Geometric Sequences

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Objective

SWBAT represent geometric sequences verbally, visually, in lists/tables, graphically, as a recursive rule/pattern, and as an explicit rules.

Big Idea

Students will represent geometric sequences with various models using an engaging and effective cooperative learning activity.

Quick Write

5 minutes

The lesson quick write, posted as students enter the classroom is:

"Now that you know that an arithmetic sequence changes by a common difference, what other differences do you think they can change by?  Make a guess as to what a Geometric Sequence is."

My intent is for students to build on the knowledge developed from the previous sequences lesson and to further understand the difference between arithmetic and geometric sequences.

I have a required binder with a Quick Write section which I grade but you may choose to have students regular paper and decide whether or not to assign grades for Quick Writes.


 

 

Introduction

10 minutes

I used this time to introduce the concept of an exponential relationship versus the linear relationships we had been talking about thus far in the previous lessons.  I first show them this link of a ball bouncing simulation.  We discuss that the ball is bouncing back up at a common ratio of 3/4 each time as opposed to a common difference.  I then show a screen capture that I have saved.  I connect the vertices of the heights and we have a discussion that this does not form a line, rather it forms an exponential curve, which we will be discussing today.

Discussion/Notes

20 minutes

During this section I use the previous knowledge of different modeling situations to teach geometric sequences.  See attached Geometric Sequences Notes for a description of how this is done.  I tie all of the modeling (MP#6) that has been done in  previous lessons to this lesson, including incorporating the same problem with the penny drop from the Grand Canyon from the arithmetic series lesson , making it into a geometric sequence problem by adding up the time it takes to reach the bottom.

Geometric Sequence Round Table

25 minutes

During this section students will work collaboratively in groups of 4-6.  During this round table activity I hand out the attached document and have them individually fill out the visual representation of the first problem.   Once complete with this section only the students put their pencils down until everyone in their group is done and then rotates their papers.  They then critique and/or agree with the representation and initial that they have reviewed and agreed with the original or revised version (MP#3).  They then work on the new piece of paper they have in front of them to work on the next section.  This process repeats over and over until the paper is complete.  Explanation of this process can be found here. I ask  questions that can be asked to encourage appropriate conversation and discourse as follows:

"What is this an appropriate picture/table/graph for this problem?"

"What other way can you represent this pictorally/graphically/etc...?"

"What does this rule represent to you?" 

"What is changing each time and by how much?"

"What are the factors involved and what which is dependent on the other?"

"Where would this data start on graph?"

"Does this result make sense?"

 

Once complete, the teacher is called over to check the activity.  If incorrect, the teacher can give the same scaffolding questions as above.  The teacher is also walking around during this acitivyt checking that these questions are being asked.

"What does this rule represent to you?" 

"What is changing each time and by how much?"

"What are the factors involved and what which is dependent on the other?"

"Where would this data start on graph?"

"Do these pieces make sense together?"

Home Work

This home work is designed to expand the students ability to represent geometric series in different models.