Curricular Unit: Bone Mineral Density and Logarithms

Contributed by: VU Bioengineering RET Program, School of Engineering, Vanderbilt University

A black and white x-ray shows the skeleton of a small animal.
A mouse image made from a cabinet x-ray system.
Copyright © Dr. Edwin Donnelly, VUMC; used with permission


Students examine an image produced by a cabinet x-ray system to determine if it is a quality bone mineral density image. They write in their journals about what they need to know to be able to make this judgment. Students learn about what bone mineral density is, how a BMD image can be obtained, and how it is related to the x-ray field. Students examine the process used to obtain a BMD image and how this process is related to mathematics, primarily through logarithmic functions. They study the relationship between logarithms and exponents, the properties of logarithms, common and natural logarithms, solving exponential equations and Beer's law.

Engineering Connection

Biomedical engineering consists of the application of engineering principles and techniques to the fields of medicine and life sciences. The process of calculating a small specimen bone mineral density image using a cabinet x-ray system us useful in determining the effectiveness of medications and treatments in medical research.

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Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (

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.

  • (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
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Unit Overview

This "legacy cycle" unit is structured with a contextually-based Grand Challenge followed by a sequence of instruction in which students first offer initial predictions (Generate Ideas) and then gather information from multiple sources (Multiple Perspectives) . This is followed by Research and Revise as students integrate and extend their knowledge through a variety of learning activities. The cycle concludes with formative (Test Your Mettle) and summative (Go Public) assessments that lead students towards answering the Challenge question. See below for the progression of the legacy cycle through the unit. Research and ideas behind this way of learning may be found in How People Learn,(Bransford, Brown & Cocking, National Academy Press, 2000); see the entire text at

The "legacy cycle" is similar to the "engineering design process" in that they both involve identifying an existing societal need, combining science and math to develop solutions, and using the research conclusions to design a clear, conceived solution to the original challenge. Though the engineering design process and the legacy cycle depend on correct and accurate solutions, each focuses particularly on how the solution is devised and presented. See an overview of the engineering design process at

Unit Schedule


Throughout the lessons and activities, homework assignments cover conversion between exponents and logarithms, logatithm properties, common logarithms, natural logarithms, change of base formula, Beer's law and solving exponential equations. All these serve as assessment tools. The concluding assessment (in the final lesson, lesson 4) is students' answer to the challenge question, which may be in the form of a pamphlet, poster or presentation.


Kristyn Shaffer ; Megan Johnston


© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University

Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University


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