(New question of the week) Now that we are receiving questions at the new site, we will be periodically posting some of those questions and answers, in addition to going back to particularly interesting questions from the old archive. This question (which I have slightly edited) came to The Math Doctors earlier this month, from …
Why Must the Number of Variables Equal the Number of Equations? Read More »
(Archive question of the week) One day back in November, as I entered the community college where I teach, a man came in behind me with his son, probably about 3 years old. It was the first snowy day of the year, and the father stood on the mat wiping his feet, while the little …
Last time, we discussed how you know whether to add or multiply (or something else) in compound probability problems (like finding the probability that you will flip heads and roll an even number). But as I’ve said before, it’s often easier to remember a formula if you know why it is what it is. I’ll …
Probability is a subject in which it can be quite difficult to see at a glance what method to use. There are many ways in which its ideas can be combined. Two principal ways are “or” and “and”: What is the probability that I draw an ace or a spade? How about an ace and …
Students often ask about formulas for areas or volumes. Sometimes they are just overwhelmed by the number of formulas they need to know; other times they are curious about how we know they are true. The answer to both questions is, in part, the same: if you know at least something of where they come …
Early in our history, we answered many questions about geometric proofs, particularly the “two-column” variety. Many of these were collected into a FAQ page. I want to briefly survey just some of what we have said about the big picture – an overall view of how to approach a proof, and how to work your …
While pondering issues that often give students trouble in algebra, I decided to check what we have said in Ask Dr. Math about inverse functions. I discovered four answers (all, as it happens, written by me – I tend to be attracted to certain topics!) to essentially the same question, spread over 13 years. It …
How to Find an Inverse Function: Conflicting Approaches Read More »
Last time, I talked about students’ difficulties carrying out divisions that involve a zero, which reminded me of another issue that sounds almost the same, but is quite different: division of a number by zero. Students often either forget the rule they were taught, or they don’t believe the rule, or they just wonder about …
I was going to move on from arithmetic to algebra, but the discussion of long division led me to think about some of the more ordinary difficulties students have asked about in that area. Here I will show several questions about the process of long division in which zero caused trouble. Zero in the quotient …
I now want to start looking more deeply at some specific questions at various levels, starting with arithmetic, then algebra and geometry. Students learning arithmetic (and their parents) tend to think in terms of following some rote procedure, just because “that’s the way it’s done”. Modern educators try to focus more on deep understanding, so …