Jul 25, 2016

Mathematics in the Modern World: Functions

Notes will be due on Tuesday.  Please make an effort to come to class every day for the remaining two weeks.  There will be in class extra credit assignments and those of you with marginal grades really cannot afford to miss them.  

The Mathematics of Chaos

Domain and Range Graphically

Relation vs Function (Domain and Range)

Understanding Functions more deeply

Iteration of Functions

Functions and Limits

Jul 26, 2016

More on Functions and Sets

Today, we will further discuss functions, transformations, iterations, and limits.  We will discuss elementary groups of symmetry operations and their relationship to transformations.

Archimedes Documentary

Function Transformations

Function Transformations 2

Symmetry Groups of Triangles

Rotational and Reflectional Symmetry

Yesterday's Class

Jul 27, 2016

Symmetry Groups on Solid Objects

We will again discuss groups.  We will talk about group operations with a focus on symmetry groups.  There will be an activity where you will be asked to figure out the size of particular symmetry groups.  

Platonic Solids

Solids of Constant Width

triple gear

Archimedean Solids

Great Rhombicosadodecahedron (Small)

Great Rhombicosadodecahedron (Large)

Great Icosadodecahedron (Large)

Great Icosadodecahedron (One piece)

Great Rhombicuboctahedron (Small)

Great Rhombicuboctahedron (Large)

Student Evaluations

Jul 28, 2016

Homework 10

More On Symmetry and Geometry

We will return to a discussion of groups and geometry.  We will consider orientability as well as the objects we have already examined.  Additionally, we will consider the surprising properties of objects of constant width.  

Rigid Motions

Sphere Inversion

Weird Objects that Roll

Nonorientability

Super Conducting Mobius Strip

Mobius Bagel

Klein Bottle

Prior Class Videos 1

Prior Class Videos 2

Prior Class Videos 3

Prior Class Videos 4

Prior Class Videos 5

Jul 29, 2016

Quiz on Functions, Iterations, and Groups

2 questions on transformations of functions

2 questions on documentaries

1 question on iterations and/or limits

1 question on groups (symmetry groups)

1 queston on platonic solids/archimedean solids/or orientability

Dimensions, Functions, Iterations

Today, we will explore the definition of dimension.  We will discuss the dimensionality of points, curves, surfaces, and solids.  We will discuss the role of boundary and that objects needs not have a boundary.  We will once again talk about orientability.  We will define fractals as the product of an iterative process.  We will reconsider the Cantor Set from this point of view.  We will briefly discuss the idea of embedding and a famous theorem that states you may always embed an n-dimensional object in a 2n-dimensional space.  We will also consider the definition of dimension in terms of degrees of freedom (or variables).  

Hunting the Hidden Dimension

Definition of Dimensions

Simple Diagram of Dimensionality

Escher Mobius Strip

Aug 02, 2016

Add some more info about this item...

Fractals

Last set of documentary notes on Hunting the HIdden Dimension  are due.  We will define fractals as the result of an infinite iterative process.  We will generate the fat cantor set.  We will examine the famous Mandelbrot set.  We will then consider Hausdorff Dimension in terms of scaling factors and number of resulting self-similar segments. There are broader ways to define this quantity, but for our purposes, this is sufficient.  We will use this to calculate dimensions of the original cantor set, the square, the line, the point, the cube, the koch snowflake, and the menger sponge.

Mandelbrot Set

Mandelbrot Self-Similar Set

Koch Snowflake

Hunting the Hidden Dimension

Sierpinski Triangle

Dragon Fractal

Various other Fractals

List of Other Fractals by Dimension

Menger Sponse

Aug 03, 2016

Review for the Quiz and the Exam

Finish Fractals and Begin Reviewing

We will finish the remaining portion of the worksheet.  We will briefly discuss box-counting as a method of determining fractal dimension.  We will review for the quiz and the exam.  

Some Natural Examples

A Fractal To Play With

Box Counting Example

Calculating Some Fractal Dimensions

More Explanation of Self Similarity

Sierpinski Self Similar

Video Lecture 1

Video Lecture 2

Video Lecture 3

Aug 04, 2016

Quiz 6

Quiz 6: Fractals, Iterations, Symmetry

2 questions from the documentary

1 question about complex numbers

1 question about fractal dimension

1 question about self-similarity

1 question about dimension

Aug 05, 2016

Final Exam

Final Exam

Jul 11, 2016

Math as a Method for Evaluating Truth: probability and statistics

Exam 1 Key

https://www.youtube.com/watch?v=ZWib5olGbQ0      Important real life statistics applications

https://www.youtube.com/watch?v=XIOoCKO-ybQ     Extended video with mathematical proof- what kind of proof is                                                                                             Ria Symonds generating?

http://www.dailymotion.com/video/x2idhlw                   The Cosmos with Neil Degrasse Tyson- Episode 7

https://www.youtube.com/watch?v=8HQ06pTN9yo     Tails You Win- A BBC documentary

Extra credit extended to Wednesday.  2 single pages of notes on both documentaries at .5 points a piece- you may email them to me before then or show me in class on wednesday.  Someone pointed out the link was incorrect.  I do not know what it went to instead and I am sorry for the mistake and possible offense this may have caused. 

Today, we will cover probability, statistics, factorial, combinatoric, sample space, independent probabilities, dependent probabilities, outcomes, experiments, conditional probability, probability distributions, expectation value and mutual exclusivity.  It is likely that we will not completely finish this in one class period and will continue into tuesday.  I will use dice, cards, and other games of chance as examples.  If you are unfamiliar with these then you need to make this known so that we can rectify that as quiz and exam questions will depend on a knowledge of this.  I've put some links for Tuesday on the website with reviews on these things and sample questions.  

Jul 12, 2016

More on Probability

http://www.mathgoodies.com/lessons/vol6/intro_probability.html       good review for probability with practice problems

http://www.mathgoodies.com/lessons/vol6/conditional.html               conditional probability

https://youtu.be/ZQ53IMEL1qY                                                           yesterday's lecture part 1

https://youtu.be/j6pG6-hyXJ4                                                              yesterday's lecture part 2

https://www.youtube.com/watch?v=0TjRjqVNAYE                             birthday paradox

https://www.youtube.com/watch?v=lCjspXB5F4A                              Hannah Fry on Probability and Terrorism

Today, we will cover statistics, factorial, combinatoric, conditional probability, and expectation value.  

Jul 13, 2016

More Probability and Some Statistics

We will discuss population, sample, mean, median, mode, standard deviation and mathematical symbology that will make these easier to write down.  We will discuss distributions, measurements, the normal distribution, and general principles for discerning when you have a measurement.  The quiz questions will largely be altered versions of the questions found in the practice below.

Basic Probability

Independent  and Compound Probability

Depedent Probability

Lisa Goldberg and The Monty Hall Problem

Permutations

Combinations

Expected Value

Combination and Permutation Practice

Poisson Distribution

Normal Distribution Rule of Thumb

https://youtu.be/0Jl91T7EtxQ                                  Yesterday's Lecture Part 1

https://youtu.be/sTePE0-nxT8                                 Yesterday's Lecture Part 2

Jul 14, 2016

Statistics Continued

We will discuss standard deviation, histograms, the standard (normal, or gaussian) distribution, the poisson distribution, how to determine when a measurement is real, and review homework problems.  There are review problems and sample data sets to work with.  I suggest the last link for practice in elementary statistics computations like mean, median, mode, and range.

Yesterday's Video lecture part 1

Yesterday's Video lecture part 2

Poisson Graphs

Statistics Review: Mean, Median and Mode

Statistics Review: Mean, Median and Mode Part 2

Practice Problems

Comparing Distributions

Sample Data 1

Sample Data 2

Sample Data 3

Simple Sample Data

Jul 15, 2016

Quiz 3: Probability and statistics

2 questions from the documentaries.

1 question on expectation value

1 question on basic probability (cards, dice, ect)

1 question on permutations or combinations

1 question on independent or depedendent probability

1 question on statistics (mean, median, mode sort of thing)

You will need a calculator.  

Jul 18, 2016

Statistics and Distributions

The Joy of Stats with Hans Rosling

BBC Horizon: The Age of Big Data

Edge Bias in Coins

Binomial And Normal Distribution

Binomial Distributions Explained

Longer Explanation of Binomial Distribution

Take a page or more of notes on each video for .5 points of extra credit.  The notes are due Tuesday- you may email them to me or show them to me beforehand.  Homework 8 is up.  9 will be up later tonight and it will focus on normal distributions and expectation values.

Jul 20, 2016

Quiz 4: Distributions, Probability, and Documentaries

2 questions from documentaries- 1 from each

1 on probability of AND/OR situations

1 question on normal distribution

1 question on histograms

1 on expectation value 

1 on binomial distribution

Solutions to HW8

Jul 21, 2016

Review is listed

Exam Review

There will be the following alotted to particular topics on the exam.  The exam is out of 23 points.

5 points documentary**

3 points expecation value*

3 points binomial distribution*

3 points sample data (mean, standard deviation, differing measurements)*

2 points normal distribution

2 points elementary probability

2 points and/or probability*

2 points calculating size of sample space*

1 point reading/making a histogram

* problems will appear in some form as free response, though may also appear in the multiple choice section

** means these will include previous questions and some new ones

Previous Lecture Video 1

Previous Lecture Video 2

Previous Lecture Video 3

Previous Lecture Video 4

Previous Lecture Video 5

Jul 22, 2016

Midterm Exam 2

Exam 2: Statistics and Probability

Jun 27, 2016

The Tools of Mathematics: Elementary Logic

Jul 08, 2016

Midterm Exam 1

Exam 1: Logic and Proof

Multiple Choice (1 point each):

6 questions from the 3 documentaries

2 questions on simplifying logical propositions 

3 questions on proofs

3 questions on arguments

3 questions on sets and infinity

Free Response (2 points each):

1 question on converting English to propositions or vice versa

1 question on truth tables

1 question on converting mathematical symbols to English or vice versa

Jul 07, 2016

Review for Exam

The key to the previous quiz was uploaded under the Quiz 2 link.  The video of yesterday's lecture is uploaded.  It takes between 4 and 6 hours to upload a video with the school's internet or from my place.  If anyone has suggestions for faster alternatives, let me know.  I already tried uploading the video with lower quality, but it really affected image and sound quality, so I am not sure that is a good option.  

https://youtu.be/LqfuOdiq-Z0                           The Review is uploaded here

Jul 06, 2016

Proofs by Induction and the Infinite

We will revisit induction with the proof that the sum of squares yields n(n+1)(2n+1)/6.  We will return to the definition of sets and introduce the idea of the power set.  We will examine the properties of the power set.  We will discuss how this pertains to the particular infinite sets we have learned about, Paul Cohen, and the continuum hypothesis.  We will outline the difference between countably infinity and uncountably infinite.  We will briefly return to the Cantor Set and discuss why it seems plausible that the Cantor Set is uncountable.

This quiz will cover

-2 questions on types of proofs: induction, direct, contradiction, and counterexample

-a proof of some sort: it will be something we have seen in class or the homework or something extraordinarily similar to it

-3 questions from the 2 documentaries that you watched over the weekend

-a truth table question of some sort

Here is the bank of mathematicians that will be considered for the quiz from the videos: 

 

Albert Einstein

Isaac Newton

Albert Einstein

Paul Dirac

Stephen Hawking

David Hilbert

Georg Cantor

Henri Poincaré

Grigory Perelman

Kurt Gödel

Paul Cohen

Julia Robinson

Evariste Galois

Nikolai Bourbaki

Alexander Grothendieck

https://www.youtube.com/watch?v=9HbfNoNOKOQ        The video of the lecture can be found here.

Jul 05, 2016

More on Truth Tables, Infinity, Proofs by Induction, and Direct Proofs

We will begin the class by discussing the two documentaries for 5 or 10 minutes.  Then we will cover proofs by induction and direct proofs.  Particularly, we will examine the proofs of summing the first n integers and the geometric series of 1/2.  We will also consider the direct proof of the latter.  If time permits we will discuss infinite limits and the cantor set today- otherwise we will discuss it tomorrow.  There will be a homework set and detailed solutions to both quizzes are posted under the link for quizzes from last friday.  We will also review what truth tables are, how to make them, and how to use them as that and the documentary questions were, by far, the most missed things on the quiz.  Questions about the documentary will appear on the exam and truth tables will appear on the next quiz AND the upcoming exam.  The average score for the quiz was a B-.  

https://www.youtube.com/watch?v=wRMC-ttjhwM                                         Truth Table Example

https://www.youtube.com/watch?v=OLGVhszBlq4                                         Further Truth Table Examples

http://turner.faculty.swau.edu/mathematics/materialslibrary/truth/                  Truth Table Generator

http://www.exegetic.biz/blog/wp-content/uploads/2013/11/cantor-set.png      The Cantor Set

Jul 04, 2016

Holiday

http://www.veoh.com/watch/v17489279SyAaxXze        The Story of Maths Part 4

https://www.youtube.com/watch?v=rJO-rEIjP0U            The Story of Maths

These are two different documentaries.  They are both about 50 minutes.  We will discuss them both in class on Tuesday.  For the first, take notes and connect the individuals with their major contributions.  For the second, connect the the individuals mentioned with the major equations that are highlighted and try to understand what the equations are saying.  Do this on separate sheets of paper and I will collect this for .5 bonus points each.  

Jul 01, 2016

Proofs

Note that this is a good time to communicate with me via email or in person if you are feeling overwhelmed or  how the course is going for you so far.  You are also going to have homework over the weekend in the form of another documentary to watch!  It will be linked to monday.  Joy!  That is your only homework for this weekend since I hope you do enjoy the 3 day weekend.  On the other hand, please try to find time to watch it over the weekend as we will discuss some of the points it made in the first 15 minutes of class on Tuesday.  

Things to study especially hard for the quiz:

There will be 2 questions from the documentary

HW 1.1, 1.2, 1.4, and 1.9

HW 2.1, 2.3, 2.6

HW 3.7, 3.9

Jun 30, 2016

Finite and the Infinite

https://www.youtube.com/watch?v=dDl7g_2x74Q                    Getting comfortable with infinity

https://www.youtube.com/watch?v=elvOZm0d4H0                   Infinity and relative sizes.

https://www.youtube.com/watch?v=_DkFYA9A_1I                   Infinite sets in greater detail

http://www.rapidtables.com/math/symbols/Set_Symbols.htm   Glossary of set theory symbols

In class, we will discuss the infinitude of the natural numbers and the primes.  You won't be asked at any point to reproduce any of the proofs present in these videos, but it would be nice if you would put forth an effort in trying to understand them.  Math is not a spectator sport- you have to engage your mind and try to reproduce the lines of reasoning on your own with a pencil and paper.

         

Jun 29, 2016

Conditional Statements (Implications) and de Morgen's Laws, Important sets and math symbols

https://www.youtube.com/watch?v=afAt0jEF74c       Statements, Converse, Inverse, Contrapositive

https://www.youtube.com/watch?v=g9SHF0iS0Dw   de Morgan's Laws

https://en.wikipedia.org/wiki/List_of_logic_symbols   list of common logical symbols and their meanings

Jun 28, 2016

Truth Tables, Connectives, and Tautologies and Contradictions

https://www.youtube.com/watch?v=OLGVhszBlq4  More on Propositional Logic

The first homework is available and underlined questions are very likely to appear on a quiz or exam.  Click on the link under the date marked This is Homework 1. The other questions may appear, but the likelihood is much lower.

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Department of Mathematics

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