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

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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)
mental activity!
mental activity!
Top view of a
PYRAMID!
• 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
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

• 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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 views: 7 posted: 5/20/2012 language: English pages: 51