Sequences as Functions (Part 1 of 2)
Lesson 10 of 11
Objective: SWBAT correctly identify the terms of sequences and graph sequences as functions.
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:
T1 = 1
T2 = 3
T3 = 6
The notation T3= 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 T4 and T5 on the board.
Finally, I ask the class what a term in the sequence like T20 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
Application: Shared practice
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.
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.