Trillion Triangle Problem
Feb 23rd, 2011 by corricelli
Link to article: http://www.sciencedaily.com/releases/2009/09/090922095651.htm.
Please respond meaningfully and uniquely to one of the following questions. Use good grammar and complete sentences. Think about QUALITY, not QUANTITY.
1.) How is this related to topic(s) you have seen this school year in mathematics?
2.) In what way is this meaningful to you in school?
3.) How is this related to you/your life?
4.) How is this connected to another subject (beyond mathematics)?
5.) Can you find a related article/discussion online? What light does this article shed on this news story?
NOTE:
At this time, Saturday, February 26, comments are closed for this article.
Thank you!
Mrs. Corricelli
1) The Trillion Triangle problem is related to many topics discussed in class. First this problem uses right triangles, something focused on and explored by us for a large portion of this year. Also the Trillion Triangle problem shows a practical application of programming. This creates a strong link to what we are currently doing as we discussed a more simple type of programming in class last week. The Trillion Triangle problem is an example of how programming can be a substantial aid in mathematics and can allow mathematicians to solve problems which previously have gone unsolved. Understanding more about programming will not only simplify math for us as students but also continue to assist professional mathematicians.
3. This is related to me and my life because it shows the importance of teamwork. Two heads, or more, are definitely better than one. Learning to work together, with people from all over the world, can make solving problems easier. For example, the article says that Fibonacci stated, but did not prove, that one is not a congruent number, but that Fermat provided the proof. Both guys were needed to discover this, and by working together they were able to come to this conclusion
2. This is meaningful to me in school because it makes people realize that math can be applied to the outside world. It’s not just “memorizing formulas” and plugging numbers into the equation, but actually playing around with certain numbers and seeing how they are related to each other. Personally, I like this message because sometimes, school can get a little repetitive and I don’t see how certain things we learn can be applied to life and our future careers…in this case, future math careers. People are always solving math problems and looking at things from different angles, which motivates us to think outside of the box, too.
Alrite well i cannot lie at first when i read the article i was wondering if it was some kind of trick problem and that you were going to have us try and answer some extremely old question but in response to the questions that you have posed, this question is meaningful to me in the way that even though this question was thousands of years old, mathematicians never gave up on it. It shows that even when a question is unable to be answered at a specific time, the technology of our day and age is improving and revolutionizing at an amazing rate and if you keep tying this technology to the question at hand, there are many questions that will continue to be answered technology will help make it possible. I feel that as seen in this document that working with technology is extremely important and this is what this article meant to me.
4) This problem goes way way far beyond mathematics. Even though the problem is literally thousands of years old we still have people going back and trying to solve it, taking new steps with technology and with the bravery of taking on a problem that has taken years to solve. Clearly these computers that can compute such huge numbers and equations and the incredible people who write them can be applied to other subjects. The thing that I keep forgetting, especially when I am frustrated with math, is that it’s freaking everywhere. In new medicine, in biomedical chemistry, engineering, paying your taxes, making statistics, absolutely everything can somehow be related to it. So if we as people are able to use our heads to create incredible computer programs, and form teams and solve problems such as this one, whose to say there is anything stopping us from overcoming any problem, whether that be involving math directly or indirectly. It’s just an incredible feat, and in this we can find confidence in our wish to solve problems no matter how old or difficult, and in any subject.
1. This is related to the simple programming that you showed us a few days ago because both are applications where laws of mathematics were applied to computer programs to interpret or evaluate the expressions. This instance of the Trillion Triangle being worked on using a computer is an example of what mathematics will evolve into over the next decade or so as technological advances occur so rapidly. Mathematicians of the past have had neither the knowledge nor technology to solve complex problems. The systematical application of advanced technology to mathematics will result in more discoveries and advances in mathematics in the next few decades than have occurred over the past thousand years. We have the knowledge and technology to do essentially whatever we would like to do with numbers. All of the unsolved theorems of the past can be solved with the technology of now, opening the doors for more and more advanced topics in mathematics to be examined. Math will look a lot different in the next few decades.
2) This article is very meaningful to me in school. First of all, with applying our knowledge to program our calculators, as we did the other day, we are acting very similarly to how these mathematicians and computer technicians are solving this trillion triangle problem. Their need for certain software and programs shows how these skills we use to generate programs in school can be utilized on a larger scale to solve thousand year old problems like this one. Furthermore, the usage of technology by these mathematicians to solve this complex problem, shows how technology is a vital part of our lives and jobs. Learning about new technology and being able to experience it in school (through computer programs, calculators,etc.) provides an advantage for us in the future for when we are in college and then progress to get jobs. This article really shows how technology is fundamental for modern careers. All in all, the skills that we learn in school like using technology and learning how to generate programs provide us with the building blocks to life beyond school, and this article really depicts that.
1) The Trillion Triangle problem is definately related to topics that we have seen in math class this school year. This article talks alot about the application of these congruent numbers in the context of a right triangle. We went through a whole unit about right triangles and they continue to be of use to us as we apply their properties to solve more complex problems. This article also puts particular emphasis on the importance of programming. Just the other day we programmed our own calculators. Programming is an important skill that I am sure we will continue to use because it make the tedious work easier so that we can really focus on the part of the problem that requires the most brain power. Even though our programming was a more simple task, it is still the same basic concept. In the Trillion Triangle article, the immense importance of program is shown in the fact that for hundreds of years this problem went unsolved, but now with the aid of computer programming mathmaticians were finally able to achieve their goal. I’m sure that we will do more programming this school year and that this concept will remain an important part to help us solve problems; however, it will also be interesting to see the application of programming by professionals to types of problems similar to the Trillion Triangle problem. Who knows what we will be able to solve with the help of programming!
In response to part one, this article was obviously related to all of the fun we have had this year with triangles, especially right triangles. Besides the 3 4 5 triangle and maybe the 8 15 17 triangle, I myself rarely used or knew about the multitude of “congruent” triangles.
As it pertains to my life, this article scares me about the the massive calculations that computers can now complete. It was said in the article that if this algorithm was done on paper, it would have went to the moon and back. As you would say, “That’s beasty”.
Finally, I’m just throwing it out there, I think we should try as a class to solve a millennium. Even though we would never be able to do it, it might fun just to try. And who knows, we could wind up $1,000,000 richer.
After reading this article, it really made me think about how absolutely ginormous(ly?) people can think. I’d like to think I can relate with these mathematicians, because I really like thinking about and solving difficult problems when they are presented. When I was reading, I was basically thinking, “wow, this is so amazing how these people are able to take such an advanced problem and simply create a program to find the solution.” That just blows me away. I would really love to learn more about programming and learning new ways to look at problems, which is why I am considering engineering as a future career. Also, I like the fact that the article mentioned that these mathematicians collaborate in order to solve problems such as this. They go to conventions all over the world in order to work together to solve a common problem, and the beauty of math is that there are no language barriers! Go math! Anyway, I also like working in groups in order to solve problems. It is just really fun (yes, I’m serious) to toss ideas around with some friends. In the end, the problem gets solved and you can also gain important skills and learn different ways to interpret problems. Overall, I thought this article was really interesting and it also made me think; If one thousand years after the problem was presented, we can solve for one trillion answers, just think about what we are going to accomplish in one thousand more years! wahhh!
4. This is connected with the field of engineering and computer processing. Not so much in the applications of the problem, but in the manner the problem was solved in. The use of computer programs to do enormous amounts of calculations and compile the results is very important in engineering. For example, in simulated tests of a weight-bearing structure, it is necessary to do a lot of calculations in order to find the cheapest way to hold the most weight, based on numerous variables. If you write a computer program to do this for you, then you don’t have to do all of the work out yourself, you just input the data and the results come out. Too bad we can’t do that for chemistry homework.
3) The Trillion Triangle problem relates to my in that it shows the importance of teamwork. The problem was first introduced by Persian mathematician al-Karaji. Who was greatly influenced by Greek mathematician Diophantus. Although his intial wording for the problem is different from today’s they are both essentially the same question. The problem mystified mathematicians and only made slow progress over the first 1,000 years. In 1225, Fibonacci was able to prove that 5, and 7 are congruent numbers. He also stated but didnot prove that 1 was not a congruent number. Later on Fermat supplied the proof that 1 is not a congruent number in 1659. When 1915 roled around congruent numbers less than 100 had been determined. Then in 1952 Kurt Heegner introduced different mathematical tachniques and proved that all prime numbers in the sequence 5,13,21,29….. are congruent. However by 1980 many cases smaller then 1,000 still remained unsolved. Many other mathematicians individual built on these ideas which finally led to the creation of two team, The team of Bill Hart, and Gonzalo Tornaria and the team of Mark Watkins, David Harvey, and Robert Bradshaw. Separetly they created algorithms which they ran through different computers tiwce to insure the validity of their results and finally solved the Trillion Triangle Problem.
How does this show team work?
Well, if not for having been inspired by Diophantus’ work al-Karaji may have never introduced the problem. Secondly the work of all the mathematicians, though done separtely, built on top of each other to lead to the creation of the two teams. Then in the teams mathematicians worked together, throwing ideas around, and brainstorming inorder to create the separte algorithms.
As a class we similarly communicated ideas and helped one another through difficult problems by using the blog, facebook, wiki, and face to face chats outside and inside class.
1. this article reminds me of the applications we did in class. for instance, when they were talking about using ideas from number theory and physics to predict the pattern of the congruent numbers, it reminded me of applying trig to complex numbers.also, in the article, they kept on talking about actually developing the program/code to solve the problem. isn’t this similar to how we did a programming lesson in class? we developed a code to add vectors, like how they made a code to find the congruent numbers up to a trillion.
2. Obviously this is related to my life because this is what I go home and do with my free time everyday. No really it’s related to my life because we worked on building a code or computer program in our calculators the other day and in class we discussed a little about combining programs and that is what we attempted over the four day weekend. This is also related to my life because it is a challenging problem and I enjoy working towards understanding problems like this and what they mean as well as their answers. We also deal with a lot of triangles in precalc so it relates to all of out school work because in the short story of this problem it is a 3-4-5 triangle which we have come to know. Also this relates to our class because it isnt just figured out by one person, we work as a team to solve problems in class and help everyone understand.
This article is important in regards to my life because of the importance of technology in solving the problem. As our generation goes out into the workfield, it is likely that almost every job will deal with technology in some way. In order to work today, being familiar with, being comfortable with, and knowing how to manipulate technology are skills that are implied as necessary for many jobs. These mathematicians show that working with this new technology is necessary, and leads to a greater chance of success. They had to create entire new programs in order to solve their problems, and it is likely that we will have to (on a less complicated scale) use computers to our advantage as well. Our generation is one of a completely new era, in which computers and other advances seem to dominate our lives. In order to succeed, we must take a page from these mathematicians, and jump head first into the technology craze.
3) This article relates to how technology and human innovation go hand in hand. In the article, it states that the first thousand years after al-Karaji proposed the problem were relatively unfruitful in finding solutions to his problem. By 1980, there were cases smaller than 1000 that have not been solved. However, mathematicians all over the world did not give up, and with the use of the computer, have been able to find over 3 billion congruent numbers. Technology helps people find ways to get to a problem; Selmer and Sage used two different algorithms to get the same answers. If we can master using technology for engineering, science, and history (to name a few), I’m sure the unsolved mysteries out there will be solved with the help of computers. It definitely helps to learn tech skills at school.
5) I found a website that explains congruent numbers in great detail and also allow readers to blog about the information given on the site. The URL is “http://bit-player.org/2009/congruent-numbers”. The site provides the equation on how to figure out if a number is congruent. The equation is as follows: Let n be a squarefree natural number. If n is odd and the number of solutions of 2x^2+y^2+8z^2=n is equal to the number of solutions of 2x^2+y^2+32z^2=n, then n is a congruent number. If n is an even number and the number of solutions 2x^2+8y^2+16z^2=n is equal to the number of solutions of 2x^2+8y^2+64z^2=n, then n is a congruent number. In addition, the website shows the connection between congruent numbers and elliptic curves, which is a very complex and confusing process. The article from class introduced the topic of congruent numbers and this site went into great detail and delved deeper into the mystery of congruent numbers.
1. This is related to school since it is about right triangles and we use them every day in class. This article just goes further on right triangle and shows how big they can get . It is also related to the programming that we did last week. We did it on our calculators but since their problem was much bigger, they had to use a computer that would be able to solve this massive problem that could only be handled by better technology. This also shows much computers can do since they were used to solve a problem that was proposed so long ago.
1. There are obvious relations of this article to what we have learned in math this year, such as triangles, their areas, sides, etc. However, I have trouble understanding how this discovery relates to the world of math in a practical way. Sure, its a very cool discovery, but how can it be used to further human technological capabilities? On a related note, it could be argued that numbers are arbitrary, because they are of human invention, and not found in nature. I would say that the original purpose of numbers is to use them as a tool, to find answers and predict outcomes, keep track of figures and their change over time. So then aren’t numbers most useful if they have a practical application? How can discovering the nuances of humankind’s greatest tool be used to make it more powerful?
3. This article showed me the importance of working together to solve our problems. In the very beggining it says “Mathematicians from North America, Europe, Australia, and South America…”. This particular math problem dates back to the 10th/11th centuries. Over 3 billion of these “congruent numbers” were found by a team of mathematicians from around the world. Had they not worked together, who knows how much longer it would have taken for all these numbers to be discovered. This can be related to Mrs. Corricelli’s blog. Many of her students work together to solve our homework problems. Helping each other is probably the best way for us to learn.
And I agree with Joe, solving a millennium would be a great way to end the year!
1.)This article about the Trillion Triangle Promblem reminded me a lot about what we have gone over in class about the importance of new technology. With out those super smart comupters those mathematician’s would not have calculated over 3 billion congruent numbers. Technology today allows people to solve math problems that most couldn’t solve in their heads. I thought it was cool that those mathematicians used a program to find out many of the congruent numbers. It’s wierd how a computer or a calculator can help us solve fairly easy math problems yet people like Bill Hart and Gonzalo Tornaria use basically the same technology to create and run a program that helps them find billions of solutions a problem over thousands of years old. Also when I read this article I thought about Watson the Jeopardy supercomputer. It is unbelieveble that technology today can be so smart. It will be cool to see in the future how the new technology will change they way we think and tackle problems like this.
3) This is related to my life because it involves people in the present day learning from the work of people from the past. I take Ancient Greek with Mr. Coleman, and in class we translate texts from Ancient Greece. Ancient Greek Society gave history some of its greatest minds: poets, philosophers, and, of course, mathematicians. Their works are read and discussed deeply today, including the Trillion Triangle problem, which al-Karaji based on the work of the greek mathlete Diophantus. Through my Ancient Greek studies, I have developed an appreciation for the works of the Ancient world.
1.) This problem, particularly the methodology used to find the solution, is related to the lab we did on radar guns . In this lab, we used something similar to a radar gun which then graphed distance and time of the movements of the person in front of the computer’s sensor, the two key components of calculating miles per hour. What might take me two hours (or longer) to graph was done instantaneously by the computer, which is madness, because it’s crazy to think that we have devices which can work at such fast speeds. In math there are various ways to attack a problem, and in this problem, the mathematicians attacked it with the use of two computers, in order to double-check their work. However, the point is that they couldn’t have done it without a computer, and neither could we (in the time allotted) in the lab. Even with the help of every mathematician in the world, this problem could not have been solved in the short time it took the computer. In less than one month, the problem was resolved not because the mathematicians now-a-days are smarter than those back in 1000, but because the technology has come so far that problems even of this enormity can be solved in a period that has decayed exponentially.
3.) How is this related to you/your life?
The ‘Trillion Triangle’ Problem is related to my life in ways I never would have thought of. The Trillion Triangle Problem seems at first so abstract and obscure that it could not possibly have any connections to my life, but I was surprised to find that it did, not simply in the actual mathematics of the problem, but in the manner in which it was solved. The article states that this problem was posed more than a thousand years ago. That fact that time didn’t discourage mathematicians is simply astonishing. Each mathematician made a contribution, Fibonacci contributed in 1225. And because of Fibonacci’s work, Fermat was able to add on to Fibonacci’s work in 1659, and together they were able to formulate a proof on congruent numbers. And even further, in 1952, Kurt Heegner added to the knowledge on congruent numbers as did Jerrold Tunnel in 1982. This made me realize that some problems require such immense brain power, too much for one person to supply. Time should not be a discouraging factor, sometimes the most challenging and lengthy problems are the ones that are most worthy and applicable. The Congruent number problem is already finding its way into modern technology and programming. There is no doubt we will see more of it. The determination, hard work and focus demonstrated in the Trillion Triangle problem is admirable and relates to my life because such determination and hard work should be emulated. Anything good requires hard work.
The Trillion Triangle problem is related to my life mainly because of the technology that is necessary to do the math for the problem. The problem does seem like it is already challenging enough to get whole numbers as the area by only using fractions as the sides. The technology needed to do this is very advanced if it has to deal with numbers that are so big, dealing with the trillions. It is amazing how even in history how the Persian mathematician was able to complete his math problems without knowing any of the information that we know now. Through time, the knowledge has increased along with the advancement of technology, and I believe that the math has improved because of the technology. The technology is relevent every in the world now too. For nearly every job, to find information, and it is what people use in their free time as well, technology is everywhere. So what the moral of the article is, is that with more technology, more mathematical results will be produced.
3) This problem is meaningful to me in school because it shows how that you can’t always be reliant on technology. The mathematicians, even today, still do not have the computer power to be able to calculate some of the numbers they are working with. This problem shows how it really comes down to being able to use what you got such as the knowledge you bring to the table along with being able to work together with a team of different backgrounds to reach a common goal. It also is meaningful because it really looks back on the history of the problem for today’s mathematicians to solve it. The article brings up that knowing the history and learning from past mistakes and accomplishments can help you in the long run. It is meaningful because it shows you need to work together will others when you can’t do something all by yourself and learning from your own or others accomplishments and failures will get you farther than you thought you would go.
This article was really intriguing because it explains how mathematicians used what they learned throughout their life about math and took it a step farther to eventually come up with new problems and solutions. It was great to see students growing up to eventually know more then their teachers knew becuase it shows how their is no limit to knowlege. There is always a different way to look at a problem and a connection to a different connept that might turn into a solution to a major problem like this article discusses. If this progress continues then their is really an infinite number of new solutions and new technologies for the future. As a future teacher I hope to see this type of passion to learn and dig deeper into simple or abstract problems from students. It is amazing how far math has already come and I cannot even imagine the next amazing discovery.
4). This “Trillion Triangle” Problem has multiple connections to real-life situations; especially right triangles. The most concrete (no pun intended) example I can think of is bridge-building. This is because the construction of bridges implements triangles for stability and support. Triangles have been proven to be the strongest of all the shapes, and right triangles in particular are utilized in bridge-building. The exact specifications for these triangles is absolutely crucial because the lives of incalculable people depend on bridges, not just for the safety of their own lives, but for the convenience that bridges account for in their utilization. This is why the triangular structure of the bridge is of the utmost importance and is why such emphasis is put on the study of geometry and trigonometry to make these calculations as exact as possible.
4) This problem can be related to life in general. Every issue no matter how simple, or how difficult, can be solved vast amount of ways. This article showed that you can always change direction and produce results. Think about it, this problem is ancient, and new ways to solve are found it little by little every day. Numbers are infinite, and so are their solutions, which is what makes them so amazing. This article only touches on a small part of a bigger picture, you can find relations to everything through numbers. Everything has a pattern and when things are considered random it is only because we have not been able to solve the pattern due to a lack in technology or lack in the ambition to solve the problem. I think this article really represents that everything has a solution, and it can be found in many different ways. I feel like I’m ranting more then anything now so I am just going to stop now before my head explodes.
4. This article is about how an ancient math problem concerning the congruent numbers of right triangles leads to people today still trying to find methods for calculating the seemingly endless possible answers. The methods and various theories for predicting congruent numbers reflect that there is definitely more than one way to solve an issue. In the case of the Trillion Triangle problem, various mathematicians have discovered so many ways to find the congruent numbers. Bill Hart, team member in solving this problem, noted that he developed “a fast general library of computer code for doing these kinds of calculations.” Michael Rubinstein, mathematician from the University of Waterloo, used a combined idea of number theory and physics to predict the statistical trend of congruent numbers. These were two out of many approaches towards predicting the trillion congruent numbers that are enormous enough to reach the moon. The two methods show that mathematics can relate to computer programming, engineering, and science. The article is one of many that shows how mathematics is so important for its relation to other topics, such as science, engineering, and architecture. We also see that in any issue or problem, there’s never only one way to find the answers.
One thing that caught my attention in the article was that one of the mathematicians in the article had found a connection between congruent numbers and elliptical curves. Although I don’t know too much about elliptic curves, what was interesting to me was that a mathematician would be able to find a visual, graphical representation of an abstract problem. Last week, I saw a video on Youtube of a girl who would write numbers in certain ways (for example, she’d write from 1 to 100 in a counterclockwise spiral on graph paper) and then color in certain patterns: for instance, she’d fill in every square with a prime number in it. As she kept doing it, she found that by filling in every square with a prime number in it would result in a visual pattern, like a spiral, that repeated as the spiral kept moving outwards.
Elliptic curves also made me think of when I went to the Cathedral of the Sacred family in Barcelona, Spain. It’s a cathedral designed by an architect named Gaudi. He was known for using a lot of organic forms in his architecture. However, he was also known for analyzing those forms with math: for instance, he found that certain plants have evolved in such a way that they have leafs stacked vertically on the stem. In order for each leave to get enough sunlight, they all make a 60 degree angle with each other, making an equilateral triangle if one looks at it from above. He was also known for, instead of building roman arches, which are square, with a semi circle opening above, he would make parabolic arches, because of their structural soundness.
Anyways, in short, what I thought was interesting was the correlation between graphical representations of number patterns and their behavior.
This problem can be related to life in general because every issue no matter how easy, difficult, time consuming or impossible a problem is, it can be solved in numerous ways. This article demonstrated that you can always change direction and produce results. Numbers are infinite along with their solutions which is what makes them so amazing and useful in everyday life in general. This article is a small part of how math truly affects our lives. People can find relationships to everything through numbers because just about everything has a pattern. If things are classified as “random” it’s because people have been hindered by technology and cant find an alternate way to s. This article really shows how everything has a solution, no matter the difficulty, and the solution can be found in a variety of different ways.
This problem is related to not only my life at the present, but also the future life of human beings. I see a lot of similarities between the way the mathematicians and scientists solved this problem to the ways we deal with problems in school. None of the mathematicians were able to solve the problem on their own, it started with the original question asked by al-Karaj. Mathematicians then began work on it, doing as much as they could with the brain power and resources they had at the time. Over the years, generations of people were able to build a solution to the problem while also finding solutions to other problems and mysteries in the world on their way. This is a large scale application of what we do in class everyday. When we began algebra we learned theorems that were developed a long time ago, often not knowing where they came from or why they worked. However, as we progress deeper and deeper into the complexity of the ‘math world’ we find connections between the different layers of applications and can draw our own conclusions. Just as the mathematicians were assisted by acquiring a database of coding software and sophisticated technology to guide them, we are able to move along faster and delve more in depth in our studies with the use of our graphing calculators. This new technology is allowing humans to progress and solve problems at a much faster rate than was possible before, and each new development is based on the last, just like the trillion triangle problem’s solutions. I think that it is important to pay attention to advancements in the developing technology around you, so that you can make use of these resources and make new discoveries. As technology grows, it may take over many jobs, however, if we use it to discover more solutions, we could also create more jobs with it’s help.
1. This is related to topics I have seen this school year in mathematics because in triangle problems, sides like 3,4, and 5 are often given in the questions. In order to solve for the unknown side/angle, sine, cosine, and tangent are involved which are the trigonometric functions. The article, Mathematicians Solve ‘Trillion Triangle’ Problem, it says “For example, the 3-4-5 right triangle which students see in geometry has area 1/2 × 3 × 4 = 6, so 6 is a congruent number. The smallest congruent number is 5, which is the area of the right triangle with sides 3/2, 20/3, and 41/6.” This sounds like something that we will maybe cover in our class, or even some other math classes in Conard High School have already gone through this. Scientists started with small numbers like 3,4, and 5. Then, boom! a trillion!
This article is very related and meaningful to me in school. It showed me how math is not a single person subject, it is a group effort. You need people around you to help you understand math and its concepts. You can’t understand math on your own and you definitely can’t solve complex problems like this on your own. All of these mathematicians from all across the globe came together to help solve an ancient math problem. This showed me that if I want to continue in math throughout my life I need to use my fellow peers for help. I will use this in my everyday life at school, not only for math, but for all of my other subjects. This article showed me a lot and i could easily relate to it.
Hello!
At this time, comments are closed for this article. Thank you to all participants. Team 2 has the closest to 100% participation, with only two people electing to not participate!!!
Almost there! Hopefully next time!
Overall, Team 2 is very, very close to 100% participation for third quarter!
Have a nice weekend!
Mrs. Corricelli
Ms. Corricelli, I already posted on the blog on Friday, during my free period -pd 5 ( i actually wrote out the answers to all 5 Qs instead of copying and pasting the post I originally had on the Chapter 6 Review)…and now I don’t see my post anymore?! AND i DID post it….im not making this up!