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Geometry concepts related to Earth and

Space

Prepared and presented by:

Souheil ZekriWandaliz 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!

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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

Altitudes

┴ bisectors

Angle bisectors

Triangle geometry

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

Median

Apex

Midpoint

Triangle geometry• A perpendicular

bisector are segments emerging perpendicular to the midline of one of the bases and ends on the opposing triangle side

• Draw a similar triangle and its altitudes using a right triangle and a ruler

┴ bisectors

Midpoint

Triangle geometry

Angle bisectors

• An angle bisector is a segment that divides an angle in two equal angles and ends on the opposing triangle side

• Draw a similar triangle and its altitudes using a protractor

Triangle geometry

Altitudes

• An altitude is a segment that emerges from one of the 3 apexes and 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

These are equalThese are equal

Sum of these is 180o

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 isEvery reference frame has an origin.There are 2 different type:• Cartesian frame• Polar frame (cylindrical, spherical)

Cartesian reference frame

In 2-DIn 3-D

Example

X

Z

Y

X1,Y1,Z1 X2,Y2,Z2 X3,Y3,Z3

Time lapsed coordinates from earth to a newly discovered planet called 2003UB313

Polar reference frame

In 2-D cylindrical In 3-D spherical

Example• You can see in this

case how it is easier to use polar coordinates rather than Cartesian because the length is the same and all we have to do is vary the angle instead of measuring the x and y for each point on the mantle surface.

X

Y

angle

length

Example

Φ

Y

X

Z

Latitude angle

Longitude angle

Let’s locate objects in space

• Using the provided reference frame and strings, find the Cartesian coordinates of different objects in the room

Cartesian Coordinates

x y z

Object1

Object2

Data Sheet

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

X

Y

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.

http://www.usf.edu/ur/logos.html

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

Surface Area {SA}

(m2)

Volume {V} (m3)

Cube

Sphere

)/1( mVSARatio

Data Sheet

Make it work… didn’t work in class for some reason

So why is the universe oval

shaped?Discussion!