Puzzles involving two or three properties of people or objects, commonly solved by Venn diagrams, are popular with teachers, and perhaps not so popular with students! Several have asked about them recently, leading me to catalog our past answers about them, to find the most useful examples to point to. Some of these are very …
(A new question of the week) In many areas of math, an answer can come in several forms, which can make it hard to know if you are right when you compare your answer to the answer in the back of the book. Even worse is when the problem is multiple-choice, and your answer has …
Last time we looked at the question of why we have to have postulates, which are not proved, rather than being able to prove everything. Often, this question is mixed together with a different question: Why do different texts give different lists of postulates, so that what one calls a postulate, another calls a theorem? …
(A new question of the week) In Monday’s post about fallacies in calculus, one of them used the definition of the derivative (or rather, misused it). Today we’ll look at a short question about applying that same definition, that came in last month.
(An archive question of the week) In searching for answers to include in Monday’s post on calculus fallacies, I ran across a long discussion that illustrates some important aspects of methods of integration. In particular, there are often multiple ways to find an integral (the best not necessarily being the one taught in your textbook); …
Integration: More Than One Way, More Than One Answer Read More »
(An archive question of the week) Last time I discussed issues that arise in solving a simple algebraic equation. In researching that, I found a discussion of solving a formula for a variable (which in some countries is called “making x the subject”, that is, changing an equation involving x into the form “x = …
Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.
(A new question of the week) Having discussed trigonometric identities on Monday, let’s make this Trig Week, by looking at a discussion from two months ago in which we were asked about alternative routes to a proof.
Proving trigonometric identities can be a major challenge for students, as it is often very different from anything they have previously done. Often they confuse this concept with solving an equation. But also, they may be give overly rigorous standards to comply with. Here, I will look at several discussions we have had about different …
Different Ways to Prove a Trigonometric Identity Read More »
(New Question of the Week) This recent question involves an ordinary differential equation (ODE) and the relation between different solutions. It illustrates common difficulties in interpreting what a problem is asking for, as well as some communication problems involving language and notation.