Introduction to Sequences

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Objective

SWBAT find the next three terms of a given sequence and create different categories based on the patterns.

Big Idea

Students will be able to write sequences from patterns, words, and given information.

Categorizing with Partners

20 minutes

In this lesson, I introduce students to sequences in a Partner Activity.  The idea for the first part of this lesson comes from another teacher of the Master Teacher Project, Colleen Werner.  I have linked her lesson here.  I hand each pair of students 10 Sequences cut apart individually.  Each pair of students is to complete the sequence, and then categorize the sequences into groups of their choice.  I do not introduce any of the sequence terminology at this time. I allow each pair of students to place each sequence into a category and explain to their partner why they are placing it there based on their observations. I supply students markers that they may write on the tables with if needed.  

As students are working, I walk around to monitor their progress, and to observe the different categories that are being created.  Most students are categorizing a sequence by adding/subtracting or multiplying/dividing. Most students have created an other category for the quadratic functions and the Fibonacci Sequence.  If some students are struggling, I encourage them to complete the sequences first. Then, try to place sequences in categories based on similarities.

After allowing students about 10 minutes to complete and categorize the sequences, I randomly call on students to share their work. I have drawn four columns on the board to write categories that the class agrees on as we review their work.  After reviewing a few sequences that the class sees as adding or subtracting, we create that category.  I have each student that I call on, share their completed sequence under the document camera. Then verbally explain the category they placed it and their reasoning.

After reviewing the Partner Work, the class created the following categories, with the given sequences.

Adding/Subtracting:  D, H, I, J

Multiplying/Dividing: B, E, F

Other:  A, G, C

 

 

Graphing Activity

20 minutes

Then I have students graph each equation on an individual graph, still keeping the categories.  I use these Grids for students to graph. After graphing the individual graphs, I call the class together again.  I have students again, share a few examples of each category under the document camera. While students are sharing, I ask the following questions:

  • Is it a function? Why or why not?
  • What type of function is it?
  • How do you know?
  • Are all of the graphs in each category similar?

Students recognize after graphing that adding/subtracting sequences model Linear Functions, and multiplying/dividing sequences model Exponential Functions.  However, when graphing the other category, students recognize another category needed, which is Quadratic Functions.  Students then place the Fibonacci Sequence in an other category by itself.  As a class, we agree to change the categories to the following:

Linear: D, H, I, J

Exponential: B, E, F

Quadratic:  A, G

Other: C (Fibonacci Sequence)

There are a very few students in some classes that recognize the Quadratic Function category early on in the lesson.

Closure

10 minutes

After the students have completed the Graphing Activity, I have students get out their notebooks to summarize what we have learned today. I allow students about five minutes to answer the following questions on their own:

1.  What is a sequence?

2.  How can we categorize sequences?

3.  Can you think of a another sequence besides the Fibonacci Sequence that would be in a different category besides Linear, Exponential, or Quadratic?

4.  Create a sequence that would result in a Linear Function.

5.  Create a sequence that would result in a Exponential Function.

 

Then we review the answers as a class.  I introduce the term Arithmetic Sequence for adding and subtracting a common difference in the sequence.  I also have them write down Geometric Sequence for sequences that increase or decrease by a factor or multiplying/dividing. I go into more detail of the differences of these sequences in the next two lessons of this unit.