Title III Technology Literacy Challenge Grant
Learning Unit
Overview | Content Knowledge | Essential Questions | Connection To Standards | Initiating Activity | Learning Experiences | Culminating Performance | Pre-Requisite Skills | Modifications | Schedule/Time Plan | Technology Use
| LU Title: Exponential and Logarithmic Functions the Fun Way! | Author(s): Donna H Briedis |
| Grade Level: 11 | School : Heuvelton |
| Topic/Subject Area: Mathematics | Address: Heuvelton CS Heuvelton, NY 13654 |
| Email: dbriedis@yahoo.com | Phone/Fax: 315-344-2414 |
This is a unit designed to help Course 3, or soon to be
Math B, students discover
Exponential and
Logarithmic Functions
using TI-Graphing Calculators and
Calculator
Based Rangers (CBRs).
|
Declarative |
Procedural |
|
Students will be able to identify the similarities and differences between exp equations and log equations. |
Students will graph exp. and log. equations via a TI-GC and then record results on a worksheet to visibly identify similarities and differences between the functions and the values of b. |
|
Students will be able to use the TI-GC to graph exp and log equations. |
Student will create real-world examples of exp. and log equations. |
|
Students will be able to display exp and log functions in multiple representations. |
Students will find coordinates of given values for both the exp and log equations. |
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Students will know the domain and range of both exp and log functions. |
Students will discover domain and range by plotting both types of graphs with different values of b. |
|
Students will be able to set up and use the CBRs. |
Students will transfer programs from the CBR to the TI-GC. |
|
Students will be able to interpret findings on the CBR and find regression equations that best fit the data. |
Students will run a CBR experiment and calculate the heights of a bouncing ball then translate the data onto the TI-GC and find the equation of best fit. |
ESSENTIAL QUESTIONS
What are the characteristics and importance of the exponential function?
What are the characteristics and importance of the logarithmic function?
What real-world examples will model an exponential function and a logarithmic function?
How can we model an exponential function or a logarithmic function using the CBR.
CONNECTIONS TO NYS LEARNING
STANDARDS
List Standard # and Key Idea #: Write out related
Performance Indicator(s) or Benchmark(s)
MST #5 Standard 5: Technology: - Students will apply technological knowledge and skills to design, construct, use and evaluate products and systems to satisfy human and environmental needs.
MST #3 Mathematics: 1- Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
MST #3 Mathematics 7 – Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simple and efficiently.
Exponential and Logarithmic Functions
Discovery1
(2 days)
With the TI-Graphing Calculator graph the function y = bx . Have the students graph a sketch of the function for each of the following values of b:
2, 3, 5, 8, .5, .2, 1/3, .125.
The students should then tell whether the function is increasing or decreasing and state the y-intercept for each sketch.
Have them answer the following questions:
What single point do all the graphs have in common? Why?
As b increases, what happens to the graph?
As b gets closer to one, what happens to the graph?
Describe the graph for b=1.
Describe the graphs for 0<b<1.
Describe the domain of all the functions f(x) = bx
Describe the range of all the functions f(x) = bx
Predict what the equation would be for the curve that results from reflecting f(x) = bx across the x-axis then test your prediction with your calculator.
With the TI-Graphing Calculator graph the function y = b-x . Have the students graph a sketch of the function for each of the following values of b:
2, .5, 4, .25, 5, .2, 10, .1, 3, .3333333.
The students should then tell whether the function is increasing or decreasing and state the y-intercept for each sketch.
What is the relationship between the values of b in each pair of equations?
How do the graphs of each pair compare?
What point do all the graphs have in common?
If b > 1, what can be said about the graph?
If 0 < b < 1, what can be said about the graph?
Describe the domain of all the functions y = b-x?
Describe the range of all the functions y = b-x?
White another equation that has the same graph as the one produced by y = b-x and as the one produced by y = b-x?
With the TI-Graphing Calculator graph the function Log b (x) . Have the students graph a sketch of the function for each of the following values of b:
2, 3, 5, 8, .5, .2, 1/3, .125.
The students should then tell whether the function is increasing or decreasing and state the x-intercept for each sketch.
What single point do all the graphs have in common?
If the graphs are increasing, what do we know about the value of b?
If the graphs are decreasing, what do we know about the value of b?
Describe the domain of y = Log b (x)?
Describe the range of y = Log b (x)?
If we wanted a graph that was flatter than the graph of y = Log b (x), what would be an appropriate value for b?
What is the relationship between pairs of graphs of the form y = Log b (x) [base b] and y = Log b (x) [base 1/b]?
Redraw the graphs if necessary.
LEARNING EXPERIENCES
In
chronological order including acquisition experiences and
extending/refining
experiences for all stated declarative and procedural
knowledge.
The Initiating Experience should give the students a good grasp of what exp and log functions look like, their domain and range, their intercepts, and what changes as the value of b changes.
The rest of the week should be filled with:
Rational Exponents (flashcards), (2/3)^2, (2/3)^-2, 4^(3/4)
Exponential Functions, 2^x Make T-charts, graph then check on TI-GC. Talk about Domain and Range
Exponential Equations (TI-82)2, Solve and then check on the TI-GC. 4^x in y1 and 2^(x+2) in y2. Solve for intersection. Check the table. Try math solve
Inverses of Exponential Functions---> Logarithmic Functions, Have them graph an exponential function and then reflect it over the y = x line and then talk about inverses. Put 2^x into y1 graph then graph inverse in the draw function. Talk about similarities and differences. Range, Domain Show Logs.
Logarithms, Use flash cards to quiz them on domain and range of different graphs and different equations. Use slates5 to have them draw out functions and solve exp and log equations
Properties of Logarithms (flash cards), Show them the log rules on the TI-82, how log 2 x 4 is equivalent to 4 log 2 x.
Logarithmic Equations (TI-82). Solve and then check on the TI-GC. Log (xy) into y1 and log x + log y into y2. They are the same. Check the table.
Followed by the culminating performance, Exploring Exponential Functions in a Real Life Situation (use the CBL or CBR3 and the TI-82), and then a day of summing up the Exploration of an Exponential Function.
I use the AMSCO Course III 2nd edition, text book. I assign problems from the Exponential Functions and Logarithmic Functions chapters.
CULMINATING PERFORMANCE
Include rubric(s)
(2 days)
Exponential Exploration:
Course III Project Expectations:
Collect data according to packet instructions.
Use your calculator to determine the regression equations for your data: Find three different models: Linear(LINReg), Logarithmic(LNReg), and Exponential(EXPReg). Construct a graph that includes all three functions and the actual data points. Compare each model(function) to the actual data. Write a description of how well each model fits the given data. Be specific about where it fits well and where it doesn’t.
Construct the Best Model: Break the data into sections and choose which type of model is best for each section. For each new section, find a new regression equation that fits. Construct a new chart that uses the different equations for the different sections of the curve. (Thus creating a perfect fit for your data.) Describe how well your new model fits the data. Discuss the decisions you made on where and how to break up the data. Describe alternate solutions you tried.
Interpolate and Extrapolate: Choose a point between two of your actual data points (half-year data point) and compare the values given by the different models. Which values make the most sense, which would your throw out? Choose a data point after the values on your curve and compare the values given by the different models. Which values make the most sense, which would your throw out?
Rate of Change: Describe how the rate of change of the rebound of the ball. Describe how this is reflected in each of your models.
End Behavior of Models: Describe what is happening to ball at the end of your curve. Compare this to the behavior of each of your models if the curve continued. Discuss why your model will or will not give accurate values for an eighty year old.
Summarize: What function(s) would you recommend be used to model your percentile curve? Justify why this is the best model for the data. What unique observations, applications to the real world, or general rule do you feel would fit this project? Elaborate
Day 1
You need to have CBL’s with motion detectors or CBR’s, TI-82 or TI-83 or TI-83 plus, the recommended packet(*), and rubber kick balls.
Students will set up their calculators to do the experiment. Bounce the balls and take readings until they get a nice graph. After they get their graphs, they need to record the bounces and the heights of each bounce into the Statistics List 1 and List 2. From this information they can find the regression equation by hand and via the calculator expreg function in the Stat-Calc menu.
Use CBl Experiment #134
“In this exploration, you will use a CBL or CBR system motion detector to investigate the motion of a bouncing ball. Specifically, you will find a function that relates the height of a bouncing ball to the number of times the ball bounces. Generally, when a rubber ball is dropped from rest and bounces repeatedly on a hard, level surface, the height to which the bball rebounds is less each time it bounces. Each time the ball bounces, it rebounds to a height that is a percent of the previous height. This percent is called the rebound rate. Because the ball bounces to a height that is a percent of its previous height and because this percent is a constant, you can model the heights of a bouncing ball with an exponential function.
y-Hpn
H is the ball’s initial height.
P is the ball’s rebound rate.
Y is the ball’s height on the nth bounce.”*
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4.60469*.73324^x
Day 2
Students will make sure that their graphs and data are accurate. They should graph the equations that they found to match the data. Then they should answer a series of leading questions from the packet4:
Print off the LCD of your bounce heights and regression equation using the TI-graph link.
What does the y intercept represent?
What is the domain and range?
Find the total vertical distance the ball will travel after 10 bounces. Explain how you obtained your answer.