Now

1
Now
Showing
SHAPES
Presented by
In 2-
and
3-D
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Reach Into The Bag
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Polygon Task Triangles Lined Up in A Row
Name ____________________________________
Shapes, Functions, Patterns
Learning and Teaching Linear Functions Nanette
Seago, Judith Mumme and Nicholas Branca
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Polygon Task Triangles Lined Up in A
RowWORKSPACE
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Shapes Functions, Patterns
Predict the volume of building 100. Find a rule
for any building (n).
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Block Structure What is the volume of building
10?WORKSPACE
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Name ____________________________________
Spatial Visualization Station

1. Build the exact models using the appropriate
   colored linking cubes. We use the linking cubes
   so we can keep it together when picking it up and
   moving it around. Remember the cubes cannot hang
   in space.
2. Draw the design on Grid paper or Isometric Dot
   paper.
3. Determine the volume, number of faces, number of
   edges, and number of vertices of each model.

1. 2. 3. 4. 5.
6.
Volume _______ _______ _______
_______ _______ _______ of
Faces _______ _______ _______
_______ _______ _______
of Edges _______ _______ _______
_______ _______ _______ of
Vertices ______ _______ _______
_______ _______ _______
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• 1. 2. 3.
• 4. 5. 6.

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Designing the Largest BoxFunctions from Formulas
The owner of a large factory has 100 rectangular
sheets of metal 11 feet by 8.5 feet. These
sheets need to be made into vats that will hold
the greatest amount possible of the waste from
the plant. These vats must all be of equal size
and will be constructed by turning up the sides
and welding. You own a metal shop and can get
this job if you convince the owner that you can
build the vat with the greatest volume.
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Designing the Largest BoxFunctions from Formulas

• Begin with a rectangular sheet of 8 ½ x 11
  cardstock.
• From each corner, cut a square of assigned size.
• Fold up the four resulting flaps, and tape them
  together to form an open box.
• The volume of the box will vary, depending on the
  size of the squares. Write a formula that gives
  the volume of the box as a function of the size
  of the cutout squares.
• Use the function to determine what size the
  squares should be to create the box with the
  largest volume.

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Trim the paper to the grid and cut out a 4 x 4
square from each corner.
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Trim the paper to the grid and cut out a 3 x 3
square from each corner.
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Trim the paper to the grid and cut out a 5 x 5
square from each corner.
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Trim the paper to the grid and cut out a 6 x 6
square from each corner.
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Trim the paper to the grid and cut out a 2 x 2
square from each corner.
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Investigating Nets
Cube pattern followsSee word document for
Cylinder pattern
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Name ____________________________________
Nets Wrap Up Activity
Investigating Nets
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Nets Wrap Up Activity
Investigating Nets
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Using Polydrons
Investigating Nets
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Name ____________________________________
Using Polydrons
Investigating Nets
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Using Polydrons
Investigating Nets
ANSWER KEY
Tetra- hedron Hexa- hedron Octa- hedron Dodeca- hedron Icosa- hedron
What shape are all the faces? triangle square triangle pentagon triangle
How many faces? 4 6 8 12 20
How many edges? 6 12 12 30 30
Each face touches how many vertices? 3 4 3 5 3
Each edge joins how many faces? 2 2 2 2 2
Each vertex touches how many faces? 3 3 4 3 5
If the edge measures one linear unit, find the approximate surface area of the polyhedron. 1.72 un2 6 un2 3.44 un2 20.64 un2 8.60 un2
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Area and Volume of 3-D Shapes
Use Power Solids to compare the area and volume
of 3-D shapes.
Investigating Using Power Solids Ordering By Area
of the Base Working with your group, put your
power solids in order from smallest to largest by
the area of their bases. Check the order by
tracing the shape onto grid paper and counting
the squares or by measuring the dimensions and
computing the base area of each
solid. Investigating Using Power SolidsOrdering
by Volume Working with your group, put your
power solids in order from smallest to largest by
their volume. Check the order by filling and
pouring rice or sand.
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Area of 3-D Shapes
Name ________________________________________
Use Power Solids to compare the area of 3-D
shapes.

• Investigating Using Power Solids
• Ordering By Area of the Base
• Working with your group, put your power solids
  in order from smallest to largest by the area of
  their bases. Check the order by tracing the
  shape onto grid paper and counting the squares or
  by measuring the dimensions and computing the
  base area of each solid.
• Label the base area amounts on the shapes you
  drew on the grid paper.
• List any two solids which have the same base.
• Which rectangular prism has a base congruent to
  the base of the square pyramid AND has the same
  height as the square pyramid?

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Volume of 3-D Shapes
Use Power Solids to compare the volume of 3-D
shapes.

• Investigating Using Power SolidsOrdering by
  Volume
• Working with your group, put your power solids
  in order from smallest to largest by their
  volume. Check the order by filling and pouring
  rice or sand.
• How many square pyramids does it take to fill the
  rectangular prism with the same base? __________
• How many triangular pyramids does it take to fill
  the triangular prism with the same base?
  _________
• What could we write about the volumes of pyramids
  and prisms that have congruent bases and the same
  heights?
• How many cones does it take to fill the cylinder?
  ____________
• What could we write about the volumes of a cone
  and a cylinder having the same height and
  congruent bases?
• How many cones does it take to fill the
  hemisphere? _________
• What could we write about the volumes of a cone
  and a hemisphere if the base of the cone is
  congruent to the great circle of the hemisphere
  and the height of the cone is the diameter of the
  sphere?

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• Activity Tripyramidal Box
• Cut out the patterns on the solid lines.
• Fold back on the scored lines.
• 3. Close the nets to make 3 pyramids.
• 4. Now put the 3 pyramids together to make a box
  or tripyrmidal.
• 5. DiscussVolume of CUBE lwh.
• The area of the Base can be Blw, so the Volume
  of the CUBE could also be written as V Bh
• 6. We can now develop a formula for the Volume a
  Pyramid? The volume of one of the pyramids is
  given by the formula V (1/3)lwh
  (1/3)Bh.

V lwh

or V Bh
h


(1/3)Bh
(1/3)Bh
(1/3)Bh
(1/3)lwh
(1/3)lwh
(1/3)lwh
w
l
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Finding the Formula for the Surface Area of a
Sphere
Geometry/Science Connection
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2.
1.
3.
4.
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Find the Formula for Surface Area of a Sphere
Name___________________________ C
ircumference, Area, Surface Area
1. What part of a planet or sun would the
circular ring represent? __________________ 2.
When we look at a 2-dimensional picture of a
planet or sun what does the circle represent?
__________________________________ ______________
__________________________________________________
_________________________________________________
3. What is the formula for the area of a flat
circular surface? ______________________ INVEST
IGATE Use string, a nail, and a styrofoam
hemisphere and cover the flat surface with the
string. Mark off the amount required to cover.
Now cover the outside of the hemisphere (not
including the flat surface). Compare your
measures. 4. Describe what you found.
__________________________________________________
______________________________ __________________
__________________________________________________
___________________________________ 5. From
what you have found write a formula for covering
the entire surface area of the sphere.
________________________
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Find the Formula for Surface Area of a Sphere
Name___________________________ C
ircumference, Area, Surface Area
Great Circle
1. What part of a planet or sun would the
circular ring represent? __________________ 2.
When we look at a 2-dimensional picture of a
planet or sun what does the circle represent?
__________________________________ ______________
__________________________________________________
_________________________________________________
3. What is the formula for the area of a flat
circular surface? ______________________ INVEST
IGATE Use string, a nail, and a styrofoam
hemisphere and cover the flat surface with the
string. Mark off the amount required to cover.
Now cover the outside of the hemisphere (not
including the flat surface). Compare your
measures. 4. Describe what you found.
__________________________________________________
______________________________ __________________
__________________________________________________
___________________________________ 5. From
what you have found write a formula for covering
the entire surface area of the sphere.
________________________
The inside of the sphere sliced through the Great
Circle or the base of a hemisphere
A ? r2
It takes twice as much string to cover the
outside of the hemisphere as it does to cover
the base of the hemisphere.
SA sphere 4 ? r2
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