1 1 1 Earth and Space Sunshine State Standards Geometric Concepts Souheil and Wandaliz

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1 1 1 Earth and Space Sunshine State Standards Geometric Concepts Souheil and Wandaliz Powered By Docstoc
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Geometry concepts related
   to Earth and Space


           Prepared and presented
           by:
               Souheil Zekri
              Wandaliz Torres
              Objectives
• Introduce geometry concepts that will
  connect visual observations of earth and
  space and the scientific concepts behind
  the observations.
• Provide simple computational
  examples
  – hands-on component of the session.
   Sunshine standards covered
• The student measures quantities in the real
  world and uses the measures to solve
  problems. (MA.B.1.2)
• The student estimates measurements in
  real-world problem situations. (MA.B.3.2)
• The student describes, draws, identifies,
  and analyzes two- and three-dimensional
  shapes. (MA.C.1.2)
    Sunshine standards covered
• The student visualizes and illustrates ways in
  which shapes can be combined, subdivided, and
  changed. (MA.C.2.2)
• The student uses coordinate geometry to locate
  objects in both two and three dimensions and to
  describe objects algebraically. (MA.C.3.2)
• The student uses expressions, equations,
  inequalities, graphs, and formulas to represent
  and interpret situations. (MA.D.2.2)
             Session Layout
• Triangle geometry (angles, bisection,
  ratios).
• Reference frames (Cartesian, cylindrical,
  spherical):
  – Hands-on solar system geometric
    measurements.
• Introductory vector concepts.
• Shape optimization and surface area to
  volume ratios.
• “Mathematics is the cheapest science.
  Unlike physics or chemistry, it does not
  require any expensive equipment. All one
  needs for mathematics is a pencil and paper.”
• "Geometry is the science of correct reasoning
  on incorrect figures."

          »    George Polya (1887-1985)
Let’s start with a
mental activity!
Let’s start with a
mental activity!
Top view of a
 PYRAMID!
Let’s start with a mental activity!
• Picture of three dimensional objects will be
  shown, drawn on pieces of papers. Each picture
  will be shown for 10 seconds and
  students/teachers will have to draw the exact
  picture out of memory afterwards.
• A discussion about the way students/teachers
  pictured the object in their memories will follow.
• Students/Teachers will be shown the actual
  three dimensional object made out of gum drops
  and sticks.
  Fundamental concepts in
        geometry
• Point: no size… just location!
• Line: no edge…just direction!
• Plane: no volume…just area!
Triangle Geometry


 Median         ┴ bisectors




 Angle              Altitudes
 bisectors
Triangle geometry
  Apex

                 • The median is a
                   segment that starts at
                   one of the 3 apexes
           Median of the triangle and
                   ends at the midpoint
                   of the opposing base.



Midpoint
  Triangle geometry
                        • A perpendicular
                          bisector are segments
                          emerging
                          perpendicular to the
           ┴   bisectors midline of one of the
                          bases and ends on
                          the opposing triangle
                          side
                        • Draw a similar
Midpoint
                          triangle and its
                          altitudes using a right
                          triangle and a ruler
Triangle geometry
              • An angle bisector is a
                segment that divides
                an angle in two equal
                angles and ends on
                the opposing triangle
                side
    Angle
    bisectors • Draw a similar
                triangle and its
                altitudes using a
                protractor
Triangle geometry
            • An altitude is a
               segment that
               emerges from one of
               the 3 apexes and
     Altitudes ends perpendicular to
               the opposing triangle
               side
            • Draw a similar
               triangle and its
               altitudes using a right
               triangle
       Classifying triangles
• By angle
        Classifying triangles
• By sides
            Classifying triangles
• By size
 • By size
      Some triangle properties
• Sum of the interior angles in any triangle is 180o
• Equilateral triangles have 3 equal sides and 3
  equal angles
• Isosceles triangles have 2 equal sides and 2 equal
  angles
• The sum of any two sides is greater than the third
  side
• Area of a triangle is ½ base times height
     Some triangle properties
• Sum of the interior angles in any triangle
  is 180o

                                     Sum of these is
                                     180o



      These are equal      These are equal
     Some triangle properties
• Equilateral triangles have 3 equal sides
  and 3 equal angles
     Some triangle properties
• Isosceles triangles have 2 equal sides and
  2 equal angles
     Some triangle properties
• The sum of any two sides is greater than
  the third side
     Some triangle properties
• Area of a triangle is ½ base times height




             Area = ½ bh
 Add an activity on
     Triangles
 Suggestions: Using
paper and folding it
            Reference frames
•   Cartesian, cylindrical and spherical
•   The right hand rule
•   Vectors
•   Application of all previously introduced
    concepts in earth and space
Reference frames: the math way
  to know where everything is
Every reference frame has an origin.
There are 2 different type:
• Cartesian frame
• Polar frame (cylindrical, spherical)
Cartesian reference frame




 In 2-D
                   In 3-D
Example
        X1,Y1,Z1              X2,Y2,Z2          X3,Y3,Z3



    Z



                   Time lapsed coordinates from
           X
                   earth to a newly discovered planet
                   called 2003UB313
Y
     Polar reference frame




In 2-D cylindrical   In 3-D spherical
Y                    Example
                         • You can see in this
                           case how it is easier to
    length                 use polar coordinates
                           rather than Cartesian
                           because the length is
             angle         the same and all we
                        X have to do is vary the
                           angle instead of
                           measuring the x and y
                           for each point on the
                           mantle surface.
                Example
        Z




                Latitude angle
               Φ
    Longitude angle

                                 Y
X
   Let’s locate objects in space
• Using the provided reference frame and
  strings, find the Cartesian coordinates of
  different objects in the room
          Data Sheet
              Cartesian Coordinates

          x            y              z


Object1


Object2
          Vector concepts
• What is a vector (geometrically and
  analytically)?
• What are they used for?
• How do we apply vector concepts to earth
  science?
       Vectors or scalars:
      what’s the difference?
• Some physical properties, such as temperature
  or area, are given completely by their magnitude
  and so only need a number are called scalar
  values.
• There are other physical quantities, such as
  force, velocity or acceleration, for which we
  must know direction as well as size or
  magnitude in order to work with them. It is often
  very helpful to represent such quantities by
  directed lines called vectors
       Vectors: General Rules




• Two vectors are equal if and only if they are equal in
  both magnitude and direction
• If c is a vector, then - c is defined as having the same
  magnitude but the reverse direction to c
• Multiplying a vector by a number or scalar just has the
  effect of changing its scale
    Adding vectors




• Using the parallelogram rule
Using reference frames to
    measure vectors
               Y




                                                X

  So if we can write the vector Q as a sum of the unit
  vectors s and t in the following matter: Q = 2.5s + 1t
  How about vectors P and R?
     How do we apply vector
    concepts to earth science?
• Combining the reference frame concepts
  and vector concepts we can easily see
  how much easier it is to locate objects
  (galaxies, stars, planets, satellites,
  comets, space ships, etc…) and calculate
  the speed and acceleration of any of these
  objects.
 Add an activity on
 vectors, relate the
concepts with similar
       vocab
        Shape optimization
    and surface to volume ratios
• What is surface to volume ratio?
• How is a shape optimal?
• Why is the Universe oval (close to being
  spherical) shaped?
       What is surface area
        to volume ratio?
• It is the ratio (or division) of the surface
  area by the volume.
• The larger this ratio is, the more surface
  there is for a specific volume.
• This allows more useful area (for
  physical or chemical reactions) for a
  fixed volume.
More examples
     How is a shape optimal?
• The higher the ratio of surface area to volume,
  the more optimized the shape is.
• Let’s use the following websites to compute the
  ratio for a sphere and a cube. (Volume is the
  same)
  – For the volume calculation use the following website:
    http://grapevine.abe.msstate.edu/~fto/tools/vol/
  – For the surface area calculation use the following
    website:
    http://www.csgnetwork.com/surfareacalc.html
                  Data Sheet
           Surface                                       SA
                   Volume {V}                    Ratio 
            Area                                         V
                      (m3)
          {SA}(m2)                               (1 / m)

Cube


Sphere


       Make it work… didn’t work in class for some reason
So why is the universe
    oval shaped?

       Discussion!

				
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posted:5/20/2012
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