On mathematics
Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that…
Read more
This morning, I’ve been watching YouTube videos. I started with Tarleen Kaur’s video on Middle Term Splitting. What I find interesting about Kaur’s Chapter to Chapter videos is that, because she’s a student in India, her methods are often different…
Read more
They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples. Here’s a good definition of absolute value: “the magnitude of a real number without regard to…
Read more
I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6,…
Read more
A persistent topic in mathematics education is whether to focus on conceptual or procedural knowledge. After reading Kris Boulton’s recent post that argues, “It depends,” I found myself thinking about the disconnect between arithmetic and algebra. What is needed to understand…
Read more
Dan Meyer’s latest post is on an exercise involving using a gridless coordinate plane to place fruit along two dimensions. The goal is a worthy one: To give students the opportunity to explore what the coordinate plane is without getting tied…
Read more
Most high school geometry textbooks will proclaim that there are four basic transformations. Three of these (translations, reflections, and rotations) are rigid transformations; the resulting copy (image) is congruent to the original version (pre-image). Here are examples: The first example…
Read more
Discussing the properties of similar triangles today, I derived a simple proof of the Pythagorean Theorem that uses ratios. (I do not claim this is original to me; I’m sure it isn’t.) Consider the diagram. \(\Delta ADC \sim \Delta BDA \sim \Delta…
Read more
In this entry, I’m going to start with a concrete problem and develop an abstract generalization. The starting problem: Given isosceles trapezoid \(ABCD\) with an altitude of 6. Point \(E\) is on \(\overline{DC}\) such that \(DE = 3\), \(EC =…
Read more
I was recently asked for an elegant proof of the following problem. It’s based on a construction challenge from Euclidea. Given: Circles A, B, and C, such that point C is on circle A, point B is on circles A…
Read more
What is the equation of a line that is secant to a circle with radius \(r\) and center \((0,0)\)? This question started as a challenge with a student. She wanted to draw a pentagram on a graphing calculator, and while…
Read more
A friend of mine, a father, recently posted this item on his Facebook feed. It’s from Pearson, and he was struggling figuring it out. I also had to read it several times to figure it out. This is a large…
Read more