# LessonApplications of Linear Functions

### Quick Look

Time Required: 45 minutes

Lesson Dependency:

Subject Areas: Algebra

### Summary

This final lesson in the unit culminates with the Go Public phase of the legacy cycle. In the associated activities, students use linear models to depict Hooke's law as well as Ohm's law. To conclude the lesson, students apply what they have learned throughout the unit to answer the grand challenge question in a writing assignment.

### Engineering Connection

The engineering applications that are useful to graph can be represented in a variety of ways. Particularly of interest is direct variation, which is the form many engineering applications take, including Hooke's law and Ohm's law. Electrical engineers use Ohm's law frequently for it states the most basic relationship between voltage, current and resistance. It is used in almost all calculations to analyze and test circuits in any electrical piece or equipment or device.

### Learning Objectives

After this lesson, students should be able to:

• Explain some of the linear relationships that exist in the real world, including Hooke's law and Ohm's law.
• Revisit and respond to the challenge question.

### 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 (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
• Solve linear equations in one variable. (Grade 8) More Details

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• Analyze and solve pairs of simultaneous linear equations. (Grade 8) More Details

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• Use functions to model relationships between quantities. (Grade 8) More Details

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• Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). (Grades 9 - 12) More Details

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• Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12) More Details

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• Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Grades 9 - 12) More Details

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• Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (Grades 9 - 12) More Details

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###### International Technology and Engineering Educators Association - Technology
• Explain how knowledge gained from other content areas affects the development of technological products and systems. (Grades 6 - 8) More Details

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• Use devices to control technological systems. (Grades 6 - 8) More Details

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###### National Council of Teachers of Mathematics - Math
• Use mathematical models to represent and understand quantitative relationships (Grades Pre-K - 12) More Details

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### Pre-Req Knowledge

Students must know about linear functions, including domain and range, slope, intercepts, and how to graph them using a variety of forms.

### Introduction/Motivation

This unit, we have been learning all about linear functions, including important characteristics and how to graph them from a variety of forms. Recall our guiding grand challenge question in which we're trying to figure out the best equation that fits the data from the research lab. In this lesson, you will learn about real-world applications, such as the research from where the challenge question data comes. We will also take a final look at how to answer the challenge question.

### Lesson Background and Concepts for Teachers

Linear relationships exist often in the realm of science. Three examples of note and accessible to the students at this age are (1) speed, (2) Hooke's Law, and (3) Ohm's Law.

1. The relationship between constant speed, distance and time is a linear relationship that students explored in Lesson 4. In this lesson, students showed the linear relationship using a motion detector. For example, as speed stays constant, distance and time are directly related to one another. The solutions to the Matching the Motion activity worksheet can now be reviewed as a class and explained to reiterate the concepts.
2. Hooke's law characterizes the relationship between a spring's length and the amount of force placed on it. The more force, the longer the spring is. Students can use the Spring Away! activity to better illustrate this relationship. For each spring, an associated spring constant exists, k, that acts as a multiplying factor. Hooke's law in equation form is where F is the force on the spring (in newtons), delta x is the change in the length of the spring (in meters), and k is the spring constant (in N/m).
3. Ohm's law also shows a direct variation relationship between the voltage (potential) across a wire and the current in the wire. The constant factor in this relationship is the resistance in the wire (or resistor, or other component). Refer to the hands-on Can You Resist This? activity to have students create their own circuits to model this relationsip. Ohm's law in equation form is where V is the voltage (in volts), i is the current (in amperes), and R is the resistance (in ohms).

At lesson end, have students get out their notebooks/journals with the challenge question information written in it. Have them spend a few minutes collecting their ideas on how to solve the question.

Have students work to compile their journal entries regarding the challenge question and assign them the "Answer the Challenge Question" post-lesson assessment. Additionally, administer the End of Unit Test over the content.

### Associated Activities

• Spring Away! - Students use springs and masses in attempt to solve for the proportionality constant of a few springs using Hooke's law.
• Can You Resist This? - Students set up their own circuits and solve for the current running through a wire while noting the change in brightness of the lamp attached to the circuit when resistors are added to the setup.

### Assessment

Post-Lesson Assessment:

• Go Public: Answer the Challenge Question: Using this handout, have students respond to the challenge question by formulating a written response in light of what they have learned in this unit, thus fulfilling the Go Public phase of the legacy cycle.
• End of Unit Test: Administer this test to evaluate what mathematical content students have learned through this unit.

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### Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University

### Acknowledgements

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.