##
* *Reflection: Modeling
Introduction to Rational Functions with Real-World Applications - Section 1: Warm-Up

What I love about this unit is that it involves some different kinds of thinking--there is more estimation and reasoning about numbers, than there are actual calculations. In order to understand rational functions, it is really more important that students can reason and use number sense than it is for them to actually do the calculations--for instance, rather than actually finding the output of the function y=4/(x-2) when plugging in x = 1.99. It would be more important to use number sense to conclude that it would be a really big output. I would prefer that a student could explain this, than that they could do the numerical calculation, without understanding.

To this end, it is essential to model the kinds of thinking and language a student would use to understand these functions. This means that I spend a lot of time "thinking aloud" while talking about these problems with students. I say, "Well, if I have a really huge denominator, then I know I will dividing the numerator by lots and lots of people, so everyone will get a really tiny piece. This means that on the graph, when the *x*-value is really big, the *y*-value will be really close to zero." I found that when I used this kind of language throughout the unit, my students were able to start thinking this way themselves, and rather jump for their calculators when it came to graphing a function, they would be able to use this thinking to generate graphs and understand functions.

*Developing Ways of Thinking*

*Modeling: Developing Ways of Thinking*

# Introduction to Rational Functions with Real-World Applications

Lesson 1 of 10

## Objective: SWBAT graph and write rational functions to model real-world situations and to describe the behavior of these functions.

## Big Idea: How long will a trip take if you travel at a certain speed? Use the relationship between speed and time to explore rational functions and discover asymptotes in the real world.

*70 minutes*

#### Warm-Up

*30 min*

The goal of this Warm-Up is for students to discuss the ideas of the lesson with their partners in as much depth as possible.

The front (page 1) of the warm-up is the essential page. The second page is included for students who master the ideas of the first page more quickly and are ready for more depth. During this long opening, I circulate and ask students the following questions about the three_key_problems.

During this time and this entire lesson, it is way more important to take the time to talk about the real world example than it is to write the equations. The abstraction of the equations is not essential today.

*expand content*

Students will now work on the Applications of Rational Functions Problems. The goal of this time is to solidify the key ideas of the warm-up. In today’s class, the emphasis is not on equations. I am okay if students do not work abstractly. In fact, it is almost better if students don’t use equations, but rather think about the real-world situation in order to complete the data table. If students don’t figure out the equations today, I won’t provide them. They can learn them from peers who do figure them out, but as the teacher I will not ask students for or give them equations. If they come up with equations, I'll ask them: “Does this equation make sense based on the situation? If you use the equation, do you get the same outputs as your partner?”

It is important that all students understand the approach statements. One great question to ask is: “Will your total time ever be exactly 3 hours? Will it ever be less than 3 hours?”

**Instructional Note**: Your decision is how much to do as a whole class versus in small groups. I spend time discussing the context of each problem and making sure that students know how to fill out the data tables *without* using equations. It might make sense to briefly discuss each situation as a whole class and make sure that students understand the real world context of the problem (with actually doing any math.) Another option would be to assign different students the 3 key problems and do these in a jigsaw format.

The appliance situation requires the most explanation because it is the least intuitive, especially for inputs between 0 and 1 year. I finally figured out that the best way to explain it is to say that the refrigerator only lasts for a certain amount of time. For instance, if the refrigerator only lasts 0.5 years, you will have to buy two refrigerators in the one year in order to have a refrigerator for the year. Alternatively, if your refrigerator lasts for 1,000 years, the initial price of buying the refrigerator will be spread out over all those years.

While students work through these three problems, my main focus will be on the behavior of the function and on making connections between the graph, the data table, and the real world situation. For instance, a student may say, “As the speed increases, your times gets closer and closer to 0 because you are going faster.” This is a great example of a partial understanding. It is true that the time gets closer and closer to 0, but really the time is not getting that close to 0. It is getting closer to 3 hours. Why would this be true? The graph shows this using the horizontal asymptote. The data table shows this because if you use a huge number for the speed, the time will be close to 3. This makes sense based on the situation because no matter how fast you go, you will still be sitting still for 3 hours, so even if you go a million mph, you will still be stopped for 3 hours. These are the types of conversations I will try to facilitate during this time.

If I notice students getting bogged down in the details of the calculations or the graphing (especially setting up the axes), this is when I provide them with more scaffolding, like setting up the axes or using a calculator for the calculations. I think it is worthwhile for a student to be able to determine that if your speed is 0.5 mph, you will take 160 each way because it will take you 2 hours to go one mile. I do want students to understand that calculation without a calculator, but they don’t need to do all calculations without a calculator. Also, students can copy each other’s data tables. The goal is to make sure that they don’t spend the whole class period doing calculations and plotting points, because they will then miss the whole point of the lesson.

*expand content*

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Introduction to Rational Functions with Real-World Applications
- LESSON 2: More Applications of Rational Functions
- LESSON 3: Graphing y=1/x
- LESSON 4: Transformations of y=1/x
- LESSON 5: Graphing y=a/(x-b)+c
- LESSON 6: Writing Approach Statements from Graphs
- LESSON 7: Matching Graph Transformations to Equations
- LESSON 8: Compare and Contrast Graphs of Rational Functions
- LESSON 9: Rational Function Review
- LESSON 10: Rational Function Summative Assessment and Portfolio