##
* *Reflection: Intervention and Extension
Sequences as Functions (Part 1 of 2) - Section 2: New Info

Triangular numbers have many interesting properties. For example, a triangular number sequence can never end with the digits 2, 4, 7, or 9.

I like to ask my students if they can find any other interesting properties of the triangular numbers. One possible answer is that triangular numbers are either divisible by 3 or have a remainder of 1 when divided by 9. This can be an open question that students can complete in class today, or over the next few days.

*More about Triangular Numbers*

*Intervention and Extension: More about Triangular Numbers*

# Sequences as Functions (Part 1 of 2)

Lesson 10 of 11

## Objective: SWBAT correctly identify the terms of sequences and graph sequences as functions.

## Big Idea: Sequences can be thought of as functions with their own special qualities that are obvious on a graph.

*55 minutes*

#### Launch

*15 min*

At the start of the class I project the Launch: Lesson 9 Functions and ask each student to complete the table displayed in their notebooks. I have created a video explaining the implementation of this launch:

**Video Source Link**: http://www.screencast.com/t/G97dGVCV

#### Resources

*expand content*

#### New Info

*15 min*

As students complete their work on the Launch, I write on the board:

**This sequence is the sequence of Triangular Numbers and defines a function**. **What is the domain of this function?**

I give my students to recall the meaning of **domain** and to discuss the answer with their elbow partners.

Next, I introduce my students to a notation for sequences. Here, I write on the board:

**If we call the function resulting from this sequence T, using subscripts to indicate the term in the pattern:**

** T _{1} = 1**

**T _{2} = 3**

**T _{3} = 6**

**The notation**** ****T _{3}= 6 is read “T sub three equals six.” The subscript number is often called an index because it indicates the position of the term in the sequence. **

I then ask the class to help me write T_{4} and T_{5} on the board.

Finally, I ask the class what a term in the sequence like T_{20 } would equal. I expect some students will try to extend the table. After a couple of additional entries, I plan to stop them and ask if there is a pattern. I say, "If there is a pattern, maybe there is a quicker way?" At some point during this activity I will introduce the term, **Explicit Formula**. I try to wait until it is needed to describe a students' work or suggestion.

This video narrative shows how I like to introduce this idea:

**Source Link for Video: **http://www.screencast.com/t/SEvETdQLB

*expand content*

#### Application: Shared practice

*20 min*

For this **Shared Practice** I pair students and hand them a copy of Application Shared practice. Students will need to use the **Explicit Formula** from the New Info section to answer Questions 1 and 2 algebraically.

As students graph the data in Question 3, I check around to see if they start connecting the points. If they do, I point out Question 4 which asks students to reflect on this issue. The mere fact of asking whether the points are supposed to be connected makes students wonder and inquire. This inquiry should lead them to understand that the domain includes only positive integers.

After students complete the shared practice I inform the class that a sequence is an example of a **Discrete Function**, meaning that it has gaps, or intervals, between successive values of the domain. Therefore, the graph must consist of unconnected points.

#### Resources

*expand content*

#### Closure

*5 min*

To close today's lesson I hand each student an Exit Ticket which should only take a couple of minutes to complete. Assessing these will give me an idea of whether I should point out a few things to the class before going into Part 2 of the lesson, tomorrow, where students will be involved in more independent practice.

For homework I ask my students to complete Sequences as Functions 1 to review today's learning and prepare for Part 2 during tomorrow's lesson.

*expand content*

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Relations that Function (Part 1)
- LESSON 2: Relations that Function (Part 2)
- LESSON 3: Functions Rule (Part 1 of 2)
- LESSON 4: Functions Rule (Part 2 of 2)
- LESSON 5: Crickets Tell Temperature
- LESSON 6: Linear? Yey or Nay
- LESSON 7: Comparing Linear and Exponential Functions
- LESSON 8: Time-Distance Graphs
- LESSON 9: Rates of Change
- LESSON 10: Sequences as Functions (Part 1 of 2)
- LESSON 11: Sequences as Functions (part 2 of 2)