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 intentionally vague problem I start with is this one:

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

Linear Model of Bacon Prices
Student model of bacon prices when we assume linear growth. Horizontal scale 20 years, vertical scale $1

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.  That means what they have so far is great!  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 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

Geodesic Dome Summer Boyscout Project

20140626_114234-EFFECTSIt 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  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.     14 - 1My 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.20140624_211212  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.  20140626_114150Keeping 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 cedars20140710_170653-EFFECTS 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.14 - 2

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)

This pipe needs cut at a 45 degree angle
This pipe needs cut at a 45 degree angle

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. 45 degree pipe cut 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:

Screen Shot 2013-11-27 at 1.27.08 PM

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.  20131115_144525Here 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. 20131125_150616 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.

2013112695112738 2013112695145429

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.

11:30-12:30 Getting Started
12:30-1:30 Algebra 1 & 2 Applications and Personalizing to your Classroom
1:30-2:30 Geometry Applications and Personalizing to your Classroom

2:30-3:30 Trigonometry and Calculus Applications and Personalizing to your Class

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.
  • Better Size picture for Geogebra
    Better Size picture for Geogebra

Orienteering sectional image










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.

The red and blue line segments are exactly one mile.  We took headings at each endpoint and created two problems.  With these measurements it is possible to find the distance between the two thumb tacks.
The red and blue line segments are exactly one mile. We took headings at each endpoint and created two problems. With these measurements it is possible to find the distance between the two thumb tacks.








Arapahoe Public School seniors calculating the distance between two towers.
Arapahoe Public School seniors calculating the distance between two towers.


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.

Find My location in Western Nebraska – Orienteering

In this video I traveled out to western Nebraska to lose myself.  Use the sectional map, Geogebra, and a little trigonometry to find my location in the video below.  Click on the map and you can download and print a larger image.

The three landmarks I have chosen for this orienteering video are Scottsbluff, Chimney Rock, and Jailhouse Rock monuments.

And to the right is the map image that you could copy and inset into a Geogebra file.


Teaching More Students Online

I am currently uploading all of my video library to Youtube.  Last night I noticed that I teach more students online than I teach in my classroom.  I average more than 100 views per day (with a 60% retention rate)  so 60 people actually watch the entire video out of those 100 that initially click, and that number is growing steadily.  I have been watching some of my early videos and they are pretty bad, but I need to make some more.  It is pretty cool to get comments from students all around the world.  Apparently I am very good at explaining the conversion of Degrees minutes and seconds to decimal degrees.  I get about 10,000 hits every 30 days on this video alone.  So if you are a teacher or want to show some skill you have to the world I encourage you to place your videos on the web.  It does not have to be youtube, but it could be anywhere.  There are becoming a ton of options out there for spreading your talents around the world.

GeoGebra Needs an App for the iPad

Math educators that use java based programs on the internet in their daily teaching are unfortunately being held hostage by Apple.  I teach in a school district that is in the middle of its 8th year of a Apple one-to-one initiative.  Because of this initiative we no longer had funds to purchase graphing calculators, so we have been using Geogebra in the classroom as a tool to show about everything mathematical.  Lets face it, although Graph it on Apple is OK, this program can’t even come close to Geogebra’s capabilities.

This year when our school talked about iPads instead of computers . . . .  my blood froze.  I new that the iPad does not support Java, and all of the online apps I have built on Geogebra and placed on my moodle course would not work with the iPad.  Granted there are many free apps out there that graph on the iPad, but none of them can hold a candle to Geogebra.

Geogebra is asking for your assistance to develop the iPad app.  Here is their kickstarter link. HELP GEOGEBRA!!!!

and video.