Returning to a paper cutting context from earlier in the unit, this Warm-up asks students to write a recursive definition from a description and a table. Unlike students’ previous work writing recursive definitions, however, this sequence starts with a first term of 0 instead of 1. This is purposeful, since the decision on how to write an equation to model a situation is up to the person doing the modeling (MP4). Students will have several more opportunities to think critically about the starting term of a sequence in future activities. Later in this lesson, students will consider non-recursive models for the same situation.

Select students to share their definitions and to pay particular attention to the starting term, , and the use of . An important takeaway from looking at this recursive definition is that the domain of a sequence is something that should be based on the situation and that starting with doesn't always make sense. Luckily, we can use the function notation to make clear that we are starting with , by beginning with , and then we change to to match.

Building on their thinking from the Warm-up, students work with non-recursive definitions of two different sequences, one geometric and one arithmetic, based on different cuts of a sheet of paper with an 8-by-10 grid. Students express regularity in repeated reasoning (MP8) by using their understanding of how the values of specific terms are calculated before explaining or expressing how the term is calculated.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students who have trouble visualizing what's happening to the paper in each sequence may benefit from drawing the paper at each step and labeling it with dimensions, or cutting paper themselves and calculating the areas. In particular, if students don't see why Kiran removes 8 square inches each time, encourage them to write down the dimensions of the paper for the first few steps and calculate each area (and draw the paper at each step if needed).

The purpose of this discussion is to encourage students to make connections between what they know about arithmetic and geometric sequences and their equations written non-recursively.

Display the two definitions from this task for all to see: and . Begin the discussion by asking, “How can you tell which of these defines a geometric sequence and which defines an arithmetic sequence?” (A geometric sequence has a constant growth factor between terms, so must represent a geometric sequence since each term is half the value of the previous term. An arithmetic sequence has a constant rate of change, so must represent an arithmetic sequence since each term is 8 less than the previous term.)

Tell students that the way the expressions for and are written are examples of defining as sequence by the term, which is the type of definition students are likely most familiar with from previous courses. These are sometimes known as a closed-form or explicit definitions, but students do not need to use these terms. In future activities, if students are asked to represent a sequence with an equation for the term, then they are being asked for an explicit definition, not a recursive definition. In later lessons, students will have practice writing equations for linear and exponential from a variety of situations.

Conclude the discussion by asking students to calculate which is larger, or ? ( is larger, since and , or, there can’t be a comparison because does not exist since a 10th cut is not possible.) Select 1–2 students to share their reasoning. If no student points out that does not exist due to the constraints of the context when is the number of cuts, bring this point up. Students will have more opportunities to think directly about domains given a specific situation in a later lesson.

The goal of this activity is for students to understand that the way an equation is written to define a function depends on how the domain of the function is interpreted. In the previous activity, starting the sequence with made sense since was defined as the number of cuts, while  was the area of a paper square after the cuts. In this activity, students consider two equations written for a sequence represented by a visual pattern. One of the equations assumes the sequence starts at Step 0, and the other assumes the sequence starts at Step 1.

To help all students understand that both equations generate the same sequence and correctly represent the visual pattern, during the activity and whole-class discussion students should be encouraged to use precise language as they explain why an equation is valid (MP6). Confusion is likely to arise unless students are clear about the definition for each equation is assuming. Some students may find the idea that both equations are equally valid challenging, so it is important to take time during the discussion for these students to understand that part of mathematics is recognizing what assumptions we are making and addressing them appropriately.

Monitor for students with clear definitions to select to share during the whole-class discussion. In particular, select students that create different representations, such as tables, to highlight that the two equations result in the same list of numbers as students' recursive definitions.

Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

Students may assume that at least one person has to be wrong because their equations don’t look the same. If this happens, consider asking:

• “How did you decide whose equation is correct?”
• “How could making a table for each help you decide if the equation is correct?”

The goal of this discussion is for students to understand why Andre's and Lin's equations are both representations of the visual pattern due to the interpretation of . Display the visual pattern for all to see. Invite previously selected students to share their reasoning about why either Andre or Lin is correct, recording student reasoning for all to see. If students still think starting with 0 is unusual after the discussion, give some more time for students to reason about Andre's point of view, and then select students to share. (For example, Andre could be thinking about as the breaking apart phase, so, for the solid black triangle, that happens 0 times.)

Conclude the discussion by telling students that identifying an appropriate domain for a function is partly dependent on the situation and partly dependent on how they see the relationship. There are often many correct equations that represent a function. An important takeaway for students is that, when they write an equation to represent a situation, they need to be clear what domain they have identified so other people can correctly interpret what they've done.