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Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public SchoolsBackground InformationSubject:Algebra IIGrade Band:9-12Duration:90 MinutesOverview:Students will discover patterns within logarithmic functions, thus discovering logarithmic properties. They will analyze and apply logarithmic functions to sound frequencies and earthquakes. A STEM Specialist will help students interpret their findings with the properties of logarithms as well as engage students in hands-on learning experiences that allow them to draw connections between course content ant the work performed by STEM professionals. Background Information:Teachers should familiarize themselves with the Properties of Logarithms:logAB=logA+logBlogAB=logA?-logBnlogA=logAnlogb1=0logbb=1logbbx=xblogbx=xlogx1xa=-aAdditionally, teachers should understand the scales for both decibels and magnitude and how they relate to logarithms.dB1102030405060708090100Sound Intensity1001011021031041051061071081091010The?Richter scale was developed to assign a single number to quantify the energy released during an?earthquake. The scale is a?base-10?logarithmic scale. The magnitude is defined as the logarithm of the ratio of the?amplitude?of waves measured by a?seismograph?to an arbitrary small amplitude. An earthquake that measures 5.0 on the Richter scale has a shaking amplitude 10 times larger than one that measures 4.0, and corresponds to a 31.6 times larger release of energy.STEM Specialist Connection:The STEM Specialist can:Help students decipher patterns between logarithmic expressions. Engage students in hands-on learning experiences on the application of logarithms in the workplace. STEM Specialists can be found at Enduring Understanding:Logarithms allow us to solve for exponential equations in which the variable is in the exponent.Logarithmic functions can be used to model seismic activity, sound intensity, magnitude of earthquakes, compute compound interest in terms of time, calculate population growth, radioactive decay, etc. Logarithms were developed to multiply large numbers without the use of a calculator.Essential Questions:How can logarithms be used to solve exponential functions in which the variable is found in the exponent?How do STEM professionals apply logarithmic functions to develop solutions to problems?Student Outcomes:Students will be able to:Convert exponential functions to logarithmic functions and vice versa.Solve base 10 logarithms.Apply the properties of logarithms to simplify and expand different logarithmic functions.Interpret and solve real world situations in which logarithms would be necessary to solve the problem.Product, Process, Action, Performance, etc.:Students will work independently to derive the properties of logarithmic functions and apply them to real-world problems. Audience:?Peers ?Experts / Practitioners ?Teacher(s)?School Community?Other______Standards Addressed in the Unit:Domain: Modeling with FunctionsCluster: Construct and compare linear, quadratic, and exponential models and solve problemsStandard: F.LE.A.4?For exponential models, express as a logarithm the solution to?abct=d?where?a,?c, and?d?are numbers and the base?b?is 2, 10, or?e; evaluate the logarithm using technology.?Cluster: Build a function that models a relationship between two quantities.Standard: F.BF.A.1: Write a function that describes a relationship between two quantities.Suggested Materials and Resources:Equipment:computer with internet accessprojectorScientific or Graphing calculatorComputers for each student (optional)Earphones (optional)Website** Students are linked to an online site for the engagement activity. The site has been chosen for its content and grade-level appropriateness. Teachers should preview website before introducing the activities to students and adhere to their school system’s policy for internet use.People, Facilities:STEM SpecialistComputer lab (optional)Materials (rubrics, worksheets, PowerPoints, answer keys, etc.):Logarithms PowerPointDiscovering Properties of LogarithmsHomeworkHomework (Answers)CalculatorComputer (with internet access)ProjectorLearning Experience5E ComponentIdentify the 5E component addressed for the learning experience. The 5E model is not linear.DetailsStandards for Mathematical Practice?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Computer with internet accessProjectorLogarithms PowerPointIndividual Computers for students (optional)Earphones (optional)Preparation:Explore on your own. Understand how the website works and make a decision whether you want your students to explore the site individually or as a class.Familiarize yourself with decibels and sound intensity. Be able to explain the different scales to your students.Use PowerPoint slides 1-4 as a guide through the engagement portion of the lesson.Facilitation of Learning Experience:Begin your lesson with the Dangerous Decibels Activity (). Have students—individually or as a class—explore the website. You can choose to do as many (or as little) activities as your class time allows. Be sure not to leave out the “measuring sound” section as that is the piece that explains the decibel scale. Students should be carefully monitored during this activity since the sounds can be pretty loud.If you choose to allow the students to explore the website on their own, you will need to monitor their use of time in each section.As a class, discuss the results from each section of the Dangerous Decibels activity. If you are completing the entire activity as a class this should be done throughout. If the students are doing this activity individually, you will need to discuss each element of their exploration. Compare and contrast how each student did with the activity.Continue your discussion on decibel levels by showing the next 3 graphics on the PowerPoint—infographic (bubbles), lesser water boatman graph of noise levels, and the frequencies heard by different animals.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.?Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:ComputerProjectorLogarithms PowerPointPreparation:Use PowerPoint slides 5-11 as a guide through the explanation section. Facilitation of Learning Experience:Discuss the origin of logarithms—where they come from, what they were (still are) used for, etc.Be sure to indicate how you switch from logarithmic to exponential form and vice versa. The equation for this process is found at the bottom of slide 8. Continue this discussion by working through a few examples with your students (found on the next two slides).Next, discuss the common bases used in everyday life (Base 10, 2, and e)Ask the students when they have used logarithms (such as in chemistry class) and why logarithms were used.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Discovering Properties of Logarithms worksheetPencil/PenCalculatorComputerProjectorLogarithms PowerPointPreparation:Use PowerPoint slides 12-17 as a guide through the exploration plete the Discovering Properties of Logarithms worksheet in advance to familiarize yourself with the patterns the students should discover.Facilitation of Learning Experience:Provide each student with a copy of discovering properties of logarithms worksheet. Allow students to work in partners to complete the assignment. Students should notice the patterns that develop in the first column between addition and multiplication. They should also notice the pattern between subtraction and division in the second column. Finally, they should notice that exponents are directly related to multiplying by a coefficient of the same numerical value.Discuss as a class the discoveries that were made among the pairs in the class.Continue that discussion with slides 12 - 15 of the PowerPoint. The properties the students should have discovered are present here. There are a few other properties that should be noted in the student’s notes that were not discovered with the activity. Be sure they are included in their notes as shown on slide 15.Use these properties to work through the next two examples as a class. These examples are worked through on slides 16 & 17.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Logarithms PowerPointComputerProjectorPreparation:Use slides 18 – 23 as a guide through the extension section of the lesson. Facilitation of Learning Experience:As a class, work through the next two examples (slides 19-23) together. The first example revisits the idea of decibels and asks the students to determine just how much louder one decibel level is than another in terms of relative intensity.Review the earthquake graph with your students. Point out different magnitude levels and show them the comparable energy equivalent. Many will be surprised by the amount of energy emitted from an earthquake and the comparable events.The earthquake example compares two different earthquakes magnitudes and asks the students to solve just how much more energy one earthquake had than the other.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Logarithmic Functions HomeworkLogarithmic Functions Answer KeyPencil/PenCalculatorPreparation:Contact the STEM Specialist in advance to co-plan the learning experience for students. STEM Specialist can be found at w. A description of the ability level of the students, as well as some of the prior knowledge your students may have of logarithmic functions may be helpful to the STEM Specialist prior to the visit.Facilitation of Learning Experience:Have the STEM Specialist engage students in the predetermined learning experience emphasizing how logarithmic functions are used in the workplace.After the STEM Specialist has engaged students in the learning experience, provide each student a copy of the homework assignment to work on independently after class. At the start of the next class, you can review answers or collect papers for grading. ?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.Supporting InformationInterventions/EnrichmentsIdentify interventions and enrichments for diverse learners.Struggling LearnersStudents should be provided with copies of notes or a note guide to follow along with the PowerPoint. Additionally, students could work with the teacher or STEM specialist in the discovery activity to better guide them to the patterns present. A list of the different properties of logarithms can (should) be provide to struggling learners to help aide them in solving difficult equations.English Language LearnersSimilarly to struggling learners accommodations should be provided. A vocabulary list may be helpful to ELL students (logarithm, exponent, magnitude, decibel, etc.) Many words in this lesson are comparable to European languages because of the origin of the words therefore, vocabulary may be helpful but is not necessarily needed in all ELL students.Gifted and TalentedExpand student knowledge by providing more complex, challenging questions. Begin with the basics that all students are learning. Give them more advanced questions as practice problems if they should finish early. Match these students with some of the struggling learners for the discovery lesson to aide those students as well as enhance the understanding of your gifted and talented students. As an additional extension, have the gifted and talented students research magnitudes of different earthquakes around the world and write their own example problems (providing the answers and work along with it).Investigation Part OneUse your calculator to find the following common logs:Investigation Part TwoUse your calculator to find the following common logs:Investigation Part ThreeUse your calculator to find the following common logs:Write down any patterns or connections you recognize between your answers. What relationships can you identify?Develop a general rule to write as a single logarithm, for all positive values of A and B.Develop a general rule to write as a single logarithm, for all positive values of A and B.Develop a general rule to write as the logarithm of a single constant, for all positive values of A and all real values of n.Test your general rule on several other values for A and B.Test your general rule on several other values for A and n.Logarithmic Functions HomeworkWrite the following equations in logarithmic/exponential form53=125log100.00001=-5777615=6log328=35Solve each equation.log4x=32log1327=xlogb1024=5log7x+6=log78x+20log45+log4x=log460log2q-log23=log27log2x+4-log2x-3=3log510+log512=3log52+log5aAn equation for loudness, in decibels, is L=10log10R, where R is the relative intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the relative intensity of 120 decibels? An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading m is given by m=log10x, where x represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than on the measures 3.5?Rob is solving a problem involving logarithms. He does everything correctly except for one thing. He mistakenly writes log2a+log2b=log2(a+b). However, after substituting the values for a and b in his problem, he amazingly still gets the right answer! The value of a was 11. What must the value of b have been?-287655-20256500An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading m is given by m=log10x, where x represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than on the measures 3.5?41148471745434137527127023359212719830329956071461032476127186423234544779503240268721018321553672220632226647676231751447557431639487319834043567-70123909287-815323746567-1020523346967-1261722832167-102052Rob is solving a problem involving logarithms. He does everything correctly except for one thing. He mistakenly writes log2a+log2b=log2(a+b). However, after substituting the values for a and b in his problem, he amazingly still gets the right answer! The value of a was 11. What must the value of b have been?295096718162833505679511332360871188733111167-2015272855927-5932726186872495532583047142633220288773 ................
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