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# Ultramarathon Pacing and Rational Functions

Lesson 6 of 12

## Objective: SWBAT investigate properties of rational functions.

*55 minutes*

#### Quick Quiz

*25 min*

Today we start class by taking a Quick Quiz (Quick Quiz - Polynomial Functions) to go over some of the important concepts of polynomial functions. We spent the last four days studying these functions, so I want a quick formative assessment before we move on to rational functions. This gives students a gauge of how well they understand the concepts before we take our test at the end of the unit. I am not allowing graphing calculators for this particular quiz because I want to assess their conceptual understanding, not their ability to use their graphing calculator. Both are important skills and I will assess both on the unit test.

#### Resources

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#### Explore

*25 min*

After the quiz, this task is a great way to get students thinking about rational functions. Specifically, this task does a nice job of introducing a context where asymptotes make a lot of sense. My inspiration for this task was my daily warm-up on the treadmill at my gym; I always ran the first minute slower than the remaining portion of the run and I would always picture this graph in my mind. Sometimes you just can't turn the math off! (I recommend completing the task on your own before giving it to your students.)

Give students the worksheet and have them work on it with their groups for about 20 minutes. It is straightforward enough that they should have no problems working through the first five problems. Writing the equation and graphing may be a little more challenging. The last two Mathematical Practices (**MP7** and **MP8**) should come in handy as they are using the structure and repeated reasoning for the first five questions to come up with a generalization to find the average pace at mile *x*. If they get stuck, remind them to look at the structure of how they came up with the first five answers.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Quadratic Function Jigsaw
- LESSON 2: Sketching Graphs of Polynomial Functions
- LESSON 3: Roots of Polynomial Functions - Day 1 of 2
- LESSON 4: Roots of Polynomial Functions - Day 2 of 2
- LESSON 5: Polynomial Function Workshop
- LESSON 6: Ultramarathon Pacing and Rational Functions
- LESSON 7: Homecoming and the Five Pound Gummy Bear
- LESSON 8: Graphing Rational Functions
- LESSON 9: Inequalities: The Next Generation
- LESSON 10: Rational Functions and Inequalities Formative Assessment
- LESSON 11: Unit Review Game: Pictionary
- LESSON 12: Polynomial and Rational Functions: Unit Assessment