Number diagrams may seem pretty simple, until you come to integers. In the diagram you have to find to difference of two numbers, fractions, or integers, and type them into the box. If you get the wrong answer, the box will not light up. Figuring out the whole numbers was pretty simple, and didn't require scratch paper. I really needed the scratch paper when I came to the fractions. Some fractions were simple and easy and others were a bit more difficult. I found that the ones with different denominators were more difficult then the ones that had the same denominator. The integers was better than the fractions, in my opinion. For example, you would have a positive and negative number and you would need to find the difference.
            The diagram was specifically shaped like a diamond with a square. You would first need to figure out the questions in the square, and inch your way into the diamond. The boxes would glow up if you got the right answer, and when your finished, the whole pattern glowed up. Completing the first one was very easy, but as you went into the integers and fractions, it became more difficult. 

               When graphing inequalities you have several signs that you need to use. The signs consist of greater than, less than, greater than or equal to, and less than or equal to. The greater than or equal to sign is the same sign, except you have a line, or half of an equal sign underneath. When graphing a greater than or equal to sign you leave the dot closed, because you have to include that number. If you simply had a greater than sign you would leave it closed, because you aren't including that number. This concept applies to the less than or less than or equal to signs also. For example, if you had an equation such as m is greater than or equal to 78, you would first draw a line down the greater than or equal to sign and solve. Than you would graph with the closed dot and arrow pointing to the right.
                 Inequalities can look very confusing, but you would just solve it as a normal equation. Just look and interpret the signs. You should always be aware, though. Although the symbols might be a bit confusing, so can the signs. Making sure you never get confused with each individual sign is extremely important. To find the right equation is one step away from creating your graph, and making it completely accurate.

          In math we are currently learning how to tell the difference between a rational and irrational numbers. A rational number is a number that repeats it's self over and over and an irrational number would be like pi or a number that doesn't have a square root. For example the fraction nine tenths would be broken down into 10 divided by 9.0. Ten goes into 9.0 .9 times so your final result would be .9. Pi is an irrational number because the the numbers in the sequence don't repeat themselves., but a number like .3333333333........ would be rational because it repeats it's self over and over again. Whenever you have a number such as .8989898989898...... it would still be rational, because it still repeats the two numbers, 8 and 9, over and over again. When you have a fraction you always divide the top number by the bottom number and you get the result. 
          Repeating decimals are rational, and Pi is  not, simply because the numbers don't repeat themselves. An irrational number is also a square root of a non perfect square. For example, if you have something such as x=4.44..., then you would add a 10 in front of the x, so the equation would then be 10x=4.44... You would divide 10 on both sides, and come out with 4/9. The literal denotation of a rational number is: any number that can be written as a fraction, who's decimal is equivalent; either terminating or repeating.

        So the question that everyone asks is "Why do the decimal shrink with the denominator?" To get the decimal you divide the denominator by the numerator. For instance, if you have the fraction one half, you divide 1.0 by 2 and you end up with .5. If the denominator continues to progressively get bigger, the decimal will get larger. If you have a fraction like pi, seven over twenty-two, you'll end up with a large decimal that goes on forever. The larger the the numerator will most likely stay the same, but the decimal will not. If you come with a fraction such as two thirds, you divide two by three, and you end up with .33333333 and so on. And as we all know the decimal and fraction is all connected somehow. 
         The denominator merely acts as the dividend. You would take the top number, or the numerator, and put that number on the outside of the the sign. Then divide the bottom number by the top number. If the number contains a terminating decimal, it will stop at one point, but if it is such as pi, the number will seem as if it is bigger. If the denominator shrinks, so will the decimal.

          In the number system there is an infinite amount of numbers. If you had a fraction such as 1 over 20 the fractions wouldn't stop there. Since there are an infinite amount of numbers the fractions would be infinite too. Fractions are basically one number dividing by another, or you would write it as one over something. You always divide the top number by the bottom number. The number on the bottom would be infinite between 1 and 2 or 0 and 1. The bottom number would continue to go on forever, because there are an infinite number of numbers. Infinity will continue to go on and that also goes for our numbers in the number system. The number on the bottom, as known as the denominator, has to be a whole number, not a mixed number or decimal. And so the numbers in the number system will remain infinite.
         Infinite numbers go on forever, and the same applies to numbers in the number line. The fractions in between each whole number are what stand as the infinite numbers. In between 1 and 2, there is 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, etc. The same type of rule applies for each number, and each one in between.

        When most people see an equation such as 2x=18, they automatically try to isolate x by dividing both sides by 2. That method does work, but a more efficient and shorter way would be to group. You could take 18 and see how many groups of 2 would fit into there. In this case, there would be 9 groups of 2 in 18. Grouping will take less time and is a more efficient way. Mental math is another way, but you might make some wrong calculations. Grouping would be a more efficient way to do this, because if you group you can actually do more complicated equations such as 2x=35. 2 doesn't go into 35 evenly, so instead of having a large decimal, you would have 17.5 or 17 and a half.
      This method should be taught even before you do division. Using the grouping method, should be used simple math, and might occasionally be used for complex math. Normally complex math actually follows the technique of dividing by both sides, merely because of remainders, or decimals coming after it. That would work in grouping, as you would have several ones left, so they would stand as the left overs.

          Different mathematical equations can be solved differently by different people. Most of us, by now, know how to solve a simple equation such as 6x+3=5. We simply take the three and do the inverse operation by subtracting the three on both sides. If, however, you have something such as 4x-3=5x, you have to find the variable on both sides. Adding, subtracting, multiplying and dividing are the basic concepts to learn how to figure out equations. The variable in any equation is there to represent a number. You can use and letter as a variable, and the answer can come out to be the strangest. Ratios can help and also proportion. By taking the proportion you take a section and break it down.
         Grouping is also an efficient way to add or subtract as well as multiply and divide. For adding, we can do that by heart, but adding integers is a bit of struggle for some of us. If you, however, have a problem such as -4-3, most people try to subtract, but don't know which way to go on the number line. Since you're subtracting, you go to the left of the number, and your answer should be -7.