Lesson Plan: Geometric Sequences by 3SKfe0tk

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									Lesson Plan: Geometric Sequences
Objective:

         -Students will become familiar with geometric sequences through investigations
         with fractals.

Prerequisites:

         -students will be familiar with sequence notation
         -students will have had some experience with fractals
         -students will have at least average skills with Geometer Sketchpad.

Introduction:
       Consider the following sequences, what relationship do you observe between
successive terms of these sequences?

                    3n
1,3,9,27,81,243,...,    ,...
                     3
 2,4, 8,16, 32,64,...,(  2) n ,...
                       n
1 1 1 1       1
 , , , ,...,   ,...
2 4 8 16      2

Sequences with this property are called geometric sequences. The ratio between
consecutive terms is called the common ratio.

Review of Fractals :

    1.       What is a fractal?
    2.       Using the National Archive of Manipulatives students will review the properties of fractals

Investigation of Geometric Sequences:

Question: Is there a shape that has an infinite perimeter, and an area of
zero?
To answer this question, students will do investigation 1 and 2, found at the end of the
lesson plan?

-Students will find the common ratio of a sequence.
-Students will find the nth term of a sequence.
-Students will determine if sequence is geometric.
-Students will investigate sequence as n approaches infinity.
The Sum of Geometric Sequence

Let    S n  a1  a1r  a1r 2  ... a1r n  2  a1r n 1
and rS n  a1r  a1r 2  a1r 3  ... a1r n 1  a1r n
by    finding the difference we have:
S n  rS n  a1  a1r n     so
S n (1  r )  a1 (1  r ) and
                       n


          1 rn 
S n  a1            r  1.
          1 r 

Investigation 3:           The sides of a square are 16 inches in length. A new square is
                           formed by connecting the midpoints of the sides of the original
                           square, and two of the triangles are shaded (See figure in GSP
                           applet). If this process is repeated 9 times, determine the total
                           area of the shaded region. What if the process is repeated an
                           infinite number of times, what is the total area of the shaded
                           region?

-Students will use all information they have learned from investigations 1 & 2, and in
                        addition will need to sum their sequence in investigation 3.
-Students will also investigate the sum of an infinite geometric sequence in investigation
                        3.
Investigation 1

    1.       Construct triangle ABC, with midpoint D, E, F.
                     B




         F                   E




A
                 D
                                 C




    2.       Construct the interior of triangle ABC.
                             B




             F                       E




     A
                         D
                                         C

    3.       Iterate the construction of triangle ABC to triangle AFD, and then
             to triangles FBE, and DEC (Hint: Use Add New Map in structure
             menu to iterate all triangles at once, and choose Final Iteration
             Only from the display menu). Once this is done, hide the original
             interior. Your image should look like the following after
             1iteration:
                             B




             F                       E




     A
                         D
                                         C
 4.     Use the + and – keys to explore other iterations of the Sierpinski
        Gasket.

 5.     Suppose a stage 0 gasket (a single triangle) has area 1. What
        would the shaded area of stage 1 (1 iteration) be? Justify.

 6.     Fill in the rest of the table:

Stage         0            1             2      3           4
Area          1

 7.     As the stages increase, what happens to the shaded area?

 8.     Does the area at each of the stages give us a geometric sequence?
        If so what is the common ratio, and what would the area of a stage
        n gasket be?

 9.     Using Geometer Sketchpad, define the coordinate system and plot
        the function between A(n) (area of shaded region at stage n) and n.
        What do you notice about A(n) as n gets really large? What would
        the area of the gasket be at stage infinity?

 10.    Suppose a stage 0 gasket has a perimeter of 3, fill in the following
        table and determine what the perimeter will be at stage infinity?

Stage         0            1             2      3           4           n
Area          3

 Investigation 2

 1. Using the GSP file Geometric Sequences. Click on the page marked
    square gasket.

 2. Find the geometric sequence for its area using each of the stages, and
    write an expression for the nth term. Suppose the area of the square is
    1 at stage 0.

 3. Find the geometric sequence for its perimeter using each of the stages,
    and write an expression for the nth term. Suppose the perimeter of the
    square is 4 at stage 0.
4. At stage infinity, can you draw the same conclusions as you did for
   the triangle?

5. What does the triangle and square gasket sequence for area have in
   common that would make it behave the way it did at stage infinity?
   What about the sequence for perimeter? Using these observations,
   make a conjecture about the behavior of geometric sequences at stage
   infinity? Be ready to defend this conjecture?


Investigation 3

The sides of a square are 16 inches in length. A new square is formed by
connecting the midpoints of the sides of the original square, and two of
the triangles are shaded (See GSP page extension). If this process is
repeated 9 more times, determine the total area of the shaded region.
What is the area of the shaded region at stage infinity?

1. Open the GSP file Geometric Sequences, and open the page labeled
   extension.

2. If I create a sequence of total shaded area for each stage, is it
   geometric? Why or why not?

3. Fill in the following table:

           Stage                     1          2         3        4           n
Area of new shaded triangles

3. Find the total sum of shaded region at stage 9 using the sum of a finite
                                              1 rn 
   geometric sequence formula:       Sn  a1           r  1 . Verify your
                                              1 r 
   result by adding the first 9 terms.

4. What is the area at stage infinity? Verify this with an algebraic
   model?

								
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