##
* *Reflection:
Fun with Functions: Basic Inverse and Function Operations - Section 2: Exploration: Inverses

Only 30% of my students answered the “*Through which of the following points would the inverse function have to pass?” *question correctly. With the most popular answer, by far, being letter A. So this tells me that my students feel that inverses have something to do with switching signs from positive to negative. Not one of my students mentioned anything about a sign change when they explained their definition of inverses. So where did this come from? In this question, students swapped the input and output and then made it opposite. But why opposite? The best answer I could pull from my students was that they felt the notation in the question should make it negative. The inverse notation with an exponent of -1 made students think negative. Which concerns me on a whole other level. Negative exponents should make students think ‘fractions’ not ‘opposite.’

This made me begin to ponder my planning of this lesson. Was the question poorly designed? Did it lead students to thinking signs would become opposite? Maybe if I switched and just asked which point was on the inverse with no inverse notation then students may be less confused. Or was the task poorly designed altogether? Did it not give students an opportunity to develop a definition of inverses for themselves? When I gave the formal definition, I was careful in noting that inverse notation does not mean reciprocal in the definition, but I did not explicitly state that it was not an opposite. Should I have? Or is the issue simply that more practice is needed on MP8?

In conclusion, I really think it’s the later. I think students continue to need more work on making conclusions based on repeated reasoning and extending a conclusion to apply it to a new situation. Many of my students missed the mark here, and I honestly think it is their lack of skill in MP8… not in their misunderstandings of inverses. They may have a few misconceptions of inverse notation and negative exponents, but I think they understand inverses. So the question on page 7 remains a great assessment for MP8, but I think a better assessment of students understanding of inverses would be their definitions they wrote or their responses to the question on page 6 of the flipchart.

*Mathematical Practice 8*

*Mathematical Practice 8*

# Fun with Functions: Basic Inverse and Function Operations

Lesson 9 of 16

## Objective: SWBAT combine functions using a variety of operations, including compositions and inverses, and will review inverse functions and function notation.

#### Warm-up

*3 min*

As a warm-up today, I am going to have students complete the two clicker questions at the start of the flipchart. These will require students to demonstrate their current understandings of piecewise functions.

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#### Closure

*7 min*

To wrap-up today’s learning, have student individually answer the 5 clicker questions on the last few pages of the flipchart. It is important that students answer these questions without the help of their teammates so we can identify who understands the material and who may need some extra help. These question do require students to extend the learning a bit from today and to piece their new knowledge with prior knowledge. For example the question on page 12 of the flipchart require students to apply their past learnings of shifting of graphs.

During these last few minutes, I also plan on reminding students about our mathematical practice of the day, **Mathematical Practice 8: look for and express regularity in repeated reasoning**, while they work on these problems. Hopefully students are able to apply the concepts of inverses and are able to go backward to answer the question on page 15. Once all students have submitted their answers to this question, I want to talk with students about the regularity we see with inverses. They undo one another. The inverse of a function is the inverse of its own inverse. Being able to see this pattern and make this generalization is one example of how this mathematical practice can help train us to be better mathematicians. Being able to identify a pattern or a repeated conclusion and make a generalization about it really helps to make math easier.

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Getting to Know You, Getting to Know All About You...
- LESSON 2: Ahoy team! What can you see? Finding functions.
- LESSON 3: Function Zoo - Basic Function Families
- LESSON 4: Parent Functions
- LESSON 5: Shifting Functions: How do they move?
- LESSON 6: Shifting Functions: How can we describe them?
- LESSON 7: Dicey Functions Day 1: Piecewise functions are basic functions... just cut up!
- LESSON 8: Dicey Functions Day 2: Piecewise functions are basic functions... just cut up!
- LESSON 9: Fun with Functions: Basic Inverse and Function Operations
- LESSON 10: Compositions in Context
- LESSON 11: Inundated with Inverses: Restricting the Range of an Inverse (Day 1 of 2)
- LESSON 12: Inundated with Inverses: Algebraic Inverse and Composition to Verify (Day 2 of 2)
- LESSON 13: Jeopardy: Basic Functions
- LESSON 14: Review Day
- LESSON 15: Test Review
- LESSON 16: Basic Functions Test