The reason inequalities with an equals sign need a closed dot is the fact that a closed dot symbolizes including, on a graph. If something is equal to something, it obviously includes something, i.e. 1+1=2, the answer includes two. If there was an open dot for an equals sign that would be the same as saying nothing. Open dots are used for actual inequalities that have a range for an answer. If x>5, the range does not include five, and needs an open dot, but if x>5, than the range includes everything from five up and would be represented with a closed dot. A general rule of thumb is that < or > will have an open dot, while =, <, or > are represented with a closed dot. The reason this works is that the ones that have a closed dot have at least the opportunity to equal something, resulting in an including, or closed dot, while the ones without the opportunity to equal will have an unincluding, or open dot.
When talking about a specific range that something must be no more than x and no less than y, closed dots generally talk about the correct range, while open dots suggest the incorrect range.
Of many different math concepts, one of my favorites is the factorial. The factorial is one of the more obscure math concepts and is symbolized by an exclamation point. The factorial symbolizes that the number being factorialized needs to be multiplied by all of the whole numbers less than it, and greater than zero. For example, 7! is 7*6*5*4*3*2*1, or 5040. Needless to say, factorials are generally used with smaller numbers. For example, a factorial doesn't change the value of a two or a one. Likewise, 3 only becomes 6 and 4 only becomes 24. I learnt of the factorial in fifth grade, where, every morning, my class tried to get an answer using speific numbers with any math means possible.
Another of my favorite math concepts is a more recently learnt one: Inequalities. While most people learn basic inequalities in third grade, such as 5>2, my algebra class has been dealing with more complex ones, like 7.2<x<7.6. Some inequalities must be solved like two sided equations, such as 14<x+1.5<16.
To solve a divison problem with another operation, proving there is "no such thing as division" there is a series of steps to take. First, you must take the two numbers being divided, and write them as fractions. For example, 8/1 / 4/1. Then you would reverse, or find the reciprocal of, the second fraction, and change the division sign to a mulitplication: 8/1*1/4. Then solve like a normal mulitplication problem, cross cancel, and you would get: 2/1*1/1 which is 2, the same answer the could be gotten from 8/4. The reason that this is better than dividing, is that when the problems get more and more advanced, it is simpler and easier to mulitply by the inverse, instead of dividing.
One more advanced math concepts is the theory of the infinity between 0 and 1. The way that this is possible is that the decimals directly after zero, such as 0.1, 0.01, 0.001 and so on, can go on forever. Even if it was possible to find an end of the 0.1's, you would immediately go to 0.11 or 0.2, which ever way you look at it. Another idea that supports the infinity between zero and one is a paradox. In the same way that the decimals can keep getting smaller and smaller, you cannot possible leave the room you are in, because each step necessary to leave can be divided smaller and smaller. For instance, if you are halfway through a 6 foot room, the necessary 3 foot step could be two 1.5 ft steps, and that could be four .75 foot steps etc. The infinity between two whole numbers is an interesting and complex concept.
The fact that the larger a denomintator is, the smaller the fraction, is a major sticking point for elementary teachers. Because that fact is somewhat illogical, they repeat it over and over again to make sure their students do well. What I'm blogging about today, though, is why. The answer is simple. The denominator tells how many pieces you divide whatever into. For example, if you took a pie and divided it in two, you would be dividing it in half, which is the same thing as 1/2. In that case, the denominator is two, and that is how many pieces the pie is in. But, if you took the same pie and divided it into fourths, or 1/4, then one of those pieces would be smaller than just splitting the pie once. On the other hand, if you increase the numerator, that would be increasing the number of pieces, and it is possible then, for a fraction with a larger denominator to be larger than one with a smaller denominator, such as 3/4 and 1/2.
In Algebra this week, we are using the theme of "Pirate Treasure" to be able to solve a problem for x. First off, Mr. Erickson rolled out a number line mat with palm trees and integers on it. On one of the sections, a PostIt was stuck. The only way to find the "treasure" was to solve the problem backwards using inverse operations and the number line itself. After two volunteers, Mr. Erickson rolled the mat back up, and gave each table individual problems and a mini number line. Then we split into two teams. In my group, everyone got everything right except for the girls team missed one problem, so the boys won. Mr. Erickson taught the lesson of "solve for x" by using a creative scenario that we needed to apply the lesson to.
