(A new question of the week) A set is closed under an operation if, whenever that operation is applied to two elements of the set, the result is still an element of the set. It’s straightforward … until you look closely at some details! In the course of the discussion, we’ll dig into different definitions …
Is {0} Closed Under Division? Thoughts, and Second Thoughts Read More »
(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.
(A new question of the week) I recently had a pleasant discussion of factoring, with the kind of student for whom I returned to teaching: one who has been away from math for a while, and with greater maturity has the determination to succeed. We’ll see several examples of the “ac-grouping” method of factoring a …
(A new question of the week) Though students often think piecewise-defined functions are unnatural, they are actually quite common in real life – after all, the world is not all once piece! In particular, they show up in various financial situations, such as taxes and pay rates. Inverses, likewise, are commonly needed in real life, …
(A new question of the week) Some topics are hard to find information about at a basic level, because they are usually dealt with in advanced math courses, and yet the basic ideas can be understood without all the trappings. That is the case for the Affine Tangent Cone, which involves tangents to an algebraic …
A couple weeks ago, while looking at word problems involving the Fibonacci sequence, we saw two answers to the same problem, one involving Fibonacci and the other using combinations that formed an interesting pattern in Pascal’s Triangle. I promised a proof of the relationship, and it’s time to do that. And while we’re there, since …
(A new question of the week) We’ve looked at domain and range problems before, but some have more interesting details than others. Here is a superficially basic radical function (and the answer is extremely easy when you just use a graphing tool), which raised some interesting issues while solving it algebraically.
Last week we looked at several basic word problems for which the Fibonacci sequence is part of the solution. Now we’ll look at two problems that take longer to explain: a variation on the rabbit story, and an amazing reverse puzzle.
Here and next week, we’ll look at a collection of word problems we have seen that involve the Fibonacci sequence or its relatives, sometimes on the surface, other times only deep down. The first set (here) are direct representations of Fibonacci, while the second set will be considerably deeper.
Continuing our look at the Fibonacci sequence, we’ll extend the idea to “generalized Fibonacci sequences” (with different starting numbers), and see that the ratio of consecutive terms is the same in general as in the usual special case. Then we’ll look at the sum of terms of both the special and general sequence, turning it …