Last week we looked at a set of special rules for working with proportions, which have been largely replaced by the more general “tool” of algebra (the “Swiss army knife” of problem solving, which can do the job of many specialized tools), though the latter can still be useful. We still find that many students …
A new question of the week A simple question about the Basic Proportionality Theorem (which goes by several other names, including Thales’ Theorem) and its relationship to similar triangles leads to some helpful ideas about how to choose a suitable manipulation at each step in a proof, a skill central to good problem solving. In …
Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?
(A new question of the week) With few new questions of general interest available this week, I thought I’d go back a few months to a couple little questions on a topic we haven’t dealt with lately, combinatorics. We’ll have one question each on permutations and combinations, showing some subtlety in both the methods we …
(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.
(A new question of the week) Two recent questions involved using trigonometric functions to model real-life (or nearly so) situations, one about breathing, the other about a Ferris wheel. Both can be done by writing a sinusoidal function; the second can be done in other interesting ways as well.
(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …
Trigonometry can be a powerful tool for solving sides and angles in triangles. But you have to be careful with it! We’ll look at a classic type of error in solving an SSA triangle, get three explanations, and then see how knowing the context of a question can change our answer – to the point …
(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …
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 …