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 …
(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 …
(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.
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.
(New Question of the Week) Last month we had a question from a Czech student asking about a geometry problem. The discussion illustrates language issues that can arise, and how we try to guide a student to solve a problem himself. I will fill in some gaps as we examine how to approach an interesting …
(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 …
From Histograms to Probability Distribution Functions Read More »
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 …
(New Question of the Week) From time to time we get a question that is more about words than about math; usually these are about the meaning or origin of mathematical terms. Fortunately, some of us love words as much as we love math. But the question I want to look at here, which came …
(Archive Question of the Week) An interesting question that has been referred to many times since it was written in 1999 deals with averaging angles. At first the question seems trivial; then almost impossible; and then we end up with a rather simple formula that is totally unlike what we started with. And further applications …
Recall that the domain of a function is the set of all valid input values (x), and the range is the set of all possible output values (y). It is reasonably easy to find the domain: look for what could make it impossible to evaluate, such as dividing by zero or taking the square root …