Exploring Exponential Equations and Graphs



Exploring Exponential Equations and Graphs

Lesson Lab Plan Summary Page

Developed by Jenny Walls (Akron Firestone High School)

 

Subject: Algebra I/Algebra II

Grade: 9th/10th grade

Strands: Algebra and Functions

Data Analysis and Probability

Topic: Exploring Exponential Equations and Graphs

Objectives:            

Strand:  Algebra and Functions

·        Students will be able to represent a mathematical relationship using a table, graph, symbols, and words, and describe how a change in the value of one variable affects the value of a related variable.

·        Students will be able to create and analyze graphs of linear and simple non-linear functions.

Strand:  Data Analysis and Probability

·        Students will create, interpret, and/or analyze tables, charts, and graphs involving data.

·        Read, interpret and use tables, charts, maps, and graphs to identify patterns, note trends, and draw conclusions. 

Materials: Graphing calculators, worksheets

Expected Time: 2 class periods

 

 

 

 

 

 

 

Exploring Exponential Equations and Graphs

Lesson Lab Plan

 Developed by Jenny Walls (Akron Firestone High School)

 

Concepts/Learning and Proficiency Objectives

Strand:  Algebra and Functions

·        Students will be able to represent a mathematical relationship using a table, graph, symbols, and words, and describe how a change in the value of one variable affects the value of a related variable.

·        Students will be able to create and analyze graphs of linear and simple non-linear functions.

Strand:  Data Analysis and Probability

·        Students will create, interpret, and/or analyze tables, charts, and graphs involving data.

·        Read, interpret and use tables, charts, maps, and graphs to identify patterns, note trends, and draw conclusions. 

Task Overview

            Students will use real-life situations to explore exponential growth and to make connections between graphs and equations of exponential functions.  This lesson should be used as an introduction to a chapter on Exponential functions. 

            This cooperative learning lesson revolves around activities outlined in the attached worksheet.  Students should work in teams of two, three, or four.  Students should follow the instructions on the worksheet and discover the answers as a group.  The teacher should act as a facilitator while students are completing the activity.  At the end of the class period or the next session, the teacher should highlight conceptual ideas that the activity covered.              An extension to the lesson is provided for students who finish early.  The use of graphing calculators is required.  When designing this lesson, the assumption was made that students already know calculator skills using lists, drawing scatterplots, and graphing equations. 

Integration Learning Strategies

            Separate students into groups of two, three, or four.  Each group member needs a copy of the worksheet activity and a graphing calculator.  In groups, the students should elect a coordinator, recorder, and reader.  The coordinator keeps the group on-task and working.  The coordinator also asks the teacher the group's questions.  The recorder writes the group's responses and reads the group's results and responses when requested by the teacher.  The reader reads the problems and activities aloud for the group.  The reader also attempts to clarify questions before consulting the teacher.

            The teacher should ensure that group members are on-task and monitors results.  The teacher will also wrap-up the lesson, highlighting new mathematical concepts that connect to future and past concepts. 

Classroom/Information Management

            The activity itself is a self-directed task.  In other words, once the students have the worksheet, they should as a group, proceed through the tasks on their own.  The teacher needs to monitor student behavior as well as checking results as the groups are working. 

Assessment

Appropriate assessment is type II in nature.  A suggested assessment exercise is to have students individually write their own story problems involving exponential growth.  The students should trade problems and then individually sketch an accurate graph and corresponding equation.  The story problems should involve some sort of interpolation or extrapolation (predicting other results).  Grading would be based on the problem the student designed and then on the solution of the problem they were given. 

Tools and resources

            These sites could be used by students to explore other exponential growth situations, design exponential growth problems as required exercises of as an extension. 

- The fractory: An interactive tool for creating and exploring fractals



Exponential growth and decay

World population and demographic data - Mining Co

All you every wanted to know about global warming



 Worksheet

            Activity worksheet is on the next page.

 

 

 

 

Exponential Growth Activity

 

Name: __________________________________   Group Number:______

Date:________    Period:_________

 

Directions:  Work on the following problems as a group.  You need a graphing calculator.

 

Problem One:

            One single bacteria lands on a kitchen counter.  It divides into two parts every 5 minutes.  Fill out the chart below to show how many bacterium are on the counter:

 

|# of 5 minute |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |

|periods | | | | | | | | | | |

|$ |.01 |.02 |         |> |> |> |> |> |> |> |

 

How much money will your parents owe you on June 30? _______

Will you get it? ______    Why or why not? _______

 

Express the amount of money earned on June 30 using exponents.  ____

Now write an equation for x days.  Remember, the starting point is not 1 this time.  y = __________

 

Graph the scatterplot on your calculator and copy it below.  Also graph the equation you wrote above.  If the scatterplot and rule do not match, adjust your equation until they do.

 

               

 

Problem Three:

            You get an email that school will be closed tomorrow.  The email instructs you to forward it to three of your friends.  Your friends will send it three of their friends and so on and so on.  How many people must forward the email so that all 500 students enorlled at school stay home?

 

Fill in the table:

 

|# of email |0 |1 |2 |3 |4 |5 |6 |7 |8 |

|senders | | | | | | | | | |

|# of students |1 |3 |9 |  |s]>  |s]>  |s]>  |s]>  |s]>  |

   

Make a scatterplot of your data.  Copy it below:

 

         

How does it compare to the other two problems?

 

   

Write a y = equation for this situation.  Remember, the numbers are not being doubled.    _____________

 Graph your equation on your calculator.  Does the graph match your scatterplot?  Adjust your equation until it does.

 Summary:

            Explain the roles of a and b in the equation below.  How do the values of a and b affect the table and graph of the equation?

 

y = a (b^x)    Extension:   Set the window of your calculator to:

                        x min:  0

                        x max: 100

                        xscl:   10

                        y min: 0

                        y max: 3000

                        yscl: 100

 

Explore the following equations.  Describe in words a situation that would match the equation.  Use your imagination!

 y = 2^x

 

y = 5 x 2^x

 

y = .5 x 2 ^x

 

y = 3^x

 

y = .5 x 3^x

 Exponential Growth Activity/Answer Key

 

Name: __________________________________   Group Number:______

Date:________    Period:_________

 

Directions:  Work on the following problems as a group.  You need a graphing calculator.

 

Problem One:

            One single bacteria lands on a kitchen counter.  It divides into two parts every 5 minutes.  Fill out the chart below to show how many bacterium are on the counter:

 

|# of 5 minute |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |

|periods | | | | | | | | | | |

|$ |.02 |.04 |.08 |ras]> |ras]> |ras]> |ras]> |ras]> |ras]> |ras]>  |

|  |  |  |  |  |  |  |  |  |  |  |

 

How much money will your parents owe you on June 30? $10737418.24

Will you get it? No    Why or why not? Too much money!!

 

Express the amount of money earned on June 30 using exponents.  .01(2^30)

Now write an equation for x days.  Remember, the starting point is not 1 this time.  y=.01(2^x)y

 

Graph the scatterplot on your calculator and copy it below.  Also graph the equation you wrote above.  If the scatterplot and rule do not match, adjust your equation until they do.

The rule and the scatterplot should match. Some students may use 2^x as the equation and forget the starting point. Studens should label the x-axis with dates and y-axis as money.

           Problem Three:

            You get an email that school will be closed tomorrow.  The email instructs you to forward it to three of your friends.  Your friends will send it three of their friends and so on and so on.  How many people must forward the email so that all 500 students enorlled at school stay home?

 

Fill in the table:

 

|# of email |0 |1 |2 |3 |4 |5 |6 |7 |8 |

|senders | | | | | | | | | |

|# of students |1 |3 |9 |27 |s]> |s]> |s]> |s]> |s]> |

    Make a scatterplot of your data.  Copy it below:

Students should label the x-axis as # of email senders and y-axis as # of students that know. Again both axes should be labeled with an appropriate numerical scale.

   How does this situation compare to the other two problems?

 The numbers are being tripled, not doubled. The numbers increase very quickly.

 

  Write a y = equation for this situation.  Remember, the numbers are not being doubled.   

Y = 3^x

 Graph your equation on your calculator.  Does the graph match your scatterplot?  Adjust your equation until it does.

  Summary:

            Explain the roles of a and b in the equation below.  How do the values of a and b affect the table and graph of the equation?

 y = a (b^x)  a is the starting point and b is what the numbers are multiplied by each stepa a  Extension:   Set the window of your calculator to:

                        x min:  0

                        x max: 100

                        xscl:   10

                        y min: 0

                        y max: 3000

                        yscl: 100

 Explore the following equations.  Describe in words a situation that would match the equation.  Use your imagination!

 y = 2^x

 

y = 5 x 2^x

 

y = .5 x 2 ^x

 

y = 3^x

 

y = .5 x 3^x

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