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
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Fun with Functions: Basic Inverse and Function Operations

Unit 1: Basic Functions and Equations
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.

Big Idea: Students learn different function operations with a colorful teacher-led presentation that engages students with tons of practice.

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