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Lesson 2: Introducing Geometric Sequences
• Calculate missing terms in geometric sequences.
• Use tables, graphs, and equations to understand geometric sequences.
Lesson Narrative
The purpose of this lesson is to learn what makes a sequence a geometric sequence. A geometric sequence is characterized by a common ratio. For example, this is a geometric sequence: 128, 32, 8, 2, 0.5, . . . The common ratio is . Here are two ways to think about how you know the sequence is geometric:
• Any term is multiplied by to get the next term. • The ratio of any term and the previous term is .
After seeing some examples of geometric sequences in the warm-up, the term common ratio is introduced. Then students encounter a common ratio in context,continually cutting a piece of paper in half. Students develop a sequence describing the number of pieces, and a di erent sequence describing the area of the pieces, using a table and graph. Geometric sequence is de ned. In the last activity, students are presented with several examples of geometric sequences and asked to nd missing terms and the common ratio for each.
In a previous course, students encountered the idea of functions and studied exponential functions speci cally. 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.” Students are also asked to describe how graphs representing each quantity in the paper cutting context are the same and di erent, and in so doing may recall prior knowledge of the behavior of exponential functions.
Some notes on language and notation:
• In previous 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 and is introduced deliberately in the warm-up.
• A geometric sequence has terms, and each of those terms has a value. However, it is often less cumbersome (and not confusing) to just say term when we really mean value of the term. For example, “each term is three times the previous term.” This shorthand language is used in this lesson and noted when it rst occurs.
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Teacher Guide Algebra