16May 2017 by Vincent Summers No Comments

Simple Algebra II Graph Symmetries Discussion and Examples

Education, Mathematics
[caption id="attachment_17854" align="alignright" width="480"] Typical functions in two variables.[/caption] College preparatory classes in high school often include Algebra and Algebra II. Perhaps the most memorable aspect of Algebra II is the two-dimensional (2D) graphing of mathematical functions in two variables. This is typically introduced beginning with the Cartesian coordinate system. The generic function is written y = f (x). This reads y equals a function of x. See the illustration for some examples of functions. Cartesian Coordinate System In the Cartesian system, two variables, often x and y, are assigned their own line, one horizontal (x), one vertical (y). The intersection between the two axes is called the origin, and is assigned the value (0, 0). The value of x is the value written on the left in the brackets;…
Read More
07Mar 2016 by Vincent Summers No Comments

What are Radians? Where Do They Come From?

Mathematics
Consider a simple equilateral triangle (a triangle that has 3 equal sides and 3 equal angles). Most high-school students know the three angles of such a triangle are 60 degrees (60°) each, for a total of 180°. But degrees is not the only unit used to quantify an angle. Alternately radians can be used. What are radians? Are they just another number? Where do they come from? Degrees Before we get into radians, however, let’s consider where degrees came from, and why it may not be the best choice for the measurement of an angle. If you are facing north and turn to the east, you have turned 90 degrees. Now turn south and you’ve turned another 90 degrees. Turn west, another 90 degrees. Continue the turn so you once…
Read More
15Jun 2015 by Vincent Summers 8 Comments

Is a Circle a Polygon or Not? Implications for Calculus

Mathematics
[caption id="attachment_16285" align="alignright" width="380"] A 12-sided dodecagon.[/caption] What do you think? Is the circle a polygon, or not? As a result of watching a child’s video, I previously wrote a brief piece about the “corners” of a circle. The video was designed to teach children the various shapes—how many sides does a shape have, and how many corners? The video maintained a circle has no corners. I called that into question. I still do. And yet, I do not. I now think it’s all in how you look at it. Or, you might say, it’s all in the mathematical perspective. Consider. The Circle by Definition One can define the two-dimensional circle as the complete collection or “set” of points equidistant from a set point, not part of the circle. In…
Read More
14Mar 2014 by Vincent Summers 4 Comments

Approximate Calculus: Area Under a Curve

Mathematics
Is it possible to calculate the area under a curve with any degree of accuracy? If we have a strong mathematical background, we may say, "Oh, that's easy. It's a matter of calculus." But what if you didn't take calculus? In fact, what if you never even attended high school? Is it possible to achieve an answer? Is it possible to use reason to come up with the principles of calculus? The answer is, indeed it is. I met a man who did just that. He asked me how I would figure the area under a curve. But he did more. He wanted to show how me how he had succeeded in solving the matter himself on his job in construction work. I was so impressed I exclaimed, "You've just…
Read More