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In this lesson students develop and analyze exponential models for the behavior of light passing through water. Students begin by considering how light intensity changes from near the surface to the bottom of the ocean.
In this lesson students develop and analyze exponential models for the behavior of light passing through water. Students begin by considering how light intensity changes from near the surface to the bottom of the ocean. They model incremental depths using layers of tinted Plexiglas as layers of water. They use a CBL to take light readings under increasing layers of Plexiglas. Then they graph light intensity as a function of depth. Students repeat the experiment using water in a tube. Several methods of analyzing the data are suggested. (author/sw)
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In the Arts and Communications, Construction, and Transportation career fields, one of the concerns is the transmission of light through film, lenses, or filters in photography, or glass or plastic when dealing with windows or lighting in buildings or motor vehicles. This lesson could profitably be used as an integrated lesson taught by the career-technical teacher and either a mathematics or science teacher, or, better still, both academic teachers.
In the Arts and Communications, Construction, and Transportation career fields, one of the concerns is the transmission of light through film, lenses, or filters in photography, or glass or plastic when dealing with windows or lighting in buildings or motor vehicles. This lesson could profitably be used as an integrated lesson taught by the career-technical teacher and either a mathematics or science teacher, or, better still, both academic teachers.
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computer
Internet connection
overhead display device |
| QuickTime Player |
| This technology integrated lesson provides an opportunity for the introduction or reinforcement of the following technology skills: |
| Grades 9-12 |
Use multimedia presentations to extend content knowledge.
Explore Mathematical Concept |
online video
online module
structured activity
website |
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| Mathematics Academic Content Standards |
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| Patterns, Functions and Algebra Standard | |
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| Benchmarks (8 - 10) |
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| E. | Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros. |
| I. | Model and solve problem situations involving direct and inverse variation. |
| J. | Describe and interpret rates of change from graphical and numerical data. |
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| Benchmarks (11 - 12) |
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| A. | Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior. |
| C. | Use recursive functions to model and solve problems; e.g., home mortgages, annuities. |
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| Grade Level Indicators (Grade 9) |
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| 5. | Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of change, maximum or minimum. |
| 13. | Model and solve problems involving direct and inverse variation using proportional reasoning. |
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| Grade Level Indicators (Grade 10) |
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| 10. | Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. |
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| Grade Level Indicators (Grade 11) |
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| 1. | Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. |
| 2. | Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years. |
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| Grade Level Indicators (Grade 12) |
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| 10. | Use the concept of limit to find instantaneous rate of change for a point on a graph as the slope of a tangent at a point. |
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| Data Analysis and Probability Standard | |
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| Benchmarks (11 - 12) |
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| A. | Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators. |
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| Grade Level Indicators (Grade 11) |
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| 4. | Create a scatterplot of bivariate data, identify trends, and find a function to model the data. |
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| Grade Level Indicators (Grade 12) |
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| 2. | Transform bivariate data so it can be modeled by a function; e.g., use logarithms to allow nonlinear relationship to be modeled by linear function. |
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| Mathematical Processes Standard | |
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| Benchmarks (8 - 10) |
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| B. | Apply mathematical knowledge and skills routinely in other content areas and practical situations. |
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| Benchmarks (11 - 12) |
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| H. | Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations. |
| J. | Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation. |
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| Principles and Standards for School Mathematics |
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| Algebra Standard | |
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| Understand patterns, relations, and functions |
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| Expectations (9 - 12) |
| analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior; |
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| Represent and analyze mathematical situations and structures using algebraic symbols |
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| Expectations (9 - 12) |
| analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior; |
| use a variety of symbolic representations, including recursive and parametric equations, for functions and relations; |
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| Use mathematical models to represent and understand quantitative relationships |
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| Expectations (9 - 12) |
| analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior; |
| use a variety of symbolic representations, including recursive and parametric equations, for functions and relations; |
| identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships; |
| use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts; |
| draw reasonable conclusions about a situation being modeled. |
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| Analyze change in various contexts |
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| Expectations (9 - 12) |
| analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior; |
| use a variety of symbolic representations, including recursive and parametric equations, for functions and relations; |
| identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships; |
| use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts; |
| draw reasonable conclusions about a situation being modeled. |
| approximate and interpret rates of change from graphical and numerical data. |
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| Data Analysis and Probability Standard | |
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| Select and use appropriate statistical methods to analyze data |
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| Expectations (9 - 12) |
| identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled. |
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| Connections Standard | |
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| Recognize and apply mathematics in contexts outside of mathematics |
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| Representation Standard | |
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| Use representations to model and interpret physical, social, and mathematical phenomena |
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| RESOURCE TYPE |
| Instructional Resource |
| PRACTICE LEVEL |
| Best Practice |
| STANDARDS ALIGNMENT |
| Grade 9 - Postsecondary |
| CAREER FIELDS |
Arts & Communication;
Construction Technologies;
Transportation Systems |
| TOPICS |
Mathematics --
Algebra;
Graphing;
Nonlinear functions;
Calculus, precalculus;
Geometry;
Models, construction;
Measurement;
Connections, applications;
Statistics;
Data display;
Technology |
| FOUND IN |
COR
Standards First |
| KEYWORDS |
CBL;
exponential decay |
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Publisher: National Council of Teachers of Mathematics
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