Measurement of Prisms, Pyramids, Cylinders, and Cones

Transcription

Measurement of Prisms,
Pyramids, Cylinders,
and Cones
Developed and Published
by
AIMS Education Foundation
This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating
Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit
AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding.
The foundation also sponsors a national program of professional development through which educators may
gain expertise in teaching math and science.
Copyright © 2010 by the AIMS Education Foundation
All rights reserved. No part of this book or associated digital media may be reproduced or transmitted in any
form or by any means—except as noted below.
• A person purchasing this AIMS publication is hereby granted permission to make unlimited copies of any
portion of it (or the files on the accompanying disc), provided these copies will be used only in his or her
own classroom. Sharing the materials or making copies for additional classrooms or schools or for other
individuals is a violation of AIMS copyright.
• For a workshop or conference session, presenters may make one copy of any portion of a purchased
activity for each participant, with a limit of five activities or up to one-third of a book, whichever is less.
• All copies must bear the AIMS Education Foundation copyright information.
• Modifications to AIMS pages (e.g., separating page elements for use on an interactive white board) are
permitted only for use within the classroom for which the pages were purchased, or by presenters at
conferences or workshops. Interactive white board files may not be uploaded to any third-party website or
otherwise distributed. AIMS artwork and content may not be used on non-AIMS materials.
Digital distribution rights may be purchased for users who wish to place AIMS materials on secure servers for
school- or district-wide use. Contact us or visit the AIMS website for complete details.
AIMS Education Foundation
1595 S. Chestnut Ave., Fresno, CA 93702 • 888.733.2467 • aimsedu.org
ISBN 978-1-60519-030-3
Printed in the United States of America
PRISMS, PYRAMIDS, CYLINDERS, AND CONES
© 2010 AIMS Education Foundation
M
easurement of Prisms, Pyramids,
Measurement
Cylinders, and Cones
Table of Contents
Welcome to the AIMS Essential Math Series!
BIG IDEA:
All solids (boxes) can be folded from a flat form called a net. This net forms
the surface of the box and its measure is called surface area.
Lesson One: A Solid Review
Day
1
Investigation A Solid Review................................................................. 9
Students sort and classify a set of six solids to recognize and review
their attributes.
Comics A Solid Review............................................................................. 17
Reviews the attributes of six solids and helps clarify associated
vocabulary.
Video Introducing Solids
Provides background of the general concepts related to solids
including classifying, naming, and types of surfaces.
Lesson Two: Box Building
Day
2
Investigation Box Building ................................................................... 19
Students use a net to build a rectangular solid to understand its
characteristics and to develop the meaning of surface area as a sum
of the areas of the faces.
Animation Folding a Net and Unfolding a Cube ................................... 21
A box unfolds to expose its net. The concept of surface area is
illustrated.
Comics Box Building .............................................................................. 22
Covers what a net is and the meaning of surface area.
5
BIG IDEA:
The space inside a solid is its volume. By finding the area of a layer that covers
the base (B) and multiplying by the number of layers (h), one determines what it
takes to fill the solid.
Lesson Three: Filling Boxes
Day
3
Investigation Filling Boxes .................................................................. 25
By filling a box with cubes a layer at a time, students recognize the
volume as a measure of filling and the relationship of volume to
cubes in a layer and number of layers.
Animation Boxes, Bases, and Blocks .................................................... 27
Seeing a rectangular solid built by row and layer provides a visual
memory of the meaning and formula for finding the volume of a
rectangular solid.
Day
4
Comics Filling Boxes ................................................................................ 28
Emphasizes that volume is a measure filling and is reported in cubic
units. It also helps students understand the formula for finding the
volume of a rectangular-based prism.
Practice: Box Building and Filling .................................................................. 30
Provides an opportunity for students to apply their understanding of
calculating surface area and volume to larger nets and solids.
Lesson Four: Special Box Building
Day
5
Investigation Special Box Building...................................................... 33
Students use nets to build a triangular-based prism and a cylinder
in order to understand their characteristics and to develop the
meaning of surface area as a sum of the areas of the surfaces.
Animation Unfolding Prisms and Cylinders ......................................... 36
Prisms and a cylinder unfold to expose their nets. The concept of
surface area is illustrated.
Comics Special Box Building ................................................................... 37
Reviews formulas for finding the area of a triangle and circle,
then progresses to summing the areas of all component parts to
determine surface area.
6
Lesson Five: Filling Special Boxes
Day
6
Investigation Filling Special Boxes ...................................................... 39
A prism and a cylinder are filled with cubes a layer at a time to show
volume as a measure of filling. Students see that the number of
cubes in a layer has to be determined by formulas for area.
Animation Filling Special Boxes ........................................................... 41
Seeing prisms and a cylinder filled a layer at a time provides a visual
model of the formulas or procedures for finding the volume.
Day
7
Comics Filling Special Boxes .................................................................... 42
Reinforces the use of the formula V = B · h for finding the volume
of prisms and cylinders.
Practice: Special Box Building and Filling ...................................................... 44
Students apply what they have learned to more abstract and larger
situations.
BIG IDEA:
The volume of a pyramid or a cone is one-third the volume of a prism or
cylinder with the same base (B) and equal height.
Lesson Six: Building Pointed Boxes
Day
8
Investigation Building Pointed Boxes .................................................. 47
Students build a rectangular-based pyramid, a triangular-based pyramid,
and a cone to understand their characteristics and to develop the
meaning of surface area as a sum of the areas of the surfaces.
Animation Unfolding Pointed Boxes..................................................... 50
A rectangular-based pyramid, triangular-based pyramids, and a cone
unfold to expose their nets. The concept of surface area is illustrated.
Comics Building Pointed Boxes, Part 1 .....................................................51
Compares finding the surface areas of a square-based pyramid
and a triangular-based pyramid.
Building Pointed Boxes, Part 2 ...................................................................... 53
Looks at the process of finding the surface area of a cone.
Video Building Solids
Demonstrates with physical models how to determine the surface
area by determining the type, size, and number of the polygons and
shapes that compose the surface.
7
Lesson Seven: Filling Pointed Boxes
Day
9
Investigation Filling Pointed Boxes .................................................... 55
Pouring cereal from pyramids and cones into corresponding prisms
and cylinders allows the discovery that the volume of a pyramid or
cone is one-third of its corresponding prism or cylinder.
Animation Pouring Pyramids and Cones ............................................. 57
BBs transferring from pyramids and cones to prisms and cylinders
reinforce the one-third factor in the formulas.
Comics Filling Pointed Boxes ................................................................... 58
Stresses the one-third relationship of the volume of a triangularbased pyramid to a triangular-based prism with equal bases and
heights, also the one-third relationship of the volume of a cone to a
cylinder with equal bases and heights.
Video Filling Solids
Demonstrates with physical models how to determine the volume
of a solid by layering or referring to other solids.
Day
10
Problem Solving Pyramid Puzzle........................................................61
Filling a cube with three pyramids reinforces the one-third
relationship of a pyramid to its corresponding prism.
Comics Pyramid Puzzle ........................................................................... 64
Verifies that the formula for the volume of a pyramid is one-third the
volume of a cube with the same height and base.
Video When Will I Use This?
Shows students how surface area and volume are applied in realworld problems.
Day
11
Assessment Prisms, Cylinders, Pyramids, and Cones Assessment ...... 65
Two assessments provide familiar contexts for determining surface
area and volume from drawings.
Glossary ................................................................................................................................. 69
National Standards and Materials......................................................................................... 70
Using Comics to Teach Math ................................................................................................. 71
Using Animations to Teach Math .......................................................................................... 72
The Story of Measurement of Prisms, Cylinders, Pyramids, and Cones ................................. 73
The AIMS Model of Learning ................................................................................................. 79
8
A SOLID
SOLIDReview
How can you use the similarities and differences of the geometric solids to identify each type?
Comparing and contrasting the different solids requires repeated examination, thereby providing a
review of the names of the solids and a more extensive definition of each one. To measure surface area
and volume of the solids, students will need to know the numbers and shapes of faces/surfaces along
with how the solids are formed from nets.
For
demonstration
purposes, Build the
six solids from card
stock. These models can
be used in all the
investigations.
n
iioon
t
t
a
a
iigg
Materials
t
t
s
s
e
e
v
Scissors
nv
IIn
Geometric Solid Templates
Card stock
A SOLID
SOLIDReview
tig
Have each
group of
students cut
apart a set of six
illustrations.
Complete the information for each picture by referring to the sample solids or illustration.
Name
Number
of
Surfaces
Shape of Faces
Rectangular Solid
6
Rectangles
Picture
Triangular Prism
5
Cylinder
3
Square-Based
Pyramid
Triangular-Based
Pyramid
Cone
5
4
2
Triangle bases (2)
Rectangle sides (3)
Round bases (2)
Curved surface (1)
Square base (1)
Triangle sides (4)
Triangle base (1)
Triangle sides (3)
Circle base (1)
Curved surface (1)
1. Explain how you could pair all of the solids so each pair shares the same attribute.
• Rectangular bases, triangular bases, circular bases
• Curved surfaces, flat tops, pointed tops
i’d like to review
my favorite attributes
of the cylinder.
2. Explain how you could sort the six solids into two groups by attribute.
• Flat tops, pointed tops
• Curved surfaces, flat faces
By
referring to
the illustrations
and, if necessary, the
demonstration models,
students determine the
shape and number of
all faces and
surfaces.
A SOLID
SOLIDReview
Cut apart
the six cards and use them
to compare, contrast,
and sort.
Have each
group sort the
six illustrations
into pairs that share
a common attribute.
Have groups explain
their reasoning for the
pairings. Share the
variety of pairings
and explanations.
Have each group
© 2010 AIMS Education Fou
sort the six illustrations
into two groups to encourage them
to recognize all the attributes. Have
groups share their solutions
and explanations.
PRISMS, PYRAMIDS, CYLINDERS,
Have
students
identify all
the solids they
know. Identify any
they do not know
or have identified
incorrectly. (Refer
to local standards
for appropriate
labels.) Record
the labels on the
record page and
illustrations.
AND CONES
11
ics
m
Reviews the attributes of six solids and helps clarify associated vocabulary.
Co
eo
d
i
V
Provides background of the general concepts related to solids including classifying,
naming, and types of surfaces.
9
A SOLID
SOLIDReview
Complete the information for each picture by referring to the sample solids or illustrations.
Picture
Number
of
Surfaces
Name
Shape of Faces
1. Pair all of the solids so each pair shares an attribute. Explain your pairings.
i’d like to review
my favorite attributes
of the cylinder.
2. Sort the six solids into two groups by attributes. Explain your sorts.
10
A SOLID
SOLIDReview
Cut apart
the six cards and use them
to compare, contrast,
and sort.
11
2
3
Let’s start building
and see what solid
we can make with this
template.
1. Cut out along the bold lines.
2. Fold along the dotted lines. (Use the
edge of a ruler or the straight edge of
a desk to get a sharp crease.)
3. Fold up into a solid using the tabs to
tape or glue it.
4. Attach the top by gluing or taping
the tab.
TAB
TAB
What solid will this net make?
BA
SE
3
4
1
2
Geometric Solid
Template
TO
P
TAB
Enlarge to 103%
when copying.
5 cm
4 cm
3 cm
1
3
12
2 cm
1
4
1 cm
TAB
TAB
TAB
ch
lun
You
construct
the solid and I’ll
take a lunch
break.
I have an idea!
1
3
2
3
TAB
TAB
Geometric Solid
Template
B
TA
E
S
A
B
1
4
1
2
3
4
attach
centicube
here
attach
centicube
here
R
E
Y
A
L
1 cm
2 cm
3 cm
4 cm
5 cm
attach
centicube
here
2. Fold along the dotted lines. (Use the edge of a ruler or
the straight edge of a desk to get a sharp crease.)
3. Fold up into a solid using the tabs to tape or glue it.
4. Attach the top by gluing or taping the tab.
(This
piece is used
in Filling
Special Boxes.)
this should be
easy!
P
13
TO
B
TA
TAB
TO
P
1 cm
2 cm
3 cm
4 cm
5 cm
attach
centicube
here
attach
centicube
here
(This piece is used in Filling Special Boxes.)
attach
centicube
here
attach
centicube
here
1
4
1
2
3
4
B
TA
BA
SE
TAB
1
3
2
3
TA
B
2. Fold along the dotted lines. (Use the edge of a ruler or the
straight edge of a desk to get a sharp crease.)
4. Attach the top by gluing or taping the tab.
TAB
Geometric Solid
Template
Match Line. Do not fold.
TAB
TA
B
14
TA
B
B
TA
does
anyone have
a phillips
screwdriver?
TAB
TA
B
I think i constructed
this solid wrong.
2. Fold along the dotted lines. (Use the edge of a ruler or
the straight edge of a desk to get a sharp crease.)
Geometric
Solid
Template
E
S
A
B
15
TAB
Match Line. Do not fold.
2. Fold along the dotted lines.
(Use the edge of a ruler or the
straight edge of a desk to get
a sharp crease.)
3. Fold up into a solid using the
tabs to tape or glue it.
Geometric
Solid
Template
m
a nu
a
Hey, this page
has two
templates!
l
TAB
16
BASE
TAB
B
A
S
E
17
it’s called
a prism. it’s the
same shape as the
glass prisms we
used in science
class.
We used
prisms in
science class?
really?
We looked
at the number
of faces and
their shapes.
in our
activity, we looked
to see which ones have
only flat faces and
which ones have a
curved surface.
Class, yesterday
we reviewed the names
of six different solids
and we tried to describe
each of them.
4. How is a cylinder similar to a prism?
2. What is the shape of a
lateral face of a prism?
Yeah, remember
you can shine a light
on a prism and see
colors like a
rainbow.
1
let’s look at
several different
prisms and talk
about what we
know about
them.
I know you’re quite familiar with
these 3-dimensional shapes from
earlier grades, but there were
For
a few words that were new
example, what
to you, right?
do we call this
3-dimensional
shape?
Remember that in math solid simply means
3-dimensional. Prisms, cylinders, cones,
and pyramids are all 3-dimensional shapes.
sometimes we call them geometric solids.
5. How is a cone similar to a pyramid?
3. What is the shape of a lateral
face of a pyramid?
ESSENTIAL MATH SERIES
1. What does the word solid
mean in geometry?
THINGS TO LOOK FOR:
A Solid Review
MEASUREMENT OF PRISMS, PYRAMIDS, CYLINDERS, AND CONES
I think
they use it in
football.
I’ve
heard of
the word
lateral.
those are the
lateral faces.
the other three
faces are
rectangles
around the sides
of the prism.
The bases are the
triangles at the
top and bottom.
Look at the prism
that’s pictured
here.
All of the
other faces of a
prism are rectangles
and are called
lateral faces.
Does that
make sense,
redmond?
Lateral??
what does
that mean?
I bet you
didn’t know
that, mark.
Sure, I
knew that.
I know
everything.
Yeah, I
think it
does.
Well, the
word lateral
means to the side
or off to the
side.
So if the base
is a triangle, it’s a
triangular prism, if
it’s a square it’s a
square-based
prism, and so on.
The shape
of the base of
a prism gives
it its name.
Yep, the
quarterback in
football sometimes
throws the ball to a
receiver who is just to
his left or right side
instead of down
That’s
field.
called a
lateral
pass.
Two of the faces are always congruent and
parallel. They can be squares or rectangles
or triangles or pentagons or any other
polygon shape. these two opposite faces are
called bases of the prism.
2
18
LATERAL
FACES
I thought the
point was always
kind of right over
the middle of the
base?
But the other
faces on a pyramid
all have to be
triangles,
right?
Thank
you, vanessa.
that’s exactly
right.
You’re right.
most of the pyramids
you’ve seen before
these were like that.
but, it’s okay for the
point to be off to the
side like it is on that
one.
And, if it’s a
pyramid, the lateral
faces are triangles
and if it’s a prism,
they’re rectangles.
So, the
lateral faces are
around the sides
no matter what the
3-dimensional
shape.
That’s
right,
shuttle.
Class, these two
3-Dimensional
shapes are pyramids.
How is a pyramid
different
from a prism?
Finally, class,
let’s talk about
the cylinder and
the cone.
Ms. Cho, I didn’t
know that the top
point on a pyramid
could be off to
one side like
that.
I get it. the
lateral faces of a
pyramid all meet at
one point, so they have
to be triangles.
Wow, red,
nice.
are those other
triangle faces of
the pyramid called
lateral faces,
too?
it’s different
because the pyramid
just has one base.
(see? everything.)
Think about the
number of bases
in a prism and in a
pyramid.
Yes, they
are. that’s a very
good question,
elora.
it’s also kind of like
a prism, because the
base of a pyramid can
be a triangle, or a
square, or some
other shape like
that.
3
The cylinder
has two bases
just like the prism
and the cone has
only one base just
like the pyramid.
I know!
You’re right,
redmond, but they
are a lot alike.
that’s all shuttle
and vanessa have
been trying to say.
Yeah, I guess they
are a lot alike.
Oh, and so are the
pyramid and the
cone, right?
ms. cho,
isn’t a cylinder
really just like a
prism that has
bases that are
circles?
That’s just
like how on the
pyramid the lateral
faces all come to
a point at the
top.
That means
that a cylinder
has two bases
that are
circles, and
one curved
lateral
surface.
Let’s see if I can answer Elora’s
question. How about instead of
calling it a lateral face, let’s
call it a curved lateral
surface.
How are the
cylinders and the
cone similar to the
prism and the
pyramid?
Or, you could say
that a prism is like a
cylinder that has
bases that are
triangles or
squares or some
other shape.
The cone is just
like it, except it
has only one
circular base.
What??? what are
you saying? isn’t a
cylinder a cylinder
and a prism a
prism.
on the cone, the
curved lateral
surface comes
to a point at
the top.
Class, this
is a good start.
we’re going to be
doing a lot more
with these
shapes.
That’s right,
redmond.
The face is
just one piece
and it curves
around the base,
right?
Ms. Cho,
it looks like
the cylinder and
the cone each just
have one lateral
face.
4

Measurement of Prisms, Pyramids, Cylinders, and Cones

Transcription

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