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					                                           Teacher Notes
                        Exponential Functions : Exponential Growth & Decay

Topic: This lesson introduces students to the concept of exponential growth and decay.

Objective: Given a geometric pattern the student will derive an exponential equation by analyzing data
for common ratios.

Time Table: This lesson will need two 45 minute periods or one 90 minute period.

TEKS Focus: A.11C Analyze data and represent situations involving exponential growth
            and decay using concrete models, tables, graphs, or algebraic methods.

Materials:      Lesson Packets (provided)
                Doubling sheets (provided)
                Halving sheets (provided)
                Letter sized paper
                Colored pencils to color doubling and halving sheets (optional)
                Graphing calculators
                T I Interactive and cable (optional)
                Printer (optional)

Lesson Overview: There are 4 major activities : Doubling Method, Halving Method, Folding paper and
Page Area. Students use the doubling sheet on Page 5 to derive an exponential growth function (y = 2x)
                                                                                1
and the halving sheet on Page 6 to derive an exponential decay function (y = ( ) x ). More exponential
                                                                                2
growth and decay functions can be derived from the folding paper and page area activities. A general
exponential representation y = abx can be derived from the activities. Students summarize and describe
each parameter from their understanding. Graphic representations of exponential growth and decay are
                                                                   1
compared using the previously derived functions, y = 2x and y = ( ) x .
                                                                   2
Grouping: The doubling activity should be led by the teacher and the students can work the rest of the
lesson in groups.

Procedures: Teacher clarifies the objectives of the lesson and uses them as part of the informal
assessment (check for understanding) during the lesson. Teacher leads off by coloring squares on the
doubling sheet and filling in the doubling table. Students should be developing an exponential growth
pattern and able to derive its equation. As a class, discuss the Zero Power on Page 1. If students feel
comfortable with the doubling activity, let students work in groups on the halving, folding paper and
page area activities. As a class, discuss the summary on page 3 by asking students to brainstorm possible
                                                                                         1
descriptions of each parameter: y, x, a and b. Have students graph y = 2x and y = ( ) x , sketch or
                                                                                         2
capture their graphs using TI Interactive and paste/staple them on the provided space on Page 4.
Compare and discuss the characteristics of both graphs. Practice problems can be answered as a class or
in groups.

Homework: Interating on a Plane. This is an integration of geometry and exponential functions.
Students build patterns for numbers of squares and surface areas after iterations. They are to derive
exponential functions and graph the functions.
   MMA Name: _____KEY___________ Date: _________ Period:______




Objectives: To derive exponential functions using the doubling and halving methods.
            To differentiate between exponential growth and decay by comparing their
             graphs and algebraic expressions.



Use the doubling sheet on page 5 to fill in the doubling table.
        Number of     Number of Squares                                Growth
                                                 Exponent Form
       Doubles (d)            (n)                                      Factor
            0                  1                       20                -
            1                  2                      21                 2
            2                  4                      22                 2
            3                  8                      23                 2
            4                  16                     24                 2
            5                  32                     25                 2
            d                                         2d                 2

          The number of squares (n) is a function of the number of doublings (d).

     n = ……… 2d …….…. or y = ……… 2x ………… or f(x) = ……… 2x ………..

   When the independent variable (x) is the exponent, the function is an
       ‘exponential function’. The doubling table shows an exponential growth.



           Number of Doubles        Number of Squares       Exponent Form
                     0                      1                     20

                                    Any number to the ………………… equals ………..

                                      You are an heir to ($1,000,000)0. How much will
                                      you inherit?    …………$1……………
Use the halving sheet on page 6 to fill in the halving table.

        Number of         Number of Squares                           Decay
                                                  Exponent Form
        Halves (h)              (n)                                   Factor
             0                   1                        ( 1 )0         -
                                                             2
             1                   1
                                     2                    ( 1 )1         1
                                                                             2
                                                             2
             2                   1
                                     4                    ( 1 )2         1
                                                                             2
                                                             2
             3                   1
                                     8                    ( 1 )3         1
                                                                             2
                                                             2
             4                  1
                                  16                      ( 1 )4         1
                                                                             2
                                                             2
             5                  1
                                  32                      ( 1 )5         1
                                                                             2
                                                             2
             h                                            ( 1 )h         1
                                                                             2
                                                             2
            The number of squares (n) is a function of the number of halving (h).
                      h                              h                           h
               1                           1                         1
      n = ……   ….…. or y = ………   ……… or f(x) = ……   ……
               2                           2                         2
   When the independent variable (x) is the exponent, the function is an
         ‘exponential function’. The halving table shows an exponential decay.



   Use a piece of letter sized paper for this activity.
        Number of         Number of Leaves       Number of Pages
                                                                   Exponent Form
         folds (f)             (L)                    (p)
             0                   1                     2                2 20
             1                   2                          4           2 21
             2                   4                          8           2 22
             3                   8                         16           2 23
             4                   16                        32           2 24
             5                   32                        64           2 25
             f                                                          2 2f

   number of leaves, L = 2 f
                           f                    f+1                   x+1
 number of pages, p = 2(2              ) =        2              OR y = 2




    Let’s start with a paper size of 8 inch wide and 10 inch long.

Number of                                                           Page Area (A)            Exponent
                  Page Width (in)              Page Length (in)
 folds (f)                                                              (in2)                 Form
     0                      8                         10                 80                  80( 1 2 )0
    1                       4                         10                    40                 80( 1 )1
                                                                                                    2
    2                       2                         10                    20                 80( 1 ) 2
                                                                                                    2
    3                       1                         10                    10                 80( 1 )3
                                                                                                    2
    4                      0.5                        10                       5               80( 1 ) 4
                                                                                                    2
    5                      0.25                       10                   2.5                 80( 1 )5
                                                                                                    2
    f                                                                                          80( 1 )f
                                                                                                    2

                                       1
                                           f                           1 x
             Page Area, A = 88( )                     or       y = …88( )……………..
                                       2                                   2




                                                                                   - independent variable
                                                                                   - domain
 - dependent variable                                                              - input
 - range                                                                           - exponent
 - amount after x events                                                           - number of events




              -   coefficient                              - base
              -   original amount                          - proportional growth if b > 1
              -   starting amount                          - decay factor if b < 1




 Exponential Growth:              b is (less than or greater than) 1.
 Exponential Decay:               b is (less than or greater than) 1.
Graph y  2x and use TI-Interactive to capture your screen. Print your graph and paste it
                                              1
in the space provided below. Repeat with y  ( ) x .
                                              2


  Graph of exponential growth                  Graph of exponential decay




  If do not have TI Interactive, have student hand sketch the graphs.


   PRACTICE PROBLEMS

   Evaluate and predict whether each of the following functions is an exponential growth
   or decay.
                                                          1
   a. y  3x when x = 4                           b. y  ( ) x when x = 4
                                                          3
       y = …81 …….                                   y = …… 0.012….

   growth vs. decay ……… growth ….                     growth vs. decay ……decay…….

   c. f (x)  2x  1 when x = 3                    d. f (x)  2 x 1 when x = 3
       y = ……9………….                                    y = ……16………….

   growth vs. decay …… growth …….                     growth vs. decay …… growth ….

   e. P  ( 2 )q  1 when q = 2                    f. f (x)  ( 2 )x 1 when x = 2
             3                                                   3
      y = …1.44………….                                    y = ………0.296……….

   growth vs. decay …… decay …….                     growth vs. decay … decay ……….

                                  Doubling Sheet
                   (This answer shows one of the many ways of successive doublings.)

No. of Doublings




                                                                                       No. of Squares
                                                                                       32
      5




                                                                                       16
       4




                                                                                       8
   3




                                                                                       4
      2




  1                                                                                    2



                                                                                       1
Halving Sheet

(This answer shows one of the many ways of successive halvings.)




       0th halving                                   1st halving




        2nd halving                                 3rd halving




        4th halving                                 5th halving
MMA                                  Name:      Solution
Exp Growth and Decay                 Period: ___ Date: ________
               Exponential Growth and Decay – Iterating on a Plane




1. If the area of the original square (stage 0) is 9 units2, what is the area of the
new figure at stage 1? 5

2. The area of the stage 1 figure is what fraction of the area of the stage 0
figure? 5/9

3. When the figure goes from stage 0 to stage 1, does the area increase,
decrease, or stay the same? ______________________

4. What is the perimeter of the stage 0 square? 12 units

5. What is the perimeter of the stage 1 figure? 20 units

6. When the figure goes from stage 0 to stage 1, does the perimeter increase,
decrease, or stay the same? ______________________

An iteration rule has three distinct parts: a reduction, a replication, and a
rebuilding. Each time the rule is applied, the result is a new and more complex
figure.
 Fill in the tables. Graph your results. Remember to label your axes
 and scale.

 Number of Squares
                                              Number of
 Stage             Process
                                               Squares
    0                      50            1

         5                51             5
    1
    2    5(5)             52             25
    3    5(5)(5)          53             125
    4    5(5)(5)(5)       54             625
    n                     5n             5n



From stage to stage, the number of squares increases/decreases by a factor of 5.

 Total area (units2)

 Stage                Process                  Total area
                                    5
    0                           9( )0          9
                                   9
           5                        5
    1    9( )                    9( )1         5
           9                        9
           5 5                     5 2         25
    2    9( )( )                9( )
           9 9                     9            9
           5 5 5                   5           125
         9( )( )( )             9( )3
    3      9 9 9                   9            81
           5 5 5 5                 5           625
    4    9( )( )( )( )          9( )4
           9 9 9 9                 9           819
           5
    n    9( )n
           9


From stage to stage, the total area increases/decreases by a factor of 5/9 .
   MMA Name: ________________                       Date: _________ Period:______



Objectives: To derive exponential functions using the doubling and halving methods.
            To differentiate between exponential growth and decay by comparing their
            graphs and algebraic expressions.



Use the doubling sheet on page 5 to fill in the doubling table.
       Number of         Number of Squares                                 Growth
                                                Exponent Form
       Doubles (d)             (n)                                         Factor
           0                    1                                            -




             d

           The number of squares (n) is a function of the number of doublings (d).
               Which can be written using different notation but are the same
             n=                 or      y=                 or     f(x) =

            When the independent variable (x) is the exponent, the function is an
         ‘exponential function’. The doubling table shows an exponential growth.


   Look at the first row in the table above.

            Number of Doubles        Number of Squares       Exponent Form
                     0

                                     Any number to the            power equals

                                        You are an heir to ($1,000,000)0. How much will
                                        you inherit?
                                                     $
Use the halving sheet on page 6 to fill in the halving table.

        Number of       Number of Squares                                   Decay
                                                  Exponent Form
        Halves (h)            (n)                                           Factor
            0                  1                                              -




             h
            The number of squares (n) is a function of the number of halving (h).

             n=                  or     y=                  or     f(x) =

             When the independent variable (x) is the exponent, the function is an
           ‘exponential function’. The halving table shows an exponential decay.




   Use a piece of letter sized paper for this activity.
        Number of       Number of Leaves         Number of Pages
                                                                     Exponent Form
         folds (f)           (L)                      (p)
             0                 1                       2




             f

   number of leaves, L = 2

   number of pages, p = 2(2           ) =    2             or         y=2
 Let’s start with a paper size of 8 inch wide and 10 inch long.
 Each time you fold only fold the width (shorter side) in half.



Number of                                                 Page Area (A)   Exponent
                Page Width (in)       Page Length (in)
 folds (f)                                                    (in2)        Form
     0                 8                     10                80         80( 1 2 )0




    f

                                  1
             Page Area, A = 80( )             or     y = ………………………..
                                  2




 Exponential Growth:       b is (less than or greater than) 1.
 Exponential Decay:        b is (less than or greater than) 1.
Graph y  2x and use TI-Interactive to capture your screen (or sketch below). Print your
                                                                 1
graph and paste it in the space provided below. Repeat with y  ( ) x .
                                                                 2


  Graph of exponential growth                        Graph of exponential decay




   PRACTICE PROBLEMS

   Evaluate and predict whether each of the following functions is an exponential growth
   or decay.
                                                          1
   a. y  3x when x = 4                           b. y  ( ) x when x = 4
                                                          3
       y = ……………….                                    y = ……………….

   growth vs. decay ……………….                       growth vs. decay ……………….

   c. f (x)  2x  1 when x = 3                   d. f (x)  2 x 1 when x = 3
       y = ……………….                                    y = ……………….

   growth vs. decay ……………….                       growth vs. decay ……………….

   e. P  ( 2 )q  1 when q = 2                   f. f (x)  ( 2 )x 1 when x = 2
             3                                                  3
      y = ……………….                                      y = ……………….

   growth vs. decay ……………….                       growth vs. decay ……………….
                                                 Doubling Sheet

No. of Doublings




                                                                                        Draw the Number of Squares
      5




       4




   3




      2




  1



    0
                   Shade twice the number of squares each time. Record data in table.
                              Halving Sheet




     Number of
   Squares (n) = 1




      0th halving                       1st halving – shade half of
                                        the original square




2nd halving – shade half of                    3rd halving
the previous square




       4th halving                              5th halving
MMA Name: ________________ Date: _________ Period:______




                                  Iterating on a Plane
                        3 units




      3 units




1. If the area of the original square (stage 0) is 9 units2, what is the area of the
new figure at stage 1? ______

2. The area of the stage 1 figure is what fraction of the area of the stage 0
figure? _____

3. When the figure goes from stage 0 to stage 1, does the area increase,
decrease, or stay the same? ______________________

4. What is the perimeter of the stage 0 square? _____

5. What is the perimeter of the stage 1 figure? _____

6. When the figure goes from stage 0 to stage 1, does the perimeter increase,
decrease, or stay the same? ______________________

An iteration rule has three distinct parts: a reduction, a replication, and a
rebuilding. Each time the rule is applied, the result is a new and more complex
figure.
            3 units



  3 units




 Fill in the tables. Graph your results. Remember to label your axes and scale.

 Number of Squares
                                  Number of
 Stage                Process
                                   Squares
    0
    1
    2
    3
    4
    n

From stage to stage, the number of squares increases/decreases by a __________ of ___ .
Equation: y = ______________     or     f(x) = ______________


 Total area (in2)

 Stage                Process      Total area
    0
    1
    2
    3
    4
    n



 From stage to stage, the total area increases/decreases by a __________ of ___ .
 Equation: y = ______________      or    f(x) = ______________

				
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