##
* *Reflection: Diverse Entry Points
Compare and Contrast Graphs of Rational Functions - Section 2: Investigation and New Learning

I have now tried teaching this lesson two different ways. The first year, my students compared and contrasted the graphs of the functions without access to graphing technology. The second year, they used graphing technology to make the comparisons. To be honest, I still am undecided about which is better, mostly because I think there are advantages to each strategy.

When students did not have access to graphing technology, this task took much longer (obviously) and students spent much more time figuring out how to choose inputs, and calculating outputs, and plotting points, which took forever. On the other hand, I think that they do get a deeper understanding of how the function actually works by going through this process, but I have no real evidence of this. I am still curious about whether or not that is true.

When students were able to use graphing technology, they could figure out how the different parameters affected the graph in almost no time, so this was much more efficient. However, I am not sure that they would be able to graph one of these functions without the technology and I feel that to truly master a skill, you would be able to do it on your own, without requiring technology.

I am still torn about this--I want to use the advantage that technology gives us, but I don't want to lose any of the engagement with the actual numbers that might foster a deeper understanding.

*With or Without Technology?*

*Diverse Entry Points: With or Without Technology?*

# Compare and Contrast Graphs of Rational Functions

Lesson 8 of 10

## Objective: Students will be able to compare and contrast the graphs of rational functions in the form y=a/(x-b)+c and describe the differences and similarities of these graphs using academic vocabulary.

## Big Idea: Engage students in the higher-order thinking task of comparing and contrasting rational function graphs using academic vocabulary while reviewing all essential skills of the unit.

*70 minutes*

#### Warm-Up

*30 min*

There is an extended warm-up provided for today, which you can choose to use or not. The idea is that students can use the day to begin preparing for the summative assessment of this unit. The extended warm-up includes problems related to each key skill of this learning unit in the same sequence that they were covered during the unit. Direct students to choose the problems that they want to focus on because obviously 30 minutes is not enough time to tackle all of the problems.

Some students may have fully mastered all of these skills during the unit, so the challenge warm-up is provided for them. Creating graphs to match the approach statements turns out to be quite difficult. If they accomplish this with graph sketches, you can ask them to find equations that would match those approach statements as well. This is incredibly challenging for some graphs, and will require some piece-wise functions.

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#### Closing

*10 min*

These questions require students to generate pairs of functions that fit the given requirements. It is like working backwards from the classwork. Some students might refer to problems on the classwork to find their examples. I would encourage them to try to generate their own examples first, and only refer to the classwork for confirmation or if they get stuck.

It is important to discuss these problems as a class to make sure that students end the class with correct examples. Have them share with each other, or put up two sets of examples for each problem and ask them to decide which pair of functions matches the given requirements.

The purpose of this discussion is two-fold: (1) to have students practice looking at the equations and identifying the key features of the graphs without actually graphing and (2) to use the key terms to describe the graphs of these functions more abstractly.

#### Resources

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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