(A new question of the week) I had a long discussion recently about the Cartesian product of sets, answering questions like, “How is it Cartesian?” and “How is it a product?” I like discussions about the relationships between different concepts, and people who ask these little-but-big questions. We’ll be looking at about a quarter of …
(A new question of the week) In an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why \(a^2 = b^2 + c^2\). But in the hyperbola, \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the equivalent relationship, \(a^2 + b^2 = c^2\), is not nearly as natural, nor …
(A new question of the week) While I was looking through recent questions to choose one to post, I ran across one that deals with an error we see very commonly – in fact, a student I had worked with that very afternoon in face-to-face tutoring had done the same sort of thing. The context …
Sometimes the more basic an idea is, the harder it is to define it. It is also very hard to understand a definition in English when you are not a native speaker! We have had some interesting discussions of such issues recently with a student who asks very basic and yet very challenging questions of …
(A new question of the week) We’ll look at a very complicated logarithmic equation, which leads to quartic equations and some very interesting graphs. We won’t find a fully satisfying solution method, but we’ll have some fun trying – and reveal the fallibility of at least one Math Doctor!
(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …
(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …
(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.
(A new question of the week) We’ve looked in the past at place values, but here we’ll see some tricks for doing multiplication and division with both decimals and large numbers by moving the decimal point around. The first question is primarily a matter of arithmetic, then the second extends it to the algebraic concept …
(A new question of the week) It is not unusual for mathematicians to define a concept in multiple ways, which can be proved to be equivalent. One definition may lead to a theorem, which another presentation uses as the definition, from which the original definition can be proved as a theorem. Here, in yet another …