## 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

Unit 6: Rational Functions
Lesson 1 of 10

## 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.

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Standards:
Subject(s):
Math, Precalculus and Calculus, inverse variation, modeling, rational function, proportional relationships
70 minutes

### Hilary Yamtich

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