Category: Mathematics
-
[mathjax]Consider the following task: \[1.\quad \text{Simplify the expression }\frac{3}{4}\div\frac{2}{5}.\] It is very common for students to struggle with this sort of task. A common teaching approach is “Keep Change Flip,” but too often that’s presented as a mechanical trick without any deeper understanding of why it works. In proper mathematical language, “Keep Change Flip” translates…
-
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…
-
[mathjax]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…
-
[mathjax]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…
-
[mathjax]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…
-
[mathjax]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…
-
[mathjax]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…
-
Don’t get caught up on the concept of “fractions”. [mathjax]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…
-
[mathjax]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…
-
[mathjax]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…