Monday, December 16, 2013

Metric Spaces defined by the Hausdorff Metric

This blog post is a partial requirement of my final project for Mathematical Computing.

Document: Metric Spaces defined by the Hausdorff Metric [PDF]

Matlab Files:

Animation #1 is the well-known Sierpinski affine contraction maps.  It rotates this from 0 to 360 degrees for both theta and phi.
Animation #2 is the Koch Curve affine contraction maps.  It rotates this from 0 to 360 degrees for both theta and phi.

Thursday, December 12, 2013

Social Genius

If you were on Facebook or YouTube over the past 3 days then at some point you probably encountered and maybe even watched the WestJet Christmas Miracle [video -].  If you haven't go watch it now and then come back.

Many people's first reaction beyond who is WestJet is wow how generous of the company (it was my initial reaction).  I just happened to see the commercial again when I was on YouTube and noticed that since it was first posted on Dec. 8, 2013 (4 days ago) it has already received 16,654,700 views!  In addition to that it has received 109,383 (0.66%) thumbs up, and only 2,823 (0.017%) thumbs down.

Think about it.  This is 16,654,700 clicks not just basic impressions.  I'm curious as to how much their entire production including gifts actually cost them?  $1.00 per click would be $16,654,700 which seems drastically high but what do I know about the cost of productions.  At $0.10 per click then it would have cost $1,665,470 and that still seems high to me.  If we drop the cost to $0.01 per click then it would have cost us $166,547 which seems like a reasonable budget.

At the rate of $0.01 how much does an acquisition cost them?  To come up with a starting point I chose 700 (0.004%) acquisitions from the 16,654,700 which gives us a rate of $237.92.  I picked 700 going towards the very low end and believe the rate will only go down from there especially considering acquisitions should continue to rise as the click numbers rises.  Now if we only knew how much an acquisition is worth to them?

Seems to be a very effective and affordable marketing and advertising plan that utilizes the power of social media.

UPDATE: On Dec. 18, 2013 the youtube views of the video are at 30,260,070.

Saturday, September 21, 2013

Using to investigate sqrt(2)

Here's a simple lesson that can be used in an Algebra 1 class to investigate irrational and rational numbers.  It integrates both math and technology into a quick 10 min information packed lesson.

The lessons objectives are for the student to observe an irrational number, sqrt(2), on the x-axis, and be able to identify its corresponding digits to a certain level of precision accurately.  A set of follow-up objectives to this lesson would be to have students analyze a rational number, such as 1/3 or 25/99, and speculate as to are there logical differences on a graph between rational and irrational numbers?  If so, what are they?  Can we develop a set of rules that will explain the difference so we can easily determine if a number is rational?

Tools is an online graphing application that I place as one of the best online math tools for math students that have come out in the past 5 years.  You can create a free account or link it to any Google account for single sign-on purposes, which give you added ability to save your graphs to Google Drive.

Create a new graph at  Enter the coordinate (sqrt(2), 0) on the graph.  Consider this our destination.
Graph of (sqrt(2), 0)

Next create an additional coordinate that only has one decimal place and is a best estimate of sqrt(2). Create it using the color black with the goal that it covers the orange dot.

Graph of (1.4, 0) covers (sqrt(2), 0) at this level of precision.
Ask your students to zoom in on the point.  Help them observe that as our precision increases our accuracy decreases.  Once the student can clearly see two distinct points they should repeat the above process of estimating the next decimal values until the two points become one again.

Zoomed in at 10^4 precision

Coordinate (1.414, 0) at this precision covers (sqrt(2), 0)

Does it ever stop?

Questions that will make your students think.
  1. Does this ever stop for irrational numbers?  Why or why not?
  2. Are there any rational numbers that act in a similar way? If yes, which ones? If no, why not?
  3. What is the difference between the rational and irrational numbers that act in a similar way?
  4. Can we describe a rational number in a different way besides as the ratio of two integers? 

Tuesday, August 20, 2013

Last words.

While reading the news today that famed writer Elmore Leonard passed away, my first response was to visit his twitter page and see some of his last words.  Your last public words if you use twitter could possibly be less than 140 characters.  Make those words count it may be the last words you leave the world to represent who you were.  Take care.

Friday, August 2, 2013

What Pope Francis Can Teach Us About Leadership

It has been just over five months since Pope Francis stepped into his new role as Bishop of Rome and what an exciting five months we have seen.  Pope Francis stands out as a true leader who seems to on an almost daily basis break the chains of tradition while maintaining respect for his position and the Church.

He is a great role model for educational leaders.  Our daily call embraces all that Pope Francis is currently achieving and modeling for us.

The desire to break the chains of tradition for a fuller and more sincere educational experience embracing new communication methods, advancing learning in our classrooms with new technology tools, and transforming our lessons to embrace students prior and future experiences.

Maintaining respect for our positions by not throwing theories and years of research to the side for immediate shallow returns on investment that last only a school year or two.  At the same time not settling for the same old because it is safe or provides a guaranteed minimum.

The Pope has all already debunked one of the biggest fears of leaders that change or pushing against tradition will destroy the institution.  We have learned quickly that the institution remains even stronger than before and that breaking tradition becomes a new tradition itself.

It is exciting times in the Church under Pope Francis' stewardship and in the world of education under our current leaders' stewardship.  I continue to pray that all leaders remember on a daily basis they are just stewards of something bigger than any one of us and that together we can all change the world. 

Wednesday, June 26, 2013

21st Century Teachers

Everyone is talking about 21st century teaching, which is good considering we are 13 years into the 21st century.  What I find, unfortunately, is for every single amazing idea there are at least two bad ideas that represent that many people even decision-makers are confused by what 21st century teaching actually is.

Since locking down a definition for 21st century teaching will be difficult and cause for debate, it is easier to list what you will find in a 21st century teacher.  I'm going to break this into two lists because there is a developing trend to only focus on the student and their relationship to 21st century learning.  When it comes to learning there are usually two players, teacher and student.  The teacher is just as vital to this discussion and their relationship to 21st century learning.  For now let's just focus on the 21st century teacher.
[Note: When it comes to 21st century education systems there are at least three players: 21st century student, teacher, and administrator.]

Here's are qualities and features that describe a 21st century teacher.
  1. Lifetime learner - enthusiastic about continuing education opportunities and advancing their knowledge base in their subject matter and educational theory.
  2. Condensed course load - 21st century teaching is a lot of work and requires significant preparation and continuous dialogue with students.  Teachers of the 20th century were bogged down with many preps and sometimes 6 classes a day, this isn't feasible for 21st century teachers.  A course load of 4 classes and 2-3 preps max are necessary.  This can be challenging because many unions and administrations are stuck in a 20th century operational mindset.  These are the standard challenges were hear in the daily news all over the country.
  3. You will find 21st century teachers in classes with less than 20 students (15 and less per teacher preferable).  This doesn't imply that classes need to be less than 20 but that when there are more than 20 you need to have a second teacher in the classroom. 
  4. A 21st century teacher is a master of their subject.
  5. They are also not scared or intimidated by new technologies; they approach them with open minds and assess the value it adds or takes away from a classroom.  They are responsible enough to know when to say something will be used or not used within their classroom and back it up with defensible reasons.
  6. A 21st century teacher has the control of a maestro so much so that it seems like there is absolute freedom within the classroom.  This is a good point because many interpret the 21st century teaching method as "all goes", no time to wait, technology changing to quickly for theory and analysis.  This is false.  There is no reason why we should not continue referencing contemporary learning theories and implementing proper experimental methods with data collection and analysis.  The theories exist and are connectible.
  7. A 21st century teacher is a master of time.  Time invested in planning and practice pay off in dividends, this will not change.
  8. A 21st century teacher knows exactly the days they will be teaching their students and has a master plan prior to the start of their course, very similar to how college professors operate.  This requires the cooperation of their schools administration in guaranteeing academic time.  Administrations would be best referencing college models where students are expected to always attend classes on a normal frequency yet are deeply involved in their school community outside of class.  Balance, inclusion in decisions, and respect works best here for developing academic scheduling.
  9. [Updates] Here's a good one that was sent to me.  21st century teachers have taken an online class and either teach an online class or utilize online class management software in their classroom.

Think about it.
  1. Can you develop a definition for a 21st century teacher now that you have seen some of the qualities that describe them?
  2. How about the relationship between technology and a 21st century teacher?  Many times the noise surrounding technology and what is allowed or is not sometimes overtakes the real discussion on 21st century teaching.
Let's continue adding more to this list.  Think you have something that belongs here, let me know through comments or email.  Can you develop a definition for 21st century teacher now that you've seen some of the qualities that describe them?

Thursday, June 13, 2013

CCM Summer Institute Day 4

Another day of some excellent information and hands-on experience.  We did a great small group activity on the differences between amplifying and transforming a classroom.  One of the most interesting concepts taken from this was that this flipped classroom concept really doesn't revolve around technology; it can completely exist on its own but is clearly enhanced when we use technology as it prepares students for their careers.

Our last set of work was individual work and gave us each an opportunity to get a few lessons created for our courses coming in the fall.  This worked out well for me because this course runs every semester and in two years will run again at the high school I teach at too.  The assignment was to create 3 separate activities exemplifying an absorb, do, and connect activity.  I have taught this topic many times over the past 10 years and have tried lots of methods that have worked but taking it from this perspective of absorb, do, connect really helped me write a deeper lesson that gives significant hands-on experience.

Here's the link to my three activities, check it out!

Wednesday, June 12, 2013

Notes from Summer Institute Day 3

I used my iPad sticky notes today to take some quick notes. It was perfectly appropriate for this type of content, I wish there was a similar sticky note concept for math.

Tuesday, June 11, 2013

CCM Summer Institute

So I signed up for a week long course at CCM in New Jersey offered to their professors.  Course title "Engaging the 21st Century Student" = awesome!  The course is hybrid with both face-to-face and online requirements, hosted through their Blackboard implementation.

The pre-reading that was required quickly identified this as a class where I would gather lots of information.  Almost immediately I learned a word I had never heard before, which embarrasses me as a teacher, but also reminds me there is so much information out there that we must be lifelong learners.  The word is andragogy and you want to think of it in terms of pedagogy vs. andragogy.  Simply stated andragogy is the study of how adults learn.

We really focused a lot of our discussion towards this theory as our leaders modeled 21st century teaching through large and small group discussions, small group work and protocol discussions (this was challenging but interesting!)

At the end of session they shared three online web services they highly recommend.
  2. Lucid Chart
  3. Google Apps
There was an open discussion where everyone who had prior experience with some of these sites could testify to their usefulness.  Reflecting back at this experience you clearly see one of the principles of Knowle's theory appear. 
Adults bring life experiences and knowledge to learning experiences.
Start your discovery of andragogy with the Awesome Chart on "Pedagogy vs. Andragogy"

Friday, April 26, 2013

1-to-1 Learning - Goals & Objectives Part 1

Here begins a brief series on decisions to make and options you have available when designing your 1-to-1 learning environment.  This information comes from 10+ years of firsthand experience as a Network/Systems Administrator in a 1-to-1 learning high school and 9 years experience as an adjunct professor of network engineering.

Lesson 1 - What are your goals and objectives?

The process begins by you building a clear and well-defined map that you will use as a guide and reference along your journey.  Things you should decide before starting your 1-1 deployment are:
  1. Who are your users?
  2. What are their intentions?
  3. What are the restrictions your users want?
  4. What are your users expectations?
  5. Which devices will be utilized?
  6. What is your budget for devices, training, staff?
  7. What are your security requirements?
Today's lesson will cover the users, topics 1 - 4.  As I write this blog, I just received the following message from a fortune cookie, which is perfectly appropriate for this lesson.

Great things are accomplished only by the perfection of minor details.
Who are your users?
Clearly, the user plays an important role when building your 1-to-1 initiative.  In a high school your answers may be: student, teacher, administrative staff.  How about guests and the public?  Just a few years ago guests may not have been a consideration, today many people assume that businesses are similar to Starbucks, so at the cost of the school some institutions are making this available.  If you choose that option you need to make sure you keep their network separate from the rest of your corporate network and that you secure it.  It can be a wise investment to support guests especially if your building is utilized by many outside groups or you host many public events.

What are their intentions?
Your users intentions are critical and sometimes the most difficult thing to get from decision makers.  It is your job to impress upon them the importance of planning and that planning will lead to a secure, stable and scalable network, which is not built from a hodgepodge of ideas but from data collection, thoughtful reflection, and clear objectives.  Here are some example intentions that you should be aware of:
  • Students should have some access to the Internet whenever they walk into the building.
  • Students will only access the Internet from labs or while in a monitored classroom environment.
  • Teachers will have access to the Internet from any device they own.
  • Student, teachers, staff will have access to the Internet from only their school provided device.
  • Your local students news station will broadcast over your Internet connection.
  • VoIP will be supported across this connection.
  • Security cameras will be supported and accessible outside of the building through your connection.
Your list of intentions may be long and that is a good thing!  Do not shoot anything down, consider this a brainstorming exercise.  Later in the process, budget, user restrictions, and security will be natural constraint variables that will help filter out unrealistic requests based on the resources of the organization.

What are the restrictions your users want?
Are there any?  Do decision makers assume some that are not mentioned?  Also what are federal and state restrictions?  As an example, for schools to have access to E-Rate funding schools must block students from access to certain types of content on the Internet (E-Rate funding can be significant).  Schools in the end have the decision whether they want this funding or not so it is their choice.

Finally, do not forget to ask your stakeholders: students and teachers.  In the end it only comes down to these two groups.  What do your teachers need to fully support your students and what do your students need to unleash their creativity within their classrooms?  If you do not ask them then who are you doing this for and how do you know the answers to those questions?

Again, treat this like a brainstorming activity; later in the process you will use your constraint variables to help identify what is currently possible with today's technology and within your budget.  I highly recommend involving the teachers and students in this process that cannot be said enough.

What are your users expectations?
At this point you may be saying, I think you have covered it all what else can there be related to the user?  Their expectations are more about the end user experience and what will qualify as poor, satisfactory, and exemplary service. 
Rule of thumb: If it is not written down and shared with everyone then it does not exist.  
If your decision makers or your students and faculty have expectations but have not put them down on paper then they do not exist.  Similar to the classroom experience a teacher lists their daily objectives and expectations for their students.  Your job is to gather these expectations, help filter through them and then work with the stakeholders to identify the ones that are obtainable and which ones are not. 

Expectations could look like this: 
  • As a teacher, I expect to be able to show an online video to my class without Internet disruption. 
  • As a student, I expect to be able to communicate with other students through skype while in school without losing my connection. 
  • As a teacher, I expect to be able to move freely around my classroom and teach without having to monitor which websites my students visit.
  • As a member of the administrative staff, I expect to be able to view security logs from any device accessible to the Internet.
  • As a student, I expect that the use of technology in the classroom will give me more learning opportunities not less.
These are few that you may run into.  Notice they may not all seem advisable for every situation.  Your job is to help manage expectations, which translates into a written set of expectations that can be referenced by all involved.

Next week
This ends the first part.  My intention is to have the next part available at the end of next week.  As always, I appreciate your feedback so feel free to comment or email me.

Monday, April 22, 2013

Markov Chains: Steady-state Probabilities App

My research project for Operations Research is using Markov Chains so I decided to create a Maple App for their new Mobius Project (  Anyone can view this app all you need access to is one of the 3 (preferential order):
  1. Maple program
  2. Maple Player (free)
  3. Web browser (
Once you have the application running you have the ability to enter an initial state vector along with a probability matrix.  You can ask for specific states to be solved or a range of states (even with interval jumps).  Computation is extremely fast, using the power of Maple.  I've used it for small probability matrices such as 3x3's but would love to hear about anyone using it for larger data sets.

Symbolically you can enter both vectors and matrices in any form acceptable to Maple (it uses a type check).  I recommend using the following:

[a0,a1,...,aN] - for a vector
[ [x11,x12,...,x1N],[x21,x22,...,x2N],...,[xM1,xM2,...,xMN] ] - for a matrix

I plan on adding additional problems to the app as long as it keeps it functional and readable.  If not I will create them as separate apps.  Any feedback is appreciated!

Download Maple Worksheet:

Download Maple Player:

View embedded Maple Worksheet:

Sunday, April 14, 2013

Introduction to Affine Maps

Affine Maps

An affine transformation of R[n] is a function t:R[n]->R[n] of the form t(x)+Ax+b where A is an n x n invertible matrix and b is in R[n].  Our current focus is on R[2] of two dimensions but all the important properties coming up hold for all dimensions.

So what makes affine transformations so special?  Affine transformations have 3 nice properties:

1. t maps straight lines to straight lines
2. t maps parallel straight lines to parallel straight lines
3. t preserves ratios of lengths along straight lines

Let's say you are interested in scanning a physical picture into your computer.  Your picture is the input, the scanner with it's configuration is the function, and the resulting digital image is your output.

Second example would be opening a digital image in photo editing software and then scaling it down and/or rotating it.

In both examples the image looks the same just smaller or larger and in a different position, with affine maps the straight lines will always be straight but the angles can change and distort the image.  A single affine map instruction covers the movement/change of an individual image.

How do I write an affine map instruction?
Without getting into how these equations are developed let's focus on how to write the set of instructions.  The standard geometric equation is

affine(r,s,θ,φ,e,f) -> (x*r*cos(θ)-y*s*sin(φ)+e, x*r*sin(θ)+y*s*cos(φ)+f)
  • r - controls the x scale
  • s - controls the y scale
  • θ - controls the left angle
  • φ - controls the right angle
  • e - controls the x shift
  • f - controls the y shift
Each value has a specific control on the starting image.  Your starting image can be a set of points that define an image or even a single point.  You send these points through the function defined by the affine map and then view the result.  Let's take a look at examples of desired movements and how to represent that in affine geometric notation.

Affine Structure  
The set of all affine maps is a group under function composition. So affine mappings are closed, there exists an identity affine map and an inverse map that undoes your changes.

Note: The above image uses Mr. Face, a 1x1 square with left open eye and right closed.  I was introduced to Mr. Face along with dynamical systems by Ken Monks at the University of Scranton during my undergraduate years.

Saturday, April 13, 2013

Goodbye to winter fractal

Weekly Fractal Problem
Here's a fun fractal I just generated that will not be as easy as last weeks.  It reminds me of a partial snow flake or one that is half melted.

Remember your goal is to determine the fractal's IFS.  Good luck!

Last weeks solution
The name of last weeks fractal was correctly answered as Cantor dust with the following IFS'

Friday, April 5, 2013

Name my IFS

For the inquisitive I'll post a new fractal every Friday along with the question of what is the fractal's IFS?  What I mean by that is, what are the set of affine maps that make up this fractal.

I'm starting off with a very easy one visually.  See if you can determine all of the affine maps that make up this IFS.  The answer will be posted on next Fractal Friday's post.

Write it out as Affine(r,s,theta,phi,e,f) where r,s are scalars, theta, phi are rotations, and e,f are shift transformations.

Tuesday, April 2, 2013

Affine Contraction Mappings

I took full advantage of my spring break by spending the day writing a program that allows the creation of geometric affine maps.  You can then pass a set of points through the map instructions and your output will be a deterministic fractal.

A geometric affine map can be written in this way:

map = Affine(r,s,theta,phi,e,f) where,
i. r,s are scale factors between 0 - 1 for the x & y-axis
ii. theta, phi are rotations along the x & y-axis
iii. e,f are transformations along the x & y-axis

Here's my first example and test to see that the program works.  I defined the famous Sierpinski affine mappings.

map1 = Affine(.5,.5,0,0,0,.5)
map2 = Affine(.5,.5,0,0,.5,0)
map3 = Affine(.5,.5,0,0,0,0)

Let your starting set of points be only (0,0) and iterate this 8 times and you will end up with 3^8 points (6561), plot those points and you will see the following image.

Here's another set of affine maps that are related to the Sierpinski triangle except for one rotational value in just one of the three maps.  This highlights the significance and sensitivity one minor variable change can have on an outcome.

map1 = Affine(.5,.5,-180,0,0,.5)
map2 = Affine(.5,.5,0,0,.5,0)
map3 = Affine(.5,.5,0,0,0,0)