# The Recursive Process with Arithmetic Sequences

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## Objective

SWBAT write a recursive formula from an arithmetic sequence, graph it, and describe the rate of change verbally and in writing.

#### Big Idea

This lesson takes students from the simple concrete problem of seating 4 people around a square table to the more abstract problem of finding the nth term.

## Warm up

10 minutes

Today's Warm Up  is intended to take about 10 minutes for students to complete and for me to review with the class.  I use this Warm Up to introduce students to an Arithmetic Sequence. A simple table problem provides us with a concrete example to discuss. My students can easily see a pattern of "adding two."

In this problem, students use Mathematical Practice 1 to persevere and select a method to solve the problem without guidance of where to start. I also use Mathematical Practice 4 by using the table problem as a concrete example to model the mathematics of finding the recursive formula and the nth term.

An important part of today's warmup is the discussion of the students' strategies. Here are some examples:

• Student 1 recognized the pattern was increasing by two each time and continued the pattern to 17, the correct number of tables to seat 36 people.
• Student 2 used the formula 2x + 2.
• A third student incorrectly reasoned that since 9 tables seated 18 people, then 18 tables would seat 36 people.

After several different methods have been shared by students, I plan to build on the content of the presentations to review the vocabulary for this lesson.  I model this in the video in the Guided Practice segment of the lesson.

## Guided Practice

15 minutes

At this point in the unit I feel that my students are not ready to work independently with arithmetic sequences. Although my students have previously worked with slope and y-intercept, the vocabulary and the notation for working with recursion is new.

To move the class towards working independently, I will lead a Guided Practice session. In the Guided Practice, students repeatedly use the formula (MP7) to find the nth term of an Arithmetic Sequence.  The students learn to use it to solve one unknown of the four different parts of the formula. The given information in the problems allow the students to substitute for three of the four parts of the formula.  The four parts of the formula are:

1. The first term of the sequence
2. The common difference of the sequence
3. The nth term of the sequence
4. The number of terms minus one

I provide students with two different formulas to find the nth term of an Arithmetic Sequence.  I explain the difference in the two formulas in the video below.

## Independent Practice

15 minutes

After the Guided Practice, I hand students an Independent Practice sheet to complete on their own. I will walk around the room to monitor progress as students are working.  I have provided a mix of practice problems for students to complete.  There are four different types of problems for students to solve:

1. Solving for the nth term
2. Solving for the Common Difference
3. Solving for n
4. Completing a sequence given the first and last term.

## Exit Slip

10 minutes

I often use an Exit Slip often as a formative assessment to check for student understanding of the lesson. However, in this lesson, I use the Exit Slip as an extension.  I present students with a grocery store problem in which they are to find the sum of an arithmetic sequence.

The grocery store problem requires students to find the number of cans needed for a triangular display.  Students are provided with the first term and the 21st term of the sequence, and asked to identify the total number of cans needed to create the display.

I expect that my students will create the sequence and then add all of the rows of cans together. However, I will be looking for different methods. I want to see if any students come up with the formula for the Sum of an Arithmetic Sequence.  If students do not come up with the formula, I will present the formula by comparing it to their method.  This is a good problem to show students a simple application of finding the sum of an Arithmetic Sequence.