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).

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.