Some students may believe that function notation "distributes," that is, that expressions like are equivalent to . If students do this, consider asking:

• “Can you explain how you evaluated the function?”
• “What is the same and different about and ?”

The purpose of this activity is for students to contrast three different types of sequences and to introduce the term "arithmetic sequence."

Monitor for students using precise language (MP6), either orally or in writing, during work time to invite to share during the whole-class discussion.

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing that students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including, especially, everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

If students need help identifying a pattern for the two non-geometric sequences, consider asking:

• “What can you say about the change between consecutive terms in the sequence?”

• “How could using a table for the term number and term value help you identify the pattern?”

Encourage students to use any language they wish to describe the pattern—they do not need to use an equation at this time.

The purpose of this discussion is to compare different types of sequences and introduce students to the term “arithmetic sequence.”

Direct students’ attention to the reference created using Collect and Display. Ask students to share how sequences and are alike and different. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. They might offer things like:

• is geometric but is not.
• In , we always add 10 to get from term to term, but in , we always multiply by 2.
• In , the growth factor is 2. In , we get the next term by adding 10 to the previous term.

Tell students that sequence is an example of an arithmetic sequence. Here are two ways to know a sequence is arithmetic:

• We always add the same number to get from one term to the next.
• If we subtract any term from the next term, we always get the same number.

Share that the constant in an arithmetic sequence is called the rate of change or common difference. In sequence , the rate of change is 10, because , , , and .

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

Use this discussion to make sure students understand how to determine the rate of change of an arithmetic sequence.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their reasoning, displaying any graphs created for all to see. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. If the idea of slope or average rate of change does not come up, ask students how they could have found the missing points for an arithmetic sequence given only and . Draw in the slope triangle, and show the computation for the average rate of change.