(A new question of the week) A recent question asked for the connection between two different ways to use determinants geometrically: to find the area of a triangle, and to find the volume of a pyramid (or the area of a parallelogram and the volume of a parallelepiped). Last time we looked at what a …
How Can 3×3 Determinants Give Both Area and Volume? Read More »
Here is a little question about making a formula to dilute a solution; we’ll see how to do the algebra, and also how what we teach in math classes isn’t quite real.
(A new question of the week) Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.
(An archive question of the week) In preparing the last couple posts, on recurrence relations, I ran across an answer to a much harder question, that illustrates what it can take to solve one that doesn’t fit the convenient forms. It’s linear, but the coefficients are not constant as they have been in all our …
A Challenging Homogeneous Second-Order Recurrence Read More »
Last week we looked at a recent question about recurrence relations, and I realized it needs a companion article to introduce these ideas. So here we will look at some answers from Ask Dr. Math about the simpler case, including general methods, why they work, and applications.
Here is a short discussion of a common type of problem in trigonometry classes: finding a trig function of the sum or difference of two angles, given minimal information about them.
Last time we looked at what a degenerate conic section is, and how it relates on one hand to actual cones, and on the other to the general equation of the conic. Here we’ll look at the parameters of conic sections (focus, directrix, axes, and especially eccentricity) and how they apply to degenerate cases. Does …
Degenerate Conics II: Are Their Parameters Meaningful? Read More »
I have a very short problem this week: How deep will you go if you dig a straight tunnel through the earth, how long will it be, and what angle do you have to start at?
Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, \(1^2+2^2+3^2+\dots+n^2\), for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …
Here is a recent question about arithmetic sequences and series (specifically, reversing the process to find the number of terms given the sum), that nicely illustrates a common type of interaction with a student: gathering information about both problem and student, then guiding them to use what they know, or giving new information as needed. …