Date: Teacher: Vildan Sertkaya Number of Students: 21 Grade Level: 10 Time Frame: 70 Exponential Equations

Transcription

1 Date: Teacher: Vildan Sertkaya Number of Students: 21 Grade Level: 10 Time Frame: 70 Exponential Equations 1. Goal(s) Students will understand how to solve exponential equations. Students will understand how to solve logarithmic equations. 2A. Specific Objectives (measurable) Students will be able to solve exponential and logarithmic equations by applying the Oneto-One and Inverse Properties. Students will be able to solve common logarithmic problems by applying the properties of logarithms 2B. Ministry of National Education (MoNE) Objectives İD Üstel ve Logaritmik Denklem ve Eşitsizlikler Terimler: Üstel denklem, logaritmik denklem İD Üstel ve logaritmik denklemlerin ve eşitsizliklerin çözüm kümelerini bulur. Üstlü denklemlerin logaritma kullanılarak çözümü yapılır. İD Üstel ve logaritmik fonksiyonları gerçek/gerçekçi hayat durumlarını modelleme ve problem çözmede kullanır. Nüfus artışı, bakteri popülasyonu, Moore yasası, bileşik faiz, radyoaktif maddelerin bozunumu (yarı ömür), fosil yaşlarının tayini, deprem şiddeti (Richter ölçeği), ph değeri, ses şiddeti (desibel) vb. örnekler bağlamında üstel büyüme/azalma veya logaritmik ölçek ile modellenebilecek problem durumlarına yer verilir. 2C. NCTM-CCSS-IB or IGCSE Standards: Numbers and Operations Algebra Develop a deeper understanding of very large and very small numbers and of various representations of them. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions. 3. Rationale Students will know how the big numbers calculate. The logarithm link to biology, chemistry, physic subjects with the numbers of bacteria, molecules, cells and atoms. 4. Materials Board markers Smart Board or projector.

2 TI-Calculator. TI- Smart Viewer GeoGebra 21 Copies of Worksheets 5. Resources Mathematics Standard Level, Course Companion, OXFORD. Mathematics for International Students, Mathematics SL, Hease & Harris Publications. 6. Getting Ready for the Lesson (Preparation Information) Before the class time starts, the teacher will write the agenda on the left side of the board. At the beginning of the lesson the teacher will want students to come closer and sit nearer the board. Then the teacher will distribute the name cards which were prepared before the class time. Before the teacher distributes the worksheet she will give the instructions and wants students to study with nearer them and their groups must consist of two students. The teacher will distribute all worksheets to the students one by one. 7. Prior Background Knowledge (Prerequisite Skills) Students know the definition of a logarithm and properties of exponents. Students know how to use TI-Calculator and how to calculate logarithm on it. Students know linear equations and solving linear equations and finding x. Students know laws of logarithm. Lesson Procedures Transition: According to my observations last week you told about the laws of logarithms and change of base rule. Let's do some review and remember what the rules are. First of all do you remember any laws about logarithm? 8A. Engage (10 minutes) The teacher will write the title of Review on the board and start to remind the laws of logarithm and changing base rule. While reminding, the teacher will ask questions to the students and construct each steps of laws with students: " The laws of exponents lead to the following laws of logarithms. Here we assume x and y are positive real numbers The teacher will mentioned about the countries demography and how it can calculate. The teacher will mention about the France and England population and their population growth rate and with which function the population can calculate.

3 Transition: Let's look at the problem. Now, read the problem and understand what it says and underline some words which is important by solving the problem for you. B. Explore (15 minutes) The teacher will reflect the problem about population growth rate on the Smart Board: " According to a research the population in France is approximately 60 million and in England it is about 65 million. The population growth rate in France is 0.57% while in England it is 0.23%. Considering the same growth rate will not change by then, after x years later than 2014 the population of France could be found out through model and population of England could be figured out through model. Based on this information find how many years later the population of England and France will be the same? " After students read the problem themselves the teacher will want one of the students to tell the problem. Then the teacher asks " What the problem asks us? " "How can you calculate the year the population of two countries are same." " Can we plot these two functions?" "If we plot the graph of these two functions what can we expect?" " If they are intersecting how can we find the intersection points?" The teacher plot the functions of and on the Geo Gebra. Then the students will see these two functions are intersecting. After that the teacher will ask " How can you find this intersection point algebraically?" If the students' response is " we must equalized the two functions because on the intersection point x and y value at the intersection point are the same and we need to find the year so we need to find the x value and for this reason we should equalize two functions". If students cannot give this response the teacher will give a clue and "Look at the intersection point and see the x and y value of these two functions are the same at intersection point and you want to find the year so which value should you find? x or y?". Then students will recognize they need to find x value and so they need to equalize these two functions. After students equalize the functions they realize that they should find x and so they need to bring down the powers and the teacher will attract the attention of students on the part of review and ask " Which law must you apply to bring down the powers? " and students realize they need logarithm functions. When students realize they need the logarithm functions students will estimate they need to take logarithm both sides of the equations. Most probably students will ask in which base they need to take logarithm both sides. When they ask this question the teacher will remind them the "e" in the equation and ask " what do you think about taking "ln" for both sides of the equation. After that the teacher will give 3 minutes to the students to solve the equation. Students will come up with the solution of. Then the teacher will want students to use their TI-Calculator and simplify the x. After students find the answer the teacher will show the intersection point on the

4 GeoGebra and students will check their answer and ensure they find the write answer Transition: As you see in this problem you can use logarithms to solve exponential equations. Let's do some exercises. C. Explain (15 minutes) The teacher will reflect the first exercise on the smart board. " Solve " The teacher will give 2 minutes to students to provide them to understand, think and solve the question. While students were solving the question the teacher will move around the classroom and observe them. If they need help the teacher will help them but work only as a guide. For the first question the teacher solves it on the board but asks students each steps " What I need to apply now?". For each steps the teacher will write what she does near the steps by using parenthesis. Then the teacher will want students to check their answers on the graph calculator. By using their TI-calculator, the teacher wants them to plot the graphs of both sides of the equation and find the intersection point. For providing students to plot the graphs and find the intersection point the teacher will show each steps on the TI-Smart Viewer. The teacher will reflect second question on the smart board. " Solve " The teacher will give 2 minutes to the students to solve the question and move around the classroom and observe students' studies. According to observations, the teacher will want one of the students who find the right answer to solve the question on the board. Then the teacher will want students to check their answers on the graph calculator. By using their TI-calculator, the teacher wants them to plot the graphs of both sides of the equation and find the intersection point. While students plotting the graphs and finding intersection points the teacher will observe them and after 2 minutes the teacher will again show each steps on TI-Smart Viewer. The teacher will reflect the third question on the smart board. " Solve, giving an exact answer. " The teacher will give 3 minutes to the students to solve the question. While the students are solving the question the teacher will observe them and after three minutes the teacher will want one of the students to solve the question on the board. For the exact answer, the teacher will remind students to leave their answer in log form since it requires the exact answer. The teacher will reflect the last question on the smart board. " Solve, giving your answer in the form, where a, b Z." The teacher will give 3 minutes to the students to solve the question. While the students are solving the question the teacher will observe them and after three minutes the teacher will want one of the students to solve the question on the board. After student solves the question the teacher will remind they cannot simplify any further,.

5 Transition: I shall distribute you worksheets but before I distribute them make groups with your desk mates who sit nearer you and solve the questions together and do not forget to check your answers. D. Extend (20 minutes) The teacher distributes the sheets and while they are studying the teacher will observe them. If students need help the teacher help them but only as being a guide. Transition: Okay guys, I prepared you another sheets and on these sheets you have only one question and I want you to do it individually. You have 5 minutes and after 5 minutes I will take your sheets. Use your time carefully. E. Evaluate (10 minutes) The teacher will distribute the sheets and observe students. 9. Closure & Relevance for Future Learning After the teacher take the evaluation sheets, the teacher will ask " What did you learn today? " "What do you think about the lesson?" " Can you remind me the laws of logarithm?" " When you come across the exponential equations what do you do to solve it?" 10. Specific Key Questions: Can you explain what the problem asks us? [Comprehension] How can you calculate the year which the population of two countries are same? [Analysis] Can we plot these two functions? [Application] If we plot the graph of these two functions what can we expect? [Synthesis] If they are intersecting, how can we calculate the intersection points? [Analysis] How can you calculate this intersection point algebraically? [Analysis] Which law must you apply to bring down the powers? [Application] What I need to apply now? [Application] 11. Modifications If students cannot use the calculator the teacher will show the steps. If students cannot do the exercises the teacher will give clues and help. If students cannot remember the laws or properties of logarithms the teacher will remind. If the students cannot understand the difference between equations and functions the teacher will tell them.

6 EXPONENTIAL EQUATIONS WORKSHEET 1. Solve to find the value of x to 3 significant figures. 3. Solve for x. (a) 2. Solve these equations to find the value of x in the form, where a, b Q. (a) (b) (b) (c) Mathematics Standard Level, Course Companion, OXFORD. Mathematics for International Students, Mathematics SL, Hease & Harris Publications.

7 Name: Date: Exponential Equations 1. (a) Solve these equations to find the value of x in the form, where a, b Q. (b) Write at least three sentences about the lesson for example, what did you learn? What did you remember from your previous knowledge? Which part was difficult for you and how can you overcome these difficulties? What do you think whether this lesson was helpful for your understanding?

8 PROBLEM: According to a research the population in France is approximately 60 million and in England it is about 65 million. The population growth rate in France is 0.57% while in England it is 0.23%. Considering the same growth rate will not change by then, after x years later than 2014 the population of France could be found out through model and population of England could be figured out through model. Based on this information find how many years later the population of England and France will be the same? PROBLEM: According to a research the population in France is approximately 60 million and in England it is about 65 million. The population growth rate in France is 0.57% while in England it is 0.23%. Considering the same growth rate will not change by then, after x years later than 2014 the population of France could be found out through model and population of England could be figured out through model. Based on this information find how many years later the population of England and France will be the same?