/ Characteristics
Examples / Non-Examples
Exponential Functions Menu Choice Board
Appetizer: Graphs of Exponential Functions - Choose one!
/ “Breadsticks”—Exploring with Computer AppletThis station will have you investigate exponential functions using a computer applet. Visit the “Breadsticks” station for detailed instructions and retrieve a laptop. Visit http://www.analyzemath.com/expfunction/expfunction.html
Click on the interactive applet using Java and follow the instructions for investigating how the graph of an exponential function changes based on each variable. Make sure to answer the follow-up questions. When finished, take the completed guide to your teacher.
/ “Salad”—Calculator Investigation with your Teacher
Your teacher will lead you through a calculator investigation of exponential functions. Before going to this station, grab a graphing calculator, clear-board, and dry erase pen with eraser. Once the investigation is complete, you will use your clear-boards to respond to the first three questions available at this site:
http://www.zweigmedia.com/RealWorld/tutnonlinear/frames2_2.html
Main Course: Lab Investigations - Choose one!
/ “Spaghetti”— Exponential Growth LabFor this lab you will work as a group to solve the “King’s Chessboard” problem. Go to the “Spaghetti” station, to get a copy of the task. You may use manipulatives in the room, chart paper, graph paper, graphing calculators, and other supplies to help you generate a solution. Your solution must also include an explanation of how you determined your answer and which option is the better option. When finished, submit your solution to your teacher.
/ “Chicken Parmesan” – Exponential Decay Lab
For this lab, each member of your group will do data collection for a different item (thumb tacks, pennies, un-popped popcorn). Go to the “Chicken Parmesan” station for a copy of the lab instructions and supplies for the lab. Complete the lab guide, making sure to graph each of the three models on the same graph. Answer the follow up questions and submit the completed lab to your teacher.
Dessert: Presenting Findings – Choose one!
It is now time to demonstrate your understanding of exponential growth and decay functions. Each dessert option will give you a different possibility for presenting what you have learned about exponential functions.
/ “Chocolate Cake”—Glogster PosterWork with your group to develop a “glog” of what you know about exponential functions. Make sure to include the following:
· Compare/contrast growth and decay
· Graphical behavior and trends
· Patterns in tables and equations
· How do you know if a function is exponential?
/ “Cannoli”—Develop a skit/song/Blabberize/Voki presentation
Work with your group to develop a presentation on what you know about exponential functions. You may choose to use technology resources to assist with the presentation or you may sing/act out your presentation. Make sure to include the following:
· Compare/contrast growth and decay
· Graphical behavior and trends
· Patterns in tables and equations
· How do you know if a function is exponential?
/ “Gelato” – Written Reflection
For this option you may work individually or with your group to develop a written reflection on what you know about exponential functions. Make sure to include the following:
· Compare/contrast growth and decay
· Graphical behavior and trends
· Patterns in tables and equations
· How do you know if a function is exponential?
Note: All images are available through Creative Commons http://creativecommons.org/
“Breadsticks” Station: Exploring Exponential Graphs with Computer Applet
Once you are assigned a computer station, visit the website: http://www.analyzemath.com/expfunction/expfunction.html
Click on the Interactive Applet using Java and complete this lab investigation.
This applet can be used for any transformation of an exponential function
For the purpose of this lab, we will keep the following constant throughout the lab: b=1, c=0
Part I: Exponential Growth
To begin, set a=1, B=2, and d=0
This is a basic example of an exponential growth function
Sketch the graph below and complete the table of values:
Identify key features:
Domain: / Range:What happens as x increases? / General shape of the graph:
Other observations:
Now, increase the value of a. How does changing this value impact the graph?
What happens to the graph if a is set to a negative value?
Reset a = 1. Now, increase the value of B. How does changing this value impact the graph?
Reset B = 2. Now, increase and decrease the value of d. How does changing this value impact the graph?
Part II: Exponential Decay
To begin, set a=1, B=0.5, and d=0
This is a basic example of an exponential growth function
Sketch the graph below and complete the table of values:
Identify key features:
Domain: / Range:What happens as x increases? / General shape of the graph:
Other observations:
Now, increase the value of a. How does changing this value impact the graph?
What happens to the graph if a is set to a negative value?
Reset a = 1. Now, change the value of B to other values between 0 and 1. How does changing this value impact the graph?
Reset B = 0.5. Now, increase and decrease the value of d. How does changing this value impact the graph?
Part III: Summary
On the back of this page, write a paragraph explaining your understanding of exponential growth and decay functions. Be sure to include information relating to key features on the graph and how these features are determined.
EXPONENTIAL GROWTH
A wealthy king was rescued from danger by a quick thinking and brave soldier. The king wanted to honor the poor soldier, so he offered a very generous reward: a beautiful chessboard made of ivory and ebony and a set of gold chess pieces.
While the chess set was beautiful and valuable, the young man asked for a different reward. To help the poor people in his country, he asked the king to distribute rice from his storehouse – two grains for the first square of the chessboard, four grains for the second square, eight grains for the third square, sixteen grains for the fourth square, and so on. The king was pleased that he could keep his beautiful chessboard and repay the soldier with such a simple grant of rice for the poor. But he soon discovered that the request was not as simple as he thought.
Examine the request from the poor soldier to determine why the king feels that the request was not as simple as it seemed. Compare the reward of rice with the reward of the chessboard to support your findings.
*Adapted from The King’s Chessboard, Birch, David. Puffin Books, 1988.
An Exponential Investigation
Mathematics is the study of patterns. In this investigation you will explore and compare several slightly different patterns. You will conduct the experiment that follows, record your data, analyze your data, and describe the pattern using a mathematical function, if possible. As the week develops, you will understand how this important pattern relates to our study of radioactive decay.
Before you begin this experiment, make sure that you identify who, within your group, will do each job. Fill in the group member name next to each job description
______/ Tosser – This person is responsible for tossing the objects described in the experiment.______/ Remover – This person is responsible for removing the objects described in the experiment.
______/ Counter – This person is responsible for counting the remaining objects and recording this number in the data chart.
______/ Reader – This person is responsible for reading the lab directions to the group and making sure that the group efficiently works through each step.
Procedure:
1. The tosser should collect the assigned object from the teacher- either:
§ Thumbtacks
§ Pennies
§ Un-popped popcorn kernels on a paper plate with one quarter shaded
2. The counter should count and record the number of objects and then place them in the cup.
3. The tosser should shake the cup and toss the objects gently onto a flat surface, or, in the case of the plate, gently spill the kernels onto the late as evenly as possible.
4. The remover should remove the objects defined below;
§ Thumb tacks that have landed point up. (careful!)
§ Pennies that land “heads” side up
§ Kernels of corn that land in the shaded section of the plate.
5. The counter should count and record (in the data table) the number of objects remaining.
6. The tosser should gather the remaining objects in the cup and repeat steps 2-5 until there are fewer than 5 objects left to toss or until 14 trials have been completed. It is important that you do not record a zero as the number of objects remaining. (If your final trial results in zero, record that value as 0.001)
7. The reader should graph the data points on a graph whose x-axis represent the number of tosses and the y-axis represents the number of objects remaining.
Data Collection
Trial Number / Number of Objection Remaining0 (Start value)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
To prepare your graphing calculator, clear L1 and L2 and turn all plots off before you begin.
- Enter the data values into the calculator in L1 and L2.
- Make a scatter plot.
- Is your plot a function? If so, what type of function is being modeled?
- Use the graphing calculator to find the equation that best models the data. Graph the equation with the data plot.