Lesson 8: The nth Term
About this lesson
The goal of this lesson is for students to understand that how an equation is written to represent a function depends on how the domain of a function is identified. With sequences, it is common to start at either \(f(1)\) or \(f(0)\). So far in this unit, the first term has been typically cited as \(f(1)\). The exception has been when \(n=1\) is confusing given the context, which is the case when the number of pieces of paper depends on the number of cuts. This lesson gives students a chance to study the effect this choice has when writing an equation to define a sequence and is also meant to help students review how to write equations of linear and exponential functions by using a table to express regularity in repeated reasoning (MP8). In the following lessons, students will write equations for these types of functions in various contexts.
Prior to this lesson students focused on defining sequences recursively using function notation. In this lesson, students will study equations representing functions that are known as explicit or closedform definitions. A closedform definition is one where the value of the term is determined from just the term number. This type of equation is one students are familiar with from their earlier work with linear and exponential equations.
A focus of this lesson is using precise language to explain patterns and understand how a function can be represented by two different equations (MP6).
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 8.1 Warmup: Which One Doesn’t Belong: Repeated Operations (5 minutes)
 8.2 Activity: More Paper Slicing (15 minutes)

8.3 Activity: A Sierpinski Triangle (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 8.4 Cooldown: Different Types of Equations (5 minutes)
Learning goals:
 Interpret an equation for the \(n^{\text{th}}\) term of a sequence.
 Justify (orally and in writing) why different equations can represent the same sequence.
Learning goals (student facing):
 Let’s see how to find terms of sequences directly.
Learning targets (student facing):
 I can explain why different equations can represent the same sequence.
Required materials:
 Copies of blackline master
 Scissors
Required preparation:
 Use of the blackline master is optional. If using, prepare 1 pair of scissors for every 2 students.
Standards:
 This lesson builds on the standard:CCSS.HSFLE.A.1MS.FLE.1MO.A1.LQE.A.1aMO.A1.LQE.A.1b
 This lesson builds towards the standards:CCSS.HSFBF.A.2MS.FBF.2MO.A1.LQE.B.4CCSS.HSFLE.A.2MS.FLE.2MO.A1.LQE.A.3
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.