Arithmetic Sequences: Growing Dots

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Objective

SWBAT determine the growth rate of a pattern and represent the rate graphically.

Big Idea

Students will use formulas to represent growth rate in various sequences.

Open

3 minutes

The YouTube video below provides a visual representation of the growth of bacteria in a petri dish (http://www.youtube.com/watch?v=XLmk0zYIFjE):

 

While bacteria growth is exponential in nature, this video will get students thinking about something growing with respect to time.  You can then tell students that they are going to be studying some simpler growing patterns during today's class.

Launch

7 minutes

This Launch (growing_dots_launch.doc) will get students thinking about how the dot pattern is growing in each scenario.  Have students first look at the dot pattern individually and then do a Think-Pair-Share with their partner around what they notice about the growth (MP7).  Students will usually define the growth in a recursive way in Question 1 (growing by 4 each second).  When they get to Question 2, they can use their recursive concept to write an explicit formula 4x+1. Some students will have time to continue to investigate the pattern on the back of this paper, some will not.  If students are able to look at the second pattern they will see that the explicit formula will be 5x+1.

IMPORTANT: At this point, pull students back together to discuss the formula for the first pattern.  Students will probably come up with two different answers y=4x+1 or y=4x-3.  The latter formula is based on the misconception that the first dot in the sequence corresponds to the first term instead of the "0th" term or y-intercept.  Use questioning to address this misconception now so that students can see a connection between the number of seconds (input) and the number of dots (output).  Only the formula y=4x+1 will give the appropriate output based on the inputs of {0, 1, and 2}.

Investigation

25 minutes

The goal of this investigation is to have students begin to connect sequences (typically written horizontally) with tables (often written vertically) and graphs in the coordinate plane.  This connection back to something familiar will help students to make sense out of the roles sequences play in understanding linear functions.

First give students the growing dots investigation worksheet (growing_dots_investigation.pdf) and have them examine the three different patterns.  Students should try to find the differences and similarities between the three patterns.  Have students turn and talk with their partner about what they notice.  Have a few partnerships share out with the class.  Most times, at least one group will notice that the times are different for each pattern (even through the growth rate is the same).  

Once students seem like they have a decent grasp of the pattern (do not be too helpful) have them begin on the investigation (growing_dots_investigation.doc).  There is also a more scaffolded version of the investigation which will help students keep their work more organized (growing_dots_investigation_scaffolded.doc)

As students work though the investigation, they will need to make sense of how to represent the sequences in multiple ways (such as in tables and graphs) (MP1).  At the end of the investigation, students will make note of the fact that the graphs of all three sequence patterns are parallel because they have the same rate of change but have different starting points (MP2).

Closure

5 minutes

This Ticket out the Door (growing_dots_close.pdf) will not only be valuable for you to see how students are thinking but also beneficial for the student.  Anytime students can try to name the things that they are understanding and having difficulty with it pushes them to think more critically. This introspection helps the students to self-assess.  I also put a caveat on Question 2.  If students feel that there is nothing they are confused about, then they should try to write about a misconception of one of their classmates.