Month: April 2018

What Is a Trapezoid? More on Inclusive Definitions

A month ago, I wrote about classifying shapes, discussing inclusive and exclusive definitions, and variations in different contexts. I promised to return to the subject, moving on to the specific issue of trapezoids, and some other related topics. Now is the time.

Is Area of a Square a Circular Argument?

(New Question of the Week) I love it when students want to know why something has to be the way it is, and are not satisfied just being told to use a formula. Last month, Shunya asked this kind of question, which gave me a chance to refer to our archive and go beyond it.

Why Isn’t Slope Run Over Rise?

Definitions are interesting in several ways. Sometimes they are essentially arbitrary; other times there is a very good reason for them, and understanding that reason can be helpful in understanding and using them. But they are usually taught just as something to memorize. Let’s think about why slope is defined as it is, and not …

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Fun with a Quartic Equation

(New Question of the Week) Sometimes a problem leads to a very interesting discussion that brings out many good ideas – but then turns out to be something entirely different, which brings out even more (and simpler) ideas. This polynomial equation problem we helped with last week was like that. I will not be quoting …

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Ranges of Inverse Trig Functions

(Archive Question of the Week) We have had a number of questions over the years about inverse trig functions and their ranges. For today’s question, I have chosen one from 2011, which will link to a number of others that I will not quote in detail.

Empty Sets and Vacuous Truth

I’m going to start this post with a simple question about the empty set, and gradually dive deeper. There will be connections here to previous discussions of conditional statements in logic.

From Histograms to Probability Distribution Functions

(Archive Question of the Week) We occasionally got questions about Probability Distribution Functions (PDFs) from students who lacked a full picture of what they are; when I searched for references to give them, I never found one that explained the whole concept as I wanted to. When the following question came in, I took it …

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Mean, Median, Mode: Which is Best?

It seems common for teachers to ask students which “measure of central tendency” (average) is best. Sometimes they ask this in a particular context (say, to report the “average” salary at a company); other times there is no context (other than perhaps some unidentified data). In the latter case, we generally consider the question to …

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