Hack and Slash
Division was never my strongest assest in math while I was growing up. I was always intimidated by the configuration of the problems, having one number over the top of another number separated by a line. These numbers can either continue on forever or terminate and have an ending. In high school most of the problems we would work on were simple fractions and had ending but sometimes we would get fractions that did continue on but we would only continue them on for one or two digits, never did we continue them on infinitely.
Number Types by Elizabeth Stapel
There are two major types of numbers, rational numbers and irrational numbers. Rational numbers are numbers that terminate. If you have a decimal digit 0.45 or written in fraction form 45/100 that number terminates and does not continue so it is a rational number. An example of an irrational number would be Pi written as decimal digit it goes on forever 3.14159… since Pi never end or terminates it is called an irrational number. “The commonest question I hear regarding number types is something along the lines of “Is a real number irrational, or is an irrational number real, or neither… or both?” (Staple). In every day use we always use real numbers, unless we are working with number or equations that use “i” meaning imaginary. Both rational and irrational number are “real” numbers it does not matter if the number terminates or continues on forever.
Divisions and It’s Discontents by Steven Strogatz
There are a few different ways to express portions of a whole piece of something, Strogatz uses a cake to illustrate this point. “If you cut the chocolate layer cake right down the middle into two equal pieces, you could certainly say that each piece is “half” the cake. Or you might express the same idea with the fraction 1/2, meaning 1 of 2 equal pieces” (Strogatz). I didn’t make a connection to fractions when I was growing up about cutting a cake and passing it out to my relatives at a birthday party, the cake was just cake and each piece was just something to be eaten. Looking at those individual pieces and being able to visualize a numerical representation of the original whole didn’t occur to me. What Strogatz mentions in his article that I took for granted was when he mentions the “slash” between 1/2, and how its a visual reminder that something is being sliced.
Another interesting point that Strogatz’s brings up in his article is how you can have a quarter of a quarter and how people don’t understand that you can have a quarter of a quarter. The diagram to the right shows how a quarter of the circle is 1/4 of the circle. Then if you take a quarter of that you would have 1/16 of the original whole circle, meaning it would take 16 of those slices to make the original circle. He also talks about a conversation between a customer service employee and a concerned customer. The customer has an issue with his bill. His contract states that his “data usage rate is .002 cents per kilobyte, but his bill showed he’s been charged .002 dollars per kilobyte, a hundredfold higher rate” (Strogatz). The customer service agent could not see the difference in this because they didn’t think you could have .002 dollars. This mental rigidity is a problem in our society and in our school system.
Don’t Recite Digits to Celebrate Pi. Recite Its Continued Fraction Instead What’s so Great about Continued Fractions? by Evelyn Lamb
Continued fractions are a much better way to express irrational numbers than arbitrary decimal digits according to Evelyn Lamb. Lamb talks about how beautiful continued fractions are, “a continued fraction is like a fraction but more so. Instead of stopping with one number in the numerator and one in the denominator, the denominator has a fraction in it too. And the denominator of that fraction has a fraction in it, and so on.” (Lamb). Personally I like looking at the continued fractions model expressed in the diagram above. It looks a lot more elegant and sophisticated than a continued decimal that goes on forever, somehow seeing all the fractions on top of each other allows it to appear more complicated yet simple at the same time. “When we truncate a continued fraction after some number of terms, we get what is called a convergent. The convergence in a continued fraction representation of a number are the best rational approximations of that number” (Lamb). Lamb states that continued fractions are the “best approximations” for irrational numbers, but since there is no such thing as a closest rational approximation for an irrational number you can just increase the number of denominators to your fraction, and you can get as close as you want. You can also do this with decimal digits by just adding more decimal terms. Being able to do this makes any approximation of a continued fraction or decimal terms “best approximation” relative. Irrational numbers never end so we can never know its exact value, I agree with Lamb and I believe continued fractions are a prettier way of viewing irrational numbers.
Bibliography:
Stapel, Elizabeth. "Number Types." Purplemath. Available from
http://www.purplemath.com/modules/numtypes.htm. Accessed 10 November 2015
http://www.purplemath.com/modules/numtypes.htm. Accessed 10 November 2015
Lamb, Evelyn. "What’s so Great about Continued Fractions?" Scientific American Global RSS. N.p., 17 Mar. 2015. Web. 11 Nov. 2015.
Lamb, Evelyn. "Don’t Recite Digits to Celebrate Pi. Recite Its Continued Fraction Instead." <i>Scientific American Global RSS</i>. N.p., 11 Mar. 2015. Web. 11 Nov. 2015.
Steven Strogatz. "Division and Its Discontents." Opinionator Division and Its Discontents Comments. N.p., 21 Feb. 2010. Web. 11 Nov. 2015.
I though this post was a very interesting approach for explaining division and fractions. I think division and fractions are something that a lot of kids struggle with when they are first learning. I think the use of Steven Strogatz’s article was very helpful in simplifying how to use fractions. I remember learning about fractions in a way similar to how he teaches with the slices of cake. I also really liked how he explained 1/2 as one of two equal pieces. To me, this is a really understandable word form of fractions and I think it really helps visualize ow simple they can be. I was a little perplexed about the section over Evelyn Lamb’s article. I have never heard of continued fractions before, and from the explanation above I’m rather happy that I haven’t. I had to read the section a couple of times to understand that continued fractions are just the fraction version of irrational numbers. I think that it is a rather neat concept, but honestly I’m not sure where they’d ever be useful. Since irrational numbers are just accepted as they are, I don’t think that we need to complicate them more by turning them into fractions. It may be a prettier way of viewing irrationals, but I don’t find it easier to understand.
ReplyDeleteThis post is extremely interesting because I never thought about the way we truncate long division to certain digits. What happens to the infinite amount of numbers we leave off? Stapel brought up a brilliant point that all numbers, except for “I”, are real. I think once we get past elementary school math we never really think about what definition to give a number. Past early years, it doesn't matter if it belongs in the rational or irrational pile.
ReplyDeleteI personally never thought about the line between fractions as a slash, like cutting the numerator in the denominator amount of pieces. I think that terminology would help better explain math to people who struggle with the subject. I feel like I am in the unruly minority when I say I loved fractions as a kid. They always seemed to make sense to me. I was a freshman in high school teaching my brother’s girlfriend, whom was in college, how to add and subtract fractions.
I never thought of putting of irrational, or non-ending, decimals as a fraction. To me, fractions always seemed to have an end unless you put them in decimal form.
Division, decimals, and other properties of numerical values was a very interesting topic to choose. I find the topic of how division is taught in school to be an even better topic of discussion. Especially for young children, I think it is really important that fractions become a visual thing before you just throw two numbers with a bar between the two at grade school children. What often gets missed in primary education is why children are doing something. If they cannot understand something as basic as cutting a piece of cake and how it can symbolize a fraction it is not time to show them the real thing. I think the writer did a good job explaining fractions and how they should be introduced to students. The part about continued fractions was very confusing to me mainly because I have never heard of them, but after reading the blog it made sense to use them in order to express irrational numbers. Based on what I gathered from the blog a continued fraction is a fraction within a fraction. I would suppose that these kinds of fractions would look very confusing at first glance, but according to the blog they make things a little bit simpler.
ReplyDeleteI found this post to be very interesting. I would have never thought of doing my blog post on this topic. One part in particular that I liked is when you talked about rational and irrational numbers. In math, whenever we use these types of numbers we always find it to the decimal where it's "close enough". What I mean by that is we always use a decimal that will get us close enough to the answer we are looking for and tend to forget about the rest of the number. Another part I liked was using cake so visualize division. This representation does a good job of showing how division works to children.
ReplyDeleteI always found division in mathematics difficult growing up. The process was very difficult for me to comprehend and I would have much rather had a calculator. As stapel said in her article there are irrational and real numbers and some numbers that continue on forever, which is a super interesting topic in my opinion. She said the most common question she is asked is whether a number is rational, irrational, neither, or both That seems like something I would ask because that does seem rather challenging to decide which is which. This topic itself is very interesting and as I have seen personally is rather challenging to many people.
ReplyDeleteI, like Vito, was not the best at fractions when I was growing up. I didn’t understand that fractions were essentially a way of expressing division. That ½ was like saying 1 divided by two. Fractions were always more of a curse than a blessing, especially when you had to reduce them. What was funny though is that I never thought of them being used in real life outside of cooking measurement. With the cake example Vito gave, I was too focused on the fact that I got to eat cake to realize that I was using math and spatial reasoning to cut the cake. I think this is an interesting topic and I am happy that Vito chose to write about it, because I enjoyed his post.
ReplyDeleteI have never considered irrational numbers to be "beautiful" by any means - I find them inconvenient and troubling because they just never end. However, Vito captured my attention in his post and I found it interesting to consider the beauty in being able to express a never-ending decimal in the form of a seemingly clean and concise fraction (22/7 for example, which is the infamous pi [3.14159265...]). I also liked the example of splitting the cake into various pieces to represent fractions of the whole. I would just really like to know how pi in particular got its name, while other infinitely expanding decimals are nothing special I guess.
ReplyDeleteI found this topic about division interesting because I was never the best with fractions in school. When we worked on fractions in class we always used numbers that had definite ending or only had one or two remainders. Seeing that fractions can continues on forever in an almost artistic fashion inspired me to try to understand fractions better. Personally I like looking at decimals rather than a continued fraction because to me it looks cleaner and not so much is going on the page. I’m also used to seeing numbers like Pi in their decimal form so seeing it in its fraction form was a bit overwhelming and not something I would like to work with in the future.
ReplyDeleteI thought this blog post was a different and different approach to fractions. I think that most young kids have trouble with the whole concept of fractions and the different ways there are to simplify, add, multiple, etc. I think the cake example that Vito gave demonstrates how fractions work and is a real world example for kids to comprehend. I also like to use decimals rather than fractions just because its easier to use with whole numbers and more complicated formulas. It’s also easier to use when describing irrational numbers such as pi or e. Vito did a good job of describing these ideas and made this post interesting to read.
ReplyDeleteI really like the structure of this blog. Learning division for the first time can be quite a confusing concept but is also so important to mathematics. Many people seem not to see as fractions actually dividing numbers and giving you a number. I used to know how to long divide but now it's just a lost ability that has been replaced be a calculator. I find imaginary numbers to be really interesting concept and It is kind of irritating that these numbers exist, but don't at the same time. Maybe i'm overthinking it but the irrational continued divisions seems a little confusing to me. Maybe I just don't understand the terminology being given but it's not clicking with me.
ReplyDeleteA clarification on irrational vs rational numbers:
ReplyDeleteIt is not whether the decimals go on and on to determine whether it is rational or not, it is whether there is a pattern. For example in 1/3 = 0.33333..., the digits go on and on, but the pattern is obvious (keep writing 3's), so it is rational. In 1/7 = 0.142857142857142857..., the pattern is to write 142857 over and over again, so it is rational. However pi = 3.1415926535... does not have any pattern. The square root of 2 = 1.414213... does not have any pattern (it is not easy to show that no pattern exists).
Also I would like to correct the misconception that pi = 22/7. 22/7 is a very good approximation to pi, but it is not pi.