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					  Implementing the 6th Grade
Mathematics GPS via Centimeter
            Cubes




        Presented by Judy O’Neal
           (joneal@ngcsu.edu)
          Topics Addressed

• Views of solid figures (polyhedra)
• Volumes of right rectangular prisms
  (polyhedra)
• Surface area of right rectangular prisms
  (polyhedra)
• Proportional relationships (scale factors)
• Connections among mathematical topics
       A View from the Top 1
• Use the numbers on the mat and your
  centimeter cubes to construct the building
  whose top (footprint) view is shown below.
                                 back
                              2
                       left   1           3
                                              right
                              1    1      2

                                  front
         A View from the Top 2
• Which of the architectural views below
  represent the front, back, left, and right of
  your building?
    A.       B.      C.                          back
                                             2
                                      left   1           3
                                                             right
                                             1     1     2
    D.       E.           F.
                                                 front




    G.        H.          I.
           Architectural Plans 3A
   1   3   1

       3

   1   2   1


   Front            Front    Left     Right    Back


QUESTIONS FOR STUDENTS:
• What is the relationship between the front and back
  views?
• What is the relationship between the left and right views?
           A View from the Top 3
• Use your cubes to construct the building
  represented by the following mats.
      A.           B.               C.
       1   3   1        4   2   1        3   2

           3                2   3        4   3   2

       1   2   1            4            1

                                         1
      FRONT             FRONT
                                         FRONT
• On centimeter grid paper (downloadable), draw
  the architectural plans for each building and label
  the front, back, left, and right view for each.
                 Architectural Plans 3B

 4     2     1

       2     3

       4


     Front           Front   Left     Right   Back


QUESTIONS FOR STUDENTS:
• What is the relationship between the front and back
  views?
• What is the relationship between the left and right views?
           Architectural Plans 3C
   3   2

   4   3   2

   1

   1


   Front          Front       Left         Right       Back



QUESTIONS FOR STUDENTS:
• What is the relationship between the front and back
  views?
• What is the relationship between the left and right views?
        A View from the Top 4
• Use the plans below to construct a building.
  Record the height of each section of the
  building on the mat.

     Top        Front    Right     MAT
       A View from the Top 5

• Use the plans below and centimeter cubes
  to construct a building. Record the height of
  each section of the building on the mat.

         Top        Front      Right       MAT




   Keep this model intact for use later in this webcast.
           Isometric Views from a
               Footprint/Mat
• Which isometric drawing shows the view from the left
  front corner of the building represented by the footprint
  below?




* Excerpt from student worksheet (downloadable)
  “Isometric Explorations”, pp. 113-114 (Navigating through
  Geometry in Grades 6-8, NCTM, 2002)
 Sketching an Isometric Drawing
• Isometric dot paper
  has dots placed so
  that isosceles
  triangles can be
  drawn easily.
• Sketching a cube
  is much like drawing
  a pattern block
  yellow hexagon with
  three blue rhombi on
  top.
   Isometric Drawings Practice
• Using isometric dot paper (downloadable),
  sketch each of these structures.
         Volume of Building 5
  Top     Front    Right      MAT




• How many cubes are there in each layer of the
  solid (saved from earlier in the webcast)?
• What is the total number of cubes in this building
  (volume)?
 Volume of a Rectangular Solid
• Use centimeter cubes to construct solids made
  up of the following stack of cubes.
    4   3   1

    2   1   5

    1   2   2



• How many cubes are there in each layer of the
  solid?
  ____ ____ ____ ____ ____
• What is the volume of this solid (total number of
  cubes)? ____ cm3
      What is a polyhedron?
• A polyhedron is a
  three-dimensional
  solid whose faces are
  polygons joined at
  their edges (no
  curved edges or
  surfaces).
  – The word polyhedron
    is derived from the
    Greek poly (many) and
    the Indo-European
    hedron (seat).
             Regular Polyhedron
 • A polyhedron is said to
   be regular if its faces
   are made up of regular
   polygons (sides of equal
   length placed              Octahedron – 8 Triangular Faces
   symmetrically around a
   common center).


Cube – 6 Square Faces


                          Dodecahedron-12 Pentagonal Faces
Irregular Polyhedra



      Faces are a
      combination
      of different
      polygons.
               Non-Polyhedra

• Cylinder



• Cone



• Sphere



• Why aren’t each of these solids a polyhedron?
         Polyhedra in our World
• Crystals are real-world examples of polyhedra.




• The salt you sprinkle on your food is a crystal in
  the shape of a cube.
            What is a Prism?
• A prism is a polyhedron (three-dimensional
  solid) with two congruent, parallel bases that are
  polygons, and all remaining (lateral) faces are
  parallelograms.
       What is a Right Prism?
• A right prism is a prism in which the top and
  bottom polygons lie on top of (parallel to) each
  other so that the vertical polygons connecting
  their sides are perpendicular to the top and
  bottom and are not only parallelograms, but
  rectangles.

• A prism that is not a right prism is known as an
  oblique prism.
     What is a Right Rectangular
               Prism?
• A right rectangular prism
  is a right prism in which
  the upper and lower bases
  are rectangles.

• A rectangular prism has
  six rectangular faces.

• How many edges?
          What is a Cube?
• A cube is a right rectangular prism with
  square upper and lower bases and square
  vertical faces.




• How many faces? edges?
            Cubes in our World
• The world's largest cube is
  the Atomium, a structure built
  for the 1958 Brussels
  World's Fair. The Atomium is
  334.6 feet high, and the nine
  spheres at the vertices and
  center have diameters of
  59.0 feet. The distance
  between the spheres along
  the edge of the cube is 95.1
  feet, and the diameter of the
  tubes connecting the
  spheres is 9.8 feet.
           Caroline the Cube
• On Caroline’s first birthday, she
  looks like one centimeter cube.

• Help Caroline finish building
  herself on her 2nd birthday.
  (Hint: Build a cube whose length,
  width, and height are 2 cm.)

• How many blocks define Caroline
  on her 2nd birthday? (What is her
  volume?)
        Caroline’s Surface Area
  The area of the exposed surfaces of a solid
  object is its surface area.
• What is Caroline’s surface area on her 1st
  birthday?

•   On her 2nd birthday?
•   On her 3rd birthday?
•   On her 5th birthday?
•   On her nth birthday?
Volume of a Cube
   Consider the 3-cube and the 5-cube on
   the left.

 • How long is the front bottom edge?
   right bottom edge?
 • What is the area of the base (number of
   cubes in the bottom layer)?
       Recall the area of a square is (side length) 2
 • How many layers are there (height)?
 • How many total cubes (volume)?
 • Volume is area of the base * height.

 • Since all dimensions of a cube are equal,
   the volume of a cube is (side length)3 or
   V=s3.
        Surface Area of a Cube
• Suppose the length, width, and height
  of the given cube is 2 cm. What is the
  surface area?

• What happens to the surface area of a
  cube when all of the dimensions are
  tripled?

• What happens to the edge length of a
  cube when the surface area is
  doubled?

• What can be said about the number of
  edges in each of these cubes?
 Building Right Rectangular Prisms
• Using 12 centimeter cubes, build all possible
  rectangular prisms.



• Which model has the largest surface area for the
  given volume of 12 cubic centimeters (cm3)?



  Excerpt from Student Activity Sheets (downloadable) “To the Surface and
  Beyond”, pp. 112-113, Navigating through Measurement in Grades 6-8,
  NCTM, 2002.
Volume of Right Rectangular Prism
• Using centimeter cubes, build a right rectangular prism
  with front edge length of 3 cubes, right edge width of 2
  cubes, and height of 2 cubes.
• How many cubes are contained in the prism?
• What is the area of the base (front edge length * right
  edge width)?
• What is the height?
• What is
  (front edge length) * (right edge width)*(height)?
• How does this compare to the total number of cubes in
  the prism?

• In general, the volume of a right rectangular prism is
  V = length * width * height or V = lwh.
Scale Factor, Volume, and Surface
   Area of a Rectangular Prism
• Excerpt from Student Activity Sheet (downloadable) p. 119-120
  from Navigating through Measurements in Grades 6-8, NCTM,
  2002.

• Two rectangular prisms have similar shapes. The front and back
  faces are the same shape, the top and bottom faces are the same
  shape, and the two remaining faces are the same shape.
• What is the scale factor (ratio)
  of the edges of the prisms?
• What is the scale factor of the
  surface areas of the prisms?
• How does the scale factor of the
  two volumes compare with the
  scale factor of the edges?
                       GPS Addressed
M6M2
    Select and use units of appropriate size and type to measure volume
M6M3
    Determine the formula for finding the volume of fundamental solid figures
    Compute the volumes of fundamental solid figures, using appropriate units
     of measure
    Estimate the volumes of simple geometric solids
M6M4
    Find the surface area of right rectangular prisms using manipulatives
    Compute the surface area of right rectangular prisms using formulae
M6A2
    Describe proportional relationships mathematically using y = kx, where k is
     the constant of proportionality
M6G2
    Interpret and sketch front, back, top, bottom, and side views of solid figures
M6P4
    Understand how mathematical ideas interconnect and build on one another
     to produce a coherent whole
 Websites for Additional Exploration
• Learning about Length, Perimeter, Area, and Volume
  of Similar Objects Using Interactive Figures: Side
  Length and Area of Similar Figures
  http://standards.nctm.org/document/eexamples/chap
  6/6.3/index.htm
• InterMath
  http://www.intermath-uga.gatech.edu/homepg.htm
• Linking Length, Perimeter, Area, and Volume
  http://illuminations.nctm.org/LessonDetail.aspx?ID=L
  261
• Eric Weisstein’s World of Mathematics
  http://mathworld.wolfram.com/topics/Polyhedra.html

				
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