In Your Classroom: "Laser Archaeology"

by Brian McCombs     Department: In Your Classroom

In my classroom, there are no lasers to determine three-dimensional objects.  Unfortunately the monies allotted to school districts just do not cover such costs.  However, in our geometry classes we tackle the concept of three, versus two-dimensional objects.   When we begin the units covering area, surface area, and volume of geometric figures - you know the ones, formula after formula for kids to memorize - we try investigating their properties and values in a whole new way.  Instead of teaching kids all the surface area and volume formulas for all the 3-d shapes, we build our own city.  Egotistically, we call it "McCombsville."  The students, in groups of 2-3, plan out their portion of a city they will construct.  Each group needs to have a school, a medical building, 2-3 stores/commerce generating buildings, 4-5 homes/housing buildings, and 1-2 others that fit in the "other" category.  Each building, depending upon the category it fits into, also has size restrictions.  That way, the homes aren't 3 times the size of the school building.

Each group begins by building nets of the buildings.  A net is a 2-dimensional representation of the building.   The students then decorate the flat building, fold it up to form the 3-dimensional building, and tape/glue it together.  Upon completing all of their buildings, the class as a whole, lay out their city.  Many times we will have an upper East side, with its upscale homes, stores, and restaurants.  We will also end up with a lower West side, with its low-income apartment complexes and tattoo parlors.  The kids paint in rivers, playgrounds, etc.  Each item they place on the city board, must be 3-dimensional, and constructed from a net of the object itself.  After the city is created, each group needs paint, and air condition for their buildings.  They determine the surface area and volume of each of their buildings, research the cost of paint, roofing materials, and air conditioners online.  They scale their buildings to actual size, and determine the amount of paint, roofing materials and what size air conditioning units are needed for each.  In a small way, they become the city managers/consumers of "McCombsville."  It is a wonderful way of getting the kids to discover the formulas for surface area and volume of 3-dimensional objects, without ever telling them what they are.  The kids love creating and learning, and every year, I hear that this was their favorite project in my class.  I have had students create 3-dimensional baseball stadiums (even the cylindrical foul poles and dugouts), car washes (with 3-d rollers), skate parks, etc.  It is amazing watching them work on this.

Another fun idea regarding the measurement of large-scale buildings or land masses, is to build a "stadia."  A Kent State University professor of mine taught me this idea during my undergraduate years.  A stadia uses the concept of similar triangles to measure the height/width of an object.  It is constructed out of a toilet paper tube, some rubber bands, tape and a piece of paper.  The student breaks the rubber band into two pieces, and tapes them over one end of the tube.  He needs to make sure they are one inch apart and parallel.  On the other end, the student will tape a piece of paper over the entire end of the tube, poking a very small hole in the middle of the paper.  To measure something, for example a tall building, the student needs to look through the hole and align the top and bottom of the building with the two pieces of rubber band.  After doing this, they need to determine the distance from the building they are at that moment.  Similar triangles are used to measure the height of the building.  The tube creates one triangle.  The length of a tube is typically 4.5 inches and the distance between the rubber bands is set at 1 inch.  That gives the student the height and base of their triangle.  The larger triangles height is the distance away the student is standing from the building, and the base of the triangle is the height of the building we are trying to find.  It is a very creative and fun, one-day activity for kids to try. 
I found several other applications of laser technology and measurement in the school classroom.  Not all were related directly to mathematics, but they were all pretty cool.  The first is entitled Laser 3D Lessons.  A museum educator in Long Island, NY, uses the technology of holograms to educate elementary students each month at the museum where she works.  It is believed to be the only program like it in the U.S.

Another website is Hands on Math.  It is created by a woman by the name of Janine Parker, with the help of teachers and staff of West Kearns Elementary School.  It has a link within it to lessons on the mobius strip and topology lessons for the elementary age student.  It is well researched, thorough in its presentation, and an excellent resource for teachers interested in this area of mathematics education.

I also found a website called A Brief History of Topology.  It was created by Artur Gorka, from Clemson University.  It has several links along with it for those of you interested in the other mathematical ideas associated with this abstract form of mathematics.

Finally, I would be remiss if I didn't include the website created by the Colorado Tourism Agency.  It is a wonderful description of the layout and history of Mesa Verde National Park.  Maybe your students could build a 3-dimensional replica of the dwellings, instead of McCombsville.  They could then determine the volume of mud and sticks required to build the lodging structures.

As I stated earlier, albeit that I don't have the technology available to me that Ben Kaycra had in this episode, there are many projects in a mathematics classroom that could bring in the measurement of 3-dimesional objects.  My kids love them.  I think yours would too.  If you have any comments, criticisms, or suggestions, I'd love to hear from you.

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