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Visual and Intuitive Explanations to Chain, Product, and Quotient Rules

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2019 ASEE Annual Conference & Exposition


Tampa, Florida

Publication Date

June 15, 2019

Start Date

June 15, 2019

End Date

October 19, 2019

Conference Session

Mathematics Division Technical Session 3

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Paper Authors


Daniel Raviv Florida Atlantic University

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Dr. Raviv is a Professor of Computer & Electrical Engineering and Computer Science at Florida Atlantic University. In December 2009 he was named Assistant Provost for Innovation and Entrepreneurship.

With more than 25 years of combined experience in the high-tech industry, government and academia Dr. Raviv developed fundamentally different approaches to “out-of-the-box” thinking and a breakthrough methodology known as “Eight Keys to Innovation.” He has been sharing his contributions with professionals in businesses, academia and institutes nationally and internationally. Most recently he was a visiting professor at the University of Maryland (at Mtech, Maryland Technology Enterprise Institute) and at Johns Hopkins University (at the Center for Leadership Education) where he researched and delivered processes for creative & innovative problem solving.

For his unique contributions he received the prestigious Distinguished Teacher of the Year Award, the Faculty Talon Award, the University Researcher of the Year AEA Abacus Award, and the President’s Leadership Award. Dr. Raviv has published in the areas of vision-based driverless cars, green innovation, and innovative thinking. He is a co-holder of a Guinness World Record. His new book is titled: "Everyone Loves Speed Bumps, Don't You? A Guide to Innovative Thinking."

Dr. Daniel Raviv received his Ph.D. degree from Case Western Reserve University in 1987 and M.Sc. and B.Sc. degrees from the Technion, Israel Institute of Technology in 1982 and 1980, respectively.

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Visual and Intuitive Explanations to Chain, Product and Quotient Rules


Today’s students are exposed to information presented in visual, intuitive and concise ways. They expect explanations for why a subject is important and relevant, as well as for its potential use. This is most pertinent in math courses that are usually taught with little or no connection to other disciplines. In order to adapt to students’ new learning preferences, efforts must be made to further modify teaching methods.

This paper focuses on introducing three concepts in calculus, namely Chain Rule, Product Rule and Quotient Rule by linking them to daily experiences using relevant and analogy-based examples that can be used prior to delving into purely mathematical explanations. The examples are meant to help in understanding the material, and therefore we use discrete values that can help in developing good intuition for the different rules.

The paper details many examples. Among them: (a) Chain Rule: --Inflating a balloon: Change in the volume of a constantly inflated (or deflated) balloon depends on the change in its radius which changes as a function of time.

(b) Product Rule --Number of working hours in a company: A manufacturing company has X factories, Y people in each, and each one works Z number of hours per year. The change in the total yearly working hours is due to change in number of factories, change in number of people in each factory, and the change in number of working hours per employee.

(c) Quotient Rule: --Taxis and passengers in NY: A metropolitan area published a report about the number of taxis and passengers during the 2015, 2016 and 2017 years, proudly stating that both the number of taxis (“g”) and the number of passengers (“f”) grew consistently. The city also published the average number of passengers per taxi, showing a decline in the ratio. Each function is a function of time. To show the change in the number of passenger per taxi over time, the change over time of the ratio of f/g must be calculated. In this example we show a specific numerical example that shows how to calculate the change in f/g, followed by taking the case to the limit and derive the actual formula for Quotient Rule.

The material in this paper is “work in progress.” In the past, when using the above examples (and many others), students have demonstrated better, clearer understanding of difficult concepts. Even though this was not an official assessment, based on similar experience that was gained and assessed by the author multiple times in other engineering related subjects (Control Systems, Digital Signal Processing, Computer Algorithms, and Physics), it is believed that the approach has a great potential.

Raviv, D. (2019, June), Visual and Intuitive Explanations to Chain, Product, and Quotient Rules Paper presented at 2019 ASEE Annual Conference & Exposition , Tampa, Florida. 10.18260/1-2--33540

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