Summary
Students will examine a problem about driving speeds to investigate a rational function.
Essential Question(s)
How is a rational function different than other functions?
Snapshot
Engage
Students work in groups to model a real-life situation about a road trip.
Explore
Students will write a function to describe a real life scenario.
Explain
Students will explain how the function they wrote is different than previously studied functions and will be introduced to the definition of rational function.
Extend
Students will explore the behavior of the function they wrote near the vertical asymptote and will be introduced to the different types of asymptotes.
Evaluate
Students will consider other phenomena and determine if the phenomena can be described using a rational function.
Materials
PowerPoint (attached)
Hand out For Extension (attached)
Engage
Display the first slide of the attached PowerPoint for students. Place students into groups of two or three to work on the displayed problem.
As groups work, travel between them to get a sense of the methods groups are devising to find a solution. After most groups seem to have reached the solution (42 6/7 miles per hour), call on several groups to share their methods for determining the solution. Be sure to call on the groups with the less efficient methods first, then work toward groups with more efficient methods.
Explore
Once the class has reached a consensus about the solution and has seen several methods, move on to the next slide, which asks students to repeat the problem using different values, finally using variables for the different quantities in the problem. Repeat the procedure from the Engage section, this time having students share only the final function they wrote. Be sure the class comes to a consensus about what this function should be, and ensure they simplify the function as much as possible. This will help students to see that their functions are equivalent, even if they initially wrote the functions in different ways.
Explain
Move on to the next slide. Have students work in their groups to answer these questions. After several minutes, have students share their responses. Explain to students that they have just described several properties of rational functions. Then forward to the next slide to show the definition of a rational function. Discuss how the function they have examined today fits this definition of a rational function. Students should copy the definition into their notes.
Extend
Distribute the "Driving Rationally-Extend" worksheet (located under Attachments) to students. Students may complete this worksheet in groups in class or at home for homework. Once students have completed the worksheet, review their answers with them. Tell students that these questions asked them to examine values near an asymptote of the function y = 25x / (x - 25). Ask students to make a hypothesis about what the definition of asymptote may be. Then, return to slides five and six to provide students with the definition of asymptotes and some examples. Students should copy these into their notes.
Evaluate
Forward to slide nine. Have students work in groups to complete the question presented. Groups should write their responses, supporting their claims by explaining why the phenomena may have asymptotes (or not), and create a sketch of the phenomena. Evaluate students' ability to conceptualize the phenomena described.
End class by having students create predictions about when rational functions will have certain types of asymptotes. This will foreshadow future lessons about asymptotes.