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I Fought Galois and Galois Won

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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
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Hard Work Pays Off

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Good morning! In September, 2015, I began taking graduate courses at Purdue University Northwest so that I could continue to teach upper level math classes as dual credit/dual enrollment. Through a variety of grants, the courses were available for free to teachers who pursuing the Higher Learning Commission's requirements for dual enrollment instructors. If I had already had a master's degree, I would only have needed 18 graduate credits in Mathematics to meet the qualifications. Since I did not already have a master's degree, I needed to earn one. This was a problem since the universities offering graduate courses were not offering degrees. Not sure what my opportunities would be in the future, I took full advantage of the graduate courses being offered through Purdue University Northwest by earning 9 graduate credits in the first year. Then the grant money ran out. Scrambling for more options, I took advantage of another opportunity through STEM Teach and earned 9 more g
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Telling the Truth About Square Roots

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Good Morning. This is the second post in a series about topics that have traditionally not been handled honestly in Mathematics classroom. 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
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Telling The Truth About Adding Fractions

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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 (
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Location: Valparaiso, IN 46385, USA
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Online Teaching Is Not the Same as In Class Teaching. Discuss.

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Over the past 5 years I have taken a total of 10 (ten) online graduate courses through Purdue University Northwest, Indiana Wesleyan, and Indiana University all in an effort to be meet the Higher Learning Commission's criteria to teach dual credit courses. The course design has been similar for each. Students login to some learning management system, get the assignments, work on them, submit them, etc. There is one element of course design, however, that has been the most beneficial for both myself and the other students in the class; online discussion. Not every course I took had online discussions, but the ones that did were the most engaging. Inserted from GIPHY The way the online discussions would work is each week the students and the professor would engage in online discussion about the assignments for that week. The questions would be posted, someone would reply with an idea of how to start, others would offer criticism or support, or continue with the next step. Everyone wo
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I'll Go There

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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 parall
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Growth In the Time of COVID 19

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So here we are. Well, here I am. In this period of social distancing, unless you are my wife or one of my children, we are probably not in the same place. On Friday, March 13, 2020, the school I teach at closed due to the COVID 19 pandemic. The closure was originally supposed to last until April, 10. Since then, the governor has closed schools for the duration of the school year. Most, if not all schools nationwide have also closed. To continue educating students, schools are transitioning to online learning, or eLearning. This has put schools, teachers, students, and parents in a new a stressful situation. Schools are empty. Teachers teach from home using some form of a blended, online learning model. Parents, even those deemed essential workers, are left to be supervisors for their kids. Teachers who are also parents have to do both. In a regular classroom environment, everyone is working on roughly the same things at the same time. The teacher is only steps away. When there is
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