Geometric Series

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Objective

SWBAT derive the formula for the sum of the first n terms of a geometric sequence and use this formula flexibly to solve problems.

Big Idea

The first proof in Algebra 2! Students learn to derive the formula for the sum of the first n terms of a finite geometric sequence.

Warm Up and Homework Discussion

20 minutes

In Warm-up: Sigma Notation Geometric Series, I present students with 3 geometric sums to evaluate.  The partial sums are presented in sigma notation and I ask that students first write out each term of the sum before evaluating.  This allows them to practice with sigma notation as they begin to consider the most efficient way to add terms of a geometric sequence.  

I display the solutions to the previous night's homework which had students using the the partial arithmetic sum formula. As I circulate the room I check students' work using my homework rubric.

When I have finished checking homework, I project the solutions to the warm-up on the board and for questions on the homework or the warm-up.  I draw students' attention to the fact that the formula for arithmetic sums does not help us add up the terms of a geometric sequence.  If students are not convinced of this, I do an example of erroneously using the arithmetic series formula on a geometric sum to show that it does not yield the correct answer.

Direct Instruction and Discussion

30 minutes

The CCSS asks Algebra 2 students to derive only two formulas: the Quadratic Formula and the formula for summing the terms of a finite geometric series.  In my experience, the Quadratic Formula is the easier of the two because it is more familiar to students.  

I'll remind the class of Gauss' Story and method that led a formula for the sum of the first n terms of an arithmetic series, telling them we will use a similar technique for geometric series.[MP7] This topic can be somewhat difficult for students. In my video, Deriving the Geometric Series Formula, I share how I present this in the class.

Students take notes on the method and participate in the discussion during the lesson presentation. [MP3] After as a whole class exercise we complete examples of using the formula to find the sum.  As with arithmetic series, we spend some time using the formula "backwards" finding the initial term, the common ratio or the number of terms.  

 

Practicing the Proof

30 minutes

To practice the proof steps, students use the online Geometric Series Proof Sorter from NRich Maths assembling a logical proof. [MP3]  I chose this self-correcting activity to provide students immediate feedback on whether their proof makes sense.  

Often I have fewer laptops available than I have students, so I have students work in two "centers" to complete this task.  For half the room I have students open laptops and retrieve the link from our Edmodo class page.  They take about five minutes to get the applet up and then another 10 minutes on the activity.  I use an overhead timer to let them know how much time they have left.  In other half of the room, I have students practice problems from the Finite Geometric Series worksheet.  When time is up, the first group leaves the laptops open and they change positions.  The second group has 10 minutes for the activity and must use the final 5 minutes to put the laptops away properly. I organizing laptop use this way to cut down on transitioning time. 

Wrap-Up and Homework

10 minutes

Students will complete a Series Formulas Exit Ticket so I have a quick read on whether they are able to select and use the appropriate series formula.  

The homework assignment is to complete the problem set that they started in class.  This set includes a wide variety of problems that require students to use the geometric series formula flexibly and sometimes incorporate the use of sequence formulas as one step in the process.