Understanding By Design Unit Template



Understanding By Design Unit Template

|Title of Unit |Exponential and Logarithmic Functions |Grade Level |10th-11th |

|Curriculum Area |Honors Algebra 2 |Time Frame |January 22nd-February 3rd |

|Developed By |Kelly Maisel |

|Identify Desired Results (Stage 1) |

|Content Standards |

| F-IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. |

| F-IF.8b. Use the properties of exponents to interpret expressions for exponential functions. |

| F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). |

| F-BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. |

|F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a |

|table). |

|F-LE.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. |

|Understandings |Essential Questions |

|Overarching Understanding |Overarching |Topical |

|-Understanding exponential growth and decay and compound interest, and the formulas relating to them. |-What is the main connection between |-What types of real world relationships are |

|-The importance of the natural log, e, and related to interest. |logarithmic functions and real world problems? |best described using a logarithmic scale?  |

|-Logarithmic functions and their purpose. |-When is compound interest used, and what form |-What relationships exist between a function |

|-Properties of logarithmic functions relating to exponents. |of interest offers the best deal? |and its inverse? |

|-Applications of logarithmic functions in real world situations. |-What other subjects, and jobs, use exponential|-What are the similarities and differences |

| |growth problems? |between exponential and logarithmic functions?|

| | |-How do we derive the exponential growth and |

| | |decay formulas from the exponential function? |

|Related Misconceptions | | |

|Students may confuse what the initial value and growth rate are for growth and decay problems. Other confusion | | |

|would be related to switching growth problems with different measures of time. Issues may come up about | | |

|converting between exponential and log functions. More confusion about when to add, subtract, multiply and | | |

|divide the log functions. Even though students will be able to convert from exponential to logarithmic | | |

|functions, they will have trouble proving they are inverses. | | |

|Objectives |

|Knowledge |Skills |

|Students will know… |Students will be able to… |

|-Know how to determine if a graph is representing growth or decay. |-Graph growth and decay exponential functions and determine intercepts and asymptotes. |

|-How compound interest and continuous interest works in real world problems. |-Manipulate and determine values of money with various forms of interest. |

|-The significance of logarithmic functions, and be able to use their properties in model and real |-Use different properties of logarithmic functions to manipulate equations in order to find solutions. |

|word situations. |-Identify applications of logarithmic functions and produce feasible solutions. |

|-Where logarithmic functions are found in real life, and what role they play in different | |

|situations. | |

|Assessment Evidence (Stage 2) |

|Performance Task Description |

|Goal |To assess student understanding of exponential and logarithmic functions by providing a unit test to look at which skills students are proficient in, and which|

| |skills are the lacking proficiency. |

|Role |To conclude the unit on exponential and logarithmic functions, and allow students to show their proficiency in the topics covered over the time span spent |

| |learning the subject material. |

|Audience |Students will perform the tasks given on the unit test, and will be given to the teacher for grading. |

|Situation |An hour long unit test given at the end of the unit where students will be working individually. Once completed students will turn in for credit. |

|Product/Performance |Unit Test on exponential and logarithmic functions. Students will complete a unit test for exponential and logarithmic functions. Majority of the questions |

| |will be short answer responses asking students to create functions and analyze various questions based off of exponential and logarithmic word problems |

| |relating to growth and decay. Other questions will consist of the different properties of logarithmic functions. Lastly, questions involving applications of |

| |logarithmic functions will be included. |

|Standards | F-IF.7e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and |

| |amplitude. |

| | F-IF.8b. Use the properties of exponents to interpret expressions for exponential functions. |

| | F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal |

| |descriptions). |

| | F-BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. |

| |F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two |

| |input-output pairs (include reading these from a table). |

| |F-LE.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. |

|Other Evidence |

|Students will be assessed throughout the unit with quizzes on each of the topics covered. These quizzes will allow students to see what they understand and what they are struggling with. Students will also |

|complete a variety of homework assignments, and those will be turned in on quiz days for points. |

|Learning Plan (Stage 3) |

|Day in Unit |Lesson Topic |Lesson Learning Objective |Description of how lesson contributes to unit-level |Assessment activities |

| | | |objectives | |

|One |Graphing |An introduction to exponential functions and |This introduction will show different models of |Students will have the opportunity to explore |

| |Exponential |graphing exponential functions. How different |exponential graphs. It will provide information such |exponential functions with an introduction to the |

| |Functions |variables effect the graph of y=abx including |as intercepts, asymptotes and transformations of |checkerboard and rice problem. From there students |

| | |transformations. |y=abx. A definition of exponential functions will be |will receive a definition of exponential functions. |

| | | |provided. |They will then have the opportunity to explore |

| | | | |different variations on y=abx. Some groups will work|

| | | | |with graphing different a values, while others will |

| | | | |work with graphing different b values. At the end of|

| | | | |the lesson we will come together to produce a |

| | | | |general idea. |

|Two |Exponential Growth |When looking at a word problem about exponential |This lesson will provide students with hands on |To start the class, we will have a review of what it|

| |and Decay |growth and decay, students will be able to properly|experience creating exponential growth and decay |means to be an exponential function. After that the |

| | |determine when a problem is related to growth, or |models with the use of manipulatives. The lesson will|class will split into groups working on the |

| | |when it is related to decay. Also, to further |show how growth and decay occurs and how to create a |“Skittles experiment” of growth and decay. Once the |

| | |confirm what the initial value, growth rate and |function based upon the model relating to growth or |class finished the experiment we will come together |

| | |time represent in y=abx |decay. |to discuss the results. At the end of the hour there|

| | | | |will be sample problems for us to talk about as a |

| | | | |class that will be homework as well. |

|Three |Compound Interest |The idea of compound interest will be discovered, |Working with compound interest will provide students |At the beginning of the hour students will talk |

| | |along with the various ways something can be |with a sense of information regarding their future. |about what they know about compound interest. From |

| | |compounded. This will be modeled in real world |They will be able to manipulate formulas based on how|there we will compare and contrast simple interest |

| | |situations involving money and investments. The |their money is compounding. Students will be able to |to compound interest. Students will hopefully |

| | |compound interest formula will be compared and |understand how the compound interest formula branches|discover a pattern that looks similar to the |

| | |contrasted to the exponential equation of y=abx. |from the exponential equation of y=abx |exponential growth and decay problems, along with |

| | | | |the exponential equation. We will go through |

| | | | |different types of compounding to show how each has |

| | | | |an effect on a certain initial value. |

|Four |Review of graphing,|Students will receive a packet of review material |This meets the unit goals by giving a comprehensive |Students will have the entirety of the hour to work |

| |growth and interest|to work as a whole group, individually and with a |review of all the topics covered in the past 3 days |on their review and get any questions answered. The |

| | |partner to clear up any questions pertaining to |on exponential functions. |solutions to the review questions will be given at |

| | |graphing exponential equations, exponential growth | |the end of the hour. |

| | |and decay and compound interest. | | |

|Five |Natural base, e |The natural base e will be derived from working |Students will see the natural base e, and be |Students will have an example of compound growth, |

| | |with compound interest of a fixed amount. There |comfortable working with continuous growth problems. |and we will continue to grow the amount of times |

| | |will be an understanding of how the b value is now |This will lead into the natural logarithm. |compounded until it grows continuously and students |

| | |equal to e. | |will notice that it will equal the natural base e. |

|Six |Intro to |We will discover what logarithmic functions are, |When working on the unit of exponential and |Students will receive a warm up activity that has |

| |logarithmic |and how they relate to exponential functions. |logarithmic functions, it is noted that there will be|them explore logarithmic functions without them |

| |functions |Students will come to understand where the |a connection made between the two concepts. This is |realizing. They will actually perform the |

| | |relationship comes from. |an introduction to logarithmic functions, and how it |operations, and will not know what they are doing |

| | | |relates to previous material covered. |until the end of the hour where the connections are |

| | | | |made. |

|Seven |Properties of |Since logarithmic functions are exponents, the |We will discover when log functions are equal to each|Students will have a review on the properties of |

| |logarithmic |properties of log functions are similar to the |other, when you add, subtract, multiply or divide |exponential functions and their properties, and will|

| |functions |properties of exponential functions. This will |functions and the shortcuts provided that are similar|have to adapt their prior knowledge of the |

| | |connect to the ideas of how log and exponential |in exponential functions when we solve these |properties to adapt them to equations involving |

| | |functions are similar. |problems. |logarithmic functions. |

|Eight |Log functions and |Up until this point, we have not talked about an |It is important to know the true reasoning behind why|Students will have an exponential and log function |

| |their inverses |exponential and a log function being inverses. This|you change from log to exponential and back. So by |to explore. The students will not know that these |

| | |will prove that what happens when you change an |showing that these functions are inverses by |functions are inverses. They will have to use their |

| | |exponential equation to a logarithmic equation is |graphing, value table and compositions they will |prior knowledge on proving inverses to determine if |

| | |really taking the inverse. |understand the relationship. |the two functions truly are inverses. |

|Nine |Power properties of|The final property of log functions is power |This is important for students when providing them |The class will go through a variety of examples and |

| |log functions |properties, where students will discover why |with short cuts to the problems because it expands |sample problems in order for them to come up with a |

| | |exactly they are able to bring down the exponent. |the critical thinking process, and allows more |property that works for all powers when working with|

| | |Together the class will come up with a general |challenging problems for students to face. |logs. From there students will use the property they|

| | |property that allows this to happen. | |created to solve additional problems. |

|Ten |Applications of log|Various applications of log functions include |Real life examples give meaning to the unit as a |A brief introduction to why logs were created, and a|

| |functions |multiplying large numbers, measuring earthquakes |whole, when prior to learning the applications |reminder to what a log is will help students realize|

| | |and intensity of sound. Students will see real life|students were simply solving problems. Now there is a|why we work with logs. Students will explore |

| | |examples where logarithmic functions play a key |reason for log functions to exist, and students will |different earthquakes and sound intensity and |

| | |role. |explore. |discover how log functions are used every day. |

|Eleven |Review for unit |Students will receive a packet of review material |This meets the unit goals by giving a comprehensive |Students will have the entirety of the hour to work |

| |test |to work as a whole group, individually and with a |review of all the topics covered in the past 10 days |on their review packets and get any questions |

| | |partner to clear up any questions. |on exponential and logarithmic functions. |answered. The solutions to the packet questions will|

| | | | |be given at the end of the hour. |

|Twelve |Test Day |Students will test on all the topics covered in the|The test will contain questions on graphing |Students will have the entirety of the hour to |

| | |past 12 class periods. |exponentials, growth rate, compound interest, e and |complete the Unit test on Exponential and |

| | | |logarithmic functions and their properties. |Logarithmic Functions. |

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