UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS


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1 UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS Summary: This unit plan covers the basics of exponential and logarithmic functions in about 6 days of class. It is intended for an Algebra II class. The unit focuses on identifying and graphing the two types of functions, recognition that they are inverses of one another, and awareness of the types of changes in the graphs brought on by change in base and by other transformations. It also includes a review of basic rules of exponents and an introduction to evaluating logarithm, natural logs, and basic manipulations of expressions involving more than one logarithm. Activities vary considerable throughout the course of the unit, but include several instances of groups work, several short minilectures, several worksheets that encourage a discovery approach, and frequent class discussions. Homework will be assigned every day, ranging from a couple of problems to a full writing assignments. Objectives: By the end of this unit, students will be able to do the following: graph an exponential function including transformations by hand graph a logarithmic function including transformations by hand recognize an exponential function in a realworld context explain in words relationships between the various transformations and base changes in the equations evaluate log base 10 and natural logs on the calculator perform simple manipulations of logarithmic expressions according to the rules move back and forth between logarithmic and exponential forms recognize that exponential and logarithmic functions are mutual inverses SOLs: This unit plan is designed to cover in part the following SOLs: AII.3 Write radical expressions as expressions containing rational exponents, and vice versa AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators. AII.9 The student will find the domain, range, zeros and inverse of a function, the value of a function for a given element in its domain, and the composition of multiple functions. Functions will include those which have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions, including exponential and logarithmic. AII.19 The student will collect and analyze data to make predictions, write equations, and solve practical problems. Graphing calculators will be used to investigate scatterplots to determine the equation for a curve of best fit. References: The M&M activity is based directly on a problem found the workbook for the Math Connects course taught at Virginia Tech. Topics and degree of depth are based on the curriculum included in chapter 12 of the textbook Merrill Algebra II with Trigonometry; Applications and Connections published by MacMillan/ McGrawHill. Homework assignment referred to by page number are taken from this book. 1
2 DAILY OVERVIEW DAY ACTIVITIES (time) 0 HW assigned: read examples 2, 3 on p. 548 p.549 (10,11,25,27,29) 1 Entry activity: Selected students put HW solutions (5) on the board as they come to class. Any computational questions about HW problems are addressed in class discussion led by the teacher. (35) M & M activity: Teacher hands out M&Ms, cups, overheads and worksheet and assigns groups. Students spend about minutes in groups, with teacher circulating to verify participation. Students then present results of the experiment to the class. Teacher facilitates whole class discussion during and after presentation. (10) Minilecture: Teacher leads students in entering their data in lists on the TI83 and performing exponential regression to find a bestfit equation. Be sure to prevent students from entering 0 values in lists. HW: p.549 (12, 46, 47) For 46 & 47, graph by plotting points plot at least 5 points for each) 2 Check HW for completeness only while students work in groups. (30) Graphing activity: In same groups, students complete graphing worksheets (one sheet per group) and prepare to present results to the class. (about 10 minutes) Teacher hands out additional worksheets for student notetaking. Groups present in turn to the class, taking notes and teacher facilitates group discussion. It will be helpful to compare the graphs of different groups and request that students write observations down on the OH as the discussion proceeds. Teacher should be sure to refer to yesterday s activity and ask the students why it was necessary not to enter 0 as a value in the list. Also should point out that y = 1 x is not strictly considered an exponential function. MATERIALS, PREPARATION Student s names and number of problem should be on board before class starts. M&Ms, cups Worksheet 1 Group assignments (on worksheet) Blank overheads & pens Graphing calculators available Group grading forms (for teacher) Class set of TI83 s Overhead calculator display Graphing calculators available Worksheet 2 Overheads & pens 2
3 (5) Minilecture: Teacher presents minilecture reinforcing the howto s of graphing exponentials and summarizing the transformations. Should emphasize that the point (0,1) can always be used as a reference and that one additional substitution is usually enough for a rough graph. Should give several examples. HW: p. 550 (49) and graph the following w/out calculator: y = ( 1 4 )x + 3 and y = 4 x (5) (10) Check HW while students work on warmup Warm up activity 1: Students work on intro to logs warm up as teacher circulates to check HW Brief discussion: Teacher leads brief discussion of activity, leading into the definition of the log, why the students were unable to do part 2, and the relationship between logs and exponents. Teacher addresses any questions from HW assignment. Activity Allowance Battle : In same groups students work on the allowance battle activity. Group completes write up to be handed in jointly for grading and prepares OH for the following day. Group hands in OH to teacher for safe keeping. (30) HW: read examples 1 & 3 on p. 554,6. p. 556 (6,7,9,11,12,13) and p. 557 (41) Using calculator log base 10 just means use log (5) (20) (10) (15) (1) Warmup activity 2: On the board as students come in intro to graphing logs. Teacher returns worksheets from the day before. Group presentations: Groups present results of allowance battle Brief discussion: Of warmup activity, emphasizing transformational approach and switching back to exponential form to do substitution when necessary. Note that calculator only does log base 10. Review HW: In groups. Each group chooses one question to ask in class the following day HW: Writing assignment from the allowance battle activity and p.557 (43, 44) Warm up activity 3: One group representative writes group s question on the board as students come in to class. Students hand in writing assignment. OH with warm up Worksheet 3 (be sure to circle equations on sheet for groups) Excel &/or graphing calculators available Overheads & pens OH with warmup OH projector Overheads from day before OH with warmup 3
4 (10) Brief discussion: Teacher leads discussion of questions, goes over 2 homework problems. Asks for general questions about exponential, logarithmic functions and their graphs. (15) Minilecture: Teacher gives minilecture presenting rules of logarithms, examples of their use. Clarifies use of log base 10 and natural log as special cases. (15) Group work: Students work on problems in groups, to hand in at the end of the hour. Teacher circulates to verify participation as portion of grade. Problems are p.561 (4,5,6,11) p. 565 (8,9,10 only find log) and p.569 (10,11,12) HW: Prepare for quiz tomorrow and fill in feedback questions (will count as portion of quiz) 6 (3) Warmup: Lead in to next topic (5) Questions: Teacher addresses any last minute questions (20) Quiz: Collect feedback answers with quiz. (15) Minilecture: Go over warm up and lead in to next topic. Group grading sheets Handout for feedback SUGGESTED PLAN OF ASSESSMENT DAY TYPE OF ASSESSMENT TYPE OF GRADE 1 Students put HW solutions on board No grade recorded Students check own HW solutions No grade recorded Group participation and presentation evaluation, Group grade taken. collect worksheet to include as part of evaluation. 2 Teacher checks HW Check for completeness only 3 Teacher checks HW Check for completeness only Group evaluation, collect worksheet to include as part of evaluation save group sheets to evaluate presentations on day 4. 4 Continue group evaluation from day 3 Group grade taken 5 Collect writing assignment from day 4 Grade individually. Group representatives put one question on the board from day 4 No grade taken Group participation evaluation while working in Group grade taken groups 6 Self evaluation Bonus points added to quiz/test grade Quiz/test Grade individually TOTAL 2 HW checks 3 Group grades 1 Written assignment 1 quiz/test 4
5 M & M ACTIVITY (worksheet one) YOU HAVE 15 MINUTES TO COMPLETE THIS ACTIVITY! GROUP NAME: STEP ONE: DO NOT EAT YOUR MATERIALS YET!!!! STEP TWO: Get with your group. Decide on a group name and choose who will fill the following roles during this activity: Group manager keeps group on task & organized responsible for materials make sure that all participate Presenter pours M&Ms at each step will demonstrate results to the class Counter counts M&Ms at each step checks to make sure all questions are addressed keeps track of time Recorder records results of each step fills in answers to questions (below) STEP THREE: Experiment. Count your M&M s and place them in the cup. This number is the initial sample size for pour #0. Pour the M&Ms on the desk. Remove all those that have a blank side facing up and count those that have an m. This number is the sample size for pour #1. Place the remaining M&M s (the ones you counted) back in the cup and repeat the process until there are no M&Ms left in the sample. (Save the M&Ms as you may want to repeat the experiment in a few minutes!) STEP FOUR: Results Answers to the following questions should be a group consensus. (You can write answers on the back of this sheet or on your own paper.) All group members should discuss the questions before deciding what answer the recorder should write down. Your teacher will be circulating as you work and full participation will count as a portion of your grade for this activity. The group presenter should be prepared to share the results with the class, but every group member should be prepared to help answer questions and/or support the presenter. 1) Record your results in some sort of table. You may do this by hand or using the graphing calculator. Be sure to label rows and/or columns appropriately. 2) Record your results by plotting points on some sort of graph. Be sure to label your axes. You may do this by hand or on the graphing calculator. 3) What observations can you make about the table and graph? Make a list of observations. Is there a pattern there? Can you find a way to express the pattern (if there is one) mathematically? 4) Which of (1) and (2) do you think best represents the experiment? Why? Write your explanation in words. 5) What did you expect to happen in this experiment? Why would this make sense? Did it happen? If not, in what way were your results different from what you expected? You may repeat the experiment to see if you get better results. STEP FIVE: Destroy the evidence! (You may eat your materials now the edible ones only!) 5
6 GRAPHING ACTIVITY (Worksheet two) YOU HAVE 10 MINUTES TO COMPLETE THIS ACTIVITY!! INSTRUCTIONS: Get with your group from last time. Choose roles again before starting. The presenter should be someone different than the presenter from last time. Group manager keeps group on task & organized responsible for materials make sure that all participate Presenter will demonstrate results to the class checks to make sure all questions are addressed Graph maker enters equations in the graphing calculator Copies graphs out onto OH Recorder fills in answers to questions (below) keeps track of time STEP ONE: Graph the three equations that are circled below. Be sure to adjust the window size so that you can see everything that is interesting about the graphs clearly. Copy the graphs out onto one set of axes on your overhead, being sure to label which is which. y = 2 x y = 3 x y = 4 x y = 1 x 2 y = 1 x 3 y = 1 x 4 y = 2 x y = 2 x +3 y = 2 x 3 y = 2 x y = 2 x + 4 y = 2 x 4 y = 1 x y =.9 x y = 1.1 x STEP TWO: In your group, discuss the following questions and write a few sentences answering each. You may use the back of this sheet or notebook paper for additional space. Again, your teacher will be checking participation as a portion of your grade. 1. Compare the graphs, which are obviously in the same family. How are they different from one another? How are they the same? List as many observations as you can find. 2. Find the domain and range for each graph. What do they have in common? How are they different? 3. Do the observations from number 1 make sense in terms of the equations for each graph? How is this similar to what we have seen in other types of graphs (parabolas, ellipses ) 6
7 The Allowance Battle (worksheet three) YOU HAVE 30 MINUTES TO COMPLETE THIS ACTIVITY!!! INSTRUCTIONS: Get with your group from last time. Choose roles again before starting. The presenter should be someone different than the presenter from last time. You will need to prepare and turn in an OH to use tomorrow in your presentation. Group manager keeps group on task & organized responsible for materials make sure that all participate Presenter will demonstrate results to the class checks to make sure all questions are addressed OH maker Copies graphs/ tables out onto OH Recorder fills in answers to questions (below) keeps track of time SCENARIO: Your allowance was $10 a week each month (a total of $40 most months), if you do all your chores every day. (Call this plan A) Every month, however, you miss at least a day a week of chores and you never end up getting anything, which doesn t seem fair to you. This past month, for example, you missed the 3 rd, 4 th, 11 th, 22 nd, and 27 th days of the month, and you didn t receive any allowance. You suggest to your mom that instead why not make it a dollar a day (for a total of $30, even less than she was willing to pay before) each day that chores are complete. (Call this plan B) You are thinking that this way, you will still at least get something even if you miss days. Your mom insists that what she is interested in is consistency, and says instead that she will only pay a penny for a single day, but that for each day that you do chores in a row, the payment is doubled. (Call this plan C) She is thinking that this way, you will have to do at least 8 days in a row to make a dollar, which is still better than she is getting out of you now. TASK: Choose a method to answer each of the following questions: You can use a calculator, a graphing calculator, or an Excel spreadsheet, or can do work by hand. You should be prepared to explain your answers and why your chose the method you did to the class next time. Answers to all questions should be written out and handed in at the end of this class. 1) How much money would you have made this past month by each plan? 2) How much money you would make if you did your chores every day for a month under each plan? 3) How many days in a row would you have to do chores with plan C to make it worthwhile to switch to plan C? 4) What pattern do you notice for each plan? Can you find a function that represents the amount of money earned compared to the number of days for each plan? 5) Which plan is the best for you? For your mom? 7
8 WARM UP ACTIVITY ONE: (on overhead) Recall the two ways we learned to find inverses of a function, one on a graph and one in the equation. 1) Use the graphical method to graph the inverse function of f (x) = 2 x, shown here. 2) Use the algebraic method to find the equation of the inverse of f (x) = 2 x. WARM UP ACTIVITY TWO: (on overhead) As we saw yesterday, the graph of f (x) = log 2 x looked like Based on what happened when we translated the exponential functions, sketch what you would expect these three functions to look like: f (x) = log 2 x + 4 f (x) = log 2 (x 3) f (x) = log 1 2 x 8
9 INSTRUCTIONS FOR EXPONENTIAL REGRESSION ON TI83: 1. Clear old lists Press 2 nd, mem, ClrAllLists, and Enter. This will clear out any old values left in lists 1 &2. 2. Enter data Press stat, edit to get to the lists. Enter your values in L1 and L2. Be sure not to enter any 0 values in L2. 3. Regression Press stat, calc, ExpReg, enter. This should display an exponential function that fits the data in L1 and L2. 4. To graph go to y= and clear out any old equations. While at y1, press vars, Statistics, EQ, RegEQ. This will insert the equation you just found into y1. You can then press graph to see the function. You may need to adjust the window for a better view depending on your data. 8
10 Date: GROUP GRADING FORM Group Name: Group Members absent, if any. Student & role Individual evaluation E = excellent G = good P = poor U = unsatisfactory Participated Fulfilled Was prepared Supported other fully at all times designated role for activity members of group Overall Grade Group Evaluation E = excellent G = good P = poor U = unsatisfactory Worked well together. Got started promptly and observed time constraints. Presenter well prepared with OH written out clearly. Presenter (& group members) explained all steps of work. Group supported presenter as necessary. Presenter added notes as necessary. Group asked questions and supported other groups presentations. Other notes: 8
11 GROUP WORK / SELF EVALUATION: Please answer the questions fully and bring with you to class tomorrow. The answers you give will not affect your grade, but you will receive bonus points on your quiz for complete, sincere answers. 1) Explain how you felt about the way your group worked together during the activities this week. 2) Is there anything you personally would do differently the next time you work in a group? 3) What is one thing we have discussed about exponential and logarithmic functions that really makes sense to you? 4) What is one thing we have discussed about exponential and logarithmic functions that you could understand better? 5) Do you have any questions I should know about before the quiz tomorrow? 8
12 QUIZ EXPONENTIAL AND LOGARITHMIC FUNCTIONS (page 1) Name Date FOR QUESTIONS (1) AND (2) YOU MAY NOT USE A CALCULATOR. WHEN YOU FINISH THESE TWO QUESTIONS, CALL YOUR TEACHER OVER AND SHE WILL GIVE YOU THE SECOND PAGE OF THE QUIZ. 1) a) On the graph below, graph these three functions. Be sure to label which is which. Plot at least two definite points for each graph. Hint: the first one should help with the second two!! y = 3 x y = 1 x y = 3 x 5 3 b) For y = 3 x, domain =, range = 2) a) On the graph below, graph these two functions. Be sure to label which is which. Plot at least two definite points for each graph. Hint: #1 may help with this!! f (x) = log 3 x f (x) = log 3 (x +10) b) For f (x) = log 3 x, domain =, range = 8
13 QUIZ EXPONENTIAL AND LOGARITHMIC FUNCTIONS (page 2) Name Date FOR THE REST OF THE QUIZ YOU MAY USE THE CALCULATORS. 3) Find the following: 4) Change to exponential form: Log 3.5 log = 4 ln 7 Change to logarithmic form: 3 7 = ) Explain in words why it makes sense that f (x) = 1 x is not considered an exponential function. (A picture may help, but a picture alone is not enough, you should explain what it means.) 6) Use the log rules to simplify the expressions: show all steps of your work!! log 3 x log log 3 12 log 2 x + log ) Enter the following data points in your calculator and use exponential regression to find the best fit exponential equation for this data. Data: (0,58), (1,18), (2,6), (4, 1 2 ) Equation: ( round decimals to three places) bonus: What is the inverse of the function f (x) = log 1 2 x? Show all your work and explain your thinking in words. 8
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