Rockdale County Public Schools: MSP Courses

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Rockdale County Public Schools MSP Courses

• Day 3 7th Grade

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Unit 7 Slices and Shadows

• Students will be able to
• describe three-dimensional figures formed by
  translations and rotations of plane figures
  through space.
• sketch, model, and describe cross-sections of
  cones, cylinders, pyramids, and prisms.

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Part 1 Describe three-dimensional figures formed
by rotations and translations of plane figures
through space.

• Unit 7

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Rotation of a Triangle Through Space

• Rotating ?ABC about line d2 (axis of symmetry)
  produces a cone whose base diameter is equal to
  the length of side AC.
• Beginning with the cone and slicing it vertically
  provides a cross-section view, which is shown as
  ?ABC.

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Rotation of a Circle Through Space

• Rotating ?O about point O produces a sphere whose
  radius is equal to the radius of ?O.
• Beginning with the sphere and slicing it through
  its center provides a cross-section view (green),
  which is shown as ?O.

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One More Time Around

• Find the volume of the solid formed by rotating
  the shaded figure about the line XY, as shown in
  the diagram.
• Describe how you found the volume.

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Translating a Circle Through Space

• Translating ?A by translation distance BC
  produces a (slinky-like) cylinder.
• Slicing the cylinder perpendicular to the axis of
  symmetry AD generates many circles congruent to
  ?A.
• Cabri 3D rendering created by Stephen F. West,
  State University College, Geneseo, NY

C
D
A
B
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Translating a Triangle Through Space

• Translating ?ABC by translation vector CD
  produces a triangular prism.
• Recall that a prism is a geometric solid whose
  bases (green) are congruent, parallel polygons
  and whose lateral faces (white) are
  parallelograms.
• Slicing the triangular prism generates many
  triangles (gray) congruent to ?ABC.
• Cabri 3D renderings created by Stephen F. West,
  State University College, Geneseo, NY

A
B
C
D
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Translating a Square Through Space

• Consider the square CEDB and the translation
  segment HK.
• Translating ?CEDB in the direction and length of
  HK produces the green rectangular prism.
• Beginning with the rectangular prism and slicing
  it perpendicular to segment AD provides a
  cross-section view, which is a square congruent
  to ?CEDB.
• Task
• Investigate and describe how a rectangular
  cross-section view could be generated from this
  rectangular prism.

E
E
K
D
H
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Translating an OctagonThrough Space

• Consider the blue octagon and the translation
  segment HK.
• Translating the blue octagon in the direction and
  length of HK produces an octagonal prism.
• Task for Students
• Describe how the octagonal prism must be sliced
  to produce a cross-sectional view of many
  congruent octagons.
• Cabri 3D rendering created by Stephen F. West,
  State University College, Geneseo, NY

K
H
H
E
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Transforming a Hexagon in Space

• Task
• Describe how hexagon A can be transformed into
  the hexagonal prism at the right.

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Creating Prisms

• If you move a vertical rectangle horizontally
  through space, you will create a rectangular or
  square prism.
• If you move a vertical triangle horizontally, you
  generate a triangular prism. When made out of
  glass, this type of prism splits sunlight into
  the colors of the rainbow.

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Cross Section Method

• One way to determine the volume of rectangular
  prisms is by multiplying the dimensions (length
  width height). Another way to determine the
  volume is to find the area of the base of the
  prism and multiply the area of the base by the
  height. This second method is sometimes referred
  to as the cross-section method and is a useful
  approach to finding the volume of other figures
  with parallel and congruent cross sections, such
  as triangular prisms and cylinders. Notice in the
  figures below that each cross section is
  congruent to a base.

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Compute Volume using the CSM

• The formula for the volume of prisms is V B
  h, where B is the area of the base of the prism,
  and h is the height of the prism. Does it matter
  which face is the base in each of the following
  solids? Explain.
• Find the volume of the figures above using the
  cross-section method.

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Part 2 sketch, model, and describe
cross-sections of cones, cylinders, pyramids, and
prisms.

• Unit 7

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Cross Sections of a Pyramid

• Beginning with a pyramid and slicing it
  horizontally provides a cross-section view of
  concentric squares.
• Graphics animation available from Demos with
  Positive Impact
• Volumes by Section Demo Gallery

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Cross Sections of a Cube

• Which of the following cross sections can be made
  by slicing a cube
• a. Square. b. Equilateral triangle.
• c. Rectangle, not a square. d. Triangle, not
  equilateral.
• e. Pentagon. f. Regular hexagon.
• g. Hexagon, not regular. h. Octagon.
• i. Trapezoid, not a parallelogram.
• j. Parallelogram, not a rectangle.
• k. Circle.

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Cross Sections of a Cube (cont.)

• Use a plastic, transparent model of a cube that
  is open at one face. By filling the cube with
  rice or water, and tipping the cube in different
  ways, the students could demonstrate the
  different cross sections that can be made.
• Submerge a solid partially into a bowl of liquid.
  
• Learning Math Website
• http//www.learner.org/channel/courses/learningmat
  h/geometry/session9/part_c/index.html

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Crazy with Cross Sections

• Predict the possible cross-sections for these
  solids. Explain how you know that these are
  possible cross-sections.
• a. Cylinder.
• b. Cone.
• c. Sphere.
• Use models of the above solids to confirm your
  predictions.
• Sketch and describe the cross-sections.
• Name three plane figures that cannot be formed
  from cross-sections of the above figures and
  explain why they cannot be formed.

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Applications of Transformations and Cross Sections

• Hurricane creation - http//observe.arc.nasa.gov/n
  asa/earth/hurricane/creation.html
• Interior of the earth - http//pubs.usgs.gov/gip/i
  nterior/
• Earthquakes Mt. St. Helens http//www.geophys.wa
  shington.edu/SEIS/PNSN/HELENS/helenscs_yr.html

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Whats My Solid?

• Each of the following descriptions fit one or
  more solids (prism, pyramid, cone, cube, a
  cylinder). For each, describe what solid that
  might be and why. If the description could fit
  more than one solid, state what solids they could
  be and why. Sketch the solid, and illustrate the
  properties described.
• I have one circular face. I also have a curved
  surface. What geometric solid am I?
• I have six faces. All of my edges are the same
  length. Which geometric solid am I?
• I have an odd number of vertices. I have the same
  number of faces and vertices. Which geometric
  solid am I?
• I have two triangular faces. I have three
  rectangular faces. Which geometric solid am I?
• I have two circular faces. I have a curved
  surface. Which geometric solid am I?
• I have an even number of vertices. I have the
  same number of faces and vertices. Which
  geometric solid am I?
  

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Whats My Solid? (cont.)

• a) A set of my parallel cross sections are
  squares that are similar but not congruent.
• b) A set of my parallel cross sections are
  congruent rectangles.
• c) A set of my parallel cross sections are
  circles that are similar but not congruent.
• d) A set of my parallel cross sections are
  congruent circles.
• e) A set of my parallel cross sections are
  parallelograms.
• f) One of my cross sections is a hexagon,
  and one cross section is an equilateral
  triangle.
• g) I can be made by translating a rectangle
  through space.
• h) I can be made by rotating a triangle
  through space.
• i) My volume can be calculated using the
  area of a circle
• j) My volume can be calculated using the
  area of a rectangle.

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Shadows

• Hold a cube under the overhead projector so that
  it casts a shadow on the wall. Sketch the shape
  of the shadow. Can the shadow have just four
  edges? Why or why not? Six edges? Why or why not?
  Can the shadow be a regular hexagon? Why or why
  not? Can a shadow of a cube have exactly five
  edges? Why or why not?
• Repeat this activity using one or more of the
  following prism, pyramid, cylinder, cone. Be
  sure to sketch the shadows.

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Warm Up Faces, Vertices, and Edges

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

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Investigating Vertices, Edges, and Faces
How do the number of faces, number of vertices,
and number of edges relate in a cube? in a
pyramid? In a prism?

• For each of the Power Solids listed in the table,
  count and record the number of vertices, edges,
  and faces.
• Describe any patterns you observe.

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Investigating Vertices, Edges, and Faces Teacher
Notes

• One pattern that may emerge from the table is
  Eulers Formula
• V F E 2
• where V number of vertices, F number of
  faces, and E number of edges.

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Applying Eulers Formula

• Suppose you see the footprint of a prism whose
  base is shown below.
• Without actually making the prism, explain how
  you could tell how many vertices, edges, and
  faces it has.
• How would you compute the volume of the prism?

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Websites for Additional Exploration

• Math Open Reference Constructions
  http//www.mathopenref.com/tocs/constructionstoc.h
  tml
• National Library of Virtual Manipulatives
  Geometry (Translations, Rotations, Reflections)
  http//nlvm.usu.edu/en/nav/topic_t_3.html
• Demos with Positive Impact Cross Sections
  http//mathdemos.gcsu.edu/mathdemos/sectionmethod/
  pyramidcross.gif
• Estimating the Circumference of the Earth -
  http//www.k12science.org/jkoen/rwlo/Eratosthenes
  /Content20Material/PartA.shtml