Transcript Implementing the 6th Grade GPS via Manipulatives 1

Implementing the 6th Grade
Mathematics GPS via Centimeter
Cubes
Presented by Judy O’Neal
([email protected])
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
1
3
1
front
2
right
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
D.
G.
E.
1
3
1
2
1
F.
front
H.
I.
right
Architectural Plans 3A
1
3
1
3
1
2
Front
1
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.
1
B.
3
1
3
1
2
1
FRONT
C.
4
2
1
3
2
2
3
4
3
4
FRONT
2
1
1
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
symmetrically around a
common center).
Octahedron – 8 Triangular Faces
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