(An archive question of the week) Trigonometric functions are sometimes introduced without a deep explanation of their meaning; they are just buttons to push on a calculator, or names to write in an equation. Even when a textbook gives a careful presentation, there are so many facets to the concept that it can be easy …
(New Question of the Week) We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the …
(Archive Question of the Week) Having discussed various issues involving categorizing shapes, let’s take a look at a very different shape question, which didn’t fit into the last post.
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.
(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.
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 …
(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) 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 …
We get many questions about classifying shapes, from both elementary and high school students (or their parents or teachers). They often have trouble with the very idea of classifying items by applying definitions, and also with the fact that definitions can vary, both between everyday and technical usages, and from one textbook to another.