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! 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 (...

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...

There's a saying that good teachers meet their students halfway. I'm not sure that is true. Don't get me wrong. I think it is critical for teachers to meet their students academically...somewhere. I just don't believe the meeting point needs to be defined with that much specificity. If I have a student who is capable of going 80% of the way toward learning a concept, meeting that student halfway is a disservice to the student. I should only meet them at the 80-20 mark. If I have a student who is only capable of going 25% of the way toward learning that same concept, meeting that student halfway isn't sufficient. I should meet that student at the 25-75 mark. My responsibility to each student is different for the exact same concept. So why should it look the same for each student in my classroom? It shouldn't. Our classrooms are filled with students who come in with varying ability levels, various levels of self-worth, various levels of self-doubt, and...