(An archive question of the week) There are several ways to restrict the range of values you need to test when you are searching for zeros of a polynomial (using the Rational Zero Test or the Intermediate Value Theorem, for example). One of them can be quite useful for difficult problems, but can be hard …
(A new question of the week) It has been a while since I have regularly included recent questions to this site, in part due to my focus on big topics. This summer I will be focusing on individual questions, some large and some small. Today, we’ll have two recent questions that look at the same …
(An archive question of the week) Recently we have had a number of series digging into multiple aspects of an idea like area or percentages. But I am always running into miscellaneous questions that can’t fit into such a series because of their length or uniqueness. I want to spend the summer focusing on random …
We have been looking at some classic “false proofs” or “fallacies”, where a seemingly valid proof shows something clearly false to be true. The goal is to learn from these, how to distinguish a valid proof from an error. In a post from last year, What Role Should a Figure Play in a Proof?, I …
Last time we looked at some false proofs, which are often used to help students understand what does and does not constitute a valid proof, and in particular, to remind them to be careful in algebraic proofs, looking for issues like division by zero and taking square roots. This time, we’ll look at two similar …
One of the ways to show the need for careful proof, which I discussed last time, is the existence of apparent proofs of false “facts”. These are also useful for training us to think carefully. In this and the next couple posts, I will look at a few categories of false proofs, focusing on what …
One aspect of mathematics that students often struggle with, particularly in geometry (which traditionally has been where proof is introduced), is writing proofs. Why do we need to learn about proofs? Why are proofs needed in the first place? Here are a few answers we’ve given to these questions.
(An archive question of the week) While I was researching for the post on uncountable sets, I ran across a discussion that didn’t quite fit, but raises interesting questions about how countable and uncountable sets can fit together. How can the rational numbers be countable, but the irrational numbers, which are closely intertwined with them, …
(A new question of the week) A recent question from a student working beyond what he has learned led to an interesting discussion of alternative methods for solving a minimization problem, both with and without calculus.
(An archive question of the week) Recently we looked at the question of how likely a two-child family with a boy is to have another boy (or, to the contrary, to also have a girl). Searching for those questions turned up another one of interest involving the gender of a pair of siblings: How do …