I was thinking about inverse function notation, and that got me thinking about function notation, and that got me thinking about operations and how meh our notation for mathematical operations is. So, let’s start fresh. We’ll pretend we don’t have any operators, just a bunch of numbers and an equal sign. We need to make…
Category: Mathematics
Math Education and One True Wayism
This is a common criticism of Common Core (CCSS): It offers these strange new methods that students must use. Except… only the first part of that is true. CCSS does offers some new strategies, but it doesn’t say that students have to use them. This article isn’t a defense of CCSS, by the way. It’s…
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…
Functions and Domains: The Other Shoe
I have taught high school mathematics for nearly a decade. I have a BS in Mathematics. The Algebra II curriculum, which I largely built for my school, is based on “the story of functions”. And yet, it was only the other day that I noticed something that was woefully wrong about the way that I’ve…
Rate of Change and the Power Rule
I’ll keep this one short. Also, it’s on calculus, for whatever that’s worth. The Power Rule for Differentiation says that the derivative of a monomial \(ax^b\) is \(abx^{b-1}\). Last night I noticed a way to derive this for positive integers that I believe I’ve seen before (so I’m not claiming originality), but which is different…
Seriously, It’s Just Division
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…
“O Function! My Function!”
I’m going to build a simple set of functions. This set will wind up being very familiar when I’m done, but let’s pretend for a few minutes first. Function and Operators In case you’ve forgotten what a mathematical function is, there are a few quick ways to refresh your memory. A common way is to…
Maria Agnesi and the Second Derivative
I’m currently reading selected parts of “Analytical Institutions”, the 1801 edition of John Colson’s translation of Maria Gaetana Agnesi’s 1748 text. Near the beginning of her second book, she presents the following theorem. Consider the diagrams: In each diagram, H, I, and M are points along the x-axis and are equally spaced. A, B, and E…
Questioning, Not Answering
Another Rant on “Mathematical” Puzzles A few months ago, I complained about those internet memes which claim to be mathematics. My complaint about them is about the presentation, not the underlying problems: “Only 1% of people get this right!” The questions are framed to encourage people to feel stupid about math. Please don’t feel stupid about math….