10 Day Lesson Plan
Lesson Day 1

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-6-04______

Objective(s):

1. TSW define the properties of exponents (Knowledge)
   {Mathematics, Advanced Algebra, 6a}
2. TSW simplify expressions with rational exponents (Comprehension) {Mathematics, Advanced Algebra, 6a}
3. TSW solve equations containing rational exponents (Application)
   {Mathematics, Advanced Algebra, 6a}

 

Materials: Chalk, Chalkboard

1. The teacher will introduce several concepts on the chalkboard from previous courses dealing with exponents and ask the students to recall what they remember.
2. The teacher will give new exponential concepts on the chalkboard for the students to write in their notes.  The teacher will ask questions about the various properties to allow the students to recall various properties from previous courses and reuse them in the new context of rational exponents.

 

Objective 2 (approx. 15 min)

1. The teacher will show several examples of simplifying expressions containing rational exponents.   The students will copy the examples.
2. The teacher will copy several problems on the board and ask students to complete them on their own.  The teacher will go through the class checking student’s progress.  Once several students have finished, the teacher will choose several to go to the board and complete the problems for the entire class to see.

 

Objective 3 (approx. 15 min)

1. The teacher will ask one student to come to the board and solve a simple equation.  After the student solves the equation, the teacher will add a rational exponent to the equation and ask the student to solve the equation again.  He will encourage other students to give suggestions to help the student at the board.  The teacher will then explain how solving a rational exponent equation is simply applying what has been given previously in the class.
2. The teacher will copy several problems on the board and ask students to complete them on their own.  The teacher will go through the class checking student’s progress.  Once several students have finished, the teacher will choose several to go to the board and complete the problems for the entire class to see.

 

CLOSE: “Today we learned one way to express a square root as an exponent and the various exponential properties as well as how to use them to simplify and solve equations.  Someone tell me how we would express the cube root as an exponent.  Tomorrow we will be using these exponent properties to solve some real world equations dealing with compound interest.” (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 2

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-7-04______

Objective(s):

1. TSW apply rational exponents in a real world problem of compound interest (Application) {Mathematics, Advanced Algebra, 6a}
2. TSW graph exponential functions (Comprehension)
   {Mathematics, Advanced Algebra, 1c, 1d, 6a}

 

Materials: Chalk, Chalkboard, Graphing Calculators

1. The teacher will break the students into groups and tell the students that for each group they have a select amount of money. Their job is to determine which percentage and rate of compounding interest will give them the highest amount in the end.  The students will be given several choices of percentage rates and compounding rates along with the exponential equation.  The teacher will briefly explain the exponential equation and how it is used. 
2. The groups will work until they have found what they believe to be the ideal solution.  At this point, the teacher will determine if the other groups are close to finding a solution.  If the teacher sees each group is getting close to a solution he will allow one of their members from each group to go to the board and write out their solution for the other groups to see.  The teacher will then change the principle amount of money and ask the students to recalculate the money. 

 

Objective 2 (approx. 20 min)

1. The teacher will have the groups construct graphs for the various rates and percentages that they tried in the previous exercise.  The students will use calculators to aid them in their construction of the graphs.
2. The teacher will then have the students graph various exponential graphs given differing values and allow various members of each group to go to the chalkboard and sketch their graphs.  Each student will sketch a graph to show the teacher they have understood the concept.

 

CLOSE: “Today we graphed the exponential functions and solved problems involving compound interest.  Can anyone tell me what P stands for in the compound interest formula?  Tomorrow we will be talking about a similar exponential function only this one will deal with exponential growth and we will be taking a quiz on the properties of exponents and their graphs.”  (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 3

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-8-04______

Objective(s):

1. TSW apply exponential functions in a real world problem of population (Application) {Mathematics, Advanced Algebra, 3c, 6a}
2. TSW recall their knowledge of rational exponents on a quiz (Comprehension) {Mathematics, Advanced Algebra, 1c, 1d, 6a}

 

Materials: Chalk, Chalkboard, Quiz for Day 3, Calculators

1. The teacher will begin by showing the students the form for exponential growth and how it is specifically used in population prediction.  The students will take notes from the lecture on exponential growth and the formulas. 
2. The teacher will set a problem of predicting the population for an island given the rate of increase and ask the students to each work individually predicting the population over various years.  The teacher will then walk around the room and check with each student to see the progress they are making.  When one or two finish the problem, the teacher will ask those students to complete the problem on the board for the rest to see.
3. The teacher will then set up a new scenario of the JurassicPark dinosaur problem where the dinosaurs are populating the island but every so often one species eats half the other.  The students are then given new criteria to apply the exponential formula and how it is to be applied at certain times.  The teacher gives hints as necessary to allow the students to complete the assignment.  When one student is finished, the teacher allows them to complete the problem on the board for other students to see.

 

Objective 2 (approx. 25 min)

1. The teacher will pass out a quiz and the students will complete the quiz.
2. When the students are finished, the teacher will collect the quiz and grade it.

 

CLOSE: “Today we looked at another exponential function for determining population growth.  Can anyone tell me what the form of the growth function is?  What does the r stand for in the equation?  Tomorrow we will be looking at a special number used in a lot of science and engineering applications dealing with exponential functions.  We will also be looking at how we use it to find continuously compounded interest.”  (approx. 5 min)

Notes:

Name:__________________________

 

Lesson Day 3 – Quiz – Exponential expressions and their graphs

Evaluate the expressions:

1)                     2)                         3)                                     4)    

Simplify the expressions:

5)                     6)         

Solve the equations:

7)                                     8)                             

9)           

 

Sketch the exponential function using a calculator:

10)   

10 Day Lesson Plan
Lesson Day 4

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-9-04______

Objective(s):

1. TSW apply Define the number e. (Knowledge)
   {Mathematics, Advanced Algebra, 6a}
2. TSW compare the function y=ex to an exponential function in a real world situation.  (Evaluation)  {Mathematics, Advanced Algebra, 3f,  6a}

 

Materials: Chalk, Chalkboard, Graphing Calculator

1. The teacher will introduce the concept of a series and convergence.  The students will take notes on how a series works.  The teacher will then introduce the concept on the number e and how it is found through the series.  The teacher will then calculate the number using a basic approximation showing the students what the number approaches.  The students will note what the number e actually is and how to find it.
2. The teacher will explain how the number is useful in many application of science and engineering.  The students will listen.  The teacher will ask for several students to share what their future plans include and relate the number e to the various occupations as allowed by the student’s responses.

 

Objective 2 (approx. 25 min)

1. The teacher will break the class into groups and introduce the concept of continuously compounded interest.  The teacher will then give each group a certain amount of money and ask them to compare the differences compounding the same amount and interest but changing the frequency of compounding.  The students will calculate using the graphing calculators the amount earned for each scenario and share what they find.  The teacher will prompt the students with questions leading into a discussion. 
2. The students will discuss the differences between continuously compounding the interest and compounding it in steps.  The teacher will introduce the idea of a limit as a basic idea relating to the highest point achievable in earning money given a percentage.

 

CLOSE: “Today we defined the number e and used it to calculate continuously compounded interest.  Can anyone tell me what the number e is to a couple decimal places?  Tomorrow you will be taking a quiz on the number e and the other exponential functions we talked about before today.  We will also be graphing the compound interest function.” (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 5

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-10-04______

Objective(s):

1. TSW apply their knowledge of exponential functions and the number e on a quiz.  (Application) {Mathematics, Advanced Algebra, 3f,  6a}
2. TSW graph the continuously compounded interest function for various starting amounts and various rates. (Comprehension)
    {Mathematics, Advanced Algebra, 1c, 1d,  6a}

 

Materials: Chalk, Chalkboard, Quiz for Day 5, Graphing Calculators

1. The teacher will pass out a quiz for the students to recall their knowledge of the number e and exponential growth.  The students will complete the quiz to the best of their ability.
2. The teacher will collect the quiz when the students are done.  Students will sit quietly till everyone is finished. 

 

Objective 2 (approx 15 min)

1. The teacher will pass out graphing calculators to the entire class.  The teacher will then have them graph the continuously compounded interest function for various amounts of principle, rate and time.  The students will graph the functions on their calculators and sketch drawings for the teacher to check.
2. The teacher will ask basic questions for the students to answer regarding how much principle it would require to reach a certain amount in x amount of years.  The teacher will have students come to the board and sketch what they have found and point out certain point of interest on the graphs.

 

CLOSE: “Today we graphed the differences between rates and starting Principles and compared the graphs to each other.  Can anyone tell me how much of a difference the starting amount makes in the equation?  Tomorrow we will be talking about the inverse to an exponential function and how they relate to each other.”  (approx. 5 min)

Notes:

Name:__________________________

 

Lesson Day 5 – Quiz – Exponential growth and the number e

Remember

1. What does P stand for?
2. What does A stand for?
3. What does n stand for?
4. What does t stand for?
5. Find how much someone would make after 5 years if compounded daily at 5.25% given an initial investment of $5000.
6. What is the number e approximately?
7. How do you find the number e?
8. What can you calculate using the number e?

 

10 Day Lesson Plan
Lesson Day 6

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-13-04______

Objective(s):

1. TSW compose the inverse function to the exponential function (Synthesis) {Mathematics, Advanced Algebra, 1d, 3c, 6a, 6b}
2. TSW compare the logarithmic property to the exponential property.  (Evaluation) {Mathematics, Advanced Algebra,  6b}

 

Materials: Chalk, Chalkboard, Graph paper, Calculators

1. The teacher will have the students separate into groups and give each group an exponential function.  The teacher will then explain to each group they must plot enough points for the graph to make sense or justly be drawn accurately.  The students will begin by plotting points for the graph on graph paper using calculators to calculate the points but not graph the function.  Each group will be given a different exponential function.
2. The teacher will then tell the students to find the inverse function for the exponential functions by inverting the x and y coordinates and plotting the results.  The students will take the points they used previously, flip the coordinates and use the resulting data to plot a new function.  The teacher will watch the groups and ask questions of them about the newly formed function. 

 

Objective 2 (approx 20 min)

1. The teacher will introduce the ideas for the logarithmic properties.  The teacher will ask the students to recall the idea of an inverse function – that is to reverse the outcome of any function – and how it is formed.  The students will recall what they remember of inverse functions and how they are formed.
2. The teacher will ask the students to compare the logarithmic property to that of the exponential property in a discussion.  The teacher will remind them about the graphs they have just created and ask them to explain the graphs in their evaluation.  The students will evaluate the differences between the logarithmic function and the exponential function.  Through questions, the teacher will lead the students to conclude the logarithmic function is an inverse of the exponential function. 

 

CLOSE: “Today we formulated the logarithmic function through graphing the inverse relationship and we compared the logarithmic properties to the exponential properties.  What is the difference between a log and an exponent?  Tomorrow we will be defining the logarithmic properties and function.  We will also solve equations using logarithms.  (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 7

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-14-04______

Objective(s):

1. TSW define the logarithmic function (Knowledge) {Mathematics, Advanced Algebra, 6b}
2. TSW apply the logarithmic properties to solve equations containing logarithms.  (Application) {Mathematics, Advanced Algebra,  6b, 6c}

 

Materials: Chalk, Chalkboard

1. The teacher will lecture on the logarithmic properties and the various operations performed through logarithmic functions.  The students will take notes.
2. The teacher will introduce the logarithmic function and how it is used as an inverse to the exponential functions discussed earlier.  The teacher will ask students to recall what they found previously in their activity from yesterday concerning logarithms and exponents.  The students will answer various questions about what they found and how it relates to the inverse of an exponent.

 

Objective 2 (approx 30 min)

1. The teacher will go through several examples using the log properties.  The students will copy the examples. 
2. The teacher will copy problems to the board and have the students come to the board and try the problems while the rest of the class tries at their seats.  When one student is finished, they pass the chalk off to another student of their choosing.  In this way, everyone in the class comes to the board to solve a problem.  If a student has trouble, the rest of the class may offer help.

 

CLOSE: “Today we defined the properties of logarithms and their functions as well as using those properties to simplify expressions.  Someone tell me what happens when we are dividing two logarithms?  Tomorrow we will be taking a quiz on logarithms and their properties.”  (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 8

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-15-04______

Objective(s):

1. TSW generate the inverse function of a logarithm.  (Synthesis)
    {Mathematics, Advanced Algebra, 1d, 6a, 6b}
2. TSW recall their knowledge of logarithms on a quiz.  (Knowledge)

 

Materials: Chalk, Chalkboard, Day 8 Quiz

1. The teacher will copy several problems for converting between a log and an exponent and complete the problems on the board.  The students will complete the problems and check their work with the teacher’s.
2. The teacher will put a log function on the board and ask the students to find its inverse using the properties of log and the idea of an inverse function.  The students will attempt to complete the problem.  The teacher will give hints to remind students along the way for procedures on doing the inversion process.
3. The teacher will explain the steps for finding the inverse of a function and go through a specific example on the board.  The students will take notes on how the procedure is done.
4. The teacher will put several log functions on the board and ask student to complete the problems at their seats and at the board.  The students will complete the problems both at their seats and come to the board and finish the problems.

 

Objective 2 (approx 20 min)

1. The teacher will pass out a quiz.  The students will complete the quiz to the best of their ability.
2. The teacher will collect the quiz either when the students are done or when the time is up.  The students will hand in their quiz upon completion of time or the quiz.

 

CLOSE: “Today we found a way to convert the logarithmic function through an inverse to an exponential function.  Tomorrow we will be talking about the number e and how it relates to the log function.  Also, we will be looking at a different type of relationship between a log function and a hyperbola function.  Can anyone recall what a hyperbola looked like?” (approx. 5 min)

 

Notes:

Name:__________________________

 

Lesson Day 8 – Quiz – Logarithms

Convert the equations to exponential form then solve:

1. 

2. 

3. 

 

Answer in a short paragraph – What is the relationship between the logarithmic function and the exponential function? 

10 Day Lesson Plan
Lesson Day 9

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-16-04______

Objective(s):

1. TSW combine the concepts of logarithms and the number e to construct the Natural Log.  (Synthesis) {Mathematics, Advanced Algebra, 6b}
2. TSW compare the relationship between ln x and y=1/x.  (Evaluation) {Mathematics, Advanced Algebra, 1d, 6a, 6b}
3. TSW review the material in the unit for a test tomorrow.  (Knowledge)

 

Materials: Chalk, Chalkboard, Graphing Calculator, Overhead Graphing Calculator

1. The teacher will ask the students to recall the number e and the exponential growth function based on it.  The teacher will direct the students to construct an inverse function for the exponential growth function.  The students will work at their desks on finding the inverse to the exponential growth function. The teacher will prompt the students when needed for guidance.  The teacher will walk around to see how each student is doing.  When most of the students have arrived at the answer, the teacher will ask one student to go to the board and explain how they arrived at the answer.  The teacher will explain to the students they have created a special case in the logarithmic functions called the Natural Log.

 

Objective 2 (approx. 15 min)

1. The teacher will use an overhead calculator to draw out the function y=1/x.  The teacher will demonstrate to the students that if the area under the curve is calculated, it will produce the same number for the Natural Log of a number.  The teacher will ask various questions about the graph to involve the students.  The students will listen to the presentation and answer various questions as prompted by the teacher.
2. The teacher will prompt a discussion about the relationship between the Natural Log and the function y=1/x, asking students to compare the two.  The students will respond with various answers evaluating the two functions from their perspective.  The teacher will conclude the discussion with a brief introduction to the idea of finding the area under a curve.

 

Objective 3 (approx. 15 min)

1. The teacher will begin a review of the material covered from the beginning of the unit through the use of a game.  The students will divide into teams and answer questions to strategically play tic-tac-toe.  The game will cover all the concepts and applications taught throughout the unit.
2. In addition to the game the teacher will write on the board the topics to be covered on the test.  The students will write the topics down to study for the test tomorrow.

 

CLOSE: “Today, we covered the idea of a Natural Log and discussed the relationship between it and the function 1/x.  Can anyone tell me what the Natural Log uses as a base?  We also reviewed for the test tomorrow and you have all the topics that will be covered on the test.  So, tomorrow you will have a test.  If you are unsure about anything, ask today not when you get the test.” (approx. 5 min)

Notes:

10 Day Lesson Plan
Lesson Day 10

Teacher:___Jeff Spanogle______         School:__Yazoo City High School___________

Grade:_9th – 12th __     Subject:__Advanced Algebra__           Class Period:__6th _

Date:_____12-17-04______

Objective(s):

1. TSW evaluate problems given on a test of rational exponents, exponential functions, the number e and its related functions and logarithms and their related functions.  (Evaluation)
   {Mathematics, Advanced Algebra,  1d, 1c, 3c, 3f,  6a, 6b, 6c}

 

Materials: Unit test

1. The teacher will pass out a test and wait for the students to finish.  The students will complete the test to the best of their ability.
2. The teacher will collect the test from the students when they are finished.  The students will sit quietly till the others are finished with the test.

 

CLOSE: “Today you had a test.  After break you will be starting the review for your final.”  (approx. 1 min)

 

Notes:

Name:__________________________                                              Date:_____________

Exponential and Logarithmic Unit Test

Section 1:  Exponents

 

Evaluate the expressions:
1)                     2)                         3)                        4)    

Simplify the expressions:
5)                         6)          

Solve the equations:

7)                                   8)                   
           
Sketch the exponential function using a calculator:

10)   

 

Section 2: Logarithms

 

Solve the following equations:
1.                       2.                       3. 

4.      

Show how to calculate the number e then find it to the first two decimal places:

 

 

 

 

Section 3: Real World Applications

Remember   and also
Find how much someone would make after 12 years if compounded daily at 5.65% given an initial investment of $3000. 

What if the initial investment is $4000?  How much of a difference is between the two investments? 

What happens if the initial investment is $5000?  Is the relationship linear?  Why does the initial investment make such a difference?

Now calculate the investment of $3000 for a continuously compounded interest.  How much of a difference is there in daily and continuously compounded interest?

Finally, what is the difference between compounding in small intervals and compounding continuously?