Constructing
Geometric Solids
Objectives To provide practice identifying geometric
solids
given their properties; and to guide the construction
s
of polyhedrons.
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Ongoing Learning & Practice
Key Concepts and Skills
Plotting Book Heights
• Identify parallel and intersecting line
segments and parallel faces. Math Journal 2, pp. 295A and 295B
Students plot book heights on a
line plot.
[Geometry Goal 1]
• Describe, compare, and classify plane
and solid figures. [Geometry Goal 2]
• Identify congruent faces. [Geometry Goal 2]
• Construct polyhedrons; sketch
two-dimensional representations
of polyhedrons. [Geometry Goal 2]
Curriculum
Focal Points
Math Boxes 11 3
Math Journal 2, p. 296
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Differentiation Options
READINESS
Sorting Geometric Solids
30 objects of various shapes
Students sort common objects by
their properties.
ENRICHMENT
Creating Cube Nets
Math Masters, pp. 389 and 444
scissors
Students find all possible cube nets.
Use Math Boxes, Problem 3a. Key Activities
Students practice identifying geometric solids
by solving riddles about their properties.
Students construct polyhedrons with straws
and twist-ties. Students explore ways to
draw a cube.
[Number and Numeration Goal 3]
Study Link 11 3
Math Masters, p. 329
Students practice and maintain skills
through Study Link activities.
Key Vocabulary
pyramid prism polyhedron regular
polyhedron triangular pyramid tetrahedron dodecahedron
EXTRA PRACTICE
Taking a 50-Facts Test
Math Masters, pp. 411 and 414;
p. 416 (optional)
pen or colored pencil
Students take a 50-facts test. They use a
line graph to record individual and optional
class scores.
Materials
Math Journal 2, pp. 293 –295
Student Reference Book, p. 102
Study Link 112
Math Masters, pp. 452 and 453 (optional)
models of geometric solids (See Planning
Ahead in Lesson 111.) straws and
twist-ties (See Lesson 112.) blank paper straightedge transparent tape slate set of polyhedral dice (optional) dictionary
(optional)
Advance Preparation
For Part 1, you need the geometric solids from Lesson 112, plus a triangular pyramid and a cube. Use
everyday objects or make them from Math Masters, pages 452 and 453. For the optional Readiness
activity in Part 3, use cans, egg cartons, party hats, paper cups, tubes, and boxes.
Teacher’s Reference Manual, Grades 4–6 pp. 187–189
860
Unit 11
Interactive
Teacher’s
Lesson Guide
3-D Shapes, Weight, Volume, and Capacity
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
4.NF.3c, 4.NF.3d, 4.NF.4a, 4.NF.4b, 4.MD.4
Mental Math and Reflexes
Math Message
Pose problems involving the multiplication of a fraction by a
whole number. Have students find each product. Suggestions:
Open your Student Reference Book to
page 102. Solve the following riddle: I have the
same number of faces as vertices. What am I?
4∗_
=2
2
5∗_
=3
5
5∗_
= 2_
2
2
24 ∗ _
=3
8
9∗_
=6
3
14 ∗ _
= 4_
3
3
30 ∗ _
=5
6
16 ∗ _
= 12
4
10 ∗ _
= 7_
, or 7 _
4
4
2
1
1
1
3
2
3
1
1
1
3
2
2
Study Link 11 2 Follow-Up
1
Have partners compare answers. Ask:
• How many pairs of parallel faces does
the rectangular prism have? 3
• How many pairs of parallel faces does the
tetrahedron have? 0
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 102)
All of the pyramids shown at the top of page 102 of the Student
Reference Book have the same number of faces as vertices.
Pyramids are named for the shape of their base. All of the
remaining faces are triangles that meet at a vertex.
Pose another riddle: I have 6 faces. All of my faces are rectangles.
What am I? Rectangular prism Prisms are named for the shape
of their two parallel bases. Emphasize that a polyhedron
(plural polyhedrons or polyhedra) is a geometric solid whose
surfaces are all formed by polygons. A polyhedron does not
have any curved surfaces.
Display the six geometric solids from Lesson 11-2 and the
triangular pyramid and cube. Ask: Which of these solids are
NOT polyhedrons? Cylinder, sphere, cone
Student Page
Geometry and Constructions
Polyhedrons
A polyhedron is a geometric solid whose surfaces are all formed
by polygons. These surfaces are the faces of the polyhedron.
A polyhedron does not have any curved surfaces.
Pyramids and prisms are two important kinds of polyhedrons.
A rhombicuboctahedron
has 26 faces. Eighteen
of them are squares and
8 are triangles.
Polyhedrons That Are Pyramids
A polyhedron is a regular polyhedron if
each face is formed by a regular polygon;
the faces all have the same size and shape; and
each vertex looks exactly the same as every other vertex.
triangular pyramid
square pyramid
pentagonal pyramid
hexagonal pyramid
The shaded face of each pyramid above is called the base of the
pyramid. The shape of the base is used to name the pyramid. For
example, the base of a square pyramid has a square shape. The
faces of a pyramid that are not the base are all shaped like
triangles and meet at the same vertex.
Polyhedrons That Are Prisms
Ask: Which of these solids are regular polyhedrons? Cube and
triangular pyramid, or tetrahedron
rectangular prism
triangular prism
NOTE Triangular pyramid and tetrahedron are two names for the same
geometric solid. When people refer to a tetrahedron, they often mean a regular
tetrahedron. Not all tetrahedrons (triangular pyramids) are regular, however.
pentagonal prism
hexagonal prism
The two shaded faces of each prism above are called the bases of the
prism. The bases of a prism are the same size and shape. They are
parallel. All other faces join the bases and are shaped like parallelograms.
The shape of the bases of a prism is used to name the prism. For example,
the bases of a pentagonal prism have the shape of a pentagon.
Many polyhedrons are not pyramids or prisms. Some are illustrated below.
Polyhedrons That Are NOT Pyramids or Prisms
Student Reference Book, p. 102
087_118_EMCS_S_G4_SRB_GEO_576507.indd 102
3/1/11 8:49 AM
Lesson 11 3
861
Adjusting the Activity
Distribute a set of polyhedral dice. Ask students to find the name of
each die and determine the die that is not a regular polyhedron. The names of
the dice are as follows: tetrahedron, octahedron, decahedron, dodecahedron,
icosahedron. The decahedron is not a regular polygon because its faces are not
regular polygons.
A U D I T O R Y
K I N E S T H E T I C
Solving Geometry Riddles
T A C T I L E
V I S U A L
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IIN
NG
Tell students that in this lesson they will explore the properties
of polyhedrons by solving more riddles like the one in the
Math Message.
Pose additional riddles for students to solve. You might read one
clue at a time and have students guess each time. When all clues
have been given, ask a student to come up and display the correct
solid, or a picture of it in the Student Reference Book, and name it.
Ask the student to explain why it is that particular solid.
Language Arts Link Consider having students look up the
word parts tetra-, -hedron, poly-, octa-, deca-, and dodeca- in
the dictionary to give them a better understanding of the origins
and meanings of geometric terms.
Riddle 1
I am a geometric solid.
I have six faces.
All of my faces are squares.
What am I? cube
Student Page
Date
Riddle 2
Time
LESSON
11 3
䉬
Construction of Polyhedrons
102
Polyhedrons are geometric solids with flat surfaces formed by polygons.
For each problem below—
䉬 Decide what the polyhedron should look like.
䉬 Use straws and twist-ties to model the polyhedron.
䉬 Answer the questions about the polyhedron.
Look at page 102 of the Student Reference Book
if you need help with the name.
1.
2.
Riddle 3
I am a polyhedron.
I have 5 faces.
Four of my faces are formed by triangles.
One of my faces is a square.
a.
After you make me, draw a picture of me in the
space to the right.
b.
What am I?
c.
How many corners (vertices) do I have?
d.
What shape is my base?
Square pyramid
5
Square
I am a polyhedron.
I have 4 faces.
All of my faces are formed by equilateral triangles.
All of my faces are the same size.
a.
After you make me, draw a picture of me in the
space to the right.
b.
What am I?
c.
How many corners (vertices) do I have?
d.
What shape is my base?
862
I am a polyhedron.
I have the fewest number of faces of all the polyhedrons.
All of my faces are triangular.
I come to a point at the top.
What am I? Triangular pyramid, or tetrahedron
Riddle 4
Triangular pyramid, or regular tetrahedron
Math Journal 2, p. 293
I am a geometric solid.
I have two surfaces.
My base is formed by a circle.
I come to a point at the top.
What am I? cone
4
Triangle
I am a polyhedron.
My faces are pentagons.
I am useful for calendars.
My picture is on page 103 of the Student Reference Book.
What am I? dodecahedron
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Student Page
Date
Riddle 5
Time
LESSON
Drawing a Cube
11 3
I am a polyhedron.
I have two triangular bases.
My other faces are rectangles.
Sometimes I am used for keeping doors open.
What am I? Triangular prism
Knowing how to draw is a useful skill in mathematics. Here are a few ways to draw
a cube. Try each way. Tape your best work at the bottom of page 295.
A Basic Cube
Draw a square.
Draw another square that overlaps your first square.
The second square should be the same size as the first.
Riddle 6
Connect the corners of your 2 squares as shown.
This picture does not look much like a real cube. One problem
is that the picture shows all 12 edges, even though not all the
edges of a real cube can be seen at one time. Another problem is
that it is hard to tell which face of the cube is in front.
I am a geometric solid.
I have only one surface.
My one surface is curved.
I have no base.
What am I? sphere
A Better Cube
Begin with a square.
Next, draw 3 parallel line segments going right and up from
3 corners of your square. The segments should all be the
same length.
Using Straws and Twist-Ties
to Model Polyhedrons
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VIN
IN
ING
Finally, connect the ends of the 3 line segments.
This cube is better than before, but it shows only the edges
and corners, not the faces. If you want, try shading your cube
to make it look more realistic.
(Math Journal 2, p. 293)
Math Journal 2, p. 294
Remind students how the straws and twist-ties were used to make
frames for cubes and rectangular prisms in Lesson 11-2. Students
should work with partners to construct polyhedrons and answer
the riddles on journal page 293. Although students’ constructions
might differ in size, the shapes should have the same properties.
Drawing Cube Models
(Math Journal 2, pp. 294 and 295)
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IIN
NG
Have students follow the directions on journal pages 294 and 295
to draw cubes in three different ways. Tell students to practice on
blank paper and then tape their best example of each method on the
bottom of journal page 295. Ask students to share any other method
they use to draw a cube.
286-308_EMCS_S_MJ2_G4_U11_576426.indd 294
Adjusting
the Activity
ELL
If possible, have several geometric solids
available for students to handle as they work
in their journals.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
Student Page
Date
Time
LESSON
Designing a Bookcase
11 3
2 Ongoing Learning & Practice
2/15/11 6:15 PM
Stephen wants to build a bookcase for his books. To help him design the bookcase,
he measured the height of each of his books. He rounded each measurement to the
1
nearest _
8 of an inch. His measurements are given below.
1
Book Heights (to the nearest _
8 inch)
3
1
1
1
1
7
1
1
1
1
1
1
1
_
_
_
_
_
_
_
_
_
_
_
_
6_
2 , 9 4 , 7 8 , 7 2 , 8, 6 8 , 9 4 , 9 4 , 9 4 , 9 4 , 9 4 , 8 4 , 8, 8 4 , 8 8 ,
3
1
1
7
7
1
1
1
7
7
1
1
1
_
_
_
_
_
_
_
_
_
_
_
_
6_
2 , 7 8 , 9, 6 8 , 9 8 , 6 8 , 7 2 , 8, 8 4 , 9 4 , 6 8 , 6 8 , 8 4 , 8 4 , 8 4
INDEPENDENT
ACTIVITY
(Math Journal 2, pp. 295A and 295B)
Students plot book heights in fractions of an inch on a line plot.
Then they use the line plot to solve fraction and mixed-number
addition and subtraction problems.
Book Heights
Number of Books
Plotting Book Heights
Plot the data set on the line plot below.
X
X
1
62
X
X
X
X
X
X
X
1
7
X
X
X
X
X
72
X
X
X
X
X
XX
8
Height (inches)
X
1
82
9
X
X
X
X
X
X
XX
1
92
Sample number models are given.
Use the completed plot to answer the questions below and on journal page 295B.
Write a number model to show how you solved each problem.
3
98 inches
1. a. What is the maximum book height?
1
62 inches
b. What is the minimum book height?
7
28 inches
c. What is the range of the data set?
3
1
7
– 62 = 28
Number model: 9 8
_
_
_
_
_
_
Math Journal 2, p. 295A
295A-295B_EMCS_S_MJ2_G4_U11_576426.indd 295A
3/16/11 11:14 AM
Lesson 11 3
863
Student Page
Date
Math Boxes 11 3
Time
LESSON
11 3
8_4
continued
55
(Math Journal 2, p. 296)
1
What is the median of the data set?
2. a.
3.
inches
How much longer is the maximum height than the median height?
3
1
1
1
1_8 inches Number model: 9_8 – 8_4 = 1_8
b.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 11-1. The skill in Problem 5
previews Unit 12 content.
Suppose that Stephen wants to make the space between the shelves on his bookcase
7
_
8 of an inch taller than his tallest book.
How far apart should he make the shelves?
1
Number model:
a.
10_8, or 10_4 inches apart
2
3
7
2
9_8 + _8 = 10_8
5
If the thickness of the wood he uses for the shelves is _
8 inch, what will be the
total height of each shelf? (Hint: The total height is the thickness of the wood
plus the distance between shelves.)
5
7
2
7
108 inches Number model: 10_8 + _8 = 10_8
b.
Ongoing Assessment:
Recognizing Student Achievement
_
4.
Stephen decides to make the bookshelf two shelves high. He will put all the books
that are 8 inches tall or shorter on the top shelf and all the books that are more
than 8 inches tall on the bottom shelf.
_
What will be the difference in height between the tallest book on the bottom
shelf and the shortest books on the bottom shelf?
3
1
1
1
18 inches Number model: 9_8 – 8_4 = 1_8
b.
_
5.
Math Boxes
Problem 3a
Use Math Boxes, Problem 3a to assess students’ ability to find multiples of
whole numbers less than 10. Students are making adequate progress if they are
able to name the first ten multiples of 6. Some students might be able to solve
Problem 3b, which involves finding multiples of whole numbers greater than 10.
What will be the difference in height between the tallest books on the top shelf
and the shortest books on the top shelf?
1
1
1
12 inches Number model: 8 – 6_2 = 1_2
a.
INDEPENDENT
ACTIVITY
Designing a Bookcase
[Number and Numeration Goal 3]
Make up and solve your own problem about the book height data.
Answers vary.
Study Link 11 3
Number model:
INDEPENDENT
ACTIVITY
(Math Masters, p. 329)
Math Journal 2, p. 295B
295A-295B_EMCS_S_MJ2_G4_U11_576426.indd 295B
3/30/11 12:57 PM
Home Connection Students solve riddles about
geometric solids.
3 Differentiation Options
READINESS
Sorting Geometric Solids
Time
LESSON
Math Boxes
11 3
1.
Draw the figure after it is translated
to the right.
2.
Find the solution of each open sentence.
7
a. _
8
6
_
1
-s= _
8
s=
1
1
b. t + _ = _
4
2
t=
3
c. _
10
2
d. _
8
1
-m=_
5
3
+x=_
4
m=
x=
8
_1
4
1
_
10
_4
_1
8
2
, or
107
3.
Finally, have students sort the objects by use. Ask questions such
as the following:
55 148
Name the first ten multiples of each number.
18
a.
6
12
b.
86
172 , 258 , 344 , 430 , 516 , 602 , 688 , 774 , 860
,
,
24
,
30
,
36
,
42
,
48
,
54
,
60
●
Which containers would be easiest to pack together in a box?
●
Why might containers for liquids and dry materials have
different shapes?
●
Why might the word container be a good description for most
of the objects?
9
4.
Insert parentheses to make each number
sentence true.
(98.3 + 1.7)∗ 2.5 = 250
21.7 / (3 + 4)= 3.1
c. (56.3 + 3.7)∗ 3 > 5 ∗ 30
d. 13.8 - 8.3 =(26.15 - 23.4)∗ 2
5.
Gum costs $0.80 per pack. What is the
cost of
a.
a.
b.
b.
c.
d.
$3.20
$8.00
16 packs of gum? $12.80
33 packs of gum? $26.40
4 packs of gum?
10 packs of gum?
150
Math Journal 2, p. 296
286-308_EMCS_S_MJ2_G4_U11_576426.indd 296
864
15–30 Min
To investigate attributes of geometric solids, have students sort
common household items into groups based on appearance and
discuss how the objects in each group are the same and how they
are different. Then have one student sort the items according to a
different attribute and ask the other students to determine how
they were sorted.
Student Page
Date
SMALL-GROUP
ACTIVITY
2/18/11 9:17 AM
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Study Link Master
ENRICHMENT
Creating Cube Nets
SMALL-GROUP
ACTIVITY
Name
Date
STUDY LINK
11 3
15–30 Min
To apply students’ understanding of attributes of
geometric solids, have them find all possible nets, or
patterns of squares that can be folded to form a cube.
(See margin.) Students should record their nets on 1-inch grid
paper (Math Masters, page 444).
Students should eliminate any nets that are duplicates when
reflected or rotated in their unfolded state.
Geometry Riddles
Answer the following riddles.
101––103
1.
(Math Masters, pp. 389 and 444)
Time
cone
What am I?
3.
2.
I am a geometric solid.
I have two surfaces.
One of my surfaces is formed by a circle.
The other surface is curved.
q
py
pyramid
What am I? square
or rectangular pyramid
4.
I am a polyhedron.
I am a prism.
My two bases are hexagons.
My other faces are rectangles.
What am I?
I am a geometric solid.
I have one square base.
I have four triangular faces.
Some Egyptian pharaohs were buried
in tombs shaped like me.
hexagonal
g
p
prism
I am a polyhedron.
All of my faces are the same.
All of my faces are equilateral triangles.
I have eight faces.
octahedron
What am I?
Try This
5.
Write your
y
own g
geometry
y riddle.
Answers vary.
Eleven nets are possible:
Practice
6.
8.
10.
$10
7.
= $10 + (-$25)
9.
-$20 + $30 =
-$15
-$15 + (-$40) =
-$55
11.
-$70
= -$35 + (-$35)
-$100
-$400 = -$300 + (-$100)
$0 + (-$100) =
Math Masters, p. 329
327-338_EMCS_B_MM_G4_U11_576965.indd 329
2/18/11 8:37 AM
On an Exit Slip, have students describe common features of nets
that will and will not result in cubes. Sample answer: Nets that
will result in cubes have 6 squares and 14 sides. Nets that will not
result in cubes have fewer than 6 squares, have 4 squares that
share a single vertex, or have more than 4 squares in a single row.
Folding a net to form a cube
EXTRA PRACTICE
Taking a 50-Facts Test
SMALL-GROUP
ACTIVITY
5–15 Min
(Math Masters, pp. 411, 414, and 416)
See Lesson 3-4 for details regarding the administration of a
50-facts test and the recording and graphing of individual and
optional class results.
Lesson 11 3
865