Download net

Implementing the 6th Grade GPS
via Folding Geometric Shapes
Presented by Judy O’Neal
([email protected])
Topics Addressed
 Nets
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Prisms
Pyramids
Cylinders
Cones
 Surface Area of Cylinders
Nets
 A net is a two-dimensional figure that,
when folded, forms a three-dimensional
figure.
Identical Nets
 Two nets are identical if they are
congruent; that is, they are the same if
you can rotate or flip one of them and it
looks just like the other.
Nets for a Cube
 A net for a cube can be drawn by tracing
faces of a cube as it is rolled forward,
backward, and sideways.
 Using centimeter grid paper
(downloadable), draw all possible nets for
a cube.
Nets for a Cube
 There are a total of 11 distinct (different)
nets for a cube.
Nets for a Cube
 Cut out a copy of the net below from centimeter grid
paper (downloadable).
 Write the letters M,A,T,H,I, and E on the net so that
when you fold it, you can read the words MATH
around its side in one direction and TIME around its
side in the other direction.
 You will be able to orient all of the letters except one
to be right-side up.
Nets for a Rectangular Prism
 One net for the yellow rectangular
prism is illustrated below. Roll a
rectangular prism on a piece of paper
or on centimeter grid paper and trace
to create another net.
Another Possible Solution
 Are there others?
Nets for a Regular Pyramid
 Regular pyramid
 Tetrahedron - All faces are triangles
 Find the third net for a regular pyramid
(tetrahedron)
 Hint – Pattern block trapezoid and triangle
Nets for a Square Pyramid
 Square pyramid
 Pentahedron - Base is a square and faces are
triangles
Nets for a Square Pyramid
 Which of the following are nets of a square
pyramid?
 Are these nets distinct?
 Are there other distinct nets? (No)
Great Pyramid at Giza
 Construct a scale model from net to
geometric solid (downloadable*)
 Materials per student:
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8.5” by 11” sheet of paper
Scissors
Ruler (inches)
Black, red, and blue markers
Tape
*http://www.mathforum.com/alejandre/mathfair/pyram
id2.html (Spanish version available)
Great Pyramid at Giza Directions
 Fold one corner of the paper
to the opposite side. Cut off
the extra rectangle. The
result is an 8½" square sheet
of paper.
 Fold the paper in half and in
half again. Open the paper
8 ½”
and mark the midpoint of
each side. Draw a line
connecting opposite
midpoints.
4 ¼”
More Great Pyramid Directions
 Measure 3¼ inches out
from the center on each of
the four lines. Draw a red
line from each corner of
the paper to each point
you just marked. Cut along
these red lines to see what
to throw away.
 Draw the blue lines as
shown
Great Pyramid at Giza Scale Model
 Print your name along the
based of one of the sides of the
pyramid.
 Fold along the lines and tape
edges together.
Nets for a Cylinder
 Closed cylinder (top and bottom included)
 Rectangle and two congruent circles
 What relationship must exist between the
rectangle and the circles?
 Are other nets possible?
 Open cylinder - Any rectangular piece of
paper
Surface Area of a Cylinder
 Closed cylinder
 Surface Area = 2*Base area + Rectangle area
 2*Area of base (circle) = 2*r2
 Area of rectangle = Circle circumference * height
= 2rh
 Surface Area of Closed Cylinder =
(2r2 + 2rh) sq units
 Open cylinder
 Surface Area = Area of rectangle
 Surface Area of Open Cylinder = 2rh sq units
Building a Cylinder
 Construct a net for
a cylinder and form
a geometric solid
 Materials per
student:
 3 pieces of 8½”
by 11” paper
 Scissors
 Tape
 Compass
 Ruler (inches)
Building a Cylinder Directions
 Roll one piece of paper to form an
open cylinder.
 Questions for students:
 What size circles are needed for
the top and bottom?
 How long should the diameter or
radius of each circle be?
 Using your compass and ruler,
draw two circles to fit the top and
bottom of the open cylinder. Cut
out both circles.
 Tape the circles to the opened
cylinder.
Can Label Investigation
 An intern at a manufacturing plant is given
the job of estimating how much could be
saved by only covering part of a can with a
label. The can is 5.5 inches tall with diameter
of 3 inches. The management suggests that 1
inch at the top and bottom be left uncovered.
If the label costs 4 cents/in2, how much would
be saved?
Nets for a Cone
 Closed cone (top or bottom
included)
 Circle and a sector of a larger
but related circle
 Circumference of the (smaller)
circle must equal the length of
the arc of the given sector
(from the larger circle).
 Open cone (party hat or ice
cream sugar cone)
 Circular sector
Cone Investigation
 Cut 3 identical sectors from 3 congruent circles or use
3 identical party hats with 2 of them slit open.
 Cut a slice from the center of one of the opened cones
to its base.
 Cut a different size slice from another cone.
 Roll the 3 different sectors into a cone and secure with
tape.
Questions for Students:
 If you take a larger sector of the same circle, how is
the cone changed? What if you take a smaller sector?
 What can be said about the radii of each of the 3
circles?
Cone Investigation continued
 A larger sector would increase the area of the base
and decrease the height of the cone.
 A smaller sector would decrease the area of the
base and increase the height.
 All the radii of the same circle are the same length.
Making Your Own Cone Investigation
 When making a cone from an 8.5” by
11” piece of paper, what is the
maximum height? Explain your
thinking and illustrate with a drawing.
Creating Nets from Shapes
 In small groups students create nets for
triangular (regular) pyramids
(downloadable isometric dot paper), square
pyramids, rectangular prisms, cylinders,
cones, and triangular prisms.
 Materials needed – Geometric solids, paper
(plain or centimeter grid), tape or glue
Questions for students:
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How many vertices does your net need?
How many edges does your net need?
How many faces does your net need?
Is more than one net possible?
Alike or Different?
 Explain how cones
and cylinders are
alike and different.
 In what ways are
right prisms and
regular pyramids
alike? different?
Nets for Similar Cubes Using
Centimeter Cubes
 Individually or in pairs,
students build three similar
cubes and create nets
 Materials:
 Centimeter cubes
 Centimeter grid paper
Questions for Students
 What is the surface area of
each cube?
 How does the scale factor
affect the surface area?
GPS Addressed
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M6M4
 Find the surface area of cylinders using manipulatives and
constructing nets
 Compute the surface area of cylinders using formulae
 Solve application problems involving surface area of cylinders
M6A2
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Use manipulatives or draw pictures to solve problems involving
proportional relationships
M6G2
 Compare and contrast right prisms and pyramids
 Compare and contrast cylinders and cones
 Construct nets for prisms, cylinders, pyramids, and cones
M6P3
Organize and consolidate their mathematical thinking through
communication
Use the language of mathematics to express mathematical
ideas precisely
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Websites for Additional Exploration
 Equivalent Nets for Rectangular Prisms
http://www.wrightgroup.com/download/cp/g
7_geometry.pdf
 Nets
http://www.eduplace.com/state/nc/hmm/too
ls/6.html
 ESOL On-Line Foil Fun
http://www.tki.org.nz/r/esol/esolonline/primary_mai
nstream/classroom/units/foil_fun/home_e.php