This is the first Notice and Wonder activity in the course. Students are shown four sequences and asked: “What do you notice? What do you wonder?”

Students are given time to write down what they notice and wonder about the sequences and then time to share their thoughts. Their responses are recorded for all to see. Often, the goal is to elicit observations and curiosities about a mathematical idea that students are about to explore. Pondering the two open questions allows students to build interest about and gain entry into an upcoming task.

The purpose of this task is to reintroduce growth factor. (Students likely encountered this term in an earlier course in which they studied exponential functions.) Students notice and describe that each sequence is characterized by the same type of relationship between consecutive terms.

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). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the sequences. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea of each consecutive term in a sequence growing by the same factor or having a common ratio does not come up during the conversation, ask students to discuss how they would describe the change from one number to the next in each sequence. Students may describe how the “same thing” is happening with consecutive terms. Encourage students to use the word term, and to be specific when they describe what is happening, for example:

• “In the first sequence, we always multiply a term by 3 to get the next term.”
• “In the last sequence, if we divide any term by the previous term, we get .”

Tell students that this “same thing” is called the growth factor or common ratio, and that we will use the first of these. For example, in the second list, the growth factor is 4 because , , , and .

Emphasize that 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, students may want to say that the pattern of the third sequence is “divide by 2 each time.” This is true, but the growth factor is , because it is the number we multiply by to get the next term. Students will continue to focus on determining the growth factor in later activities.

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 purpose of this task is to give students practice working with geometric sequences and identifying the growth factor of a sequence. Students reason repeatedly (MP8) about the structure of geometric sequences in order to generalize how to determine the missing terms.