# Optimizing Box Volume with Marshmallows

The problem is a classic in algebra and calculus, and to be honest it really did not make sense to me until I built the boxes, when I was 35 years old.   Here is the problem:

• Given a standard 8 ½ inch x 11 inch piece of paper, determine a function which gives the volume of a box (without a lid) made by cutting squares from each of the corners and folding up the sides.   Let x be the length of a side of the square, and write the volume as a function of x. What are the units of x and v(x)?  What is the maximum volume of the box?

Granted this is an expanded version of the problem.  You couldn’t ask all of these questions on a standardized test.  But this problem is full of concepts like: domain, range, polynomials, graphing, roots, and volume just to name a few.  My classrooms have students at varying degrees of experiences in math.  For instance I will have students that are ready for calculus working alongside students who really don’t understand what volume means.  In fact if you ask some of them, they will reply, “L X W X H”.  If that is all students think about when they are asked to find volume, then they probably do not understand the concept of volume.

I have a bunch of 5X8 note cards, so I ask the kids to imagine cutting equal squares out of the corner of the note card and filling the resulting tray with marshmallows.  How many marshmallows could we contain in this resulting tray if we don’t pack them and we do not heap them above the box?  I let them fiddle around with the marshmallows.  I then give them 1 minute and tell them to write their name and their guess on a piece of small yellow post it note.  We then arrange them on the board in a makeshift line graph.  I do this so that students buy into the problem and practice estimation.  This guess cements their curiosity.   They start asking themselves.  “Am I right?” ” Is my answer the most?”  “Is my answer even possible?”  Some students will try and build the box so that it matches their answer.

As they are building these boxes some students will ask, “How big should my squares be?”  I respond, “How big can your squares be?”  I notice that the older I get the more I sound like a psychiatrist.  But here I am trying to lead students to the domain.  Eventually we will formalize the domain and the range for this problem.  But first we have to see how many marshmallows we can stuff in these boxes.

What we are finding so far is that unsurprisingly the number of marshmallows divided by 8 is just under the actual volume of each box in cubic inches.  Here we can talk about the gaps between mallows and ask questions like, does smashing them give us closer estimates?  How about piling them? (You are on your own here.  The kids ask this but I redirect them to the GeoGerba applet below:)

Finally, we talk about the function that represents the volume of the box for any square of length x cut from the corners of the 5 X 8 note card.  In this case V(x) = x(5 – 2x)(8-2x) (I will let you do the distribution to find the simplified version of that polynomial;).  And about the best place to explore the function is in Ted Coe’s GeoGebra worksheet  at http://www.tedcoe.com( His site is incredible!!!!! BTW). Here is the html applet.  I hope it works because it is awesome!!!
The power of this project comes in the form of having students tape their creations to a projected GeoGebra display of the function the class developed.  Every time I do this it brings a small tear to my eyes at its beauty. To me it shows the true power of technology in our classrooms.  The first time I saw this problem was Algebra II when I was 16 years old and our internet was floppy discs.  Along with the fact that we only used computers when instruction was complete.   I am sure my teacher at the time tried to show us the solution, but I did not register any understanding.  It was one of those problems that the teacher assigns and then when no one tries, he/she solves it for them on the board while all students nod their heads that they understand.  I saw it again in Calculus and I still don’t think I truly understood anything outside of (the algorithm) the fact that when I applied the derivative to the function and then set the resulting function to zero and solved I would get my maximum and minimum values.  The third time I saw this problem was in my masters program at the University of Nebraska-Lincoln.  Here is where I came up with the idea that maybe when we solve this problem we should actually construct the boxes. I am pretty sure that constructing the boxes actually makes this problem a Jo Boaler Low floor high ceiling  problem.  (YOUCUBED.ORG has changed my very perception of mathematics education).  I give this problem to all of my students 9 – 12 and they all love it.  Probably because they get to eat marshmallows, but I hope that at the very least some finally see the connections between stuffing marshmallows in a box and volume are the same thing.  If we get to the connections between algebra, calculus, and geometry…. that would be bonus for those that are ready to make those connections.  I am pretty sure I wish I lived right now.  Education is awesome!!!

# Who thought about that problem differently?

The question above should be asked by math teachers after almost every problem.   During my bell ringer today we had unexpected, crazy, rich discussion over two problems that  I never thought of as rich in mathematical ideas.  The problems were :

In a certain school, 45 percent of the students purchased a yearbook. If 540 students purchased yearbooks, how many students did not buy a yearbook?

and

If a machine produces 240 thingamabobs per hour, how many minutes are needed for the machine to produce 30 thingamabobs?

Solve these and see what you come up with.  Keep track of what you do.  In my classroom, I randomly choose students to come up and show their methods.  Sometimes they are wrong and we use incorrect answers as an opportunity to  show common mistakes and I  do everything in my power to show that these mistakes are helping everyone in class grow their brain (Thanks youcubed.org!!!) In this first problem, many students were blurting 243.  In this case students were multiplying 0.45 and 540 or switching up their proportional reasoning.  To handle this I told them to slow down and read the question again.  Does that answer make sense?  Also, please stop blurting answers! You are stealing someones opportunity to think.

Problem 1 method 1 The first student came up and told me that you first take 540 students divided by 45 percent to find the number of students per percent.  Then take that answer (12) times 55 percent for 660.  I asked them how they came up with 55 percent and they were able to tell me that 100 – 45 = 55 which is the percent of students who did not buy the yearbook. Then I asked, “Who thought of this differently?”  Surprisingly, many hands flew up.

Problem 1 Method 2 The next method was the straight up algebra.  If x is the total number of students, then 0.45x = 540.  Solve this and 540/.45 = 1200.  Since there are 1200 students we know that 1200-540 = 660 student that did not buy the yearbook.  Each time a new method popped up I would give it the name of the student who produced it.

Problem 1 Method 3 In the third method a student divided 45 and 540 by 9.  This student then proceeded to tell me that because of this every 5 percent is 60 students, so 11 times 60 is 660.  Here the 11 felt like witchcraft for some of the kids, so I had the student explain where the 11 came from and he was able to tell everyone that 55% has 11 groups of 5.  This method is similar to method 1 and this student may just have wanted to come to the board, but that is ok.  Kudos for thinking deeply.

Problem 1 Method 4 A student kept blurting out FISH.  At this students previous school they had been taught that proportional reasoning was a method called fish.  When you set up the proportion  45/540 = 55/x , where x is the total number of students.  We then proceed to cross multiply and divide. When you trace out the pattern of this the Christian fish symbol is produced. .

In all cases I was able to show students the connections between their solutions and use the language of mathematics.  In the word percent, per is divide and cent is 100.  Using labels helps you keep everything organized.  I had one student use 45/100 = x/540 to come up with 243.  Again, when I added the labels for each numbers 45( purchasers)/100 (entire class) = x (entire class)/540 (purchasers) several students in the class had an aha moment.

Problem 2 method 1 By the time we made it to the second problem, I had students practically running up to the front of the class to show what they did.  So I didn’t have to randomly pick them.  The first student told me it takes 8 boxes of 30 thingamabobs to package 60 minutes worth of product, so 60/8 means that it takes 7.5 minutes to produce one box.

Problem 2 method 2  On the second method a student told us that if 240 thingamabobs are produced in 60 minutes, then 120 are produced in 30 minutes.  Following this reasoning then 60 are produced in 15 minutes and 30 are produced 7.5 minutes.

Problem 2 method 3 It surprised me that the algorithm we teach is always the last method shown.  My personal opinion is that when we follow these “efficient” algorithms it slows down thinking .  In this method the student lets x = minutes for 30 thingamabobs.  Then 240(thingamabobs)/60(minutes) = 30(thingamabobs)/x(minutes).  Then students cross multiply and divide to get 7.5 minutes.  240x=1800 then 1800/240 = 7.5  I think “cross multiply and divide” is a “trick” that students don’t really understand.  I try and always show that it is really just common denominators and the multiplication of the entire equation by that denominator in order to simplify the fraction.

# Rethinking Educational Data and Comp Studies with Google Sheets and Motion Graphs

```Hans Rosling and his site gapminder.org are revolutionary and should be changing the way all of us look at local data!  I am surprised that his concept of the motion graph has not trickled down into more local applications.  Especially where these motion graphs can be made with Google spread sheets for free!!!!! (visualization may not work on phones or tablets) I have been part of negotiations since 2000.  Communicating to other teachers, community members, and school board members what is going on statistically in the negotiations process has always been our biggest problem. When I first saw Hans Rosling's 2006 TED talk, I knew that we needed this visualization in local education.```

```First off, play around with the graphic above. Notice how dynamically all the variables can be changed.  Check the box marked "Average".  This is the average for each year.  Compare it to other districts.  Are district above or below the average?  Change the variables.  This is data taken directly from district arrays.  What about teachers who are MA+18.  How do the districts compare when we change this variable?  Change the x-axis to Approximate base cost per contract day (12 X monthly insurance + base)/(contract days). Change the y-axis to ultimate max.  Now you are comparing what minimum costs districts pay per contract day and the most salary a district will pay to a new teacher.  Notice that the two large districts seem to spend less per contract day, but teachers can ultimately make higher salaries.  Does this suggest that larger districts are more efficient?  The data from Scottsbluff public schools and Gering public schools, skews the average.  But I threw those two schools in just to prove that we could create one of these visualizations that housed all Nebraska School districts.  This data is negotiations and compensation data from https://www.nsea.org/compensation.  I have spent well over 100 hours typing all of this data into a google spread sheet.  There are mistakes. Mainly because this is for instructional purposes only!!! PLEASE do not use this to make decisions if you are one of the schools in this visualization!! Although it is public data that can be accessed by anyone, I know that it is full of mistakes.  At the same time, I know that something like this would be useful in communicating data to local communities all across Nebraska. So I cannot keep it secret any longer!  It must be shared. Imagine what these visuals could do for local communities!!!! ```

IMAGINE THE POSSIBILITIES!!!  We could literally create a visualization that houses all data in the Nebraska Department of Education.   We could visualize every budgetary line item in districts across the state!!  For example, we could look at trends in transportation spending statewide and find district who are efficient and ask them what they do. How are thinning rural populations  affecting their schools transportation budgets?   We could look at standardized test scores and compare those scores to population demographics data from each district in the state.  The graphic above is simple and very narrow in focus (negotiations for a handful of small Nebraska Schools)…… Imagine the possibilities….  We could unlock the darkness of data.  We could make everything more transparent.  Although there is always the possibility that we will run into the problem that correlation does not necessarily imply causation.  But at least we would be able to see what could be going on and then look deeper into the possible correlations.  Imagine the questions we could answer.  Imagine the questions we could create!!!!

# Google Sheets, Motion Graphs, and Student Fitness Levels

Above you will see circles that represent students.  The small circles are girls and big circles are boys (sorry girls, not trying to suggest you are smaller, change it to same size if it offends).  The colors represent their grade in school.  Red is seniors and blue is 7th graders.  I have a variable called VDOT in the x-axis and the students potential mile time on the y-axis.   All variables can be changed dynamically and feel free to play around.  When showing this to my students I will only change the y axis to fitness levels.  I am not 100% sure what VDOT means and I am sure it will be a blog of its own when I have time, but for now it can be thought of as the amount of oxygen consumed in a minute for each runner.

I coach track and have always felt my running program lacked something substantial.  Then out of the blue I heard about this coach called Jack Daniels.  After reading one of his books, “Daniels’ Running Formula”, I could see why.  This book may be the first track manual I have ever read cover to cover.  Simply amazing technical information.  Not sure I understood all of it, but what I gleaned has made a huge impact on my program and the fitness of our team.   Being the graphics geek that I am, and knowing that Hans Rosling’s motion graphs are easily produced in google sheets (Check out gapminder.org), I decided to collect my own data from my athletes and track their Daniels fitness levels during the 2015 track season.  I was able to collect data for students in grades 7 – 12 from March to May.

I started out at the beginning of the season running everyone in a 12 minute run.  I told them to run as far and as fast as they could in 12 minutes.  I plugged this data into the site Training Zones/Running for Fitness.   This site produced a VDOT  which in turn I used to create a fitness level, also using Dr. Daniels’ formula.   As the season progressed I was able to use students competition times to evaluate VDOT and fitness levels.  You can see that a 12 minute run is not very motivating and the moment competition started their fitness levels blew up.  Anyone that ran less  than 800 meters had to continue the 12 minute runs in order for me to gauge their fitness levels.  Everyone on my team had to complete this.  It included my throwers.  I lost some throwers and sprinters because of this, but my plan was and always will be to improve fitness.  Gold medals produce temporary joy, but personal fitness habits create a lifetime of joy.

The real reason I share this is that I hope it generates ideas on how to represent local data with the ideas of Hans Rosling.  Every small community has data that could be tracked and monitored using this tool.  It is fairly simple.  I have some grand ideas to take this to the next level in state level data tracking, but for now this is a beginning.  Lets see where it leads.  Until next time…. God Bless.

# The story of bacon: Teaching and Modeling Growth from Linear to Exponential

I love teaching logarithms!!  I love modeling exponential decay and growth with students!  It is like the Christmas holidays around here everyone growing exponentially.  In fact I see this as essential learning.  Of all the skills and concepts I teach, logarithms and all of the math behind them are the most important for all of our children to have a deep fundamental understanding.

The average retail price of Bacon per pound in 1895 was around \$0.13.  The price of a pound of bacon in 2014 was around \$5.78.  What is the annual percentage rate of increase?

I sneak this problem at the end of a traditionally taught lesson reviewing the rules of exponents with the introduction to rational exponents.  Here is a link to that worksheet.  Usually students are grappling with their understanding of those rules from Algebra I and Algebra II that they probably forgot, when all of a sudden,Wham! problem 16 hits them.  If your students are like mine a few attempt the problem and have something written down.  Most leave it blank because there are scary words that bring on frozen fear. While another group does in fact very confidently come up with an answer.

Here is where the teaching magic happens.  I have these students that are confident come up to the board and explain their thinking about the problem. This is almost always what I get from the few who try.

5.78-0.13 = \$5.65

2014 – 1895 = 119 years

5.65/119 = .0475 price per year

At this point many do not know what APR means so they might quickly mumble 4.75% increase or something like that and suddenly sit down.  This problem is vague for a reason.  I am only giving them part of the story.   The students that I teach have spent their entire math career working on proportional reasoning.  Some will get this answer and not even realize that they found a slope, where rise is price and run is years.  Some may not even realize that they are looking at the problem linearly.   Students need to see their solution modeled for all of the years.   As a class we go ahead and model their solution with Geogebra to show what their story of bacon would look like.  Here is where we can ask the students. “Is this what the model of bacon prices over time should look like?”  You’ll get a mix of answers and probably get some pretty good discussion going. But here is the actual data according to the Bureau of Labor Statistics. Combining this data with data from www.infoplease.com we can get a true story of the price of a pound of bacon from your local grocery store.

It is time to show them the true story of bacon.   You cannot tell the story of anything with only two snapshots in time.  Bacon’s story is filled with wars, depressions, booms, busts, supply, demand…. but most importantly inflation.

I did some of the research for you and put the raw data in Geogebra.  Click here for the geogebra file.  Now what?  How can students manipulate this data?  How can we model this data?  What is the annual percent of increase for the price of bacon?  Here is where I will leave the problem for now.  I present this problem in a couple of days and hate to do more because this is where I like to leave a problem. What will students come up with?  Will they care?  Does it motivate a question from my audience?  Will they dive into it and make it their own?   These are the questions I am concerned with.  Questions that all teachers should be concerned with.  The products we as educators produce are only relevant if the students find them interesting and relevant in their own lives.   It is time to test the problem and find out.

Where would I like to have students end up with this problem?  I would like students finding out that their original linear proposal was a fine model as an initial guess and matches the simple interest formula.  But after that initial view and looking at all the data, it looks like there could be a better model.  Maybe it is y=ab^x  for compounding annually.  Maybe it is y = P(1+r/c)^(ct) for other compounding like quarterly, monthly, daily, hourly….. Maybe it is y = Pe^(rt) for growth compounded continually.

Ultimately I want students to see this model in our stock market, land prices, loans, heating, cooling, and other geometric series. The connection of this data to other naturally occurring phenomena is the beauty of mathematics.  After all isn’t mathematics just a language used to describe our world?  Shouldn’t all student speak that language?

Source data for average bacon prices 1895 – 2014

http://data.bls.gov/pdq/SurveyOutputServlet

# Geodesic Dome Summer Boyscout Project

It is summer and among the various tasks I am assigned by my wife, family, and the community, I am still finding ways to include math in these tasks some how.  I have been fascinated with geodesic domes lately and found this great geodesic dome calculator http://www.desertdomes.com/domecalc.html.  It is the first site that pops up when you Google search “geodesic dome calculator.” I love math, but am really more interested in what we as humans can do with it in a practical sense.     My fascination with these structures began when I stumbled across the site with my new phone.  After seeing several of these online I decided to build my own.  I am new to the world of geodesic domes so I started with  a 1V structure, which is a structure with one strut length.  A 2v is totally constructed with two strut lengths, 3v has 3 strut lengths, etc.

My first model began at Camp Augustine boyscout camp in Grand Island, Nebraska.  Each year I go to boyscout camp with my son and while the boys are at various classes I am left alone at camp with several thousand feet of basil twine and about 5 days with nothing to do.  This year I decided to pick up sticks around camp and make one of these structures with approximately 2 foot struts and lashings.  For a 1v structure it takes 25 sticks.  I placed five of them in a regular pentagon shape and lashed them together.  Then off of each of these struts I lashed four more struts.  So this make six vertices with five struts lashed together.  At this point I had to start raising the structure.  Then you take two of the struts and lash them to two more making five vertices with four struts lashed together.  Keeping the lashings tight is critical, but I waited until I had a rough looking structure up and then tightened the lashings.  All of this took me about four hours working alone with a pair of nippers and a knife.

Once I had this model built we burnt it in the Friday night fire, and it was time to make something bigger and more useful.  My son and I are avid hunters and I keep chickens, so I thought these would make great structures for chicken houses and hunting ground blinds.  I also have been cutting cedar trees to clean up a pasture and built fence.  I hate to waste anything, so with these cedars I make 6 foot posts out of the trunk but then I have a bunch of limbs left over.  So I cut these limbs into approximately five foot pieces.  Once I had 25 of them I drilled 1/4 inch holes in each end of the sticks and wired them together just like my boyscout dome.   So far this structure is not very rigid because of the wire and I may reinforce each vertex with something else, but for now the structure is built and ready to be covered.  For the chicken houses I will cover the top with plywood and the sides with wire mesh laying around the farm.  So it definitely will have that Mad Max look to it, but should be very useful.  I love this project because you don’t have to be a math genius to build something mathematically beautiful and useful.

# Cutting a Large Pipe at a 45 Degree Angle

I love it when a common textbook problem comes alive.   An employee of a Nebraska county road crew had to cut a culvert in two pieces so he could weld it back together and make a 90 degree angle with the pipe.  They have a device called a master marker made by Flange Wizard that can be used to mark the angle needed, so that a torch can be used to cut the pipe.   The problem with this is that the pipe they have(37 inch diameter)

was too big to use with the Flange Wizard they had on hand and to purchase a bigger flange wizard was out of the question.  The question becomes then how do we create a mark on that pipe at 45 degrees with materials available.  Before you read my solution, put some thought into this.  How would you answer the question?  My solution is not the only way.  There are many methods for getting this done.  How could you put this into your classroom to teach others? Organize it in acts like Dan Meyer or just throw the problem on the board and see what happens?  I usually prefer the second scenario, but you have to be ok with some chaos.

The solution I come up with was to cut an ellipse out of a plane (cardboard) and slide the plane onto the pipe.  I first found the major and minor axis of the ellipse.  The minor axis is the radius of the pipe, in this case 18.5 inches.  The major axis can be found by the hypotenuse of a 45-45-90 triangle with legs 37 inches.  By pythagorean theorem the hypotenuse of this triangle is 37√2.  The two red dotted lines represent a side view of the pipe.  The blue line represents the cut that needs made and the long axis of the ellipse. The major axis of the ellipse is that number cut in half.  This yields the equation of the ellipse as:

Once you have that equation it is possible to find the distance the foci of the ellipse are from the center of the ellipse along the major axis by taking the square root of the absolute value of the difference of the denominators  in the equation.  This results is a distance from the center of the ellipse to the foci of 18.5 inches.  Here the fun begins.  Once you measure from the center to the two foci it is possible to tack down a stretched piece of string between those two points and trace out the elliptical shape on the cardboard.  Cut this shape out and then slide it over the pipe.  The crew put wood slats on the cardboard in order to keep it from bending.

I missed the boat with my students on this one.  I am in the heat of battle teaching the Nebraska State Standards.  I think it would have been fun to put them in groups with the problem and see what they came up with, had them present their proposals to the actual people doing the work, and let the people doing the work decide on the best proposal.  I just didn’t feel like I could sacrifice the time. But I will put this idea on the shelf and maybe next year with better organization fit it into the curriculum some where.  Later this spring this group also needs to cut similar pipe at different angles, so maybe we will have more time.  So there is already an extension of the problem built in.  I think they need 22 degree angles cut next time so they can send water down a 40% grade.

# Scottsbluff GeoGebra Presentation Using Padlets

The Following links will take you to my padlets for each session.  You may post right onto the padlets I think.  This is the first time I have ever used them.  They were used at the North American GeoGebra Conference and I still go there and find cool stuff.  I will be sharing some of the cooler presentations from that experience.

3:30-4:00 Collaboration, Sharing, Questions and Answers

Getting Started

# Pre Post for GeoGebra North American 2013

I am going to concentrate on three problems for my two presentations.  Two of the problems are based on orienteering and finding our location using Geogebra with the files below.  The third will be constructing a model in Geogebra that fits the situation of finding the distance between two towers with compass headings and the distance between the to measurements.

These problem work best if you make a video yourself and have students guess where you are based on a location that is familiar to them.  Also your students have a connection to you.  Feel free to use mine, but your students will respond better to their own teacher driving around in the country filming themselves:)

• first file you will need is the sectional maps of southwest Nebraska for my first orienteering problem done on the bus route.

The second file needed is the Cheyenne Sectional for western Nebraska when I filmed by Chimney Rock.

For the third problem you need to watch the video and then use GeoGebra to create a model of the situation.   Again this works best if you take your students out into the country and do it with them.  Place them in teams and make it a competition.  Who’s model will come closest to a measurement from google earth?  Have them write down a guess while you are out in the country.  Who is the best guesser?  Estimation on all problems gives students ownership of the outcome.  Very motivating.

# An Unexpected Problem

So it all began innocently enough, my dad thought I should try my hand at farming.  I own land, but have always had others farm it for me and used it as a kind of rental income, like owning a house.  An investment like stocks or bonds.  All of which have their own mathematical challenges.  Between you and I it has never been that profitable.    I am basically looking at a blank canvas.   I have 60 acres of land that I can do whatever I want.  I could turn it into the next festival site like Woodstock, Comstock, or Burning Man.  None of these seem very practical and a math teacher just does not seem like the type that can pull it off.  So I am going to plant a crop and see what happens.  But what crop?  Corn, soybeans, milo, wheat, alfalfa, and oats all crossed my mind.  I even thought of turning it back to native pasture, but that is not very profitable or interesting.   So I decided on milo or sorghum.    Then the next set of questions hit you in the face.  How are you going to plant it? How much fertilizer?  How much insurance?  How much seed should I buy?  How deep should I plant it? When should I plant it?  Then these questions lead to other questions!!!  What is the population of seed per acre?  How many pounds of seed will produce that population?  Is it the same for all varieties of milo?   Does the diameter of the seed matter?  If it does what does it effect? Soil moisture is low, does this make a difference?

All of these questions have mathematical answers.  And they all take you down some pretty interesting mathematical roads from algebra to statistics, and probably calculus with all of the changing rates.  The answer ultimately comes down to what is in front of me.  What tools do I have at my disposal.  The only tool I have is a 15 foot drill with 7.5″ spacings between each row.    What is the population for my milo seed per acre? I have this handy chart in the planter manual that tells me that I can set the planter to apply 15 to 122 pounds of seed per acre.  Milo seed averages out to 14,000 seeds per pound, and from a few web sites I have browsed it is said to plant from 18,000 to 70,000 seeds per acre.  Several posts on an ag related site said to shoot for 30,000 seeds per acre.  If I just dump seed into my drill I can plant between 210,000 and 1,708,000 seeds per acre.  This is way to high, so I have to modify the drill in some way to bring this number down dramatically.  Which holes should I plug in the drill?  Every other one?  Plug two and keep one open?  Then once I decide on this how do I know that it is correct?  How far apart should the milo seeds be planted?

This I do know.  I need to answer the question of how many rows should I plant and how far apart should I find my seed in a row once  I plant.  I also need to plug my holes so that I can plant in a symmetrically.  Huge gaps between plants are bad and are not spotted until it is too late.

One thing is for sure.  Farming is for those that are mathematically strong.  Probably more to post on this with more specifics.  It is turning out to be an intellectual challenge.  I see why people enjoy farming and beginning to think I chose the wrong profession.