Connecting Polynomials to Sequences

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Objective

SWBAT write explicit polynomial equations to describe sequence patterns.

Big Idea

There are connections between sequences and polynomials. Sometimes the explicit form of a sequence is a polynomial function.

Warm-Up

20 minutes

Students will complete Manipulating Polynomial Pretest from the MAP Manipulating Polynomials task.  This is an individual, 15-minute pre-assessment that helps me gauge what type of support this group of students will need in completing the day's work.  

When students have completed the pre-assessment, I ask them to take out the homework from the previous night and complete the rest of the factoring problems.  I project the solutions to these factoring problems before moving into the next part of the lesson.  

As students complete the factoring problems, I look over the responses to the pre-assessment and make note of misconceptions and potential problem areas.  Specifically, I need to know if students are able to abstract a given situation, represent it symbolically and manipulate the representing symbols [MP2]. If they are very weak in this, I will add in a bit more scaffolding in the next section of this lesson.

Whole Group Introduction

15 minutes

Manipulating Polynomials is a MARS task that connects visual sequences to polynomial expressions.  It is a good entry-level modeling activity because the situation modeled is easy for students to grasp [MP4, MP2].

The first part of the lesson is an introduction in which I show Dot Patterns to Project using a projector.  Each student uses a small write board at their desk and a dry-erase marker.  We work a few examples of translating dot patterns to algebraic expressions and discuss how the same expression can be written in multiple ways to reveal different properties of the pattern [MP7]. The ability to abstract a situation and represent it symbolically is an important part of the group work that follows so I take time to provide some instruction in this if necessary.

Group Work

50 minutes

Students will take the rest of the class period to complete the matching activity outlined in the Manipulating Polynomials task. Students work with the Dot Sequence Card Set to match dot patterns to polynomials that represent the patterns.  I arrange students in groups of two or three for this activity.  In forming these groups, I am primarily looking for productive groups, in which students are working with people they like and can also stay on task with. 

The matching part of this activity is more complicated that it originally seems because there are various starting points provided in the polynomial expressions hand out.  I let students know that this part might be challenging and encourage perseverance [MP1].

Students need the following materials for this part of the activity:

  • A Dot Sequence Card Set. I like to copy these on colored cardstock and laminate them so that I can use them from year to year.  See the image Laminated Sequence Cards for details.
  • A Matching Record Sheet  which is where students record which dot pattern matches each polynomial.  
  • After students have attempted to match up the patterns and polynomials, one person from each group visits another table to see what decisions another group has made.  This person then returns to their own group to report what they have learned [MP3]. This serves to jump start any group that was stuck and provide another perspective to groups that thought they were done.

Throughout the activity, I use the 3-Cup System to help students feel supported and to determine where my assistance is needed most [MP1].

Wrap-Up and Assignment

10 minutes

Students return materials to the cabinet and give me their record sheet from the group work.   For homework, each student will make their own black and white dot sequence and write the polynomials that describe the pattern.  This will be recorded on Dot Sequence Blank Cards.