Summary
Students explore Hooke's law while working in small groups at their lab benches. They collect displacement data for springs with unknown spring constants, k, by adding various masses of known weight. After exploring Hooke's law and answering a series of application questions, students apply their new understanding to explore a tissue of known surface area. Students then use the necessary relationships to depict a cancerous tumor amidst normal tissue by creating a graph in Microsoft Excel.Engineering Connection
Hooke's law defines the direct proportionality between a spring's deformation and the restoring force that results. Most commonly, a derivative of Hooke's law is used in engineering applications—a relationship that directly relates stress and strain. For example, the stressstrain curve is commonly used by material scientists and engineers while selecting materials for structures. Within the linear region, the slope is defined by the Young's modulus of elasticity. Civil engineers often study the stressstrain curve when using strain hardening and other methods to increase the yield strength of a material. In this activity, particularly in the investigating questions 6 and 7, students explore the relationship between Hooke's law and the stressstrain equation. In addition, students must apply their understanding of Hooke's law to create a strain plot.
Learning Objectives
After this activity, students should be able to:
 Describe what is meant by Hooke's law.
 Apply Hooke's law relationships to analyzing tissue of a known surface area.
 Depict a cancerous tumor using graphing methods in Microsoft Excel.
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.
NGSS: Next Generation Science Standards  Science

Apply scientific and engineering ideas to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision.
(Grades 9  12 )
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This Performance Expectation focuses on the following Three Dimensional Learning aspects of NGSS:Science & Engineering Practices Disciplinary Core Ideas Crosscutting Concepts Apply scientific ideas to solve a design problem, taking into account possible unanticipated effects. Alignment agreement:
If a system interacts with objects outside itself, the total momentum of the system can change; however, any such change is balanced by changes in the momentum of objects outside the system. Alignment agreement:
Attraction and repulsion between electric charges at the atomic scale explain the structure, properties, and transformations of matter, as well as the contact forces between material objects.Alignment agreement:
Criteria and constraints also include satisfying any requirements set by society, such as taking issues of risk mitigation into account, and they should be quantified to the extent possible and stated in such a way that one can tell if a given design meets them.Alignment agreement:
Criteria may need to be broken down into simpler ones that can be approached systematically, and decisions about the priority of certain criteria over others (tradeoffs) may be needed.Alignment agreement:
Systems can be designed to cause a desired effect. Alignment agreement:

Motion and Stability: Forces and Interactions
(Grades 9  12 )
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Common Core State Standards  Math

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
(Grades 9  12 )
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Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
(Grades 9  12 )
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Fit a linear function for a scatter plot that suggests a linear association.
(Grades 9  12 )
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Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
(Grades 9  12 )
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Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
(Grades 9  12 )
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Summarize, represent, and interpret data on a single count or measurement variable
(Grades 9  12 )
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Summarize, represent, and interpret data on two categorical and quantitative variables
(Grades 9  12 )
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International Technology and Engineering Educators Association  Technology

Telemedicine reflects the convergence of technological advances in a number of fields, including medicine, telecommunications, virtual presence, computer engineering, informatics, artificial intelligence, robotics, materials science, and perceptual psychology.
(Grades 9  12 )
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Use computers and calculators to access, retrieve, organize, process, maintain, interpret, and evaluate data and information in order to communicate.
(Grades 9  12 )
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Research and development is a specific problemsolving approach that is used intensively in business and industry to prepare devices and systems for the marketplace.
(Grades 9  12 )
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State Standards
Maryland  Science

Motion and Stability: Forces and Interactions
(Grades
9 
12 )
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Apply scientific and engineering ideas to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision.
(Grades
9 
12 )
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Do you agree with this alignment?
Tennessee  Math

Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
(Grades
9 
12 )
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Summarize, represent, and interpret data on a single count or measurement variable
(Grades
9 
12 )
More Details
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Summarize, represent, and interpret data on two categorical and quantitative variables
(Grades
9 
12 )
More Details
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Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
(Grades
9 
12 )
More Details
Do you agree with this alignment?

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
(Grades
9 
12 )
More Details
Do you agree with this alignment?

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
(Grades
9 
12 )
More Details
Do you agree with this alignment?

Fit a linear function for a scatter plot that suggests a linear association.
(Grades
9 
12 )
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Do you agree with this alignment?
Tennessee  Science

Solve problems related to velocity, acceleration, force, work, and power.
(Grades
9 
12 )
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Materials List
Part 1:
Each lab group needs:
 physics lab stand
 meter stick
 spring (with hooks)
 pendulum clamp
 slotted mass set
 computer with Microsoft Excel (or other spreadsheet application)
 Hooke's Law Worksheet, one per student
Part 2:
Each group needs a computer with Microsoft Excel and the Generating a 1D Strain Plot Handout. The handout has instructions specifically for Excel, but if you adjust the instructions, another spreadsheet program could be used.
Worksheets and Attachments
More Curriculum Like This
Students are introduced to Hooke's law as well as stressstrain relationships. Through the lesson's twopart associated activity, students 1) explore Hooke's law by experimentally determining an unknown spring constant, and then 2) apply what they've learned to create a strain graph depicting a tumo...
Students calculate stress, strain and modulus of elasticity, and learn about the typical engineering stressstrain diagram (graph) of an elastic material.
In addition to the associated lesson, this activity functions as a summative assessment for the Using Stress and Strain to Detect Cancer unit. In this activity, students create 1D strain plots in Microsoft Excel® depicting the location of a breast tumor amidst healthy tissue.
As part of the engineering design process to create testable model heart valves, students learn about the forces at play in the human body to open and close aortic valves. They learn about blood flow forces, elasticity, stress, strain, valve structure and tissue properties, and Young's modulus, incl...
PreReq Knowledge
A basic understanding of the concepts of Hooke's law, stress and strain, as presented in Lesson 2.
Introduction/Motivation
Have you ever wondered how the value of the gas constant was measured/discovered, or the charge on an electron, or the Young's modulus of elasticity values we used in the problem set yesterday? Ever wondered where all these values come from? Well today we are going to solve for one ourselves.
In groups of three, we are going to experimentally find the spring constant, k, for a few springs. After collecting data, we will use the relationship given by Hooke's law to solve for an approximation of the constant.
After exploring Hooke's law and answering a few application questions, we will apply what we've learned to study a body tissue with known surface area. Because Hooke's law applies to springs, we must make a few adaptations to the expression F= k Δx, to account for area. By the end of the activity, you will be able to apply what you know about Hooke's law, stress and strain to depict a tumor amidst normal tissue using a graph in Microsoft Excel.
Procedure
Background
This activity constitutes the Research and Revise phase of the legacy cycle. Students explore Hooke's law in a handson, laboratory situation. They experimentally solve for the spring constant, k, of a given spring by measuring the spring's displacement when a mass of known weight is added. After answering some application questions on Hooke's law, students relate Hooke's law to a body tissue of known surface area. Continuing their research and revising their initial thoughts for solving the engineering challenge, students follow stepbystep instructions to depict a cancerous tissue in a graph generated in Microsoft Excel. Though students work in groups, it is expected that they complete their own activity worksheets. Students may discuss the questions but should answer the questions individually.
Before the Activity
 Provide each lab station with the necessary materials.
 Assign groups of three for the activity.
 Make copies of the Hooke's Law Worksheet and Generating a 1D Strain Plot Handout.
With the Students
 Pass out the two handouts. Use the Hooke's Law Worksheet as an instructional guide when creating strain plots. Use the Generating a 1D Strain Plot Handout as an instructional guide to the lab; each student is responsible for completing and submitting the analysis and application questions by the end of the class period.
 Have students move into their assigned groups and go to their lab benches.
 Direct students to follow the worksheet and handout instructions. Remind them that they may work together, but each student is responsible for completing and turning in their own answers and solutions.
 When students are ready to move on to the strain plot, have them remain in their groups; only one graph needs to be turned in per group. Remind them to return to their initial thoughts notes and add any new notes that may help them solve the challenge.
Vocabulary/Definitions
cancer: A malignant and invasive growth or tumor tending to recur after removal and to metastasize to other sites.
force: An influence on a body or system, producing a change in movement or in shape or other effects.
spring: An elastic body such as a wire of steel coiled spirally that recovers its shape after being compressed, bent or stretched.
strain: Deformation of a body or structure as a result of an applied force beyond limit.
stress: The physical pressure, pull or other force exerted on a system by another, producing a strain. Measured by the ratio of force to area.
ultrasound imaging: The application of ultrasonic waves to therapy or diagnostics, as in deepheat treatment of a joint or imaging of internal structures.
Young's modulus of elasticity: A mathematical constant that represents how difficult a material is to stretch
Assessment
Activity Embedded Assessment: The Hooke's law application questions and the 1D strain plot both function as means of assessment. Students must first develop an understanding of Hooke's law. Then they must relate this concept to a tissue with known crosssectional area. This concept may be used to detect a cancerous tumor where the tumor's elastic properties differ from that of normal tissue.
Investigating Questions
 How does Hooke's law and the stressstrain relationship relate? Which variables correspond?
 What do we know about cancerous tissue that allows us to use these concepts to depict it?
 What types of software would be appropriate for our imaging?
 Using these methods, will our imaging method be painless? Will it be effective and reliable? How about cost effective?
Activity Extensions
To extend the handson aspect of exploring the tissue, consider obtaining ballistic gel (such as https://en.wikipedia.org/wiki/Ballistic_gelatin) of differing stiffness. This may be used to mimic the differing tissue structure of cancerous and normal tissue as represented by varying Young's modulus of elasticity.
Activity Scaling
 For upperlevel students, remove the stepbystep instructions for generating the 1D strain plot.
 For lowerlevel students, take the time to relate Hooke's law to the stressstrain relationship as a class. Make this connection with the students, using a visual representation on the board.
References
Dictionary.com. Lexico Publishing Group,LLC. Accessed December 28, 2008. (source of vocabulary definitions, with some adaptation)
Contributors
Luke Diamond; Meghan MurphyCopyright
© 2007 by Regents of the University of Colorado; original © 2007 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: August 16, 2018
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