ratio. Students are also asked to describe how graphs representing each quantity in the paper cutting context are the same and different, and in so doing may recall prior knowledge of the behavior of exponential functions. In the last activity, the connection is made explicit.
A note on language and notation: in earlier courses, students may have learned that a ratio is an association between two or more quantities. In more advanced work, such as this course, ratio is typically used as a synonym for quotient. This expanded use of the word ratio comes into play in this lesson with the introduction of the term common ratio.
- Create tables and graphs to represent geometric sequences.
- Determine missing terms in geometric sequences.
- Determine the growth factor of a geometric sequence.
- Let’s explore growing and shrinking patterns.
Scissors and copies of the blackline master are optional for the paper-cutting activity. Provide them if students will physically carry out the paper cutting instead of observing how the paper-cutting would look. If using, prepare one pair of scissors and one 8 inch by 10 inch piece of blank paper for every two students. Blank paper may be used as an alternative to the grid on the blackline master, which is meant to make the meaning of area more clear.
- I can find missing terms in a geometric sequence.
A sequence in which each term is a constant times the previous term.