In this activity, students generate two geometric sequences from a mathematical situation. The purpose is to use tables and graphs to create representations of geometric sequences. At this time, students do not need to write equations for the situation since that work will be the focus of a future lesson.

Monitor for students sketching neat and accurate graphs to highlight during the whole-class discussion. Note that future lessons will focus on a reasonable domain for sequences when regarded as functions. In this activity, students might “connect the dots” in the graphs with lines, but it’s not crucial to address at this time whether or not that’s a sensible thing to do.

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?” The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.

The goal of this discussion is for students to identify the growth factor for each sequence and learn that sequences with a growth factor are called "geometric sequences."

Invite previously identified students to share their graphs with the class. Display the two sequences in this activity (1, 2, 4, 8, 16, 32 and 80, 40, 20, 10, 5, 2.5) for all to see.

Here are some possible questions for discussion:

• “When did you stop cutting the paper and complete the table using a pattern?” (Answers vary.)
• “What was the pattern you noticed?” (To find the number of pieces, multiply the previous number by 2. To find the area, multiply the previous number by )
• “How did you find the results after six cuts?” (Multiply 32 by 2 to get 64, and divide 2.5 by 2 to get 1.25.)
• “What is the growth factor for each sequence?” (It’s 2 for the number of pieces and for the area.)
• “How can you see the growth factor in each graph?” (For the number of pieces, the height of each plotted point is twice the height of the previous plotted point. For the area, the height of each plotted point is half the height of the previous plotted point.)

Tell students that the sequences they have seen today (in this activity and in the Warm-up) have a special name: geometric sequences. Geometric sequences are characterized by a growth factor. In a geometric sequence if we divide any term by the previous term, we always get the same value: the growth factor for the sequence.

Some students may notice the similarity between a geometric sequence and an exponential function. Tell these students to hold on to those ideas until the Lesson Synthesis.

If students do not yet identify the growth factor of a sequence, consider saying:

• “Can you explain how you completed the sequence.”

• “By what number could you multiply to get the next term?”

The purpose of this discussion is to make sure that, in the lead up to writing recursive definitions for sequences, students understand that, for geometric sequences, the growth factor is defined as the multiplier from one term to the next. That is, the growth factor is the quotient of a term and the previous term. For example, many students will want to say that the pattern of the third sequence is “divide by 10 each time.” This is true, but the growth factor is .

For each sequence, invite a student to share how they completed the sequence and determined the growth factor. Highlight methods for finding the growth factor:

• Dividing any term by the previous term. For example, in the geometric sequence 3, 6, 12, 24, . . . , we divide 6 by 3 to get 2. Therefore, the growth factor is 2.
• Determining the missing factor that, when multiplied by any term, gives the next term in the sequence. For example, 3 times the missing factor is 6, so the growth factor must be 2.

Emphasize that the presence of a growth factor is what makes a sequence geometric.