(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen.
(A new question of the week) Today I want to look at a recent question that led into both geometrical and trigonometrical solutions, and particularly a useful perspective on the Law of Sines.
(A new question of the week) Last week we looked at a question about a triangle inscribed in a semicircle. Not long after that question, the same student, Kurisada, asked a question about triangle inscribed in a circle, which had some connections to the other. As we enjoy doing, we led the student through several …
Today we’ll look at a question from a student who was troubled by the amount of guessing needed to solve certain problems. This leads to an interesting survey of different kinds of guessing, and ways to develop that skill.
(A new question of the week) Like many questions we get, this one can be solved in many ways. We like to guide a student to whatever solution will fit what they have learned; along the way, we may find various additional methods, and side trips into other topics of interest.
(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 …
(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.
The last two posts were about transformations of functions (shift, stretch, reflect) and their effect on a graph, first individually and then in combination. The next thing to look at will be how to determine the transformations when you are given a graph; but before we take that challenge in general, we need to see …
I previously wrote about finding the range of various kinds of functions. The examples there were relatively easy. A recent question raised the level of difficulty, bringing up some interesting issues.
Last time we looked at various ways to find tangent lines to a parabola without using calculus. Another standard calculus task is to find the maximum or minimum of a function; this is commonly done in the case of a parabola (quadratic function) using algebra, but can it be done with a cubic function? Yes, …