Blog Reflection

The end of the semester is here, and that means class reflection and evaluation time. I must say I feel this blog assignment added a lot to my class experience. First of all it's great to have such  unique homework assignment for a math class, it makes the class less boring, as for as math classes go. Most students don't like long lists of problems to do for required homework, I know I don't. The blogs were a great alternative. In relation to the focus of the class, the practical teaching techniques, it helps see math in a different way. Up until this point I've looked at math from the student's point of view, I know my opinions on math (not a fan) and the critical thinking the blog posts give are really helpful in thinking like a teacher. I don't know if I'll continue blogging, but I enjoyed the experience.  As far as the idea of using it in my own classroom, I feel the critical thinking behind some of the posts may be more appropriate for older students, secondary and high school. I'm more geared toward english, so I feel I may be a bit harsh.

All in all I think the blogs were a great contribution to the class. It really shows how math doen't have to be all arithmetic problems. It's a great opportunity to think differently and creatively and I feel I gained a lot, though I don't think it would be appropriate for an elementary setting.

Geometric Constructions

A compass is more than just a tool for drawing circles. I remember as a student the first time I used one for that purpose, it was so cool. Later in school I learned all the other things a compass can be used for. Who knew something that made circles, when also used with a strait edge, can also be used to make triangles. That just blew my mind. 
It is important to know how to use a compass and strait edge to create angles because with just these two tools, you can accurately make any geometric figure. As early as ancient Greece mathematicians were using these tools to solve geometric problems. In these times though, the compasses were collapsable, that is, they would close as soon as they were moved, and would not hold its radius. A Greek mathematician named Euclid formulated three famous geometry problems that for centuries went unsolved. The basis of these problems were to solve them using only a strait edge and compass.
the three problems are:
1.Squaring a circle
2.Doubling a cube
3.Trisecting an angle

It wasn't until a few thousand years later that it was proven these constructions are impossible to make using only a strait edge and compass.

In elementary geometry, we only go so far as to construct angles, triangles and bisectors, with these tools. Yet these same tools have been used for thousands of years to solve (or not solve) much more complex problems.

Why do we need to know the metric system?

Upon further investigation, after asking on the discussion boards, another student told me the blog assignment for this week was basically explaining to children why we need to know the metric system. I would tell them it's the system of measurement used by the rest of the world, and in order to translate measurement with any other country, you need to be familiar with the metric system. I would also explain that because it is a base ten system, it is easier to calculate and convert from one form of measurement to the next.

Assuming Measurement (to be refined later)

So, this is my shot in the dark blog post because I'm not sure what I'm actually supposed to write about, as I did not get the assignment because I was home with some weird flu bug. Assuming the post will relate to the chapter that was covered, I'm gong to ramble a bit about measurement a little just so I have SOMETHING to be graded, to show I made an effort despite being totally clueless. I do plan to refine this post once I know what I should actually be writing, but for now this is all I got. Fortunately, I actually dealt with a lot of measuring this week despite not going to class. For example, measuring out medicine in those tablespoon cups (yuck). There was a lot of medicine being measured. On Wednesday I made myself a detox bath hoping it would make me feel better. The recipe called for three pints of hydrogen peroxide (because people use pint ALL the time), and two ounces of ground ginger. Because I had never really used pint as a form of measurement before, I grabbed my measuring cup and got to converting. One cup is a half a pint, so needless to say I bathed in six cups of hydrogen peroxide. I then  converted ounces to tablespoons because my kitchen doesn't really have any means of measuring two ounces very well. That ended up being four tablespoons. It was a strange experience. Yay for real life relate-ability to this weeks lesson. At least I got something.

Working with small children, it's amazing how early some math concepts can be introduced, though not necessarily understood mathematically. One prime example is the concept of symmetry. As early as the "two's room" at the daycare where I work we inadvertently teach the kids about symmetry, simply by doing projects involving folding paper in half to create a cut out or to make a pretty paining (by smearing paint, folding the paper in half, and opening it again). One perfect example, especially this time of year is cutting out snow flakes, or christmas trees. If I were to teach younger children (1st, 2nd grade) symmetry, I would love to introduce it this time of year if I could, making trees with them and explaining how the fold of the paper is the line of symmetry, and how the two sides of the tree show reflective symmetry. I would then let them get creative and decorate their trees (symmetrically of course), but thats beside the point. A few lessons later, to show how some figures can have more than one line of symmetry, I would make snowflakes with them. Apparently Christmas time and winter lends itself perfectly to teach symmetry to kids. I would explain how things in nature, such as trees, flowers, butterflies, snowflakes, etc. occur symmetrically. 

Really complicated for small children, but really geometric looking. Yee.

Flatland

This week we learned all about geometric figures. Plane, point, line etc. When I was in AP Lit senior year of high school, I had the *joy* of being assigned a book called Flatland, which is literally about 2D geometric figures, living on a plane. Now every time I hear about points, lines and polygons I have horrible flashbacks to that book. Hence, the title of this post. Anyway, back to the real reason for this post. I remember back in middle school first learning about geometric figures, and doing it all on the computer, it was actually quite fun. This time around it was still online and still quite fun. One of the most interesting concepts to play with is angles. Especially when you get to or more involved. An angle is made up of two rays that stem from the same endpoint (not to be confused with an angel, which a spiritual being believed to act as an attendant, agent, or messenger of God, conventionally represented in human form with wings and a long robe). The most common angles are acute angles > 90, right angles which are 90, and obtuse angles which are < 90 and A ray is a part of a line segment in which one end has an endpoint, and the other end goes on for infinity (I always remember drawing arrows on the infinity end to demonstrate that fact.) I think for every student, the moment when you realize how angles react and compliment each other, theres a moment of logical wonder. One of the best representations of how the angles interact is when you introduce vertical angles. These types of angles are created by the intersection of two lines. When you add all the angles together, you get 180. The angles opposite each other have the same degree value. Its great to demonstrate these by having one of the lines rotated around, and watching how, despite the fact the angle values change, they still stay equal. It's always fun when you get to discover math by "playing" with it like that. Here is my vertical angle example from class:

Notice how angles ABD and CBE are opposite yet equal. The same goes for angles ABC and DBE. Also notice how angles ABD and ABC add up to 180, as well as ABE and DBE. 

Probability

This week's lesson was on probability. For example, the probability of spinning a 10 in the game of life, the amount of green skittles in a bag, rolling a 6 on a die, flipping a coin, etc. Something I found really interesting was what is called a probability tree. It's a way of representing the different outcomes of two or more "events' (ex. flipping a coin than rolling a die) and writing it out to show the probability of each set. Going with the example of a coin and a die, you start with the two outcomes, or branches, of the first event, here it would be either heads or tails, each with a 1/2 probability of being flipped. From each of those braches, you branch out the outcomes of the rolle of the dice, in this case we're only using one die, so the heads would have six braches, 1,2,3,4,5 and 6, and the tails would as well. Each of those outcomes would have a 1/6 chance in occurring. To figure out the probability of lets say getting heads and rolling a 4, you would multiply the two probabilities together, 1/2 x 1/6 = 1/12.  The tree would look something like this: