Summary
Students revisit the mathematics required to find bone mineral density, to which they were introduced in lesson 2 of this unit. They learn the equation to find intensity, Beer's law, and how to use it. Then they complete a sheet of practice problems that use the equation.Engineering Connection
Students see the realworld connection with this material right away. Beer's law and the other equations that they learn in this lesson are mathematical tools that biomedical engineers use every day when they make bone mineral density readings. Environmental engineers also use light intensity measurements when designing solar panels and other forms of energy storage. Almost every type of engineer knows and uses Beer's law.
Learning Objectives
After this lesson, student should be able to:
 Apply their knowledge of logarithms to solve Beer's law problems.
 Explain the breath of the applications of logarithms.
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Educational Standards
Each TeachEngineering lesson or activity is correlated to one or more K12 science,
technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
Common Core State Standards  Math

Interpret expressions that represent a quantity in terms of its context
(Grades 9  12)
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Interpret parts of an expression, such as terms, factors, and coefficients.
(Grades 9  12)
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Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
(Grades 9  12)
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Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
(Grades 9  12)
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Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
(Grades 9  12)
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(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
(Grades 9  12)
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For exponential models, express as a logarithm the solution to ab^{ct} = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
(Grades 9  12)
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Use the properties of exponents to transform expressions for exponential functions.
(Grades 9  12)
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Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
(Grades 9  12)
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International Technology and Engineering Educators Association  Technology

Technology transfer occurs when a new user applies an existing innovation developed for one purpose in a different function.
(Grades 9  12)
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State Standards
Tennessee  Math

(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
(Grades
9 
12)
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Introduction/Motivation
Recall from lesson 1 that you are working for a company that makes small specimen cabinet xray machines for medical research. Today you will learn how to do some calculations with the data that you might collect from such a machine.
Lesson Background and Concepts for Teachers
Calculation of Bone Mineral Density:
The basic equations for dualphoton absorptiometry can be derived from a number of underlying assumptions. First, it is assumed that the material is composed of varying amounts of only two substances (in this case bone and soft tissue). Second, it is assumed that scatter can be ignored. Under these circumstances, for any given photon energy, the number of photons striking the detector (N) can be calculated from the number of incident photons (N_{o}) using Beer's law.
Beer's law:
where μ_{s} and μ_{b }represent the mass attenuation coefficients (cm^{2}/g) of soft tissue and bone (respectively) and M_{s} and M_{b} represent the area densities (g/cm^{2}) of the two tissue types.
I = I _{o} e ^{μl}
I = Intensity.
I_{0} = Intensity with no object.
μ = attenuation coefficient (depends upon material and xray energy).
l = length of the xray path.
Lecture Information
Present to students the BMD Math Presentation, a PowerPoint® file. Then have students complete the BMD Math Practice Problems. Administer the Log Quiz 2. Then students are ready to conduct the associated activity, Light Intensity Lab.
To answer the Challenge Question introduced in lesson 1, students next must Go Public. Refer to the Pamphlet Poster Paper Instructions for how students are to present all that they have learned about logarithms and bone mineral density during the course of the entire unit.
Associated Activities
 Light Intensity Lab  In this lab exercise, students learn how to measure light intensity through transparencies.
Worksheets and Attachments
Assessment
PostIntroduction Assessment
Practice Problems: At lesson end or as homework, assign students to complete the BMD Math Practice Problems. Review their answers to assess the depth of their understanding.
Lesson Summary Assessment
Quiz: At lesson end, administer the 13question, multiplechoice Log Quiz 2, which covers all types of logarithmic problems covered in this unit. Review students' answers to gauge their comprehension.
Student Presentations: In order to answer the unit challenge question first posed in lesson 1, have students create posters, pamphlest or papers, as described in the Pamphlet Poster Paper Instructions.
Contributors
Kristyn Shaffer; Megan JohnstonCopyright
© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt UniversitySupporting Program
VU Bioengineering RET Program, School of Engineering, Vanderbilt UniversityAcknowledgements
The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.
Last modified: July 20, 2017
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