Exponential Functions Lesson PDF

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Lab to help understand how to work with exponential functions...

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STEM-Centric Lesson Exponential Functions Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public Schools Background Information Subject: Identify the course the unit will be implemented in.

Grade Band: Identify the appropriate grade band for the lesson.

Duration: Identify the time frame for the unit.

Overview: Provide a concise summary of what students will learn in the lesson. It explains the unit’s focus, connection to content, and real world connection.

Background Information: Identify information or resources that will help teachers understand and facilitate the lesson.

STEM Specialist Connection: Describe how a STEM Specialist may be used to enhance the learning experience. STEM Specialist may be found at http://www.thestemnet.com/

Enduring Understanding: Identify discrete facts or skills to focus on larger concepts, principles, or processes. They are transferable - applicable to new situations within or beyond the subject.

Algebra II 10-12 One 90 minute class period In this lesson, students learn the real-world application of exponential models. In subsequent lessons they will learn how to solve exponential and logarithmic equations as well as base e, natural logarithms, and exponential decay and growth problems. Teachers should understand how exponential functions model both growth and decay in our everyday world. Applications of exponential decay can be found with the following examples: medications/caffeine leaving the body, radioactive decay, half-life, carbon dating, depreciation of material objects, etc. Applications of exponential growth can be found with the following examples: Population growth, bacteria growth, appreciation of material objects, compound interest, etc. The STEM Specialist can:  Help students interpret the results of their dice activity.  Engage students in a discussion of how exponential growth/decay can be found in our everyday lives.  Encourage students to look for patterns in the real-world application problems.  Discuss careers in which application of exponential models is prevalent (Biology, Anthropology, Banking, Investments, Financial Advisors, Real-Estate Agents, Census workers, Food Production, etc.)  

Mathematical models are used to develop solutions to real-world problems. Exponential models carefully define the percent rate of change in real-world applications.

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STEM-Centric Lesson Exponential Functions Background Information Essential Questions: Identify several open-ended questions to provoke inquiry about the core ideas for the lesson. They are grade-level appropriate questions that prompt intellectual exploration of a topic.

Student Outcomes: Identify the transferable knowledge and skills that students should understand and be able to do when the lesson is completed. Outcomes must align with but not limited to Maryland State Curriculum and/or national standards.

Product, Process, Action, Performance, etc.: Identify what students will produce to demonstrate that they have met the challenge, learned content, and employed 21st century skills. Additionally, identify the audience they will present what they have produced to.

Standards Addressed in the Unit: Identify the Maryland State Curriculum Standards addressed in the unit.

1. How can exponential functions be used to model real-world problems and solutions? 2. How does a STEM professional use exponential functions? Students will be able to: 1. Graph exponential functions expressed symbolically and show key features of the graph by hand and using technology. 2. Determine if a graph/equation is representing exponential growth or exponential decay. 3. Explain the application of exponential models in real-world situations. Audience: ☒Peers  Students will accurately graph and explain the results of the ☒Experts / dice activity. Practitioners  Students will think through a problem and persevere in solving ☒Teacher(s) an exponential equation given bank account information over a ☐School period of time. Community  Students will analyze a situation (exponential model) and make ☐Online sense of the situation by solving and explaining the solution. Community ☐Other______ Common Core Algebra II Standards: Domain: Modeling with Functions  Cluster Statement: Interpret expressions for functions in terms of the situation they model. Standard F.LE.5 Interpret the parameters in a linear or exponential function in terms of context. 

Cluster Statement: Construct and compare linear, quadratic, and exponential models and solve problems. Standard 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 (including reading these from a table).

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Cluster Statement: Analyze functions using different representations

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STEM-Centric Lesson Exponential Functions Background Information Standard F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 

Suggested Materials and Resources: Identify materials needed to complete the unit. This includes but is not limited to websites, equipment, PowerPoints, rubrics, worksheets, and answer keys.

Cluster Statement: Analyze functions using different representations. Standard F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Equipment:  computer with internet access  projector Websites*  Magnitude of an Earthquake (http://www.youtube.com/watch/?v=nxqRQgRAe9o)  The Science of Overpopulation (http://www.youtube.com/watch?v=dD-yN2G5BY0)  Futurama video clips: (http://threeacts.mrmeyer.com/frysbank/) This particular website has been edited into video clips for convenience in the classroom. Those clips can be found as media clips in the PowerPoint and/or file folder associated with this lesson. * The sites have been chosen for their content and grade-level appropriateness. Teachers should preview all websites before introducing the activities to students and adhere to their school system’s policy for internet use. Materials:  Dice (36 per group of 2-4)  Dice Activity  Exponential Functions PowerPoint  Fry's Bank Account Video Questions  Fry’s Bank Account Video Questions Answer Key  Fry's Bank Account (First Clip)  Fry's Bank Account (Second Clip)  Exponential Functions Homework  Exponential Functions Homework Answer Key

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STEM-Centric Lesson Exponential Functions Lesson Overview: Students will discover the use of exponential function in the real-world. They will work through several examples independently as well as a class. Additionally they will view video clips explaining the real-world use of exponential modeling. Duration: 90 Minutes Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

☐Engagement ☒Exploration ☐Explanation

Details Materials:  Dice (36 per group of 2-4)  Dice Activity worksheet (1 per group)  Computer  Projector  Exponential Functions PowerPoint

☐Make sense of problems and persevere in solving them.

Preparation:  Gather enough dice needed to complete the activity for your class.  Copy the Dice Activity worksheet for each of your groups. The groups could be pre-determined or you may allow students to choose; the preference is yours based on your classroom dynamics.

☒Construct viable arguments and critique the reasoning of others.

☐Extension ☐Evaluation

Standards for Mathematical Practice

Facilitation of Learning Experience: Allow students time for experimentation/exploration. Students will start with 36 dice. Each time a 6 is rolled that die should be removed from the bunch and the remaining dice should be recorded in the table. After 15 trials (or when the dice run out), the students will be able to show you an exponential decay regression graph. They will then answer the following questions:  What do you notice about the graph?

☒Reason abstractly and quantitatively.

☒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.

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

☐Engagement ☐Exploration ☐Explanation ☒Extension ☐Evaluation

Details  What type of function do you think this is? Transition:  The questions from the activity will be discussed as a class once each group is finished with their experiment.  This discussion should naturally transition into the remaining aspects of the PowerPoint. Materials:  Exponential Functions PowerPoint  Computer  Projector Preparation: Go through the PowerPoint as the instructor before presenting the information with your students to ensure your own understanding of the material. Facilitation of Learning Experience:  Discuss, in detail the difference between linear and exponential functions. The graphs on the PowerPoint will provide clarification. Emphasize that linear graphs have a constant rate of change and exponential graphs have a percent rate of change.  Work through the job offers example together as a class. Discuss the choices present: Which is better immediately? Which is better in the long-term?  Allow students time to complete the table from the savings account example on their own; then discuss the results as a class. Help them derive the equation for the model of the bank account. Once finished with this, allow the students to be the ones to determine

Standards for Mathematical Practice

☒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.

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details

Standards for Mathematical Practice

how much money is in the account after 20 years. Compare answers as a class. Transition: Discuss how exponential models are used in the real-world. Allow students time to think about how these models are used in the realworld before giving them the answers. Or A STEM Specialist can be used to help students understand how exponential functions are used in the STEM workforce. The Specialist will engage students in a hands-on learning experiences that demonstrates how he/she employs exponential functions in his/her field. To find a STEM Specialist visit theSTEMnet.com. ☐Engagement ☐Exploration

Materials:  Exponential Functions PowerPoint  Computer  Projector

☒Explanation ☐Extension ☐Evaluation

Preparation:  Familiarize yourself with exponential equations: f ( x ) =a bx .  Know that a is the constant in each exponential equation. This relates to the y-intercept of the graph (initial value).  b is the base of the equation and tells us the rate in which the function increases/decreases.  The variable is in the exponent allowing us to calculate values per interval of time (usually time).

☐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.

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details Facilitation of Learning Experience:  Go through each slide of the PowerPoint explaining the symbolic meaning of an exponential function as well as the graphical meaning.  Show a graphical representation of an exponential function and explain how you can tell if the function is an increasing/decreasing exponential function based on the graph alone.  Break down the individual pieces of the equation of an exponential function and stress what each means.  Discuss with the class the meaning behind the mathematical language of f ( x ) =a∙ b x for b>1 (growth) and f ( x )=a∙ b x for 0< b < 1 (decay).  Challenge student understanding of the meaning behind the symbolic representation of an exponential function by asking, “Can you automatically conclude that an exponential function models decay if the base of the power is a fraction or decimal?”

Standards for Mathematical Practice ☒Attend to precision. ☒Look for and make use of structure. ☒ Look for and express regularity in repeated reasoning.

Transition: Discuss the answer to the question posed above. You cannot conclude that a fraction/decimal implies exponential decay. This is true because not all fractions/decimals are less than 1. Some fractions/decimals represent numbers larger than 1 which would imply exponential growth. ☒Engagement ☐Exploration

Materials:  Computer  Projector  Exponential Functions PowerPoint

☒Make sense of problems and persevere in solving them. ☒Reason abstractly and

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details

Standards for Mathematical Practice

☐Explanation



Fry's Bank Account Video Questions

quantitatively.

☐Extension



Fry's Bank Account (First Clip)

☐Evaluation



Fry's Bank Account (Second Clip)

☒Construct viable arguments and critique the reasoning of others.



Fry’s Bank Account Video Question Answer Key

☒Model with mathematics.

Preparation:  Preview each of the following video clips to ensure they are appropriate for your classroom audience.  Complete the Fry’s Bank Account Video Questions in advance to prepare for any questions that may arise from your students.

☒Use appropriate tools strategically.

Facilitation of Learning Experience:  Pass out Fry’s Bank Account Video Questions to each student (the supplement provided is cut into two—half sheet per student should suffice).  Allow the students to watch the video clip from the TV show Futurama. If students have not seen the show explain that the character Fry is from the past. The show takes place in the year 3000. Fry is from the year 2000. This will help with the understanding of the clip.  After Clip 1 has been shown have students guess how much is in Fry’s bank account. They should write down a number they know is too small and a number they think is too large. The actual number

☐ Look for and express regularity in repeated reasoning.

☒Attend to precision. ☒Look for and make use of structure.

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details

 

Standards for Mathematical Practice

will be bleeped out in the video clip to allow for students to think about it. Once all the students have answered take a quick poll of the class to see what kind of numbers they came up with. View video clip 2. In this clip they say that Fry’s bank account with a balance of $0.93 with 2.25% interest over 1000 years has accumulated to $4.3 Billion. This amount can be confirmed using

( nr )

n ∙t

the equation A=P 1+



. Assume the account compounds

interest once, annually. Show the students the mathematics behind this amount manually and using a calculator (reinforce the importance of the order of operations). Students should then answer the remaining questions:  It took Fry 1,000 years to get that much money. How long will it take him to double it? 

How long will it take him to get a trillion dollars?



Be sure to let the students figure these questions out independently or with a partner before discussing the answers as a class.



After the Futurama video clips there is one more real-world example in the PowerPoint. This helps students see how slow (sometimes fast) dilution of medication happens in the blood system.



This would be an interesting place to discuss how all

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details

Standards for Mathematical Practice

medications/supplements affect the human body (a great example is caffeine). Work this example through with the students as a class and discuss how to find a growth factor. 

If time allows, show students the video clip discussing the magnitude of earthquakes and how exponential models help us mathematically define the epicenter.



If additional time allows, show students the Science of Overpopulation clip. It is lengthy put does a nice job outlining how human population will one day outgrow the world’s ability to produce food.

Transition: Re-emphasize to the students how examples of exponential decay/growth can be found everywhere in the real-world. ☐Engagement ☐Exploration ☐Explanation ☐Extension ☒Evaluation

Materials: Exponential Functions Homework Exponential Functions Homework Answer Key

☒Make sense of problems and persevere in solving them.

Preparation:  Take time to complete the exponential functions homework assignment before handing it out to students. Be prepared to ask any questions the students may have about the assignment.  Feel free to alter the questions to the ability level of your students.

☒Reason abstractly and quantitatively. ☒Construct viable arguments and critique the reasoning of others.

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STEM-Centric Lesson Exponential Functions Learning Experience 5E Component Identify the 5E component addressed for the learning experience. The 5E model is not linear.

Details Facilitation of Learning Experience: Allow the students time to discover information/work through problems on their own.

Standards for Mathematical Practice ☒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.

Interventions/Enrichments Identify interventions and enrichments for diverse learners.

Supporting Information Struggling Learners     

Group students based upon ability, learning style, or other appropriate criteria, so all students can equally contribute to group work. If the questions asked in class are too vague, try guiding your questioning strategies to all...