Geometric and Arithmetic Sequences

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Objective

SWBAT create an explicit formula for a sequence of numbers.

Big Idea

Students will model arithmetic and geometric sequences by identify a common difference or ratio.

Do-Now

10 minutes

Students will complete the Do-Now on Slide One during the first five minutes of today's lesson. I will then ask four volunteers to write their answers on the board, and, to explain their responses with the class.

Next, a student will read today's lesson objective to the class: SWBAT create an explicit formula for a sequence of numbers. Following this, I will ask the class to define the word "sequence".  I will spend a few minutes introducing artihmetic and geometric sequences, then I'll ask the class to share ideas about the difference between the two types we will look at today.

Introduction

10 minutes

In today's lesson the class will briefly explore two different types of sequences and use this knowledge to calculate missing values in sequences.  We will only spend one day on this topic, so the goal of today's class is to introduce the thinking processes used in working these types of problems. We will employ these processes further in our next unit.

Using this Presentation and these Guided Notes, I will begin instruction by asking a student volunteer to read the prompt on Slide 3 and Slide 4. Next, students will work in pairs for five minutes to calculate which paycheck option is the best one to choose and why. I will circulate around the room as students are working. I will push my students to calculate as many weeks as they possibly can by hand. 

When we reconvene as a whole group I will ask a few pairs of students to share their responses. I will ask students to describe the practicality of using a step-by-step method to explore this task. And, since I expect some students will be off by a few cents, we'll discuss issues of rounding and precision.

Guided Notes + Practice

30 minutes

Students will continue to take Notes while following along with this Presentation

We will generate a class definition for Arithmetic and Geometric sequence on Slide 7 and Slide 9, and then complete the example problems on Slides 10 - 13 as a whole group. 

Slide 10: I will ask students to use the example problems to create a formula that can be used to calculate any arithmetic sequence. The majority of the class will be able to derive a formula, but may struggle during this task. I will allow the students to struggle through this portion, with minimal support from me. If a group derives a formula, I will ask them to prove that their formula is correct with a new set of numbers..

After 5-10 minutes students are able to create a formula, but may have used algebraic variables or words to describe what they are trying to say. As a whole group, we will translate the formula into one that we can use for all arithmetic sequences: tn  =  t1  +  (n - 1)d

Slide 12: I will allow students to attempt the deriving of a formula for a Geometric sequence, but the majority will not be able to. This is okay, as the arithmetic sequence is more aligned to our upcoming unit, and the Geometric sequence formula does not rely on intuition as much as the other one does. It is helpful if students have calculators during this portion of class.

With a lot of guidance and teacher probing, we will arrive with a formula for Geometric sequences:   tn = t1 . r(n - 1) 

Partner Practice

20 minutes

The class will continue to explore sequences as we complete the following two partner activities. For these activities, I think it is best to group my students into homogeneous ability pairs. The activities are not dependent on each other, and can be completed in any order. 

Activity A: Matching Activity - Students will sort the cards into 10 different groups. Each group will consist of 4 different types of cards: 5 terms, type of sequence, explicit formula, and the 27th term. Students may use a calculator during this activity.

Teacher Note: The Matching Activity cards must be cut up and shuffled prior to student use. 

 

Activity B: Sequence Task Problems - Students will answer the four questions using what they have learned about arithmetic and geometric sequences.

Closing

10 minutes

We will end today's lesson with a whole group Exit Card. Students will use Promethean clickers to answer the questions on Slides 15 - 17 of today's presentation. After each question, a student volunteer will come up to the board to explain how they got to their answer.

To close, I will ask students to Turn and Talk with a neighbor. I will prompt them to discuss the ways in which sequences can be used to solve problems in an Algebra class.