Background Mathematical conventions represent the linguistic aspect of mathematics. One of the strengths of modern mathematics is the way in which we can represent some fairly complex ideas in a shortened, rigorous symbol set. However, as a result of these abbreviations, there are some ambiguities that are generally settled democratically: Some group decides that the…
Author: Clio
10101 and 11011 are never prime
One particularly tricky aspect of number sense is being able to separate the abstract notion of value from more concrete visual representations of numbers, and the even more concrete notion of countability. For instance, some people get caught up on zero not being a value because there’s no point in counting zero of everything; after…
Just forget my Dear Aunt Sally
The purpose of a mnemonic is to make something easier to remember. Roy G. Biv represents the major colors of the spectrum (Red, Orange, Yellow, Green, Blue, Indigo, Violet); it has the weakness that most people tend to think of the spectrum having six colors these days, instead of seven, with purple in place of…
Multiplying Polynomials
The traditional way of teaching the multiplication of binomials is FOIL: First, Outside, Inside, Last. For instance: \[(x + 3)(2x – 5) = (x)(2x) + (x)(-5) + (3)(2x) + (3)(-5) \\ = 2x^2 -5x + 6x – 15 \\ = 2x^2 + x – 15\] This works well enough for binomials, but for more complicated…
Hero’s Formula and Mirror Triangles
Here’s a problem with an interesting solution. You’re given two triangles, T1 and T2. The sides of T1 are 25, 25, and 30. The sides of T2 are 25, 25, and 40. Which has the greater area? The impulsive answer is probably to say that T2 is larger, since the third side is larger. However,…
Pi aside: Degrees are even worse
Make no mistake: When I run the world, π will be set aside in favor of τ. For those who haven’t heard, there’s a movement to use 6.28318530718… (that is, 2π) as the basic variable relating to circles rather than 3.14159265359…. I personally feel that there’s a strong pedagogical motivation at the secondary education level (specifically, τ is the circumference of…
Factoring quadratics and linear equations
Factoring a quadratic equation involves finding two linear equations whose product is the quadratic equation. This is an example where mathematics teachers often act as if (a) there is one method of solving and (b) there is one solution. The AC Method The “one method of solving” strategy usually goes something like this: If the…
Listening to students (reflection)
One of the greatest benefits to my current teaching position is its small class sizes, which affords me a significant amount of one-on-one tutorial time. I know fairly well how I think about numbers; I don’t know how other people, particularly students who struggle with mathematics, do. I feel that it’s crucial that mathematics teachers…
=
When I was a lad studying mathematics, the equality sign seemed particularly simple: The stuff on the left is equal to the stuff on the right. However, I have since been developing a much more sophisticated perception of the simple little sign. It can indeed be a troublesome symbol, not because of its own meaning…
Let’s Make a Deal
The Problem In the misnomered “Monty Hall” problem, the rules are set out as follows: You as the contestant are faced with the choice of three doors, behind exactly one of which is money or something else of significant value. You choose a door. The host, who knows where the money is, then opens a…