Lesson plan

Lesson 8: The nth Term

teaches Alabama State Standards 8a-27.
teaches Alabama State Standards 8a-24.a.
teaches Alabama State Standards 8a-24.
teaches Arizona State Standards A2.F-BF.A.2
teaches Arizona State Standards A1.F-LE.A.2
teaches Common Core State Standards MP5 http://corestandards.org/Math/Practice/MP5
teaches Common Core State Standards HSF-BF.A.2 http://corestandards.org/Math/Content/HSF/BF/A/2
teaches Common Core State Standards HSF-LE.A.2 http://corestandards.org/Math/Content/HSF/LE/A/2
teaches Common Core State Standards MP8 http://corestandards.org/Math/Practice/MP8
teaches Common Core State Standards MP6 http://corestandards.org/Math/Practice/MP6
teaches Colorado State Standards HS.F-LE.A.2.
teaches Colorado State Standards HS.F-BF.A.2.
teaches Georgia State Standards MGSE9-12.F.LE.2.
teaches Georgia State Standards MGSE9-12.F.BF.2.
teaches Kansas State Standards F.LQE.2.
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Minnesota State Standards
teaches Ohio State Standards F.LE.2.
teaches Ohio State Standards F.BF.2.
teaches Ohio State Standards F.IF.9.
teaches Pennsylvania State Standards CC.2.2.HS.C.5.
teaches Pennsylvania State Standards CC.2.2.HS.C.3.

Lesson 8: The nth Term

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 closed-form definitions. A closed-form 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 Warm-up: 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 Cool-down: 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.


  • This lesson builds on the standard:CCSS.HSF-LE.A.1MS.F-LE.1MO.A1.LQE.A.1aMO.A1.LQE.A.1b
  • This lesson builds towards the standards:CCSS.HSF-BF.A.2MS.F-BF.2MO.A1.LQE.B.4CCSS.HSF-LE.A.2MS.F-LE.2MO.A1.LQE.A.3






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