Introduction to Sequences

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Objective

SWBAT differentiate among arithmetic, geometric and other types of sequences and understand that sequences and series can be used to model real world phenomena.

Big Idea

Detecting patterns in numbers helps students see the mathematical relationships that underlie real world phenomena. In this colorful lesson, students look at patterns of numbers and uncover the rule used to generate them.

Launch and Explore

30 minutes

Students have studied number patterns before Algebra 2 and in this unit they learn to use algebraic representations for these patterns.  This first lesson is specifically designed to connect our study of sequences and series to what they already know about functions and number patterns.  

There is a one-time investment in the creation of 6-10 sets of Sequence Strips (depending on your class size).  The strips are used in this and the next two lessons in this unit to increase engagement and foster flexible thinking.  Each of the sequence strips has the first 6 terms of a number sequence followed by 3 blanks for students to fill in.  I like to print these strips on brightly colored paper (one color per set) and then laminate them.  The color coding helps we keep sets together and the laminating makes it possible to reuse as students write their answers with wet-erase markers.

It's important to give students the opportunity to explore so I create random student groups of 3 by "counting off" and provide each group with a set of sequence cards to work with.  

I tell students that their task is to (1) extend the patterns and (2) sort the cards into 3 piles [MP1]. When students have completed the task, we come together as a group to discuss how they have sorted [MP.3].  

Discuss and Formalize

20 minutes

To frame the whole discussion [MP.3] either I or a student volunteer will make a chart on the board with 3 sections, one for each "pile."  I ask a group that struggled somewhat with the sorting part of the activity to share how they decided to sort the cards.  When they have explained their process (and their struggle with it) I ask "can anyone build on what <student> has offered." We come up with categories like "keep adding" or "multiply by the same number" and then typically an "other" category for sequences like the Fibbonacci and quadratic patterns.  I "cold call" students to come to the board and each write one extended sequence in the category they think is appropriate.  It is important to hold off my corrections during this part of the process so that students have an opportunity to think about what makes sense rather than what I may consider correct.   When all sequences have been displayed on the board we work as a group to check the work and move any sequence that does not fit in the category it was placed in.  During the sorting activity, I will have introduced some of the formal vocabulary, so we close the activity with a vocabulary lesson.  Students use their notebook to record the meaning of the terms listed below and some examples of each.

sequence

arithmetic sequence

geometric sequence

the nth term in a sequence (using a with n as a subscript)

Reinforce and Extend

20 minutes

After taking notes on some of the formal vocabulary we use to describe sequences, students will explore some applications of sequences.  Students work independently or with their table group to complete  WS investigating sequences handout that includes 4 contexts for arithmetic and geometric sequences and some follow up questions.  I let students know that they will have 15 minutes to complete this activity and set a timer on the screen in the classroom.  As students work, I circulate and listen for the new vocabulary terms in students discussions.  When the timer rings, we go over the answers to the worksheet.

Closure and Assignment

15 minutes

To wrap up the introductory lesson on sequences, I ask students to turn to a partner and each explain one of the two sequence types we focused on today using the new vocabulary terms they wrote in their notes [MP3].  I then pass out the homework, which is an application of generating sequences called The Devil and Daniel Webster from the NCTM Illuminations project.  I like this activity because students are generally drawn in by the story and answering the question requires students to use a recursive process, which will be important in the next day's lesson.