I've looked at a few other people's blogs. Most of them also had difficulties with proofs and the other course materials too. I've also noticed that not everyone makes very frequent posts just like me. So now I don't feel so bad......it was interesting looking at other people's proofs for their chosen questions and some motivated me when choosing my own question.
Overall I'm just happy this course is over and I haven't failed....Danny was a great prof and I learned a lot from him in class and especially during office hours but the subject matter itself was complex and the assignments were stressful @_@
Wednesday 3 April 2013
Problem Solving
FOLDING
Take a strip of paper and stretch it so that you have one fold between your left thumb and index finger and the other between our right thumb and forefinger. Fold the strip so that the left end is on top of the right end. Repeat several times, each time folding so that the left end is on top of the right end of the strip.
When you're done, keep holding the right end and unfolding the entire strip. Some of the creases point vertex up, some down. Can you predict the sequence of ups and downs for that number of times you carry out the folding operation? Can you form a convincing argument that your prediction method is correct? Can you extend the problem to two dimensions (folding first left over right, then back over front, then left over right....)?
Understand the problem:
Create a formula that can predict the sequence of ups and downs for that number of times you carry out the folding operation (for n folds).
Devise a plan:
Fold the paper and write down the folds.
Devise a formula based on n number of folds.
Carry out the plan:
Folding a piece of paper once = 2 pieces.
Twice = 4 pieces
Three times = 8 pieces
FORMULA (on n number of folds) = 2^n
Folds:
1. Down
2. Up. Down. Down.
3. U U D D U D D
4. U U D U U D D D U U D D U D D
5. U U D U U D D U U U D D U D D....(unable to continue/tell folds apart)
Look back (Conclusion):
Left side center Right side
[mirror image] Down [previous fold]
Based on folds:
There is always an odd number of folds = there is always a centerpiece (which points Down).
The pattern represents a binary number system.
FORMULA (n = number of folds): (2^n) - 1, n>=1 (cannot have negative folds, 0 folds = 0)
Acknowledge when, and how, you're stuck:
I doubt I'd be able to build a program out of this....it would start with creating an empty list of (2^n) - 1 elements...
Take a strip of paper and stretch it so that you have one fold between your left thumb and index finger and the other between our right thumb and forefinger. Fold the strip so that the left end is on top of the right end. Repeat several times, each time folding so that the left end is on top of the right end of the strip.
When you're done, keep holding the right end and unfolding the entire strip. Some of the creases point vertex up, some down. Can you predict the sequence of ups and downs for that number of times you carry out the folding operation? Can you form a convincing argument that your prediction method is correct? Can you extend the problem to two dimensions (folding first left over right, then back over front, then left over right....)?
Understand the problem:
Create a formula that can predict the sequence of ups and downs for that number of times you carry out the folding operation (for n folds).
Devise a plan:
Fold the paper and write down the folds.
Devise a formula based on n number of folds.
Carry out the plan:
Folding a piece of paper once = 2 pieces.
Twice = 4 pieces
Three times = 8 pieces
FORMULA (on n number of folds) = 2^n
Folds:
1. Down
2. Up. Down. Down.
3. U U D D U D D
4. U U D U U D D D U U D D U D D
5. U U D U U D D U U U D D U D D....(unable to continue/tell folds apart)
Look back (Conclusion):
Left side center Right side
[mirror image] Down [previous fold]
Based on folds:
There is always an odd number of folds = there is always a centerpiece (which points Down).
The pattern represents a binary number system.
FORMULA (n = number of folds): (2^n) - 1, n>=1 (cannot have negative folds, 0 folds = 0)
Acknowledge when, and how, you're stuck:
I doubt I'd be able to build a program out of this....it would start with creating an empty list of (2^n) - 1 elements...
I don't remember what week I'm on...
It's been a month...
Assignment 2 wasn't too bad. We got marks taken off for some steps we didn't mention in questions 2,3 and 4. Curious to know how many marks we would have gotten for just writing the outlines of the proofs as we seemed to lose a large amount of marks for small errors.
Test 2 was ok...The cheat sheet really helped. The first two questions were fine and we've had lots of practice with them. However, the third question with the ceiling I felt to have little practice with and I was unaware that the prof had sent us an email with a proof of a similar question already solved. If I had known I would have written it down on my cheat sheet. It would help greatly if the prof made announcements on piazza instead as I check that as my major source for course info/announcements instead of my uoft email.
Assignment 3 so far has been rather stressful, it seems straightforward but all our proofs are rather short. My partner and I are having a hard time thinking of everything that should be written, especially in the comments. We went to the prof's office hours and he was a great help. I actually understand some questions more in the 5 min he worked with us than when I'd spent hours of it on my own. This of course leads to me worrying about the exam as I'd obviously have no one there to help me along. Here's hoping my cheat sheet can save my life....and for the exam I'll make sure to check my email more often for important announcements.
Assignment 2 wasn't too bad. We got marks taken off for some steps we didn't mention in questions 2,3 and 4. Curious to know how many marks we would have gotten for just writing the outlines of the proofs as we seemed to lose a large amount of marks for small errors.
Test 2 was ok...The cheat sheet really helped. The first two questions were fine and we've had lots of practice with them. However, the third question with the ceiling I felt to have little practice with and I was unaware that the prof had sent us an email with a proof of a similar question already solved. If I had known I would have written it down on my cheat sheet. It would help greatly if the prof made announcements on piazza instead as I check that as my major source for course info/announcements instead of my uoft email.
Assignment 3 so far has been rather stressful, it seems straightforward but all our proofs are rather short. My partner and I are having a hard time thinking of everything that should be written, especially in the comments. We went to the prof's office hours and he was a great help. I actually understand some questions more in the 5 min he worked with us than when I'd spent hours of it on my own. This of course leads to me worrying about the exam as I'd obviously have no one there to help me along. Here's hoping my cheat sheet can save my life....and for the exam I'll make sure to check my email more often for important announcements.
Friday 8 March 2013
Week 6, 7, 8: Feb 17 - Mar 9
Nothing much to say again...Nothing much has changed since two weeks ago.... I'm getting better with generic proofs. Assignment 2 took some time but I think I benefited from doing it as I understand how to write different kinds of proofs now. Office hours and the TAs in the help room were very helpful. If the level of the questions on the upcoming term test are at the same level as those of the assignment I think I will do alright provided that we will only be covering proofs in the test. The biggest problem is that when given a proof that I've never seen before I almost never know how to do it the first time. If I get a similar question to one I've seen before then I don't usually have many problems....so right now I'm kinda hoping I've done enough proofs to cover all the ones that will be on the test....
On the other hand, whatever we're learning right now. I have a basic grasp of what's going on in class but when given a question I have no idea what the question is asking of me (eg: tutorial exercise/ quiz). Am I supposed to count the times it iterates? List the steps? Prove something? o.0
On the other hand, whatever we're learning right now. I have a basic grasp of what's going on in class but when given a question I have no idea what the question is asking of me (eg: tutorial exercise/ quiz). Am I supposed to count the times it iterates? List the steps? Prove something? o.0
Monday 25 February 2013
Week 5: Feb 10 - 16
Not much to write....proofs are still hard for me. I have the formatting down thankfully but the actual proving part feels more like I'm just writing random stuff down and hoping for the best. I also don't know how to prove things false. I just assume that everything I'm given is true which so far it has...
Also, the graphs we're doing in class. What is the point of them? Are they a concept we're supposed to be tested on or are they just there toconfuse us even more help clarify things? If so I don't think it's working.... if not then can we get more practice with them in tutorial or extra questions with them or something?
I'm still worried for the next assignment. I like how we get part marks for writing the structure. I also have no idea how to view or what the importance is for the assignment #2 LaTeX source file. It just looks awfully confusing. Is the formatting supposed to be like that? :S
In other news I did pretty good on the assignment and the test so I think I have a decent grasp on the previous concepts.
Also, the graphs we're doing in class. What is the point of them? Are they a concept we're supposed to be tested on or are they just there to
I'm still worried for the next assignment. I like how we get part marks for writing the structure. I also have no idea how to view or what the importance is for the assignment #2 LaTeX source file. It just looks awfully confusing. Is the formatting supposed to be like that? :S
In other news I did pretty good on the assignment and the test so I think I have a decent grasp on the previous concepts.
Saturday 9 February 2013
Week 3+4: Jan 27 - Feb 9
Been busy with midterms and essays this week. The course has definitely picked up and I'm feeling more and more behind. Proofs are confusing and I don't really understand how to come about proving anything. I can read a proof and understand it but making my own from scratch seems downright impossible at the moment. It seems very math based and math has never been a subject I do well in. I still like how the professor gives us time to discuss among others and solve a question on our own before taking it up in class. This keeps me motivated to concentrate during the lecture and try to understand it at least.
We had the midterm yesterday. I had expected more to be covered and I was also hoping for some English to symbols and vice versa translations like I'd seen on the past test as I'm sure I would have done much better on those. I had a hard time with the 2nd question where I had to determine which one was true or not. I'm a bit wary as to why all the questions involved negation because if I did it wrong for one I must have done it wrong on all the others.... Overall, other than that I think I did alright. It helped the the midterm seemed to cover only the very beginning of the course and not what we're doing now... I don't understand much of what we're doing now to be honest....
We had the midterm yesterday. I had expected more to be covered and I was also hoping for some English to symbols and vice versa translations like I'd seen on the past test as I'm sure I would have done much better on those. I had a hard time with the 2nd question where I had to determine which one was true or not. I'm a bit wary as to why all the questions involved negation because if I did it wrong for one I must have done it wrong on all the others.... Overall, other than that I think I did alright. It helped the the midterm seemed to cover only the very beginning of the course and not what we're doing now... I don't understand much of what we're doing now to be honest....
Saturday 26 January 2013
Week 2: Jan 17 - 26
This week the course material seems straightforward but I'm definitely struggling with some concepts. However, I seem to understand the basics as I can do the tutorial quiz and most of the exercise. It's just the really complicated solutions (eg: exercise 1e and 1f) that I know I would never be able to come up for a solution to correctly on my own. Given that, when I see the solution to those problems I understand how to get it. Perhaps I just need more practice with similar questions. The assignment I'm having problems with how to prove sentences using symbols. I find it much, much easier to go from symbols to words than the other way around and I need to do this for all symbol sentences before I can even begin to understand what they mean. >.<
I cannot safely say that I've mastered any concept taught in class so far, even if I have gotten full marks for my quizzes. I'm still unsteady on certain concepts and I need to look at my notes very frequently. Work is time consuming and frustrating so I can't exactly say I'm enjoying this class although I do deeply hope to endure through it.
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