12 September 2016

Test 1 review problems (Answers are on Piazza)

Test 1 is scheduled for Thursday 15 September. Here are the review problems for you to use to prepare:

the answers or links to answers are posted on Piazza, so I could use math notation.

08 September 2016

How to scan and submit your Journal pages (with important note!) - UPDATED and bumped to the top

**There are two changes to the instructions below, starting with Journal2. They are marked with double stars!**

Important note: all of the journal pages you submit each week should be contained in one single file, numbered according to the week! And make sure that you are submitting EVERYTHING that is supposed to be in the journal!

Submitting your journal pages:

  First of all, make sure that you know what your journal is supposed to contain: read the syllabus! 

Each time I ask you to submit, you will submit the pages of the journal which are new since the last time you submitted.

 The pages must be scanned, not merely photographed, and saved as one single pdf file. jpg files or any other file type are not acceptable.

 However you scan your journal pages, you are responsible for making sure that the result is clearly legible and is saved with the appropriate file name in pdf form.

 You must save the file containing the scanned pages using the following name format:
Lastname-firstname-Math1311-Journal-number.pdf

 For example, Joe Seeley would save his first journal submission under the name
Seeley-Joe-Math1311-Journal-1.pdf
and his second journal submission would be
Seeley-Joe-Math1311-Journal-2.pdf

 Important note: Your file MUST have its filename in that format. If you forgot to name it that way when you scanned it, make sure to change the name before you submit it.

Double-check that your file is  easily legible and contains all the things you intend it to contain before you post it.

Post the file containing your Journal pages to Piazza as a **Private Question** (not a Note) to me.
**Make sure that the "Summary" line includes your name and the number of the journal submission; for example Joe Seeley could use the Summary line Joe Seeley Journal 2 for his second Journal submission.**
Put it in the Journals folder when you post it.

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 There are several ways you can scan the pages:

 • There are free scanner apps available for smart phones or iPads. Two which have received consistently good reviews and which I have tried out are: CamScanner, and Genius Scan. They are both available for iPhones and Android phones and, I believe, for Windows phones as well (and for iPad). Specific instructions for each of these will be posted as I get them written up, but they are both fairly easy to learn to use.

 • There are scanners available for your use in several Brooklyn College Library locations:
The self-service scanners are located in the following areas of the Library and Library Café:
 • Lower Level
 • 1st Floor
 • New Media Center (2nd Floor)
 • Library Cafe
Please see the staff member at the appropriate service desk for assistance in obtaining and using a scanner.

Thursday 8 September class

Topics:
• We discussed some of the homework problems: the socks problems from the second day (30 August) and the algebraic problems from last time.
If you need more review of using the quadratic formula, here are some good sources:
Webpages: Regents Prep (good explanation), Algebra Lab (this has practice problems where you can enter your answer and it checks it - excellent for practice!)
Videos: Khan Academy, Patrick's Just Math Tutorials (this link is to Example 1; there are links below the video to two more examples)
New topic: Prime numbers and Prime Factorization
Most of this is from the reading: prime numbers and prime factorization of natural numbers (from Math is Fun), except when other links are provided.
 • What is a prime number?
   Note: divisibility rules are linked at the end of the reading from Math is Fun
 • Finding prime numbers using the sieve of Eratosthenes: here is the grid that I printed out
 • Finding the prime factorization of a natural number (this is done in the reading)


Homework:
Journal assignment: write a definition of prime number. Explain in your own words what the prime factorization of a natural number is: give at least 3 examples to illustrate it.

Problems:
Find (by hand, using any of the methods we discussed in class) the prime factorization of each of these numbers:
30, 48, 143, 41

Next topic: We will state the Fundamental Theorem of Arithmetic, and then prove that there are infinitely many prime numbers (here's another description of the proof). Along the way we encounter the Division Algorithm, which explains one of the steps in that proof.

06 September 2016

Tuesday 6 September class

Topics: more properties of Fibonacci sequence and similar sequences
• Activity: can you write these numbers as sums of distinct Fibonacci numbers?
• Theorem: any natural number can be written as a sum of one or more non-consecutive Fibonacci numbers.  Also, there is only one way to do this. (This is called Zeckendorf's Theorem)

• Looking at the ratios of consecutive Fibonacci numbers, we previously saw they approach a number we call "the Golden Ratio"which will turn out to be irrational (more on this later).

 • A "Fibonacci-like" sequence starting with any two randomly selected natural numbers (in place of 1 and 1) will have its ratios approach the same value 

 • A continued fraction representation of  : Here is a video which shows the continued fraction being developed, by Michael McCafferty. (You could stop watching at about 2:00 when he has finished the continued fraction, or go on as he explains more advanced properties.)

 • From this we learn that satisfies the equation

 which we can solve to find the exact value


What is important to get from each of these: (some of this I mentioned last time)
• You should know how to get the next number in the Fibonacci sequence (or any Fibonacci-like sequence) by adding together the two previous numbers
• You should make sure to understand how the algebraic formula an = an-1 + an-2 represents that rule
• You should understand how we computed the ratios of successive Fibonacci numbers and how we concluded that those ratios were approaching a single number as we went farther and farther out
• Same thing for the "Fibonacci-like" example in the reading which started with 192 and 16
• Make sure that you see how the Fibonacci numbers and the spiral are being shown in the examples from nature: the sunflower seed head and the chambered nautilus shell in particular
• Understand how we got the continued fraction representation of  and that the fact that it contains only 1's means that  is very special indeed
• Know how to solve the equation to find the exact value of  (including why we choose the + sign in the quadratic formula).

We will be going into more detail about rational and irrational numbers later in the course, and will prove that the square root of 5 is irrational, so that will prove that  is also irrational. For now it is enough to know that a rational number is a number that can be written in the form of a fraction, and that the decimal expansion of a rational number either terminates or eventually repeats.

Homework:

Journal assignment: 

Use your journal to start work on understanding each of the points above on the topic of Fibonacci numbers. Describe any part that seems unclear to you and try to work it out, possibly by discussion on the Piazza discussion board. (Don't worry if you don't get this totally understood by next time; it's just important to identify what you need to work on and to get started on it. The homework problems below may help with some things.)


Homework problems: (the first one was assigned last time, I'm just repeating it herre)
1) Choose any two natural numbers (not 1 and 1, and not 192 and 16, but some other pair, preferably not too big) and make the "Fibonacci-like" sequence starting with them. Write down at least the first 10 numbers in the sequence. Then compute the ratios of successive numbers in your sequence, as we did for the Fibonacci numbers. Do your ratios appear to be approaching ? Why or why not?

2) We can verify that
by simplifying "from the inside out":
   

Do the same to simplify



3)  Now simplify


4) Solve for x:


5) Solve for x:


Reading for tomorrow: We will start a new topic, prime numbers and prime factorization of natural numbers (from Math is Fun). This will lead eventually to discussion of public-key cryptography, a very important (and lucrative!) application of number theory. Something to look forward to.

02 September 2016

Thursday 1 September class

• Activity: Pigeonhole principle statements and mis-statements. Here is the handout. We did not finish all of these in class: you should look at the rest of them at home.

• Fibonacci numbers: beginning with the reading from Math is Fun.
 - How to get the next number in the sequence by adding together the two previous numbers
 - A spiral that is constructed on squares whose side lengths are the Fibonacci numbers
 - Subscript notation so we can write the above rule in algebraic form: $a_n = a_{n-1} + a_{n-2}$
 - Looking at the ratios of consecutive Fibonacci numbers, they approach a number we call "the Golden Ratio" $\varphi \approx 1.618$ which will turn out to be irrational (more on this later).
  - Fibonacci numbers in nature [the link I used: Math is Fun, also look at Fibonacci Numbers and Nature]
  - A beautiful video showing many real-world instances related to Fibonacci numbers, the Golden Ratio, and things that follow from them: Nature by Numbers, by Cristobal Vila


What is important to get from each of these:
• You should know how to get the next number in the Fibonacci sequence (or any Fibonacci-like sequence) by adding together the two previous numbers
• You should make sure to understand how the algebraic formula $a_n = a_{n-1} + a_{n-2}$ represents that rule
• You should understand how we computed the ratios of successive Fibonacci numbers and how we concluded that those ratios were approaching a single number as we went farther and farther out
• Make sure that you see how the Fibonacci numbers and the spiral are being shown in the examples from nature: the sunflower seed head and the chambered nautilus shell in particular

We will be going into more detail about rational and irrational numbers later in the course, and will prove that $\sqrt{5}$ is irrational, so that will prove that $\varphi$ is also irrational. For now it is enough to know that a rational number is a number that can be written in the form of a fraction, and that the decimal expansion of a rational number either terminates or eventually repeats.

Homework:

Journal assignment: 
Record any new thing you learned from the homework discussion today.

Use your journal to start work on understanding each of the points above on the topic of Fibonacci numbers. Describe any part that seems unclear to you and try to work it out, possibly by discussion on the Piazza discussion board. (Don't worry if you don't get this totally understood by next time it's just important to identify what you need to work on and to get started on it. The homework problems below may help with some things.)


Homework problem:
 Choose any two natural numbers (not 1 and 1, but some other pair) and make the "Fibonacci-like" sequence starting with them and using the Fibonacci rule to get the next number. Write down at least the first 10 numbers in the sequence. Then compute the ratios of successive numbers in your sequence, as we did for the Fibonacci numbers. Do your ratios appear to be approaching $\varphi$? Why or why not?

Reading for next time: We will find a continued fraction expansion for the Golden Ratio, which leads to an equation we can solve to find its exact value (which is, alas, irrational).  I'm still looking for sources for this, so no links yet, sorry.

01 September 2016

Using Genius Scan to scan and post Journals to Piazza

In most ways I prefer Genius Scan to CamScanner (slightly prefer, let's say), but one thing I do not like is that there does not seem to be any way to delete a document in Genius Scan! All you can do is delete all the pages in the document, but the document remains cluttering up your Documents screen. Oh well.

Note: I'm posting this now without illustrations, just so I can get it up. I will put in the illustrations when I have time to get them from my phone.

(In the illustrations, I'm scanning the syllabus for this class page by page to show you how it's done.)

Always before scanning, try to have a nice flat surface, preferably of a color other than white, to place your pages on. This is not strictly necessary but it helps.

(Note: when you first open Genius Scan , you may see a screen that asks you to sign in or register. Just ignore those choices and move on, if that happens.)

Put your first page flat on your surface. Now you will touch the camera icon at the bottom of the screen:

Position the camera so that you can see the entire page. Notice that you don't necessarily have to hold the camera parallel to the document: the scanner will take care of that later.

Notice that Genius Scan has automatically outlined the document's edges in orange. When it is ready to scan the little rotating circle icon will be totally surrounded, and the scan will be done automatically: no need for you to press the button! Notice then that Genius Scan has corrected the perspective. (THIS is a big reason why you want a scan and not a photo! It is always best to take the scan as close to parallel as possible, but it is not necessary to be precise, as it would be if you were taking a photo.)

Now I can edit the scanned page to make it clearer, but I am happy with it like this.

When I am happy with the appearance of this page, I will touch Save, save to New Document, and then add the next page: notice that the document is automatically given a name which is the date and time it was created. Just leave this for now and go back to Documents.


Set up the next page, and then touch the camera icon at the lower left of the screen to add the next page. When its image is ready, I touch Save but I will add it to the Existing Document. I continue adding pages until they are all added and ready to go.

I want to name this document "Shaver-Sybil-Math1311-Journal -1" (that's not what it really is, but remember that is the way you should be naming your journal submission) so I  touch the document in the Documents screen and then touch its name at the top of the screen. Delete the old name and type the new one in. (You don't have to capitalize, but please put in the dashes where I've indicated.) And touch "Done".


I'm now going to share it via Piazza. I touch the Upload icon (box with an arrow coming out of the top) at the bottom right of the screen. I see this:


I touch the "Other Apps" and then swipe left until I see the Piazza icon. And touch it. Log in to Piazza if necessary.

I touch the new post icon at the bottom right in Piazza, post as New Note

My Note Summary is "Journal 1". I type in something (it doesn't matter what) into the Note Details, because Piazza won't let you leave this blank even if you are posting a file. Then touch Next and add folder "journals", go back < and add file, choose the file that I just shared from Genius Scan.

Touch Done. Now I go to Post to... because I want this to be a private post. I change it to post to Individuals. I then add the individual (myself, in this case) that I want to post to: that means that I choose "All Instructors".



Then "Done" and "Post"

Don't omit any of these steps in Piazza, they are all necessary!








Using CamScanner to scan Journal and post to Piazza

Note: I'm posting this now without illustrations, just so I can get it up. I will put in the illustrations when I have time to get them from my phone.

(In the illustrations, I'm scanning the syllabus for this class page by page to show you how it's done.)

Always before scanning, try to have a nice flat surface, preferably of a color other than white, to place your pages on. This is not strictly necessary but it helps.

(Note: when you first open CamScanner, you may see a screen that asks you to sign in or register. Just ignore those choices and move on, if that happens.)

Put your first page flat on your surface. Now you will touch the camera icon at the bottom of the screen:

Position the camera so that you can see the entire page. Notice that you don't necessarily have to hold the camera parallel to the document: the scanner will take care of that later. When you are ready, push the button to take the picture:

Notice that CamScanner has outlined the document's edges. If necessary, you can correct that outline by touching and dragging the dots you see around the edge. But this one looks OK, so I will proceed.

Touch the checkmark at the lower right when the outline is correct:

Notice that CamScanner has corrected the perspective. (THIS is a big reason why you want a scan and not a photo! It is always best to take the scan as close to parallel as possible, but it is not necessary to be precise, as it would be if you were taking a photo.)

Now I can edit the scanned page to make it clearer. I see that it looks a little pale in the B&W choice along the bottom. Going back to Original I see this:

When I am happy with the appearance of this page, I will touch the checkmark, and then add the next page:


Set up the next page, and then Touch the "Add" camera icon at the lower left of the screen to add the next page. I then proceed as before to edit that page and check it off, and add pages until I have added all the pages I want. Now I have this:

My document at this point has the name "new doc 1" and has 4 pages. To change the name, I touch the name "new doc 1" and it pulls up a Rename dialog:


I want to name this document "Shaver-Sybil-Math1311-Journal -1" (that's not what it really is, but remember that is the way you should be naming your journal submission) so I type that in. (You don't have to capitalize, but please put in the dashes where I've indicated.)

And touch "OK". Now I want to preview it, so I touch "More..." at the lower right of the screen and I see this:

Touching "PDF Preview" lets me see what it will look like. (This one actually looks pretty bad, because I had my flash turned off when I took the scans. Take my advice and use your flash! Bright lighting also helps.)

I'm now going to share it via Piazza. I touch the "<Back" at the top left of the screen and then Cancel the actions to get back to the sharing screen:

Touch "Share" at the bottom of the screen and I see this:

By swiping left I can move over until I see the Piazza icon. Touch it, then sign in to Piazza if necessary,

I touch the new post icon at the bottom right in Piazza, post as New Note

My Note Summary is "Journal 1". I type in something (it doesn't matter what) into the Note Details, because Piazza won't let you leave this blank even if you are posting a file. Then touch Next and add folder "journals", go back < and add file, choose the file that I just shared from CamScanner. (You will notice at this point that CamScanner has appended a lot of numbers at the end if the file name. They are the date and time that the file was created, and I know of no way to keep them from showing up, so don't worry about them for now. This is one thing I do not like about CamScanner.)

Touch Done. Now I go to Post to... because I want this to be a private post. I change it to post to Individuals. I then add the individual (myself, in this case) that I want to post to: that means that I choose "All Instructors".



Then "Done" and "Post"

Don't omit any of these steps in Piazza, they are all necessary!