If you want to square a two-digit number, you could just use a calculator, or you could use the traditional algorithm. I’m going to talk about a different method here, but not because I think this is a particularly useful method. The point of this discussion is to look at how numbers are interrelated; if…
Category: Algebra
Synthetic Division
Synthetic Division is one of three common techniques for dividing one polynomial by another. The other two are long division and the box method. Of the three methods, students generally prefer synthetic division. It has the advantage of having minimal writing and being strictly algorithmic: Add, multiply, add, multiply…. First, let’s see how it works….
Exponents: Language
Our language surrounding exponents is confusing and, I think, misleading. Power An exponential relationship involves three values. Historically, these were called the base, the exponent, and the power. On a logarithmic scale, the base represents the step size, the power represents the target value, and the exponent represents the number of steps. For instance, if…
There Is No Spoon (QF Edition)
As I was nearing the end of my article yesterday, something creeped into my head and lingered in the shadows for a while. This morning, it came into the light. The function that allows us to find the input that corresponds to the output of another function has a name: It’s the inverse function. One…
The Quadratic Formula (Vertex Form)
I really don’t like the quadratic formula. As a teacher, it feels like one of the absolute worst examples of what’s wrong with mathematics education: An arbitrary formula with weird aspects, including what is often the first appearance for students of the plus-minus sign. Sure, we can do algebraic manipulation to show why it is…
Math: Estimating Roots
A few years ago, I developed a quick algorithm for approximating square roots. I’ve since come across a more effective, albeit slightly more complicated, algorithm. Both of these are meant to create quick approximations in an era where calculators can come up with more precise values far quicker than humans. My first method If you’re…
The Logarithmic Rules
In this item, I will show how the basic logarithmic rules, including the Change of Base formula, follow from this equivalency: \[\log_b m = n \Leftrightarrow b^n = m\] For the ease of reading, I’ll generally use the natural base (\(e\)) and the natural logarithm (\(\ln\)). However, everything here applies to all valid bases (\(b…
What’s the Deal with Logarithms?
I’m going to talk about logs here. I have more to say later, but this is a basic intro sketch. First I’m going to talk about the stuff of elementary school. When it comes to mathematics, most people find comfort in elementary school mathematics. So, consider the humble number line: We want to move along…
Reframing the Quadratic Formula
When I was in school, I was taught the Quadratic Formula. I was taught that it was the most efficient, more reliable way to find the roots of a quadratic function. This is what I was taught: Given a function in Standard Form, \(ax^2+bx+c\), its roots can be found by evaluating \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). I was instructed…
Proof: The rationality of the y-intercept
Theorem: Given a quadratic function with rational roots, the \(y\)-intercept is rational if and only if the stretch is rational. Proof: If \(f\) is a quadratic function with rational roots \(m\) and \(n\) and vertical stretch \(a\), then \[f(x) = a(x – m)(x – n) \\ = a(x^2 – (m+n)x + mn) \\ = ax^2…