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! 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...

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 (...

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...

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...

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...

One of the "revelations" I had over the past few years that has helped me change my instruction was that having someone else waste your time sucks. I enjoy having my time wasted as much as the next person. Which is to say, I do not enjoy having my time wasted at all. It is one of the things that irritates me the most. When I realized that students often feel that way in school, I knew I had to be different. I didn't want to lecture any more than necessary. And when I did, I wanted to err on the side of talking too little. If a student can learn something without me, I want them to do so. If I could assign 15 problems instead of 30, I'd assign 15. I was reminded again recently of what it feels like to have your time wasted by someone else. I am in a graduate class and one of the requirements for the class is that we post solutions to homework problems on a discussion board and others respond to them. You can't see the solutions others post until after you pos...

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...

When I was a student I seldom enjoyed reading. Anything. Don't get me wrong, I was able to read at grade level. I understood what I read. But I wasn't passionate about any of it. I never asked for books for Christmas or my birthday and consequently rarely received them. And that remained true until I got to college. Once I got to college I was forced to spend a considerable amount of time with my face buried in books because it was college and the books were required. They were also really expensive and I figured I shouldn't let that money go to waste. To this day, I have very particular tastes in the books that I read. I don't read mysteries or biographies or novels and books like that. I do have several Mathematics books that I have amassed over the years, some computer programming books, and a rather impressive collection of Theology texts including an inordinate number of books related to the Protestant Reformation. It wasn't that I didn't like reading. I...

It's time I came clean about who I really am. What I'm about to tell you may shock you. It may disgust you. But you deserve to know the truth about me. I'm a Patriots fan. And I'm proud of it. Tom Brady is the G.O.A.T. Belichick is a genius.  In Superbowl LI, the Patriots were playing the Falcons, who had built a substantial 28 - 3 lead by the waning minutes of the 3rd quarter. Things looked bleak. It looked like the haters were going to get their wish.  The usually prolific Patriots offense was sputtering and was ineffective against the Falcon defense. The Patriots punted 4 times during the game, giving the ball back to the team that was already beating them. Fans of the game are already well aware of what I am about to tell you, sometimes it's o.k. to punt. Sometimes punting is the right play. In the end, the 4 punts did not hurt the Patriots as they completed the greatest comeback in Superbowl history to win 34 - 28 in overtime.  Sometimes our studen...

So much talk about how relationships matter. So. Much. Talk. If you're an educator and you've been paying attention lately, you are no doubt familiar with the push to improve relationships between teachers and students. Relationships will get students to behave better, be more engaged, learn more, be more emotionally intelligent, and so much more. I recall a training I was at about a year ago. The training was supposed to be about teaching students who come from poverty. But for a training that was a couple of hours long, the overwhelming majority of the training was about defining poverty, characteristics of students living in poverty, and other things of that nature. At the very end, literally only the last few minutes, it was suggested that the way to reach students from poverty was to build, you guessed it, relationships. The Twitterverse and Blogosphere are much the same. Relationships are everything. They're the future. They'll fix all of your classroo...

[This past summer (2018) I had the opportunity to present at two different  Indiana Department of Education Summer of eLearning  conferences; the  South Shore eLearning Conference  hosted by  School City of Hammond  and the  eVillage  Conference hosted by  East Porter County Schools  in conjunction with eLeadNWI. The presentation that easily received the most buzz was my " How I Flipped My Classroom In All the Ways That Matter ." This blog entry is the fourth in a four part series outlining the contents of that presentation.] Part 1 Recap : Teaching the way I was taught wasn't working. I started using Khan Academy in class. That helped. But I wanted more. Part 2 Recap : I flipped my classroom for 2 years. That seemed to work better, but it wasn't great. I wanted more. Part 3 Recap : Being on Twitter introduced me to ideas in education I would not have heard otherwise. My past experience with coding gave me some perspective on the l...

[This past summer (2018) I had the opportunity to present at two different  Indiana Department of Education Summer of eLearning  conferences; the  South Shore eLearning Conference  hosted by  School City of Hammond  and the  eVillage  Conference hosted by  East Porter County Schools  in conjunction with eLeadNWI. The presentation that easily received the most buzz was my " How I Flipped My Classroom In All the Ways That Matter ." This blog entry is the third in a four part series outlining the contents of that presentation.] Part 1 Recap : Teaching the way I was taught wasn't working. I started using Khan Academy in class. That helped. But I wanted more. Part 2 Recap : I flipped my classroom for 2 years. That seemed to work better, but it wasn't great. I wanted more. Great. Now you're up to speed. I thought I had changed some big things about how my classes operated and was only seeing small changes in the outcomes. But the truth i...

[This past summer (2018) I had the opportunity to present at two different  Indiana Department of Education Summer of eLearning  conferences; the  South Shore eLearning Conference  hosted by  School City of Hammond  and the  eVillage  Conference hosted by  East Porter County Schools  in conjunction with eLeadNWI. The presentation that easily received the most buzz was my " How I Flipped My Classroom In All the Ways That Matter ." This blog entry is the second in a four part series outlining the contents of that presentation.] Part 1 Recap : Teaching the way I was taught wasn't working. I started using Khan Academy in class. That helped. But I wanted more. Great. Now you're up to speed. The following August I began the school year using Khan Academy in my Algebra 1 classes. It wasn't the only thing we used, but we used it plenty. While Khan Academy did offer students the opportunity to practice specific skills with instant feedback , ...