1-2
Drawings, Nets, and
Other Models
1-2
1. Plan
1
What You’ll Learn
2
1
2
3
4
5
• To draw nets for three-
1.
dimensional figures
Isometric Drawing
Orthographic Drawing
Foundation Drawing
Identifying Solids From Nets
Drawing a Net
To make a foundation
drawing, as in Example 3
Math Background
Geometric principles are the
basis of drawing techniques.
For example, isometric dot paper
is a pattern of the vertices of
equilateral triangles and can
easily be used to draw equilateral
triangles and hexagons that
tesselate a plane.
A
B C
. . . And Why
1
Lesson 1-1
Draw the next figure in each sequence. See margin.
orthographic drawings
Examples
GO for Help
Check Skills You’ll Need
• To make isometric and
A
To make isometric and
orthographic drawings
To draw nets for threedimensional figures
A
Objectives
C D
D E
2.
O
X
H
■
O
X
New Vocabulary • isometric drawing • orthographic drawing
• foundation drawing • net
Drawing Isometric and Orthographic Views
You will study both two-dimensional and three-dimensional figures in geometry.
A drawing on a piece of paper is a two-dimensional object. It has length and width.
Your textbook is a three-dimensional object. It has length, width, and height.
Representing a three-dimensional object on a twodimensional surface requires special techniques.
Vocabulary Tip
In Greek, isos means
“equal” and metron
means “measure.”
In an isometric drawing,
all 3-D measurements
are scaled equally.
You can make an isometric drawing on isometric dot
paper to show three sides of a figure from a corner view.
The simple drawing of a refrigerator at the right is an
isometric drawing.
More Math Background: p. 2C
Lesson Planning and
Resources
1
1. Answers may
vary. Sample:
EXAMPLE
Isometric Drawing
Make an isometric drawing of the cube structure at the left.
See p. 2E for a list of the
resources that support this lesson.
Isometric drawing:
PowerPoint
Bell Ringer Practice
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Check Skills You’ll Need
Step 1
For intervention, direct students to:
Finding and Using a Pattern
Quick Check
Lesson 1-1
Extra Skills, Word Problems,
Proof Practice, Ch. 1
E B
1 On isometric dot paper,
make an isometric drawing
of the cube structure.
See back of book.
H
10
10 cm
10 cm
Chapter 1 Tools of Geometry
10 cm
4
7 cm
Special Needs
10 cm
10
Step 3
■
A
1, 2.
Step 2
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Below Level
L1
Have students reproduce the net in Example 4 using
scissors and graph paper.
learning style: tactile
L2
Provide cubes for students to examine as they work
through Examples 1–3.
learning style: tactile
2. Teach
An orthographic drawing is another way to show a three-dimensional figure.
It shows a top view, front view, and right-side view.
2
EXAMPLE
Guided Instruction
Orthographic Drawing
Make an orthographic drawing from the isometric drawing at the left.
Isometric drawing:
2
Orthographic drawing:
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3
Top
Dashed lines
show hidden
edges.
Front
Quick Check
Careers
Architects use orthographic
drawings for floor plans. Ask: A
floor plan of your school building
would show which view? top
Solid lines
show visible
edges.
Fro
EXAMPLE
EXAMPLE
Error Prevention
Because a foundation drawing
shows the base, students may
think there is a bottom view.
Remind them that the top view
shows the base.
Right
PowerPoint
2 Make an orthographic drawing from this isometric drawing.
See back of book.
Additional Examples
1 Make an isometric drawing
of the cube structure below.
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nt
A foundation drawing shows
the base of a structure and the
height of each part. A foundation
drawing of the Sears Tower is
shown at the right.
EXAMPLE
49
89
65
109 109
89
65
49
89
The Sears Tower
is made up of
nine sections.
The numbers tell
how many stories
tall each section is.
Foundation Drawing
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Isometric drawing:
2 Make an orthographic drawing
from the isometric drawing in
Additional Example 1.
Connection
3
The foundation drawing
shows four heights in the
nine sections of the Sears
Tower in Chicago, Illinois.
2
2
1
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Quick Check
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1
Front
Front
3 a. How many cubes would you use to make the structure in Example 3? 9 cubes
b. Critical Thinking Which drawing did you use to answer part (a), the foundation
drawing or the isometric drawing? Explain.
Answers may vary. Sample: The foundation drawing; you can just
add the five numbers.
Lesson 1-2 Drawings, Nets, and Other Models
Advanced Learners
Have students research the Platonic Solids or
Archimedean Solids and draw their nets. Students can
then make models of these solids.
learning style: verbal
Have each student bring an empty cereal or other
type of box to cut into a net. Clarify that more than
one net or pattern can be drawn for a given solid.
learning style: tactile
Top
Right
3 Create a foundation drawing
for the isometric drawing in
Additional Example 1.
11
English Language Learners ELL
L4
Ri
Right
Real-World
Foundation drawing:
t
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nt
Make a foundation drawing for the isometric drawing at the left.
2
1
Right
3
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1
Front
11
Guided Instruction
2
1
Nets for Three-Dimensional Figures
PowerPoint
A net is a two-dimensional pattern that you can fold to form a threedimensional figure. A net shows all of the surfaces of a figure in one view.
Additional Examples
4 Is the pattern a net for a cube?
If so, name two letters that will be
on opposite faces.
yes; A and
A
C, B and
E, D and F
B
D
E
F
4
Identifying Solids From Nets
EXAMPLE
Multiple Choice The net at the left shows all the surfaces of a three-dimensional
figure. Which figure can you fold from the net?
C
5 Draw a net for the figure with
a square base and four isosceles
triangular faces. Label the net
with its dimensions.
10
The black triangle is shown on the largest side of the figure instead of the smallest
in choices C and D. Those choices cannot be correct. The black triangle will be at
the same end as the two diagonal lines when the net is folded into a box. Choice B
is correct.
cm
1
4
5
A
A
B
B
8 cm
m
0c
D
C
C
D
E
E
A
Test-Taking Tip
B
C
D
E
The number of surfaces
on a solid must match
the number of regions
in its net.
1
Package designers can use nets to help design containers.
5
Drawing a Net
EXAMPLE
Packaging Draw a net for the graham cracker box. Label the net with
its dimensions.
Resources
• Daily Notetaking Guide 1-2 L3
• Daily Notetaking Guide 1-2—
L1
Adapted Instruction
14 cm
GRAHAM
CRACKERS
20 cm
Closure
7 cm
20 cm
7 cm
14 cm
Explain how isometric, orthographic, and foundation drawings
are alike and how they are
different. Sample: All show
three-dimensional figures on
a two-dimensional surface;
isometric drawings show three
faces, orthographic drawings
show the outlines of three views,
and foundation drawings use
the top view of orthographic
drawings to show the height
of each part of the figure.
Quick Check
5 Draw a net for the solid shown. Label the net with its dimensions. See margin.
10 cm
10 cm
7 cm
12
12
F
E
D
C
B
E
D
C
B
A
3
E
D
C
B
A
2
E
D
C
B
A
4 The net at the right folds into the cube shown
beside it. Draw the cube and show which letters
will be on its front and top. E, C
D
8 cm
Quick Check
Chapter 1 Tools of Geometry
4 cm
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
3. Practice
Practice and Problem Solving
Assignment Guide
A
Practice by Example
Example 1
GO for
Help
On isometric dot paper, make an isometric drawing of each cube structure.
1–3. See margin.
1.
2.
3.
(page 10)
1 A B 1-10, 17-22, 28-30
2 A B
11-16, 23-27, 31-32
C Challenge
33-34
Test Prep
Mixed Review
Examples 2, 3
(page 11)
For each figure, make (a) an orthographic drawing, and
(b) a foundation drawing.
4–6. See back of book.
4.
5.
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 4, 14, 18, 22, 26.
6.
ht
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Exercises 1–3 If possible, provide
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35-38
39-45
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wooden cubes so that students
can model each cube structure.
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How many cubes would you use to make each of the following?
Example 4
7. the structure in Exercise 4 6
8. the structure in Exercise 5 6
9. the structure in Exercise 6 8
10. a model of the Sears Tower on
page 11 713
GPS Guided Problem Solving
L2
Reteaching
L1
Adapted Practice
Match each three-dimensional figure with its net.
Practice
Name
(page 12)
C
11.
A
12.
L3
L4
Enrichment
B
13.
Class
L3
Date
Practice 1-2
Points, Lines, and Planes
Refer to the diagram at the right for Exercises 1–15.
* )
1. Name AB in another way.
2. Give two other names for plane Q.
3. Why is EBD not an acceptable name for plane Q?
A
Q
Are the following sets of points collinear?
* )
* )
A.
B.
C.
D
E
4. AB and C
5. B and F
6. EB and A
7. F and plane Q
Are the following sets of points coplanar?
* )
* )
F
9. DB and FC
11. AE and DC
* )
12. F, A, B, and C
13. F, A, B, and D
14. plane Q and EC
15. FB and BD
* )
* )
C
* )
* )
8. E, B, and F
10. AC and ED
* )
B
* )
Find the intersection of the following lines and planes in the figure
at the right.
* )
* )
16. GK and LG
H
G
17. planes GLM and LPN
L
K
18. planes GHPN and KJP
M
J
19. planes HJN and GKL
N
P
© Pearson Education, Inc. All rights reserved.
* )
* )
20. KP and plane KJN
21. KM and plane GHL
Refer to the diagram at the right.
Example 5
(page 12)
22. Name plane P in another way.
Draw a net for each figure. Label the net with its dimensions. 14–16. See
back of book.
14.
15.
16.
7m
D
23. Name plane Q in another way.
24. What is the intersection of planes P and Q?
25. Are A and C collinear?
C
B
A
Q
26. Are D, A, B, and C coplanar?
27. Are D and C collinear?
* )
P
* )
28. What is the intersection of AB and DC ?
2 in.
2 in.
4 in.
30 mm
10 m
12 mm
17. Answers may vary.
Sample:
a.
6m
Apply Your Skills
17. a. Open-Ended Make an isometric drawing of a structure that can be built
using 8 cubes. a–b. See margin.
b. Make an orthographic drawing of this structure.
c. Make a foundation drawing for this structure. See left.
Lesson 1-2 Drawings, Nets, and Other Models
1.
2.
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* )
30. Are AB and plane Q coplanar?
31. Are B and C collinear?
8m
36 mm
B
29. Are planes P and Q coplanar?
13
Front
Right
b.
3.
Fr
on
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Front
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Right
Top
13
4. Assess & Reteach
For each foundation drawing, make (a) an isometric drawing on dot paper, and
(b) an orthographic drawing. 18–20. See back of book.
2
1
19.
1
Front
Use the figure below for Exercises
1–3.
2
3
2
1
20.
Front
1
3
2
3
2
1
Right
3
Right
Lesson Quiz
3
Right
18.
GPS
PowerPoint
Front
Read the comic strip and complete Exercises 31 and 32.
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ht
nt
Rig
1. Make an isometric drawing of
the cube structure.
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21. orthographic top
view
22. Answers may vary.
Sample: You may
want a bird’s-eye
view for a tourist
map showing
locations of
attractions.
ht
nt
Rig
Visualization Think about how each net can be folded to form a cube. What is the
color of the face that will be opposite the red face?
blue
green
orange
purple
23.
24.
25.
26.
2. Make an orthographic
drawing.
Front
Top
Right
3. Is the pattern a net for a cube?
If so, name two letters that will
be on opposite faces. no
27b. Answers may vary.
Sample: the net
shown in Exercise
23; it is easy to cut
and fold.
F
A
21. What type of drawing that you’ve studied in this lesson is a “bird’s-eye view”?
See left.
22. Writing Photographs of the Washington Monument are typically not taken
from a bird’s-eye view. Describe a situation in which you would want a photo
showing a bird’s-eye view. See left.
B
C
27. There are eleven different nets for a cube. Four of them are shown above.
a. Draw as many of the other seven as you can. (Hint: Two nets are the same if
you can rotate or flip one to match the other.) See margin.
b. Writing If you were going to make 100 cubes for a mobile, which of the
eleven nets would you use? Explain why. See left.
D
Make an orthographic drawing for each isometric drawing.
E
yes; A and C, B and D,
E and F
28.
GO
Homework Help
Visit: PHSchool.com
Web Code: aue-0102
4. Draw a net for the figure.
29.
30.
nline
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28–30. See back of book.
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31. Draw a net for a cylinder. (Hint: The net needs to show three regions: two
circles and a rectangle.) See margin.
Sample:
32. There are eight different nets for a pyramid with a square base. Draw as many
of them as you can. See margin.
14
Chapter 1 Tools of Geometry
31. Answers may
vary. Sample:
27. Answers may vary. Sample:
14
32.
C
Alternative Assessment
Challenge
33. Visualization Use the orthographic drawing at the right.
a. Make an isometric drawing of the structure.
b. Make an isometric drawing of the structure from part (a)
Top
after it has been turned on its base 908 counterclockwise.
c. Make an orthographic drawing of the stucture from part (b).
d. Turn the structure from part (a) 1808. Repeat parts (b) and (c).
a-c. See margin. d. See back of book.
Front
34. The net at the left is folded into a cube.
Sketch the cube so that its front face is
shaded as shown at the right. See margin.
Right
Give each student 10 cubes.
Instruct students to use all their
cubes to build a structure. Then
have students draw isometric,
orthographic, and foundation
drawings of their structures,
and have partners check that
the drawings correctly describe
the structure. Have students
dismantle the cubes, then
exchange drawings with new
partners who use the drawings
to rebuild the same structure.
Test Prep
Test Prep
Multiple Choice
35. A three dimensional figure is made with 11 cubes. The top view of the
figure shows 5 squares. Which of the following is the greatest possible
number of cubes in a stack represented by one of the five squares? D
A. 4
B. 5
C. 6
D. 7
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 75
• Test-Taking Strategies, p.70
• Test-Taking Strategies with
Transparencies
36. Which of the following shows a top, front, and right view of a threedimensional shape? G
F. an isometric drawing
G. an orthographic drawing
H. a foundation drawing
J. a net
Short Response
37. Draw a net for the rectangular box.
Label the net with its dimensions.
See margin.
7m
37. [2] correct net and
labels
1m
5m
Extended Response
1m
5m
38. Make drawings to show the top view, the
front view, and the right-side view of the
figure at the right. See back of book.
7m
[1] mislabeled net
39.
Mixed Review
GO for
Help
n
1
2
3
4
5
6
7
8
39. Reasoning What is the last digit of 345? To answer, make a table, look for a
pattern, and use inductive reasoning. Explain the pattern. See margin.
Lesson 1-1
Find a pattern for each sequence. Use the pattern to show the next two terms.
40. A, C, E, G, . . . I, K
41. 2, 6, 12, 20, 30, . . . 42, 56
42. 4, 16, 64, 256, . . . 1024, 4096
43. 100, 95, 85, 70, 50, . . . 25, –5
Last digit
of 3n
3
9
7
1
3
9
7
1
11R1
4
45
The pattern
3, 9, 7, 1
repeats 11
times for
n ≠ 1 to 44.
For n ≠ 45,
the last digit
is 3.
Skills Handbook x 2 Algebra Evaluate each expression for the given values.
44. a2 + b2 for a = 3 and b = -5 34
Lesson 1-2 Drawings, Nets, and Other Models
lesson quiz, PHSchool.com, Web Code: aua-0102
33. a.
b.
c.
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15
34. Answers may vary. Sample:
Top
Fro
45. 12 bh for b = 8 and h = 11 44
Front
Right
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15