Applications of Linear Functions

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.

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.

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

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 iswhere 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 iswhere 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.

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.

Post-Lesson Assessment:

• Go Public: Answer the Challenge Question (PDF): 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 (PDF): Administer this test to evaluate what mathematical content students have learned through this unit.