Don’t get caught up on the concept of “fractions”. There is one topic students of mathematics consistently struggle with, to the point that it has become legendary: Fractions. I teach Algebra II. Fractions don’t exist. I’m not saying, of course, that \(\frac12\) and \(\frac5{31}\) aren’t things that might occur. I mean that I encourage students…
Category: Algebra
Logarithms: The Dark Sorcery
I used to hate logarithms. They were hopelessly confusing. Sort of like this: https://www.smbc-comics.com/comic/operations This is the third year now that I’ve been teaching Algebra II. Each year, my understanding of logarithms increases, and my love increases in kind. One reason I disliked logarithms is because of the way in which we tend to compartmentalize…
Every Third Triangular Number
This is a quick proof based on an observation inspired by “Mathematical Lens” in the May 2016 Mathematics Teacher (“Fence Posts and Rails” by Roger Turton). A triangular number is the sum of all integers from 1 to n. The general formula for T(n), the nth triangular number, is \[T(n) = \frac{(n)(n + 1)}{2}\] Challenge:…
al-Jabr: Integer Parameters
I was thinking about the third scenario described in al-Khwarizmi’s al-Jabr: \(x^2 = 3x + 4\). I was curious about the integer solutions of the general pattern, \(x^2 = ax + b\). It’s easy enough to demonstrate that this will hold if \(x = b = a + 1\), since that means \((a + 1)^2…
Al-Jabr (continued)
In my previous post, I looked at the first two detailed examples provided by al-Khwarizmi in his compendium, the title of which gives us the word “algebra”. Al-Khwarizmi discussed three types of mathematical objects: Numbers (N, constants), roots (R, unknowns), and squares (S, squares of unknowns). Because he was limited to positive solutions, he was…
Al-Jabr
The word “algebra” comes to us from the title of a book, al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabalah written by Mohammed ben Musa al-Khwarizmi (there are variations in the transliterations of both the title and the author) around AD 825. He was not the inventor of algebra; indeed, his book was a compendium and extension…
Complex Numbers making Real Numbers
As a point of curiosity, I found myself wondering when a complex number to an integer power creates a real number. For the sake of completeness, I also looked at when the result is a fully imaginary number. More rigorously, define \(\mathbb{C}^*\) as the set of complex numbers \(a + bi\) where both \(a\) and…
Misadding Fractions
A common mistake students make when adding fractions is to add both the numerators and the denominators (I’ll use a special symbol to reinforce that this is not proper addition): \[ \frac{2}{5} \heartsuit \frac{3}{7} = \frac{2+3}{5+7} = \frac{5}{12} \] The general approach is to tell students that that doesn’t normally work. However, while errors in process sometimes…
Radicals and Mixed Numbers
The lesser known of two math memes currently wandering around the Internet involves an interesting equation: \[\sqrt{2\frac{2}{3}} = 2\sqrt{\frac{2}{3}}\] This has spawned at least three discussions I’ve seen so far: What other values is this equation true for? Is this example good or bad for students? What’s with mixed numbers, anyway? I’ll discuss each topic…
Forms of the Quadratic: Terminology
Because mathematical terminology developed piecemeal over time, there are many inconsistencies which prove to be a challenge to students. One of the more obvious examples is what is called the “standard form” of the quadratic. A quadratic equation has three common forms: \(ax^2 + bx + c = 0\) \(a(x – h)^2 + k =…