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.
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