Written work, due Tuesday, November 29th, in class:
Section 11.0 p178: 3,4
Sec 11.2 p187: 1,2,7 – In addition, complete Example 11.8 at the top of p182. WeBWorK – Assignment 6, due Tuesday, November 29th, at end of day. OpenLab – none

Project Deadlines:
Final Draft of paper due in class on Thursday, 12/1.

I forgot to get either of your contact information. So I figured I’d make a post and hope you guys (including Prof. Reitz) see this. If you have any comments or criticism, lay ’em on me. This is an inital draft so it’s not complete, currently juggling three essays T.T

Written work, due Tuesday, November 29th, in class:
Section 11.0 p178: 3,4
Sec 11.2 p187: 1,2,7 – In addition, complete Example 11.8 at the top of p182. WeBWorK – Assignment 6, due Tuesday, November 29th, at end of day. OpenLab – none

Project – First draft of group paper due in class this Thursday, 11/17.
Final Draft of paper due in class on Thursday, 12/1.
Group Presentations on Thursday, 12/1 and Tuesday, 12/6.

The review sheet for Exam #3, taking place next Tuesday 11/22, is posted under Classroom Resources / Exam Reviews. As always, if you have any questions or notice any errors please let me know (by email, in person, or here on the OpenLab).

This video shows the relation of simple doodling to demonstrate and compare the relationship between the Fibonacci sequence and real life.

This video demonstrates the use of symmetry and its reflection in order to create a pattern shaped as a braid. The use of this type of pattern may be used to produce a sound braid shape such that when notes are added to it there will be harmony. Mathematically speaking this sound braid is considered a an object exsting on a two-dimensional plane with space time.

This video gives details on the different types of infinity and impresses on the difference between countable infinity and uncountable infinity.

Part 2. My favorite video of the three is the Doodling in Math video. The video was eyeopening for me because I never knew how much of a connection a sequence can be related to real life. I loved the use of a pineapple and a pine cone, which already are two amazing things, the video left me wanting to know much more about the Fibonacchi sequence. I learned that in fact a pine cone of any shape still follows the order of the Fibonacchi sequence. One question I have is there a way to show that the pine cone’s leaves on the top reflects or follows the sequence as well?

Part 3. Thia video has inspired me to be creative when I teach math in the future. I believe videos like this or even examples related to this, could have helped me get a stronger foundation of math because it might have kept my attention a little but longer than the more traditional way of teaching where a teacher gives direction and leave the student to apply the concepts to solve problems. One may ask, is this way of teaching math? I declare that it is indeed math and this type of teaching may prove highly relevant in teaching students at any grade level, and it is done in a way that will help a student remember and apply it to other forms of sequences and patterns. There is a big connection between this video about the Fibonacchi sequence and the reading Lockhart’s Lament, because Lockhart professes that there is a need to teach students in a way in which they get a full understanding of the meaning of things instead of memorizing a formula. The example he used with a box whose space is shared by a triangle and using the triangle as part of the rectangle is a very useful way to show students why the formula of the area of a triangle works in turn helping them understand that the imagination is what helps you understand the notion. The author states that something as simple as drawing a straight line connecting to points on the rectangle gives a drawing more meaning. Accordingly, Vi Hart shows that she has the same sentiments by having a video where she explains the sequence but takes her time to demonstrate what a sequence meant and proved it in various ways using a pine cone and a pineapple giving the sequence more meaning. The use of real life examples is a effective way of teaching, I find that students may pay more attention and they will most likely remember what was taught without using memorization.

Doodling in Math Class: Stars
This video involves the concept of factoring using stars. The stars total points are factored and you are able to create 1 dimensional stars with those factors.

Doodling in Math: Sick Number Games
This video is about turning numbers in specific triangles and creating patterns that would make it easy to guess the numbers that would follow.

Rhapsody on the Proof of Pi=4
This video involves breaking down a square with perimeter 4 into even smaller ones until you almost get the shape of a circle. Other shapes were used and were broken down until they became almost straight lines.

Part 2
The video I chose to discuss is Doodling in Math Class: Stars which explains how you can make factors of star points useful. It was the first video I watched so it completely intrigued me at first glance. The quick drawing with precision and accuracy made the presentation that was displayed appear visually entertaining. When I then looked at what was being said I became even more intrigued since it showed at first how stars can be easily made by using 2 of the same shapes. These shapes included a triangle, which came first, then a square and finally a hexagon. Then she went into factoring these stars and made 1 dimensional stars that correlated to the factors. It was like making factoring, which she said was barely discussed, into a more enlightening scope than a formal one.

Part 3
This video reflects to my math teaching since it can help to make people think outside of the box. Factoring might be simple but there is never a fun way to teach it. By creating stars and then factoring them then factoring can lead children to see the fun in math. People usually remember entertaining teachings but tend to forget the boring ones. It can even lead you to start your own ideas since one perspective can lead to many others. By looking at this video I was able to broaden my understanding of star points where they can be made using regular shapes instead of having to draw every single point one by one. This sort of correlates to Lockharts Lament since this sort of teaching of factoring would make children think and to not see things with a formal approach but a creative one. By doing this, it might spark creativity in them as well making them real mathematicians instead of just the copy and paste ones.

The last significant group assignment for your semester project is a group presentation (there will be one more individual assignment, a reflection on the process). I’ll put the details here, followed by an outline of the grading criteria (the presentation is worth 20 points total).

Semester Project – Group Presentation

This is your chance to share your group’s work with the rest of the class. Each group will give a 5-8 minute presentation, including the following items:

State your conjecture (this should be written down, either on a slide or on the board). Give an explanation, and an example to demonstrate your conjecture.

If you were able to prove your conjecture, give a proof. If not, describe briefly some of the ideas you had and strategies you tried while trying to prove it.

Give the class at least one puzzle to work on on their own – a challenge!

Give the audience a chance to ask questions (either during the presentation, or after).

Keep in mind the following:

You must include some kind of slides (you may also put work on the board): PowerPoint, Google Slides, Prezi.com, LaTeX Beamer, or other.

You may decide as a group how to divide up the work, but each group member must present something to class.

Be aware that you will be asked at a later time to describe your own specific contributions as well as those of each group member.

Presentations will be given at the beginning of class on Thursday, 12/1 and Tuesday, 12/6. Your group must sign up for a presentation time before leaving class on 11/20.

Grading Criteria (20 points total)

_____ points (4 possible). Basics. Stay within time limits (5-8 minutes). All group members participate.

_____ points (6 possible). Conjecture. Conjecture is written down. Explanation and example are provided.

_____ points (7 possible). Proof of conjecture or proof process description.

_____ points (3 possible). Challenge the class. At least one puzzle is given for the class to work on on their own.

From watching this video I’m amazed that you can make 90 and 180 degrees from make of any size or folded. When I’m in school I would always be using a ruler or a protractor in middle school and high school. I feel so amazed from learning how much a small of piece of paper can accomplished when folded. If I ever need to draw a 90 or 45 degree angle I would know a good way to create one know. I would really recommend that others should watch this video also on how informative it is.

This video is interesting and also I enjoyed how the person was rhyming like if they were actually there during that time. The video was like a code book because the person was using the numbers from pi. This is making me think about the books that I read from high school from William Shakespeare. I think now that could there way of speaking be some kind of language from using the numbers from pi. The video was also informative for me learning about champernowne’s number and also copelano-erdos number.

I know that singing was a good way to help memorize things for class or was a good educational way to learn new things. I didn’t realize that you can memorize pi from singing it by changing your pitch while singing each of the numbers. The video helped me remember some of the numbers of pi that I forgot. This video is a good way of helping us remember the numbers of pi. Singing the number in a higher pitch for the bigger numbers while a lower pitch for the lower numbers in pi. This is a good method for helping others to remember the digits of pi other than 3.14.

Part 2:

From watching the three videos I really enjoy watching them and the information they talked about. I now know resourceful ways in creating 90, 60, 45, 22.5 degrees with folding a piece of paper 3 to 4 times in have depending on what angle you want to draw. Another video helps helps we with remembering the digits of pi with a way that everyone enjoys doing and listening in this kind of day and age. Singing can be a good way to help anyone to remember information for class or for teachers to help students learn a new topic that they are having trouble with. I didn’t realize that when the characters in William shakespeare books could be speaking in code from using the digits of pi. I knew that numbers can be used as codes for letters but I didn’t know that numbers from pi. The one question that I have is was the characters from William shakespeare plays actually speaking in pi when you decode the words that they are saying into numbers? Another question that I have is if you can sing about the digits in pi can you sing with some of the digits in pi but can you do it for all the digits in pi?

Part 3:

The videos could mean in my own teaching is that I can teach students to be resourceful for things that aren’t enough supply of in the classroom like protractors. I believe that the videos can be both math and teaching because when watching the videos you are learning a concept form math and also teaching you new information that you didn’t know until you watched the videos. I believe that the video about William Shakespeare and connecting to pi could be a prove and is still trying to prove it now currently. I can connect this to Lockhart’s Lament because students can learn from watching videos or find creative ways to help them to remember information for a new topic or information for a test.

Vi Hart’s “How to Draw a Perfect Circle” had similar traits to Lockhart’s Lament. While Lockhart emphasised trial and error and exploration into concepts, the pen’s character and voyage for adventure hearkened to that same concept. While some may have thought it was frustrating to see the pen constantly fail or go against Vi Hart’s instructions, the end result sparked a eureka moment. I think I’ll remember how to draw a perfect circle because of the zaniness of the pen’s quirky deviancy.

I thought this video was clever in explaining a paradox. The supposed pi=4 illusion is easily explained by asking “what’s a circle?” That alone was simple enough to disprove this sophistry. I can’t help but make a connection to our course in making sure our arguments while (dis)proving something must be logical and not fall into fallacies.

I found this video to be creative and amusing. Our standard way of proving something is chalk on blackboard, this is just another engaging way in learning the Pythagoras Theorem in a more hands on approach. We’re not simply told this is true but shown; that and asked to try it ourselves!

Part 2: How to Draw a Perfect Circle

Initially I felt intrigued by the video, who wouldn’t like to know how to draw a perfect circle? Then I see this weird conversation between Vi Hart and a super determined pen. After various struggles from the pen I did feel a little bored, wondering “why am I watching this again?”

Vi Hart then interrupts the pen’s crazed state and simply shows us how to draw the circle in the end. While it was funny, the actual answer kind of came about at a halt. The video still did achieve that “aha!” moment, so that actual lesson in drawing a circle did click and stick in my head.

I learned to draw a perfect circle, although it’s not too different from most techniques out there (the idea of rotating paper isn’t new). I also frowned slightly when she drew against the natural curvature of her hand to draw said circle. Coming from a slight architectural background; it’s much better to draw with the natural curve of your wrist than go against it.

I don’t have any particular questions, it was a simple how to video. I would initially ask why the quirky beginning but that adds to the charm of the lesson.

Part 3: Lockhart

This video actually taught me something important about teaching; make it fun! Despite the dorky dialogue and zany frenzy of the pen; I bet kids would love a video like this. Imagine teaching a child a simple technique for drawing (whether it be for arts or graphs) and have it click in their heads in an engaging and humorous manner. It’s definitely teaching, but is it maths? Probably not. I can’t imagine it being super relevant to the work we’ll be doing in class other than working through our proofs and concepts.

As I talked about in Part 2; this reminded me of Lockhart’s Lament where he talked about trusting our youth to make mistakes through trial, error and exploration. The pen reflects that sentiment.

Wau: The Most Amazing, Ancient, and Singular Number

In this video Vi Hart talks about a number that was discovered in East Asia and in many other ancient societies. She describes the different mathematical relationships, usually infinite fractions, that included Wau (some that just included Wau and others were equal to Wau). She related the special number to nature and other fields of study like physics.

https://www.youtube.com/watch?v=wsOXvQn3JuE

Why Every Proof that .999… = 1 is Wrong

In this video Vi Hart goes through some of the many proofs that are used to show that .9 repeating is equal to one. She goes on to prove or attempt to prove these wrong to say that the two are different numbers that are independent of each other. She invalidates these proofs and then briefly talks about why mathematicians believe it is so.

https://www.youtube.com/watch?v=CfJzrmS9UfY

Doodling in Math Class: Stars

In this video Vi Hart talks about drawing stars and connects this activity to math. She talks about stars with different numbers of points and how the number of points relates to the number of sides in each of the polygons that make up the star. She then makes this star drawing into a “game”.

I was a little confused, especially during Hart’s “Doodling in Math Class: Stars” video not because I did not understand what she was saying just because I felt that she was going a little to quick while speaking and the video recording was also sped up so while trying to process what she had just said she was already on to the next part of her speaking and all throughout her hand was moving very quickly on the paper. She did show what she was saying at the same time that she was saying it. For this reason I had to pause the video multiple times to reassess what she said. There was not anything in the video that particularly bewildered me or made me say wow except for when Hart started drawing the much larger stars with many points (when she began using the protractor): I thought that the stars that she made were beautiful and they also require a lot of talent to make. I learned that a star with p points can be drawn with 2 p/2- gons; this was a mathematical relationship that I had not been aware of. In fact I have not ever looked into the math behind stars before. I also learned that even with a set number of points in the star and a set number of polygons formed within the star, there are still multiple ways of drawing the star even with the restrictions. Although I did have to pause the video ever so often so that I did not fall behind on what Hart was saying, I for the most part understand everything that she said. So the only question I have after watching the video, is not directly about the math or the drawing in the video, it is: how does Hart discover the relationships between things like the sides of the stars and the number of polygons in the star; or rather: how is she so successful at finding things like the “game” in this video, that still relate to math, but that a majority other people would not be able to discover?

I think the video could really help me with my own math teaching, even though I did not really enjoy the fast pace of it. What I can learn from Hart is that I need to implement creativity in the classroom at least every once in a while especially in order to keep the students engaged. As Hart emphasizes in the beginning of the video while talking about how she was bored during her math class while learning about factoring, classes that are only lecture can get boring and can cause students to become distracted. So rather than have my students lose focus like Hart did from what the class was actually about: I will give my students an activity to do in class that directly relates to the topic of the day. This way they have something fun to look forward to and are then more inclined to pay attention during class so that they can preform well during the activity. This way they get the fun that they want in class, without losing focus from the actual class and making fun for themselves. Hart’s video really helped me come up with this idea for my own classroom because she expresses that class can get boring, which is true for any student at one point or another and that there are fun and thought-provoking activities to do during class that can still be mathematical and related to class. Kids have a hard time focusing on school work especially for note-taking and problem solving but they are often all for interactive activities, and so I have come to realize after watching this video that combining the two ways of learning can help to take away some of this lack of focus.

Consider the statement: P ∧ ¬P ⊢ Q where: P is any proposition, ¬P is the negation of P. Q is another proposition. Wouldn't proving both P and ¬P to be true simply lead to a new proposition Q, rather than introducing a third truth value? Even if we follow the principle of explosion, wouldn't […]

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The fundamental analysis book I'm reading has a section on the difference between the implicit and explicit definitions of functions. The implicit definition of a function f specifies what property $ P(x,y) $ links the input x with the output $f(x)$. How is the implicit definition different from an explicit one, showing how one generates […]

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Let $\text{RT}^n_k$ denote (infinite) Ramsey's theorem for $n$-tuples and $k$ colors. Let $\leq_W$ denote Weihrauch (i.e., uniform) reducibility. It is known that, for fixed $k \geq 2$, if $n > m \geq 2$, then $\text{RT}^m_k \lneq_W \text{RT}^n_k$ (pages 5 and 6 of this paper). It is also known that, for fixed $n \geq 2$, if […]

In Analysis I, Tao lists several axioms which equality, defined upon a class of objects $T$, must satisfy. The reflexive axiom he gives is Given any object $x$, we have $x=x$. However, from Wikipedia here, equality between 2 expressions asserts "that the expressions represent the same mathematical object". This makes the reflexive axiom seem superfluous- […]

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