Good Morning Blog

Good Morning! My sophomore year in high school (1991-1992), while I was learning about right triangles, geometric means, and similar triangles, I asked myself what I thought was a fairly simple and obvious question; if I continue to draw the altitudes for the new triangles formed by the original altitude, will I ever get triangles that are congruent in addition to being similar? For those who are not math-savvy, congruence is a stronger form of similarity. The following images illustrate the idea. The first altitude. The second altitude. The nth altitude. My question is about when triangle CBD 1 is congruent to triangle AD n-1 D n . Using some basics of trigonometry, over the years I was able to write a polynomial equation that I could use to answer my question. In fact, I was able to determine that there was a specific right triangle which produced congruent triangles with the first altitude. Then, using the quadratic formula, I was able to determine a specific triangle which produc...

Good Morning. This is the second post in a series about topics that have traditionally not been handled honestly in Mathematics classrooms. The first post was about adding and subtracting fractions .  In this post I would like to address two aspects of square roots that are often treated with less precision than they ought to be; square roots of negative numbers and simplifying square roots. To better understand both of these issues, it is important to have a clear understanding of what is meant when we say that a is the square root of b . We mean that a times a is equal to b . More precisely, . Square Roots of Negative Numbers When students first learn about square roots, they generally have only worked with the set of Real numbers; natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The collection of all of these numbers is called the Real numbers. But "real" in this sense has nothing to do with these numbers existing and other numbers not...

Good Morning! When I ask my students how they feel about fractions, I don't really need to listen to what they say. I just need to pay attention to their faces. They're generally not fans. When I ask them why they don't like fractions, the responses are not very diverse; they hated adding and subtracting fractions and they hated having to find the lowest common denominator. And I get it. If the fractions have a common denominator, you can just add the numerators. But if they do not have a common denominator, you have to find one first and then rewrite the original fractions as new, equivalent fractions with the new, common denominator. Perhaps you may remember your teacher saying something along the lines of, "You need to find the lowest common denominator first. If you can't find it you can multiply the two denominators." We will come back to that statement later. First, I want to discuss the topics of Greatest Common Factor (GCF) and Lowest Common Multiple (...

Every parallelogram is a trapezoid. Remember in school when your teacher told you that every square is a rectangle but not every rectangle is a square? Do you? DO YOU? Because it's literally the same thing. Every parallelogram is a trapezoid but not every trapezoid is a parallelogram. For some reason this is a debate that periodically rages in many Mathematics teacher circles. Why? Because textbooks disagree on the definition of a trapezoid. The definition of a parallelogram is basically standard. A parallelogram is a quadrilateral with two pairs of parallel sides. The definition of a trapezoid, however, is not as standard. Some textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides. While other textbooks define a trapezoid as a quadrilateral with exactly one pair of parallel sides.  It is this second definition that poses the problem for parallelograms. If a trapezoid is defined to have exactly one pair of parallel sides, then a p...

We've all seen them. For a while they were all the rage. Hack videos. You know, "47 Paperclip Hacks You Didn't Know," "12 Ways to Use a Tissue Your Mom Never Taught You," and "23 Ice Cube Hacks You Can Use Today." None of those are real, but these are; " 43 Simply Brilliant Camping Life Hacks ," " 26 Nail Hacks Every Girl Should Try ," and " 30 Crazy Food Hacks ." [click at your own risk] I think one of the reasons these types of videos became a thing was because they showcased everyday items doing unusual things. Things most people would not have thought of. In many ways, math classes should be the same way. There is a difference between learning how to solve math problems and learning how to solve specific math problems. If Algebra 1 students know the steps to solving a variety of two-step equations, but do not understand how to apply the concept of inverse operations in other contexts, they don't really u...

I often get the sense that much of what people have come to understand about teaching and learning Mathematics is based on a decades old mentality that students will need to be able to do mathematical calculations with computer-like speed and precision once they enter the workforce. And in order to meet this need, mathematics instruction should look like drilling procedures over and over again so that students become proficient and ready to join the workforce. Here's the problem with that approach; it sounds an awful lot like the goal is to program students to behave just like little math computers. Which shouldn't come as a big surprise. Decades ago, people were already being told about the wonderful world that access to computers would usher in...once they became more readily available, that is. In the meantime, we'll need employees to function in much the same way. That sounds a bit dehumanizing. But here we are, 2018, and computers are everywhere . I'm writing...