Freshman and Formal Proofs
Let's face it ...proofs can be difficult at first. But, by now, students are really shining, beautifully writing complex and elegant proofs. And to think...eight months ago, you did not know what a proof was! Keep up the good work, students!
Essential Question: How does the structure of a proof in Geometry reinforce its purpose? Please comment and give specific examples from recent parallelogram, trapezoid and kite proofs.
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| Colin, aka Co-llinear, Brown teaches Block B a tricky trapezooid proof. |
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| Colin's classmates in Block B. |
26 comments:
Hi, this is SamR from Pre-calc block C.
Proofs were the toughest, but most important subject to conquer in geometry. I think proofs help solitify the gap between numeric and visual mathematics. They teach students to not only to think about the dimentional structure of a function or shape in words, but to help understand the link between numbers and their visual representations.Proofs teach "why" something happens in math. Students tend to think of a number and it's graph as different entities. Students need to start thinking about these as one in the same, and not just something you memorize.
Hi this is Bridget C from C block. I completely agree with Sam proofs are definitely the hardest part of geometry but they are very important too. Their isn't an equation that you just plug numbers into or steps to solve for a variable you have to show why something is true in math as if you didn't know the rule. They are definitely not easy but the more you practice them the better you get at them.
Hi this is Bridget C from C block. I completely agree with Sam proofs are definitely the hardest part of geometry but they are very important too. Their isn't an equation that you just plug numbers into or steps to solve for a variable you have to show why something is true in math as if you didn't know the rule. They are definitely not easy but the more you practice them the better you get at them.
Hi, this is Ricky P from Pre-Calc block C.
I also found proofs quite challenging, and a bit tedious once you understood them. But after a while, you do come to see the importance that they contain. Math is different from most things, because there is always an absolute right answer. In Math, things can be proven to be shown that they are correct. Many times in life, students will learn a certain topic, and be able to do what is required of them competently. But many times, people don't understand the meaning behind it, and the real reason why things work. Proofs, at least in my case, was the first exposure to this kind of thinking. It is one thing to say that two triangles are congruent because of SSS, it is another to show step by step why that statement is true, and to fully understand everything that goes into the problem. While proofs tend to be the dreaded subject of geometry, its also in my opinion the most important.
Hi, this is Matt Vallis from Honors Geometry B Block.
At first this year, I thought proofs were the worst thing in the world of math. Nothing about them made sense to me and I couldn't see how they were applied to the what we were learning other than vocab. But now that we are farther into the year I don't mind doing proofs at all. It helps me work out the logic in problems and lets me see how everything relates. I like how we use old postulates and theorems in problems now too because it is like we are having constant review. This happened with a lot of the triangle concurrency proof theorems like SAS, AAS, and ASA. Those are used in proving quadrilaterals squares and rectangles now so it helps us not forget it for the final. The more proofs you do, they better you get at them, understanding the topic, and it will help you in the future too.
Hi, this is Dan S. from B Block
I remember this proof being a very tedious, albeit very important one. With this proof, like many others that we have done in Geometry, are important in order to prove the theorems and postulates that we are using in other proofs. At their core, proofs are used to prove the various laws in Geometry and why we are allowed to use them. For example, we are able to use the Base Angles Theorem in isosceles triangles because by definition, isosceles triangles have two congruent sides leading to a vertex with an altitude that bisects the base. This means that each of the two triangles that the original triangle is split into have two congruent angles, the ones formed by the altitude bisecting the vertex and the right angles formed by the altitude bisecting the base. This means that the base angles must also be congruent because both triangles must add up to 180 degrees and they have the same values for the other two corresponding angles.
Hi this is Hannah from Block B. I, like the majority of students, found proofs very difficult at first. It took me awhile to understand how to do them. I always thought of math as being all numbers so when it came time to do proofs I was confused by them. Tying numbers and visualization together didn't come easy to me. Many times proofs get me frustrated because I know that something is right but I don't know how to put it in words and explain why. Proofs open up a whole new way of thinking and although tricky at times they will defiantly help out all students in the long run with problem solving.
Hi its Amanda from B block. A proof is very important in geometry. It is because just because you know something is true you have to back it up with evidence. You cannot just says something and then if someone asks you how you know that you cannot just say because I do. You have to back it up with evidence. When you are writing a proof you must have it in the right sequence or else it wont make sense. If you have two things to prove in a proof then you need to write all the statements and reasons that go with the first thing you need to prove and then do the other steps that go with the second hing you need to prove. In the very long trapezoid proof that we had to prove angle B and congruent to angle C and angle BAD was congruent to angle CDA. TO prove angle BAD and angle CDA are congruent you have to prove they are SSIA. You would not do this in the middle of when you are proving angle B and angle C are congruent. It would not make sense if you did it that way and it would probably confuse you when you are looking over your notes. That is why having the right structure in a proof reinforces its purpose.
Hi this is Claire Walsh block A.
Being a Freshman, I had not heard much about proofs before my high school years. However, my Dad does lots of things with math and did try to explain proofs to me once or twice. Even though I was introduced to proofs before we worked on them in geometry and my Dad gave me some knowledge of how to work with them, however, I still did not know what a proof looked like or how it worked. Being the visual person that I am, I did not understand proofs when they were only explained, but needed the physical, logical set up of them to help me complete and understand them. For example, when doing a proof on a parallelogram, it always helps me to see it, and draw out on the figure what the givens represent, and to figure out visually what I have to prove. In addition, the structure of these two-column proofs really helped to reinforce the logical thinking and straight-forwardness of such parallelogram proofs. If you asked me to prove the two base angles in a triangle congruent by proving that it is isosceles, and did not allow me to use a concrete two-column proof, I would be lost. In all, the structure of a proof definitely reinforces its purpose, to help you logically and simply state why something works in math.
Hello, this is Sam K from block A.
I found this proof to be tricky, but I really enjoyed how we were using information from a parallelogram and then an isosceles triangle, to solve a problem where we were originally given a trapezoid. When it comes to proofs, I originally thought that they were ten times more challenging than I feel now.I still do think that the proofs are challenging, but I know that if I was to receive a proof to solve with twenty steps during the first quarter I would be freaking out. I found it very interesting how we used different theorems from other units as well, like ones we used with parallel lines, triangles and parallelograms. It is defiantly rewarding to use past information! finally I really ind proofs to be a great tool in geometry because they are a great way of sharing information. They not only list off the steps used to solving a problem, but they also describe the reasons behind why the steps were used. This is great because they can pass on information and require little explanation, because the proofs do that already.
Hi, this is Olivia Cronin-Golomb from B Block Geometry. Doing this proof helped me to understand why the base angles of a trapezoid are congruent. Instead of just memorizing it and accepting it as true, I had to figure out why it was true. It reinforced what we were taught. The proof also showed me that not everything in math is straight forward and that you have to think outside the box sometimes. Also, the proof showed that I can't just forget what we learned previously. Everything in math is connected. For example, I had to think back to the days of isosceles triangles and corresponding angles to solve this one.
This is Tommy W from B block. I remember when we first started doing proofs at the beginning of the year and i remember feeling so confused. You always told us as students we had to think outside the box and eventually that's what I adapted too. I began to think in pieces instead of trying to figure it out in one big part. I also find it interesting that the theorems we used such as triangle congruency theorems aren't just useful in triangles they are helpful in trapezoids, rhombus's, and almost everywhere there are triangles found. I find it funny how everything connected in the end. I used to hate proofs, but now I find them kind of interesting just because I like tp put it all together like a puzzle and see how it turns out.
Hi, this is Nat T from Block B Honors Geometry
I agree that proofs are tedious, although I never found them particularly challenging. I agree with what Ricky said that proofs are important because they show the real reason why things work. Proofs, unlike other elements of geometry, show not how, but why. They help to explain why certain key concepts or theorems that make up geometry work, and why they are important. Most people know that if the corresponding sides in two triangles are congruent, then the two triangles are congruent, but my guess would be that many people wouldn't know why. Proofs help explain this, which is their basic purpose. And while the average American citizen is not going to encounter proofs in their daily life, they still function as a learning tool, to introduce and reinforce key concepts in geometry.
Hi, this is Mia from block A. I think that doing proofs in groups is a great way to help us really learn the material. When your stuck, you have others to help get you back on track. Ideas can be talked over and fuzzy concepts can be explained. Sometimes it's hard to go over everything as a whole so this provides the opportunity to do so effectively. For me, proof work in groups is a highly, productive time.
Hi, this is Chloe M. from Block A. I do have to admit that proofs have always been tricky for me, but once you understand the logic behind them and why you have to do them, they aren't half as bad. With most math, there is almost always a definite answer that you need to find. Proofs, however, require you to dig deeper and add more thought to what you are doing. They help you understand WHY things are a certain way, and give you space to back up your reasoning with logic. The recent trapezoid proofs that we have been working on have made us use our prior knowledge of other shapes to tie in to what we already know. This helps increase our thinking on both subjects, and adds new dimension to what we learn in the future.
Hi this is Greg McHale from B Block. Proofs, as Mrs. Fitzpatrick has said many times before, are the most important part of the geometry curriculum. They force the student to think immensely in order to solve them. Critical thinking in particular is taught in proofs, a thinking skill that is extremely beneficial in dealing with life's problems. They also teach fantastic attributes such as being neat, orderly, and precise in proof writing, attributes that can be useful in occupations down the road. So, in a strange way, proofs are preparing students for life's hurtles, it's ups and downs, and the challenges it brings.
This is Natalie Rotstein from block B
Proofs were hard at the beginning of the year, and often still are difficult now, but they also help a lot with reinforcing the concepts we are trying to learn, and help to understand how a lot of different theorems and properties work for different shapes. Solving proofs to prove that theorems and postulates are true helps to show exactly how things work, which helps me understand a lot of the concepts better. Some proofs are simpler, but some are much more difficult, and really make you work to get the solution.
Juliann L, Block B
I think the structure of a proof is very logical for it's function. The two separate columns for statements and reasons helps to organize each individual step that you are taking to solve the problem while explaining the mathematical laws that allows that step to be done. Proofs are essential to fully justify how you got the answer that you did. Completing a proof in this structure can not only help to logically solve problems with accurate justifications behind your answers, but it can also help you to understand the logic behind postulates and theorems that may be used in other problems. For example, in class recently we were given proofs where the thing that we had to prove was a property of the quadrilateral we were working on, but we were not allowed to state that property while working on the proof. To do this, we had to go through a long series of steps that ended in us actually proving the property itself. This shows the importance of a proof because proofs can not only be a structured way to back up your work with reasoning, it can also be a way to understand the logic behind the reasoning that you use in your work.
Paul Chong (Block A):
I think that proofs reflect the content that they hold. Mathematics in itself is a very structured concept, as the laws and rules of mathematics demand a very structured regulated process for solving. Proofs, with their theorems, postulates, and structured format reinforce their purposes because they train the mind to think in a structured way, which in turn aid in solving other problems even without proofs. As well as reinforcing our own knowledge of the topics we are learning and implementing them into proofs, we are also learning to think in a more efficient way by putting on paper what should be going on in our heads.
Hey its Sam A from Block B, After reading this post, I do agree on many things.I really do thing that now we "are really shining, beautifully writing complex and elegant proofs. And to think...eight months ago, you did not know what a proof was! Keep up the good work, students!" It really is amazing to think that now to long ago, we just started our road through proofs, and now we can do them without hesitation. And, about the question, "How does the structure of a proof in Geometry reinforce its purpose?" the answer that i can give is that the proof asks for a statement and reason, and thats exactly what proving something is, giving a statement and then backing it up with cold hard facts and reaasons.
Cynthia Yang: Block A
Proofs are to be honest, extremely tedious, and in some cases, a seemingly waste of time. However I also think that proofs are extremely helpful when learning all the theorems that geometry includes because you have to go through the steps to explain, and figure out for yourself how the theorem works. Learning from the teacher that the two base angles are congruent is one thing, but figuring out why exactly the base angles are congruent and doing the math behind it helps the knowledge sink in deeper. It reinforces the knowledge. Also, the structure of the two columned proof really helps as we can go through the steps one by one, and it is more organized. I feel that the most difficult proof would have to be the one shown in the picture with the line drawn in, trying to prove the two base angles were congruent.
This is Jessica Strack from A block.
Since I learned about proof I have loved doing them. Whenever we learn something new I always want to know the why and how. Proofs are just that. They show us and help us to show why this shape works and how that problem is true in this circumstance. For example, say someone tells you that this quadrilateral is a parallelogram. Why? This person would then use a proof to show that the quadrilateral indeed has a property of a parallelogram and therefore is one. Proofs can be tedious and frustrating but they are an important part of math.
I think that learning proofs, although challenging, was fun and very beneficial. Doing them in groups was good so we could help each other out and teach each other new things. It's crazy to think just a few months ago, I had no clue what a proof was or how to do one. Although I didn't do so well with proofs at the beginning, I now am very good at completing Proofs.
Hi, this is Gabrielle from block a's friend. I also had to learn how to write proofs this year which was difficult to get used to at first. It was so challenging to justify everything because although in your head you knew the reason behind the logic, you didn't always know how to put it into words. I think proofs really test your understanding of the topic although they seem tedious and rather useless at first. Sure you can just memorize all the theorems and postulates, but when you write a proof you really have to know them.
Hi This is Julia from Block A.
When I first started proofs I was extremely annoyed because I did not want to remember which postulates and theorems were which and what they meant. Then I realised it is almost easier than algebraic formulas because everything has a reason and can be solved in a similar pattern. They are also challenging but in a fun and puzzling way rather than in an impossible problem that you can never solve.
Hi this is Cynthia from block A's brother. I haven't learned about proofs yet, and I think they seem very interesting. I feel like writing out all the steps and actually realizing what each theorem is about (which is what my sister said proof-writing is)would help me learn the theorem, or piece of knowledge better and when doing problems, I would actually understand why the answer is such.
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