Comparing Geometric and Arithmetic Sequences

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Objective

SWBAT identify linear and exponential function based on a table of values, a context or a sequence.

Big Idea

The values in a table that represents a function can give insight into the type of function being represented.

Launch

10 minutes

Students will work with their partners on this opening activity:

exponential_and_linear_tables_open 

I plan to give each pair of students a copy of the cards (these can be cut out as well) and the graphic organizer, exponential_and_linear_tables_organizer.  Students will be matching each situation card with the appropriate table, difference/ratio, function, and graph.  This matching activity will help students to see a connection among all of these different representations of the same function.  

Practice

25 minutes

This practice assignment exponential_and_linear_tables_practice will help students to make a connection between situations and tables with sequences.  In each case, the students will have to reason about the numbers in the sequence in order to determine if the situation is arithmetic or geometric (linear or exponential).

Students will work with their partners on this practice assignment.  I will have an answer key posted in the classroom so that students can independently monitor their understanding or get a hint if they are stuck on a particular question.  I encourage students to be precise with their formulas for the nth term.  In each case, students should choose a number from the sequence and determine if their formula is correct.  For example, in Question 6, students should derive the formula a(n)=2(2)^(n-1).  To ensure this is correct, students could substitute n=4 to determine if they get an output of 16.  

Question #12 may cause some difficulty for students.  Some will want to write the formula for the sequence as a(n)=6000(1.1)^n.  Others may write it as a(n)=6000(1.1)^(n-1).  In this case, let students make their argument for both cases.  It will be very interesting to see their rationale for why each formula would be the correct choice.  If students are looking at the situation as having $6000 in the first year then a(n) = 6000(1.1)^(n-1) makes the most sense.  If students are thinking about the situation from a function perspective (Cami has $6000 initially at time = 0) then encourage them to write the formula as a function f(t) = 6000(1.1)^t.

Closure

5 minutes

This Closing Activity incorporates a writing component so that I can very clear see how my students are thinking.  In each question, students need to reason quantitatively in order to find the missing values in each table (either additively or multiplicatively) (MP2).  The Bonus Question requires students to think outside the box.  Encourage all students to try this question.  I encourage you to try it too!  

 

Answer: If the common ratio is changed to r = -2 then the first and fifth terms would still be 6 and 96 respectively.