The Pythagorean Theorem states that, given a right triangle, the areas of squares placed along the two legs will have the same area as a square placed on the hypotenuse. This is normally written as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the leg lengths and \(c\) is the hypotenuse length. De…
Category: Mathematics
Pascal, Pacioli, Probability, and Problem-Based Learning
I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of mathematics as it was driven by Pascal and Fermat, with regards to a specific problem…
The Fourth Dimension (Thoughts)
I’ve had two recent thoughts about the fourth dimension. The first relates to Euler’s Formula, which says that the difference between the sum of the vertices and faces of a convex polyhedron and its edges is always 2 (that is, \(v + f – e = 2\)). The Number Devil presents this slightly differently: The sum…
Geometry for multiplication, division, and roots
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 it is typically referred to as Euclidean geometry. However, it is only part of the…
Factoring and long division
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 from those I’m familiar with. That’s the case in this video as well. I haven’t…
Positive numbers and absolute value
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 its sign; the actual magnitude of a numerical value or measurement, irrespective of its relation…
The smallest angle
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, and AC = 7, what is the smallest angle? (Edit for clarity: The question is simply…
Concepts vs procedures
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 algebra? The first leap that students need to be able to make is from the…
Triangular Gaps
There is an unfortunate gap in the triangle congruency theorems. It would be nice to be able to say that we can declare that two triangles are congruent based on a pair of sides and exactly two other bits of information, but we cannot. If we can match up all three pairs of sides as…
Graphing and the coordinate plane
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 down by its rigid formalism. However, the nature of the exercise highlights that there are…