Applied Science - Science and Math (6A)
Lab 

   
OBJECTIVES:
  • Investigating the relationship of mathematics and science.
  • Exploring the topology of a closed surface.
VOCABULARY:
  • geometry
  • mathematics
  • ring
  • surface
  • topology
MATERIALS:

Students use tangles to explore topology.

BACKGROUND:

Children love to play with different shaped blocks and build structures.  The inherent beauty of these three-dimensional objects is somehow lost when mathematics turns to paper and pencil arithmetic.  Keep in mind that because a child is not good in arithmetic computation, doesn't mean the child cannot conceptualize higher mathematical skills.

The least needed skill in science is computational skill because addition, subtraction, multiplication, and division are easily obtainable by a calculator.  This doesn't mean these parts of mathematics are not important, they just are not the door that opens children's minds to higher mathematics.  This activity exposes students to a different field of mathematics that is more abstract and has more direct relationship to science.

Mathematics also relates to nature.  Emphasize that the importance of mathematical relationships helps determine patterns in nature.  We give students the skills to measure straight objects, but most objects are curved.  Even a straight line must curve sometimes. (If we were a flat Earth a line could be straight, but a continuous line will eventually curve!)  Mathematics is almost mystical because the results are sometimes not easily explained.

PROCEDURE:

  1. This activity has more questions than answers and is like a mathematical puzzle.  If geometrically based puzzles are available, include them with this lab.
      
  2. This lab also tries to show mathematics as a subject that can be fun as well as complicated.  Topology is a branch of mathematics that relates the points of an object that can be distorted.  A lump of clay is a collection of points that can be squeezed without changing topologically.  Shape and size are unimportant, but the connectivity of a figure is.  This is very different from  the strict measurements we usually require of students.  Remember that the Earth, Solar System, and Universe are all somehow connected but the points keep changing in space.  Topology simply is the study of closed curves.  Just think of it, a sphere and a potato are topologically equivalent!
      
  3. In the module, there are sets of "tangles."  When closed, a tangle is a good example of how a closed surface can make many shapes.  These shapes are topologically related.  Follow the worksheet and have students work out the problems.  Remember the key objective is for students to relate to curves and discover how difficult it is to describe and quantitate curves. 
      
  4. ANSWERS:
  1. Students should make 3 shapes that are topologically related.  Notice that although the circumference of the closed curve remains the same, the area inside changes constantly.  Have students use the 16 segments of the small tangle.  Trace the curves on the worksheet.  Use the larger tangle to show unique shapes to the entire class.
      
  2. Students are asked to see if they can make a knot from the closed curve of 16 segments.  They cannot.  They are then asked to make a knot by unsnapping just one link.  If students have difficulty with this, have them think of the tangle as a shoe lace and then ask them to make a knot.  Have the students compare a knot with a closed curve.  They will notice that the knot interweaves where the closed curve does not.
      
  3. Keeping the tangles closed, have students make a coil.  This will require them to "wrap" the curves.
      
  4. Students are asked if the curved surface can be made flat if they unsnap one of the links on the tangle.  Yes, they can, if they make the tangle in a undulating wave pattern.  Students are then asked if they can measure the curves with a ruler.  The answer is no, curves are difficult to measure and have to be determined by a mathematical relationship that includes using the constant pi (determining the circumference of a circle).
      
  5. Students are asked to make as many circles from the tangles of 16 curves.  They will find that they can make 4 (4 curves make one circle).  Give them instructions of how to find the circumference of the outside  circle and the inside  circle.  Measure the diameter and multiply by pi or 3.14.  The circles are 2.5 cm on the outside and 1.5 on the inside.  The circumference is 2.5 times 3.14 (7.85) and 1.5 times 3.14 (4.71).  An easy way to refer to the circumference is 2.5 pi and 1.5 pi.  Now, can the students determine the length of the tangle.  Yes.  Some of the students may figure there are 4 complete circles.  That means the outside total circumference is 4 times 2.5 pi and inside circumference is 4 times 1.5 pi.  The answer for the total length of the outside is 10 pi or 31.4 cm and the inside is 6 pi or 18.84 cm. 

  6. You may want to use the larger tangle with 18 segments and repeat the exercises above. 
      
  7. Remember, this activity is to help students realize that mathematics is a tool to model the real world.  The higher the math they take (and the better the teacher) the more exciting math becomes.  It is part of our life, whether we like it or not!

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