Arithmetic Series

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

Big Idea

Arithmetic sums can be used for estimation. How many seats are in a stadium? How many tiles in a work of art? What's the total profit made by a business?


15 minutes

As a warm-up, students complete Warm-up Sigma Notation and Arithmetic Sums which is a set of five arithmetic sums written in summation notation.  The task is to expand and evaluate these sums.   Later in today's lesson, we will connect summation notation to the arithmetic series formula, so it is helpful to remind students how summation notation works.

Group Discussion and Note Taking

20 minutes

In the previous day's activity, Series Exploration, students derived and learned to use the formula for the partial sum of an arithmetic sequence.  Specifically, they saw that that the partial sum of an arithmetic sequence can be found by averaging the first and last terms and multiplying the result by the number of terms summed.  

In this portion of the lesson, I provide some direct instruction on arithmetic series and their application.  First, I ask students to write the arithmetic series formula in their notebooks along with a few examples of how to use it.  As I present this information, I keep students engaged using Cold Calling when necessary to guarantee participation in the discussion.  As I record the notes on the board, I ask students for ideas on how to proceed and encourage them to ask questions on anything that doesn't make sense.

I show students that they now have three methods of representing a sum of the terms of an arithmetic sequence: the terms themselves joined by addition signs, summation notation, and the arithmetic series formulas.[MP7]  We take note of a few partial sums written with all three notations.  

Students need to be very flexible with these formulas, so I show them several examples of using the formula with different given information.  The most challenging problem type is the one in which students are given the value of an arithmetic sum and they need to determine the number of terms added.  This type of problem can be difficult because solving it involves solving a quadratic equation.  Students are taught to solve quadratic equations in Algebra 1, but it is often the case that students did not master it in that course.  Here I remind them that all quadratic equations can be solved with the Quadratic Formula, and that we will review other methods in the next unit (Polynomials).



Individual/Group Practice

40 minutes

In Algebraic Work with Partial Arithmetic Sums, students work independently to solve problems involving arithmetic sums and translate among verbal, numeric, and algebraic representations of arithmetic sums. [MP1]  I set an overhead timer for 20 minutes and ask students to work quietly and independently during this time.  After this, they have 10 minutes to pair up and compare answers.  During this time, students are welcome to revise their work and ask for help from other groups or me.  

After students have worked independently and compared answers in pairs, I project solutions to Algebraic Work with Partial Arithmetic Sums and take any questions that did not get resolved in their partner discussion.  

Exit Ticket and Homework Assignment

15 minutes

To gauge how well students are able to use the arithmetic series formula, I ask them to complete  Exit Ticket Arithmetic Sums that includes two arithmetic sum problems.  The first is a straightforward application of the formula, and the second involves solving for the number of terms.  

The homework for the evening is Applications of Partial Arithmetic Sums, which is a collection of scenarios which can be modeled with arithmetic sums.  [MP4]  I tell students to complete all questions on the worksheet and check their answers on Edmodo before the next class.