Exponential Functions

Transcription

1 Mathematical Models with Applications, Quarter 3, Unit 3.2 Exponential Functions Overview Number of instruction days: (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Identify exponential patterns of growth or decay. Differentiate whether a table of data represents a linear, quadratic, or exponential relationship. Create exponential equations and use them to solve problems. Interpret key features of exponential functions from tables and graphs. Sketch exponential graphs, showing key features, given a verbal description of the relationship. Graph and manipulate exponential equations using technology. Relate the domain of an exponential function to its graph and to the quantitative relationship it describes. 1 Make sense of problems and persevere in solving them. Identify important features of exponential equations to find regularity or trends. 4 Model with mathematics. Create models to represent relationships between quantities of a situation. Interpret mathematical results to draw conclusions, and determine if your model is reasonable or if adjustments are necessary. 5 Use appropriate tools strategically. Use a graphing calculator to create exponential models to represent a relationship between quantities. Adjust the viewing window on their graphing calculators to get the information needed given exponential functions. 7 Look for and make use of structure. Use first and second differences to identify functions. Use common ratios to create exponential equations. Providence Public Schools D-1

2 Math Models, Quarter 3, Unit 3.2 Exponential Functions (22-24 Days) Essential Questions How do you determine if the relationship between variables is an example of exponential relationship? If so, how do you determine if it represents growth or decay? How can an exponential relationship be recognized given a table, graph or equation? What characteristics of the data in a given problem indicate whether an exponential equation is to be used or not? How is the domain of the graph of an exponential function related to the relationship it describes? Standards Common Core State Standards for Mathematical Content Algebra Creating Equations A-CED Create equations that describe numbers or relationships [Equations using all available types of expressions, including simple root functions] A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. * Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context [Linear, exponential, and quadratic] F-IF.4 F-IF.5 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it D-2 Providence Public Schools

3 Exponential Functions (22-24 Days) Math Models, Quarter 3, Unit 3.2 takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewise-defined] 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. F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Providence Public Schools D-3

4 Math Models, Quarter 3, Unit 3.2 Exponential Functions (22-24 Days) Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In grade 5, students learned to graph points on the coordinate plane, and interpret ordered pairs in the context of a situation. In grade 6, students began to investigate and write equations to represent relationships, and first encountered exponents. Students represented and analyzed quantitative relationships using multiple representations. They also used exponents to evaluate expressions, including D-4 Providence Public Schools

5 Exponential Functions (22-24 Days) Math Models, Quarter 3, Unit 3.2 those from geometric formulas. In grade 7, proportional relationships were a major cluster. Students learned to represent proportional relationships using graphs, tables, and equations, and interpreted proportional relationships. In grade 8, students learned how to apply the properties of integer exponents to generate equivalent numerical expressions. They used square root and cube root symbols to represent solutions to equations, and also evaluated square and cube roots of small perfect squares and cubes. The study of functions was a major cluster in this grade. Students translated between different representations of functions, and interpreted functions in the context of a given relationship. Additionally, students applied the properties of exponents to generate equivalent expressions, wrote numbers using scientific notation, and performed operations on numbers written in scientific notation. In Algebra 1, students interpreted parts of exponential expressions, and used properties of exponents to rewrite exponential expressions in equivalent forms. They also created and solved equations in one or more variables to represent exponential relationships. Finally, students rearranged exponential formulas to highlight a quantitative interest and used function notation to evaluate and reveal different properties of the exponential function (including exponential growth and decay). Students used the properties of exponents to transform expressions for exponential functions for example, identify percent rate of change in a function. They extended the properties of exponents to rational exponents and rewrote expressions involving radicals and rational exponents using the properties of exponents. They applied exponents in growth and decay problems by writing equivalent functions and interpreting the parameters of exponential functions. They also used the properties of exponents to transform expressions for exponential functions. They learned that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. They used function notation, evaluated functions for inputs in their domains, and interpreted statements that used function notation in terms of a context. They interpreted key features of graphs and tables in terms of the quantities, and sketched graphs showing key features given a verbal description of the relationship, focusing on quadratics and linear functions. They graphed functions expressed symbolically and showed key features of the graph. They compared properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). They constructed linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs. They related the domain of a function to its graph and, where applicable, to the quantitative relationship it described. Current Learning In this unit, students identify appropriate types of functions for given data, adjust parameters as needed and compare the quality of models, with and without technology. They synthesize and generalize what they have learned about a variety of function families. Students analyze the behavior of given data and its context to create a model to represent exponential relationships. Students interpret key features of exponential graphs and tables, sketch graphs given verbal descriptions, and relate the domain and range of an exponential function to its graph and to the quantitative relationship it describes. They also use properties of exponents to interpret expressions for exponential functions. Providence Public Schools D-5

6 Math Models, Quarter 3, Unit 3.2 Exponential Functions (22-24 Days) Future Learning In Algebra II, students will identify appropriate types of functions for given data, adjust parameters as needed and compare the quality of models. Students will synthesize and generalize what they have learned about a variety of function families. This work will be extended to include exponential functions solving exponential equations with logarithms. Students will continue mastering concepts related to exponential equations and functions in PreCalculus and AP Calculus. In Calculus, students will need to apply rules for derivatives and integrals of exponential functions. A variety of careers will require understanding of concepts related to exponential equations and/or functions. Scientists use exponential functions to model populations, carbon date artifacts, help coroners determine time of death. Financial analysts compute investments. Chemists use exponential functions to graph the rate at which temperatures level off. Bankers use exponential to compute interest and the government uses them to compute our national debt. If you are considering any kind of career in Science you need a solid grasp of exponents. Exponents will be used in many types of scientific experiments. Doctors may use exponents to figure out how quickly a disease is going to spread. Research Scientists often perform statistical calculations and interpret data. Additional Findings According to NCTM A Research Companion to Principles and Standards for School Mathematics, the complexity that teachers and researchers now see in function graphs flows from the fact that a graph has many potential meaning and can be interpreted in many ways.(p. 250) Among the most widely agreed upon conclusions of research on graphing in the past 25 years is that visuality is a key source of difficulties for students using graph. Iconic interpretation, interpreting a graph as a literal picture, and other inappropriate responses to visual attributes of a graph, are the most frequently and constructing graphs (Leinhardt, Zaslavsky, & Stein, 1990, p.39) Errors due to this kind of interpretation not only are pervasive but also seen highly resistant to change. NCTM A Research Companion to Principles and Standards for School Mathematics (p. 257) Assessment When constructing an end of unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS Content Standards and the CCSS Practice Standards should be considered when designing assessments. Standards based mathematics assessment items should vary in difficulty, content and type. The assessment should include a mix of items which could include multiple choice items, short and extended response items and performance based tasks. When creating your assessment you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed D-6 Providence Public Schools

7 Exponential Functions (22-24 Days) Math Models, Quarter 3, Unit 3.2 within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Math Models students should be provided with multiple, alternative methods to express their understandings of the concepts that follow: Create exponential equations and use them to solve problems. Interpret solutions to exponential equations as viable or not viable in a modeling context. Use tables and graphs to identify key features of exponential functions. Given a verbal description sketch exponential graphs, showing key features. Graph exponential functions expressed symbolically and show key features of the graph, with and without technology. Describe the effect that the domain of an exponential function has on its graph. Instruction Learning Objectives Students will be able to: Graph, analyze, and identify patterns as exponential growth or decay. Use the first differences, second differences, and successive quotients to identify and create functions. Create exponential equations to model real-world problems. Use the correlation coefficient to identify the appropriate function modeling the data. Represent constraints and interpret solutions as viable or not viable options in a modeling context. Interpret key features of exponential functions from tables and graphs. Graph exponential functions expressed symbolically and show key features of the graph, with and without technology. Sketch exponential graphs, showing key features, given a verbal description of the relationship. Relate the domain of an exponential function to its graph and to the quantitative relationship it Providence Public Schools D-7

8 Math Models, Quarter 3, Unit 3.2 Exponential Functions (22-24 Days) describes. Reflect on and demonstrate understanding of exponential functions. Resources Modeling with Mathematics: A Bridge to Algebra II, W.H. Freeman and Company, 2006 Sections (pp ) Sections (pp ) Section 7.7 (pp ) Section 7.7 Assignment (pp ) Section 7.8 Assignment (p. 454) Online Companion Website: TI-Nspire Teacher Software Additional Resources located in the Supplementary Unit Materials Section of the Binder: o o GapMinder Activity M & M project: M & Ms Project Worksheet Gallery Walk Scoring Table and Rubric o education.ti.com Characteristics of Exponential Functions Comparing Linear and Exponential Functions Shall I Double Up or Keep the Million? Applications of Exponential Functions o TEXTEAMS Algebraic and Geometric Modeling Institute, Exponential Growth, pp Exponential Decay, pp D-8 Providence Public Schools

9 Exponential Functions (22-24 Days) Math Models, Quarter 3, Unit 3.2 Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery section below for specific recommendations. Materials Graphing calculators, scissors, copy paper, M&Ms, small plastic cups, rulers, calculator based data collection device, temperature probe, data collection program, patty paper, gridded chart paper, chenille stems. Instructional Considerations Key Vocabulary compound interest Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: exponential function, exponential decay, common multiplier, successive quotients, ratio, base, exponent, power, exponential growth and decay factor. In this unit, students synthesize and generalize what they have learned about a variety of function families and extend their knowledge to include exponential functions. While studying exponential functions students interpret key features of graphs and expressions for exponential functions using the properties of exponents. Students describe and explain graphs of exponential functions as transformations of related functions, and they adjust parameters to choose a model that fits. GapMinder is an optional tool, which can either be accessed online or downloaded to students computers. This activity will provide the opportunities for students to investigate mathematical relationships using authentic data with a global perspective. The students will research historical connections to significant changes on the graph and apply their knowledge of functions to identify characteristics, make predictions, and observe trends. The M & M Project, located in the Supplemental Materials section of the binder, provides an optional kinesthetic approach to exploring the properties of exponent functions. This project can be used instead of Section 6-4 in the textbook. Teachers should purposely group students for this activity. Students work in groups to conduct the experiment and collect the resulting data. They graph the data on gridded chart paper, generate linear, quadratic, and exponential equations, analyze the related correlation coefficients, Providence Public Schools D-9

10 Math Models, Quarter 3, Unit 3.2 Exponential Functions (22-24 Days) and determine the appropriate regression equation. The activity culminates with a gallery walk where students analyze and evaluate the work from other groups. Chenille stems cut into small pieces and small plastic cups may be used as an alternative to M & Ms. Additional problems for exponential functions can be found on the companion website. Select Activities under Instructional Resources, and then select Chapter 06: Exponentials Students and/or Chapter 06: Exponentials Teachers. Section 6.7 Assignment is excluded from this unit. Limited sections from Chapter 7 have been included in this unit in order to apply real-world applications of exponential functions using a financial context. Graphing technology will assist students by displaying multiple representations of exponential functions and with modeling real world situations. Several examples supporting the integration of technology in this unit are provided below. The teacher and student pages for the activities are provided in the supplementary materials section of this curriculum frameworks binder. The activities can also be found by going to education.ti.com and searching for the activity titles. However, you will need to download the tns file to the calculators for these activities. Characteristics of Exponential Functions: This lesson involves investigating how the graph of an exponential function changes when 0 < b <1, b = 1, or b >1. As a result, students graph an exponential function and describe the domain, range, and y-intercept of the exponential function. Comparing Linear and Exponential Functions: This lesson involves comparing data from two different scenarios, linear and exponential growth. As a result, student: complete a table involving prize money scenarios, examine the table for trends in the growth of each prize money scenarios, and examine the graphs to compare the data. Shall I Double Up or Keep the Million?: This activity asks students to explore the question: If you were given the opportunity to be given a permanent monthly salary of 1,000,000 for 30 days of work or a salary beginning with a penny on day one and doubling each day for 30 days which would you choose? This activity does not require a tns file. Applications of exponential functions: In this activity, students explore applications involving bacteria growth and decay, where exponential functions are used to represent the data. They also explore the domain and range of the exponential functions in the context of the applications. Additional TI-Nspire and TI-Navigator resources related to this content can be found using the TI-Nspire Teacher Software. Activities from the TEXTEAMS Algebraic and Geometric Modeling Institute have been included in this unit in the supplementary resource section of the binder. You can also search the TEXTEAMS CD for these or other related activities. The goal of the TEXTEAM activities is for students to gain mathematical content knowledge by engaging in a four step, mathematical modeling process which involves: identifying a problem in some context, formulating a model, applying appropriate mathematical analysis, and drawing conclusions from that analysis. D-10 Providence Public Schools

11 Exponential Functions (22-24 Days) Math Models, Quarter 3, Unit 3.2 Notes Providence Public Schools D-11