Posts

Geometry - Circle Properties

Image
We can find the measures of each of the labeled angles using circle properties and basic geometric theorems. First of all, since the arcs between each pair of adjacent vertices is equal, we know that we are looking at a regular pentagon inscribed in this circle. A pentagon can be made up of three triangles, thus the interior angles must add up to 3(180) degrees. Since it is a regular pentagon, we know that angle 1 is 3(180)/5 or 108 degrees. Now, notice that line AB is parallel to line EF, so angle 1 is congruent to angle 8 by the Alternate Interior Angle Theorem. So angle 8 is also 108 degrees. Angle 2 is supplementary to angle 8, so angle 2 must be 180-108=72 degrees. Line BC is parallel to line AD, so we can use the Corresponding Angles theorem. Angle 2 must be congruent to the angle supplementary to angle 3 within the triangle also containing angles 4 and 5. Thus, angle 3 must be 180-72=108 degrees. Now, let's just back up to the top of the circle. Line BF is tang...
1 comment
Read more

A Letter To My Future Self

Dear Ms. Koch, I assume that you've gotten used to people calling you that by now, even though it sounds weird to me right now. I really hope that you are a teacher and are happy with your life. You may remember that I know what its like to work everyday at a job that isn't what you really want to do, and I want to believe that I've paved the way for you to not have to live that way anymore. On top of you being happy though, I also hope that you are a good teacher. I know how nervous you were when you started. In case you've forgotten, I've included some things for you to remember that you learned and believed in on how to be a better teacher. First, remember that you aren't just a math/physics teacher. It matters of course that the main concepts of the class come across and that your students know how to solve specific types of problems, but there's more to it than that. What you really need to teach them is problem solving skills, how to use logic to m...
1 comment
Read more

Learning How to Struggle... Productively

Image
It seems counter-intuitive to say that teachers want their students to struggle. That doesn't make any sense. It is the teacher's job to make sure their students learn the concepts, which should mean that the students aren't struggling anymore. The key piece of that is the "anymore" bit. The goal is, ideally, to get every student to the point that they no longer struggle with concepts that were once new and confusing to them. The only way to really make sure this is the outcome, however, is to let them struggle in the beginning. The concepts are new to students when you first introduce them, it's only right that they don't understand it immediately. So you start with some basic, trivial examples to get them started. Now that they have an idea of the concept you are trying to teach them, you have a choice. Keep giving them easy examples. They get more practice with the method They gain confidence because they always find the correct answer Give th...
1 comment
Read more

Math IRL

One of the biggest things that students hate about math is that it doesn't seem necessary. I don't think I made it a single day in high school without hearing someone complaining about math, specifically that they would never need to know the stuff they were learning after graduation. Even today, I hear people say that we should stop teaching higher level math classes in high school because it just isn't necessary for every  student to know that stuff. Obviously, I strongly disagree. Perhaps I'm a bit biased seeing as I'm trying to be a high school math teacher. I'm the most obvious example of someone who uses the math they learned in high school in their everyday life. I also would argue strongly for keeping as many math classes in high school as possible so that I have a better shot at having a job. If we ignore my bias for a minute, we can look at reasons for why math should stay for the students' benefit and why they should stop complaining about it (t...
2 comments
Read more

Learning To Teach Algebra

Image
I was asked to talk about an interesting algebra problem that I've done. Instead, I want to talk about an interesting problem/issue that comes up when teaching algebra that I've encountered that I think everyone (teachers and students) need to be aware of. Students learn whatever their teacher decides to show them. This can be an issue as sometimes students learn a specific method rather than the concept the method is covering. I remember thinking that math was so great because it wasn't up for interpretation. There was only one way to solve a problem and it produced a single answer. It wasn't until much later that I realized that it isn't as simple as that. Let's consider multiplying two polynomials. If you are multiplying a constant by a binomial then you distribute the constant to both of the terms in the binomial individually and then add the two products together. 3(x+1)=(3*x)+(3*1)  =3x+3  Easy enough. What about multiplying a binomial instead o...
2 comments
Read more