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					TAKS: Measurement & 3-D
• 8.7.D: Locate and name points on a coordinate plane
  using ordered pairs of rational numbers.
• 8.8.A: Find surface area of prisms and cylinders using
  [concrete] models and nets (2-D models).
• 8.8.B: Connect models to formulas for volume of
  prisms, cylinders, pyramids, and cones.
• 8.8.C: Estimate answers and use formulas to solve
  application problems involving surface area and
  volume.
• 8.10.B: Describe the resulting effect on volume when
  dimensions of a solid are changed proportionally.
Points on a Coordinate Plane
A coordinate grid
  is used to
  locate and
  name points on
  a plane.
Points on a Coordinate Plane
The x-axis
  and y-axis
  divide the
  coordinate
  plane into
  four
  regions,
  called
  quadrants.
Example
Example
Polyhedron
A solid formed by polygons that enclose a
  single region of space is called a
  polyhedron.




Separate your Geosolids into 2 groups: Polyhedra and others.
Parts of Polyhedrons
• Polygonal region = face
• Intersection of 2 faces = edge
• Intersection of 3+ edges = vertex




    face          edge           vertex
Example
Separate your Geosolid polyhedra into two
 groups where each of the groups have
 similar characteristics.
These are the only kind of polyhedra on the
 TAKS test. What are the names of these
 groups?
Prism
A polyhedron is a prism iff it has two
  congruent parallel bases and its lateral
  faces are parallelograms.
Classification of Prisms
Prisms are classified by their bases.
Right & Oblique Prisms
Prisms can be right or oblique. What
  differentiates the two?
Example
Pyramid
A polyhedron is a pyramid iff it has one base
  and its lateral faces are triangles with a
  common vertex.
Classification of Pyramids
Pyramids are also classified by their bases.
Example
The three-dimensional figure formed by
 spinning a two dimensional figure around
 an axis is called a solid of revolution.
Cylinder
A cylinder is a 3-D
  figure with two
  congruent and
  parallel circular
  bases.
• Radius = radius of
  base
• Axis = segment
  connecting centers
  of bases
Cone
A cone is a 3-D figure
  with one circular base
  and a vertex not on the
  same plane as the base.
• Altitude =
  perpendicular segment
  connecting vertex to the
  plane containing the
  base (length = height)
Sphere
A sphere is the set
  of all points in
  space at a fixed
  distance from a
  given point.
• Radius = fixed
  distance
• Center = given
  point
Sections
When a solid is cut by a plane, the resulting
 plane figure is called a section. A section
 that is parallel to the base is a cross-
 section.
Example
Example
Nets
Imagine cutting a 3-D
  solid along its
  edges and laying
  flat all of its
  surfaces. This 2-D
  figure is a net for
  that 3-D solid.
Example
Match one of the red, rubbery nets with its
 corresponding 3-D solid. Which of the
 shapes has no net?
Example
There are generally two types of
 measurements associated with 3-D solids:
 surface area and volume. Which of these
 can be easily found using a shape’s net?
Surface Area
The surface area of a 3-D
  figure is the sum of the
  areas of all the faces or
  surfaces that enclose the
  solid.
• Asking how much
  surface area a figure has
  is like asking how much
  wrapping paper it takes
  to cover it.
Lateral Surface Area
The lateral surface area of
   a 3-D figure is the sum
   of the areas of all the
   lateral faces of the solid.
• Think of the lateral
   surface area as the size
   of a label that you could
   put on the figure.
Example
The net can be
 folded to form
 a cylinder.
What is the
 approximate
 total surface
 area of the
 cylinder?
Example
Using the Formula Chart
You could just as easily compute the surface
 area using a formula.
Height vs. Slant Height
On the formula chart, h represents height
 and l represents slant height.
Height vs. Slant Height
On the formula chart, h represents height
 and l represents slant height.
Example
Find the total surface
  area of the square
  pyramid.
Volume
Volume is the measure
 of the amount of
 space contained in a
 solid, measured in
 cubic units.
  – This is simply the
    number of unit cubes
    that can be arranged
    to completely fill the
    space within a figure.
Example
Find the volume of
  the given figure in
  cubic units.
More Formulas!
The volume of a solid is also easily
 computed with a formula.




What does the B represent?
Example
Example
Example
Which solid has the greater
 volume: a cylinder 3 inches
 high with a radius of 2
 inches or a cone of the
 same radius that is 8.5
 inches high?
Example
          A pipe in the shape of a
            cylinder with a 30-inch
            diameter is to go through
            a passageway shaped
            like a rectangular prism.
            The passageway is 3 ft
            high, 4 ft wide, and 6 ft
            long. The space around
            the pipe is to be filled with
            insulating material.
Example 10
             What is the volume of the
              insulating material?
Example
Find the volume of a
  cube with a side
  length of 2 inches.
Now find the volume
  of a cube with a
  side length of 4
  inches.
How do the volumes
  compare?
Example
Find the volume of a
  cube with a side
  length of 2 inches.
Now find the volume
  of a cube with a
  side length of 4
  inches.
How do the volumes
  compare?
Volumes of Similar Figures
If two solids have a
   scale factor of a:b,
   then the
   corresponding
   volumes have a
   ratio of a3:b3.
Similarity Relationships
 For two shapes with a scale factor of a:b, each of the
 following relationships will be true.
Example
Example
Example
A breakfast-cereal manufacturer is using a
  scale factor of 2.5 to increase the size of
  one of its cereal boxes. If the volume of the
  original cereal box was 240 in.3, what is the
  volume of the enlarged box?
Assignment
10th TAKS Practice
  Workbook:
• P. 85, 97, 99, 101,
  107

				
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