# Functions and Modeling

### Due Next

Next homework assignment is due by Monday, August 21st.

## Defining functions intuitively and creatively, but also consistently:

In my experience, we don't spend enough time in the classroom talking about the role of creativity in doing mathematics. I'm confident in assuming that most of us have had the experience of being stuck on a difficult (math) problem until we came up with (or came across) a new approach or a new way of thinking about the concepts and connections at hand. One way that we might characterize the transition from ''I'm stuck and banging my head against the wall!'' to the light-bulb-moment is by distinguishing between divergent thinking and convergent thinking.

• Convergent thinking: assumes that a question has one right answer and that a problem has a single solution
• Divergent thinking: seeks multiple perspectives and multiple possible answers to questions and problems

In 1956, J.P. Guilford developed the Alternative Uses Test as a way of measuring divergent thinking. One task is as follows: "Name as many uses for a paper clip as possible in three minutes." The results are scored according to the following measures:

• Fluency: how many uses you can come up with
• Originality: how uncommon those uses are (e.g. "router restarter" is more uncommon than "holding papers together")
• Flexibility: how many areas your answers cover (e.g. cufflinks and earrings are both accessories, aka one area)
• Elaboration: level of detail in responses; "keeping headphones from getting tangled up" would be worth more than "bookmark"

This week, we're going to apply some of these notions of creative thinking to the mathematical concept of function.

## Introduction

This course is all about exploration and inquiry. We are going to revisit many familiar concepts from algebra to calculus but with a focus on ways to take simple principles and ask deep and interesting questions about them. In doing so, we will consider connections between secondary mathematics and college mathematics. All of this will be done through collaborative learning and, where appropriate, the use of technology.

## In-class Work

In class we worked on bottle calibration problems and discussed covariational reasoning.

## Homework Assignment 1

This assignment is due Sunday, July 8th. It requires a substantial amount of work to complete, so please do not wait until the last minute. Your solutions and responses should be written up in a Google Doc named HW1_LastName_FirstName, which you should share with my gmail account (papadopoulos.dimitri@gmail.com). Where appropriate, you should use Google's built-in equation editor to show any algebraic work. For your written responses and reflection, I expect good writing - i.e. correct grammar, punctuation, etc. Please label each section/problem clearly.