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Geometry
ISBN: 11
Extending 0-536-08809-8
652
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
Chapter Perspective
The beauty of mathematics is accentuated through the lens of geometry. Geometric designs in buildings, such as the Alhambra in Granada, Spain shown in this photo, reveal symmetry and give insight into this important mathematical idea. The intriguing world of patterns, designs, and tessellations brings to life some fascinating relationships among mathematics, nature, and art. Also, whether you are thinking of an astronomer describing the changes in the positions of stars in the sky, or a physicist describing the movement of particles in an accelerator, you notice that motion plays an integral part in natural events. The idea of transformations helps us give useful mathematical descriptions of motions in our world. In this chapter, we focus on gures, relationships, and patterns in space, including geometric descriptions of motion, tessellations, and special polygons to extend the development of your spatial sense.
Big Ideas
Proportionality: If two quantities vary proportionally, that relationship can be represented as a linear function. Patterns: Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. Orientation and Location: Objects in space can be oriented in an innite number of ways, and an objects location in space can be described quantitatively. Shapes and Solids: Two- and three-dimensional objects with or without curved surfaces can be described, classied, and analyzed by their attributes. Transformations: Objects in space can be transformed in an innite number of ways, and those transformations can be described and analyzed mathematically.
Connection to the NCTM Principles and Standards
The NCTM Principles and Standards for School Mathematics (2000) indicate that the mathematics curriculum in geometry for grades PreK8 should prepare students to apply transformations and use symmetry to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems (p. 41).
Connection to the PreK8 Classroom
In grades PreK2, students use their own physical experiences with shapes, such as tting pieces into a puzzle, to learn about slides, turns, ips, and symmetry. In grades 35, students are ready to mentally manipulate shapes to make predictions, learn the mathematical language to describe their predictions, then verify their predictions physically. In grades 68, students further develop spatial sense by analyzing motions of geometric gures and by putting polygons together to form patterns such as tessellations.
ISBN: 0-536-08809-8
653
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
654
C H A P T E R 11
EXTENDING GEOMETRY
Section
11.1
Translations, Rotations, and Reections Connecting Transformations and Symmetry Transformations That Change Size Transformations That Change Both Size and Shape
In this section, we investigate the geometric transformations that do and do not affect a gures shape or size. We use mathematical ideas and symbols to describe, analyze, and compare the different types of transformation. You will have opportunities to use special computer software to create and move the geometric gures.
Transformations
Essential Understandings for Section 11.1
The orientation of an object does not change the other attributes of the object. Congruent gures remain congruent through translations, rotations, and reections. Shapes can be transformed to similar shapes (but larger or smaller) with proportional corresponding sides and congruent corresponding angles. Some shapes can be divided in half where one half folds exactly on top of the other (line symmetry). Some shapes can be rotated around a point in less than one complete turn and land exactly on top of themselves (rotational symmetry).
In Mini-Investigation 11.1, you are asked to use some motions that are related to geometric transformations.
Draw a picture to illustrate each motion that you used. Technology Extension: Use geometry exploration software (GES) to create a triangle or quadrilateral. Explore the GES options for sliding, turning, and ipping the gure. (See Appendix B.) Describe how GES may be helpful in studying different ways to move gures.
M I N I - I N V E S T I G A T I O N 11 . 1
Solving a Problem
In what ways (slide, turn, ip, or combinations) can you move pentomino (a) to test whether it matches pentominos (b), (c), (d), and (e)? (Use a tracing if one will help.)
(a)
(b)
(c)
(d)
(e)
Translations, Rotations and Reections
ISBN: 0-536-08809-8
In Mini-Investigation 11.1, you may have observed that an object may be slid, turned, or ipped without changing its shape or size. With these three simple
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
T R A N S F O R M AT I O N S
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P
P B
A F I G U R E 11 .1
A gure and its translation image.
motions, or combinations of these motions, we can move a gure anywhere in space to match another gure that is congruent to it. We use these motions to describe transformations in the following subsections. Slides or Translations. You encounter the motion of slides often in everyday life: a notebook sliding across a desk, a window frame sliding up and down, or a drawer sliding in and out. In Figure 11.1, the blue pentomino shape is the image of the yellow pentomino shape after all the points in the plane have been slid in the direction and distance indicated by the slide arrow (or directed segment) from point A to point B. To visualize the slide, trace the yellow pentomino outline and the slide arrow. Then, slide point A on the tracing paper along the arrow to point B. (If you trace the entire arrow, you will not accidentally turn the tracing paper.) The pentomino shape you traced on your paper should now coincide with the blue pentomino shape, which is the image of the slide. The physical motion of sliding, mathematically called a translation, is dened as follows.
Denition of a Translation
If each point P in the plane corresponds to a unique point in the plane, P , such that directed segment PP is congruent and parallel to the directed segment AB, then the correspondence is called the translation associated with the directed segment AB and is written TAB .
The denition indicates that when a gure is translated along a directed segment AB, each point, P, of the gure slides to point P , in the same distance and direction that point A slides to get to point B. To describe further the effect of a translation, we write TAB(P) = P to say that P is the image of P under the translation associated with the directed segment AB. When considering the translation TAB, point B becomes the image of point A. If we want to indicate a slide from B to A, we write TBA.
ISBN: 0-536-08809-8
Turns or Rotations. Turns in the real world include food rotating on a turntable in a microwave oven, doorknobs turning, and a Ferris wheel turning around its large axis as the seats turn on their individual axes. The blue pentomino shape
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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C H A P T E R 11
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P
C
P
F I G U R E 11 . 2
A gure and its rotation image.
P
shown in Figure 11.2 is the image of the yellow pentomino shape after all the points in the plane have been turned around point C in the direction and angle measure indicated by the turn angle, PCP . T visualize the turn, trace the yellow pentomino outline and PC of the turn angle. o Place the point of your pencil on point C and hold point C still while you turn the tracing paper in the direction and angle measure of the turn angle until the PC drawn on your tracing paper coincides with P C on the page. The pentomino shape you traced on your paper should now coincide with the blue pentomino shape, the image of the turn. In mathematics a turn is called a rotation, and is dened as follows.
Denition of a Rotation
C P
Consider a point C and an angle with measure A from - 180 to 180. If each point P in the plane corresponds to a unique point in the plane, P , such that m PCP A and PC = P C, then the correspondence is called the rotation, with center C and angle A, and is written RC, A .
The denition indicates that when a gure is rotated around a center point C, each point, P, of the gure turns around C through an angle, a, to point P , which is the same distance from C as P. When considering the rotation R C, a, we write R C, a(P) = P to say that P is the image of P under the rotation associated with the center C and angle a. In some cases, as in Figure 11.3, the center of the rotation, C, is in the interior of a gure, and the rotation image of a point P of the gure about that center is another point P of that same gure. When this happens, the gure is rotated onto itself. We will consider this special case in more detail on pp. 663664. Flips or Reections. Flipped images in the real world appear when you look in mirrors, turn transparencies over on the overhead projector, or press an inked stamp onto a piece of paper. The blue pentomino shape in Figure 11.4 is the image of the yellow pentomino shape after all the points in the plane have been ipped over line /. To visualize the ip, trace the yellow pentomino outline and line /, marking point L on line /. Lift your tracing paper, ip it over, and replace it on the page by laying the traced line / over line / on the page with point L on the tracing paper
F I G U R E 11 . 3
RC,90(P)
P
P L
P
F I G U R E 11 . 4
ISBN: 0-536-08809-8
A gure and its reection image.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
T R A N S F O R M AT I O N S
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matching point L on the page. The pentomino shape that you traced should now coincide with the blue pentomino shape, the image of the ip. The physical motion of ipping (mathematically called a reection) can be dened as follows.
Denition of a Reection
If each point on line / corresponds to itself, and each other point P in the plane corresponds to a unique point P in the plane, such that / is the perpendicular bisector of PP , then the correspondence is called the reection in line /, and is written M/ .
P P
The denition indicates that when a gure is reected about a line /, each point, P, of the gure is ipped to point P , so that line / is the perpendicular bisector of PP . With this notation, we write M /(P) = P to say that P is the image of P under the reection in /. In some cases, as in Figure 11.5, the reection image in line / of a point P of a gure is another point P of that same gure. When this happens, the gure is reected onto itself. We will consider this special case in more detail on p. 661. In Example 11.1, you are asked to identify the motion that produced the given image.
F I G U R E 11 . 5
M/(P)
P
Example
11.1
Identifying Transformations
Identify which transformation (translation, rotation, or reection), if possible, that would change each yellow gure shown to the corresponding blue image. Justify your answers and use correct notation to show the effect of the transformation on point A.
D B C B C C
A B
A
D
A
D B
A
D C (a) B B A A
ISBN: 0-536-08809-8
(b) C C D D B C (d) A A D D B C
(c)
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
658
C H A P T E R 11
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SOLUTION
In part (a), the transformation is a reection because the orientation of the gure has changed. In the original rectangle, reading from A clockwise around the gure gives ABCD. In the blue image, reading clockwise from A gives A D C B . We write M line of reflection (A) = A . In part (b) the transformation is a rotation. Its orientation hasnt changed, but the segments that were horizontal now arent. We write R C, angle of rotation(A) = A . In part (c) the transformation is a translation. If A is connected to A , B to B , C to C and D to D , the segments formed are parallel and congruent. All the points moved the same distance in the same direction. We write TBB(A) = A . In part (d) the blue image of the yellow gure cant be produced with a single transformation. The orientation of the gure has changed, so it must have been reected. But the image doesnt appear where it would if a mirror had been placed between the gures. Thus it has been translated and reected.
YOUR TURN
Practice: Identify the type of transformation that would transform each yellow gure to the corresponding blue image. Use correct notation to show the effect of the transformation on point A.
A
A
A (a) A (b)
A
A
A A (c) (d)
Reect: Which of the transformations in the practice problem changed the orientation of the points in the gure? When given a gure and a transformation such as a reection, rotation, or a translation, there are various techniques that can be used to nd the image of the gure under the specied transformation. Example 11.2 illustrates some ways this can be done.
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
T R A N S F O R M AT I O N S
659
Example
11.2
Using Transformations
Find the image of quadrilateral ABCD under the following transformations: a. TMN b. R O,45
M A 45 D C
c. M /
N B
O
SOLUTION
a.
M A A
N B B
D
D A
C B
C
b.
O
45
D A B D
C
C
c.
A
B
D D
C C
A
B
ISBN: 0-536-08809-8
Henrys Solution: I drew ABCD on paper and then traced it on another sheet of paper. For (a) I made two dots on my tracing paper and slid one dot from M to N, using the other dot to ensure that the direction didnt change. For (b) I made a dot on the tracing paper to place on point O and another dot on the side of angle O. Then I rotated the tracing paper so that the dot on the side of the angle turned through an angle of 45. Then I marked points on the original paper for A , B , C , and D .
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
660
C H A P T E R 11
EXTENDING GEOMETRY
For (c) I marked two dots on line / and then traced ABCD and the dots. Then I ipped the tracing over line /, keeping the traced dots on the original. Then I marked points on the original paper for A , B , C , and D . Lenas Solution: I used GES and the instructions for creating translations, rotations, and reections in Appendix B.4, p. 871, to complete (a), (b), and (c).
YOUR TURN
Practice: Copy ^ ABC and use a method of your choice to nd the image of ^ ABC under the following transformations. a. TRS b. R O,90 c. M /
R A O 90 B C S
Reect: What is true about the gure and its image under each of the preceding transformations? Combinations of Motions. As indicated in the solution to Example 11.1, no single translation, rotation, or reection transformed gure ABCD to gure A B C D in part (d). It required a translation and then a reection over the line of translation. This combination motion is called a glidereection. In MiniInvestigation 11.2 on the next page, you will see the results of following a reection with a reection, a rotation with a reection, and a translation with a rotation. All two-motion combinations of rotations, reections, or translations other than the translation followed by a reection can be replaced by a single rotation, reection, or translation. You will also have an opportunity to explore combinations of motions in Exercises 39, 40, 46, and 49 at the end of this section. Transformations and Congruence. Experiences with motions in the real world suggest that translations, rotations, reections, and glidereections dont change the size or shape of a gure because the points in the plane maintain their relative distances from one another. In other words, distance is preserved. Mathematicians call a transformation that preserves distance and other characteristics a congruence transformation, or an isometry. The properties of polygons and transformations suggest that translations, rotations, reections, and glidereections are isometries. Thus, in an isometry, not only distance but also betweenness, angle measure, and size and shape are preserved. That is, when one of these properties is determined in a gure, it is the same in the transformed image of the gure. Thus a gure can be repositioned anywhere in space without changing its size or shape through the use of no more than the four types of transformation that we have discussed. As a result, we can state the transformation denition of congruence on the next page.
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
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Talk about which of those combinations of motions you could have completed in just one motion. Explain how.
M I N I - I N V E S T I G A T I O N 11 . 2
Technology Option
Use GES or other tools to describe the result of the following combination of motions on the gure shown: a. Reect ABC over line m and then reect the resulting image over line n. b. Rotate ABC 90 counterclockwise around point P and then reect the resulting image over line n. c. Translate ABC on line PQ from C to P and then rotate the resulting image 180 clockwise around point Q.
A
B P C
Q
m
n
Denition of Congruence, Using Transformations
Two gures are congruent if and only if there exists a translation, rotation, reection, or glidereection that sets up a correspondence of one gure as the image of the other.
An interesting illustration of how transformations can be used to solve problems is the crossword puzzle problem in Exercise 56 at the end of this section.
(a) Original figure
Connecting Transformations and Symmetry
Symmetry in the Plane. The idea of symmetry plays a central role in art, interior design, landscaping, and architecture, as well as in mathematics. When a plane gure can be traced and folded so that one half coincides with the other half, as shown in Figure 11.6, we have modeled the congruence transformation we have referred to as a reection, in which each point in one half of the gure corresponds in a special way to a unique point in the other half of the gure and each point on the line of reection corresponds to itself. In this case, we say that the gure has reectional symmetry. The fold line is called the line of symmetry.
(b) FoldDo the halves match?
ISBN: 0-536-08809-8
F I G U R E 11 . 6
Folding paper to illustrate reectional symmetry.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
Analyzing Student Work: Transformations
Connection to the PreK8 Classroom
Elementary school students are asked to test their understanding of important mathematical concepts, such as transformations, by applying logical reasoning to questions about the concept. Below are three student responses to the following question.
Elizabeth took the parallelogram below and performed some slides, ips, and turns with it. When she nished, she claimed she had a rectangle. Is it possible that Elizabeths claim was correct? Why or why not?
Analyze the work for students A, B, and C.
Give your interpretation of
Student A
what each student was thinking.
Which of the student
explanations shows the deepest understanding of the concepts of slides, ips, and turns? Why?
What other reactions do
you have to the student work?
Student B
Student C
ISBN: 0-536-08809-8
Source: 2006 Heinemann Publishing. www.heinemann.com/math, Question ID = 12, #1, 2, 3.
662
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
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F I G U R E 11 . 7
Testing for reectional symmetry with a plastic reector.
Source: GeoReector Mirror courtesy of Learning Resources, Inc.
A commercially produced piece of plexiglass, shown in Figure 11.7, can also be used to identify reectional symmetry. If a plastic reector can be placed on a gure so that the reection of half the gure ts exactly on the other half, the line of the reector is a line of reectional symmetry. Another type of symmetry is rotational symmetry. To test a gure for rotational symmetry, we can use a trace and turn test, as shown in Figure 11.8. The red gure is traced in blue on a piece of tracing paper or clear plastic. Then, keeping the blue gure directly on top of the red gure and holding the center point xed, turn the blue tracing until it again coincides with the red gure. Since the blue tracing ts exactly on the red gure after rotating it 90, we say the red gure has 90 rotational symmetry. The blue tracing also ts exactly on the red gure after rotating it 180, 270, and 360. Since a tracing of any gure will t back on the gure after a rotation of 360, we concern ourselves only with angles of rotation a, where a 6 360. Also, if a gure has a rotational symmetry, a tracing of it will also coincide with the gure if rotated na, where n is any nonzero integer. Because of this, we usually use the smallest possible angle a through which the tracing can be rotated to t back on the gure to describe the rotational symmetry of the gure. We say that the gure has
A
A
ISBN: 0-536-08809-8
F I G U R E 11 . 8
Using tracing paper to test for rotational symmetry.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
A
664
C H A P T E R 11
EXTENDING GEOMETRY
a rotational symmetry. The point that is held xed is called the center of rotational symmetry. When a gure has 180 rotational symmetry, we say that the gure has point symmetry about its center of rotation. In a gure with point symmetry, the center of the half turn (180 rotation) is the midpoint of a segment connecting any point of the gure and its image after the turn. To connect transformations and symmetry, we observe that the trace and turn test for rotational symmetry models the congruence transformation we have referred to as a rotation, in which each point of the gure corresponds in a special way to another point of the gure, except for the xed point, which corresponds to itself. We apply these ideas about symmetry in Example 11.3.
Example
11.3
Describing Symmetry Properties
Describe the symmetry properties of the gure on the left.
SOLUTION
The star shown has reectional symmetry. There are ve lines of symmetry, as shown below. b. A tracing can be made to coincide with the star after it is rotated 72 about its center, so the star has 72 rotational symmetry.
YOUR TURN 2
a.
Practice: Describe the symmetry properties of the following gures:
1
3
5
4
(a) Rhombus
(b) Equilateral triangle
(c) Pinwheel
Reect: differ.
Explain how the symmetry properties of a rhombus and a parallelogram
In Chapter 10, we learned that triangles and quadrilaterals could be classied according to side and angle properties. Classes of triangles or quadrilaterals thus formed were given special names, such as equilateral triangles and parallelograms. Triangles and quadrilaterals also may be classied according to symmetry properties, but it isnt common practice to give special names to the classes of gures formed. In Exercise 58 at the end of this section, you will be asked to classify triangles and quadrilaterals according to their symmetry properties.
Connection to the PreK8 Classroom
Just as symmetry is basic to an understanding of the universe, so are ideas of symmetry necessary to an understanding of mathematics. Students in the primary grades fold and cut paper to make hearts, pumpkins, and other gures having lines of symmetry. Ideas of symmetry are gradually expanded as students progress through elementary school.
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
T R A N S F O R M AT I O N S
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B A P
C
Transformations That Change Size
Size Transformations. The idea of same shape but not necessarily same size plays an important role in everyday life. For example, people enlarge photographs as shown with the buttery pictures below. They also enlarge blueprints and draw large circuit diagrams on a wall and shrink them to the size of a microchip. What guarantees that these larger and smaller images retain the essential characteristics of the originals? One way is to use a special transformation that enlarges or shrinks the size of a gure along specied lines by multiplying distances by a given factor. The physical action of enlarging or shrinking, mathematically called a size transformation, is dened as follows (with reference to Figure 11.9).
B A B A P C P
C
O F I G U R E 11 . 9
Using a size-changing transformation to enlarge and shrink a gure.
Denition of a Size Transformation
If point O corresponds to itself, and each other point P in the plane corre r (OP) for r>0, then the sponds to a unique point on OP such that OP correspondence is called the size transformation associated with center O and scale factor r and can be written SO,r .
The denition indicates that when a gure is shrunk or enlarged from a center O, by a factor r, the image of each point P of the gure is determined by multiplying OP by r to produce OP on OP . P is the image of P. With this notation, we can write SO,r(P) = P to say that P is the image of P under the size transformation with center O and scale factor r. The enlarging and shrinking lines are determined by a point called the center of the size transformation. The multiplier that enlarges or shrinks the lengths is called the scale factor. For example, the blue gure, A B C P , in Figure 11.9 is the image of parallelogram ABCP when it is enlarged by using point O as the center and 2 as the scale factor. Note that OA is 2 times OA, OB is 2 times OB, OC is 2 times OC, and OP is 2 times OP. The yellow gure, A B C P , is the shrunken image of ABCP when 1 is 2 the scale factor instead of 2. In this case, OA is one-half OA, OB is one-half OB, OC is one-half OC, and OP is one-half OP. Size transformations can also be carried out with a computer drawing program, as illustrated in Figure 11.10.
ISBN: 0-536-08809-8
Buddy Mays/CORBIS
Steve Kaufman/CORBIS
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
666
C H A P T E R 11
EXTENDING GEOMETRY
Original image
Transformed image
F I G U R E 11 .1 0
Effecting a size transformation by using a computer drawing program.
Source: 2006 Corel Corporation Ltd. Box shot(s) reprinted with permission from Corel Corporation.
Size Transformations and Similarity. To relate size transformations to similarity, we rst consider some of the size properties of transformations. Measuring the sides BC B of the parallelograms in Figure 11.9 reveals that AB = AC . This result suggests that B the ratios of lengths of sides of a gure and the ratios of the lengths of corresponding sides in its image are equal. Because of this property and the fact that size transformations also preserve betweenness and angle measure, size transformations also preserve shape. Sometimes a gure can be made to correspond to a gure with the same shape by a single size transformationbut not always. In Figure 11.11 on p. 671, the original triangleon the left in part (a)was rst reected in line r, then this reected image was subjected to a size transformation to produce the blue image of the triangle shown in part (b). Similar gures were described, with focus on proportional variation, on p. 384 and on p. 621. The idea that a combination of transformations is needed to transform a gure into a gure of different size and shape is used in the following denition of similarity.
Denition of Similarity, Using Transformations
Two gures are similar if and only if there exists a combination of an isometry and a size transformation that generates one gure as the image of the other.
ISBN: 0-536-08809-8
In Example 11.4, a size transformation along with the problem-solving strategies of draw a picture and write an equation are used to solve a practical problem.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 1
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r (a) F I G U R E 11 .11
(b)
Transforming a triangle into another triangle that is the same shape.
Example
11.4
Problem Solving: The Poster Problem
The service organization at Lonnies school is planning a fundraiser with stars as the theme. Lonnie can only nd a pattern for a star that is 8 centimeters high, which is too small for the posters. How can Lonnie enlarge the star pattern so that it is 20 centimeters high?
SOLUTION
Trace the center O of a size transformation and the 8-centimeter star pattern in the lower left corner of the paper. The height needs to be changed from 8 to 20 centimeters, so the scale factor is 20 , 8, or 2.5. Draw the rays from O through
ISBN: 0-536-08809-8
O
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
668
C H A P T E R 11
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the tips of the small star pattern and measure 2.5 times the distance from O to a small star tip along these rays and mark the tips of the larger star. Connect every other point to form the 20-centimeter star pattern.
YOUR TURN
Practice: Use a size transformation to nd an image of the pentagon that is threefourths as wide as the following pentagon.
Reect: Could you use a size transformation to nd an image that is congruent to the original shape? Explain your answer and use mathematical notation to symbolize the transformation.
Transformations That Change Both Size and Shape
Topological Transformations. Transformations that represent shrinking, stretching, or bending a curve or surface without tearing it or joining points are called topological transformations. These transformations can change both the size and shape of a gure, as shown in Figure 11.12. Note that the images have no holes or breaks that were not also in the originals. The fact that connectedness doesnt change in a topological transformation leads to other important characteristics that remain constant, such as the characteristic of having no ends. For example, squares, triangles, and all other polygons can be transformed by a topological transformation into a circle. If one gure can be transformed to another by a topological transformation, the two gures are topologically equivalent. For example, a rubber band is topologically equivalent to a compact disk, and a bagel is topologically equivalent to a coffee mug with a handle. Also, a schematic diagram of a circuit is topologically equivalent to a diagram to scale of the actual circuitry; how the components and wires are connected and their relative locations are what is preserved. However, a picture frame and a grocery sack are not topologically equivalent. If the picture frame were made of modeling clay, it could never be formed into the shape of a grocery sack by just stretching or squeezing the clay. Some parts would have to be joined or the clay broken apart, neither of which is allowed in a topological transformation.
(a) Topological transformation of a line. F I G U R E 11 .1 2
(b) Topological transformation of a circle.
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Topological transformation of a line and a circle.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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In Mini-Investigation 11.3 you are asked to analyze topological transformations further.
Make a chart showing your classication. Talk with classmates and compare your charts.
M I N I - I N V E S T I G A T I O N 11 . 3
Using Mathematical Reasoning
How would you classify the digits 09 into topologically equivalent gures?
In Exercise 54 at the end of this section you are asked to use a topological transformation to solve an interesting problem, the castle court problem.
Connection to the PreK8 Classroom
Many young children can distinguish topological properties such as connectedness, separation, inside, and outside before they can distinguish characteristics of rigid geometry such as length, angle measure, or relative position. Elementary school students extend their spatial visualization abilities by drawing on balloons and watching what changes and what doesnt change as the balloon is inated and deated. The topological concepts of a gure being connected and separating a plane are used later in classifying shapes and creating formal denitions for polygons and other geometric gures.
Problems and Exercises
for Section 11.1
8.
figure image
A. Reinforcing Concepts and Practicing Skills
1. What information is necessary for identifying a specic slide? 2. How is the information in Exercise 1 related to the mathematical notation for a translation? 3. What information is necessary for identifying a specic turn? 4. How is the information in Exercise 3 related to the mathematical notation for a rotation? 5. What information is necessary for identifying a specic ip? 6. How is the information in Exercise 5 related to the mathematical notation for a reection? In each diagram in Exercises 710, identify the motion or combination of motions that would produce the image. Justify your choices. 7.
image
9.
image
figure
10.
image
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figure figure
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For Exercises 1122, use tracing paper, graph paper, a geoboard, or geometry exploration software to nd the image of the quadrilateral obtained from each transformation. 11. TMN 12. TDC 13. TRS 14. TAD 15. R O,a 16. R C,180 17. R Q,-90 18. R O,180 19. M l 20. M r 21. M s 22. M t
t E A Q D P r C s S M O B
24. Describe the rotational symmetry, if any, of each of the gures in (a) through (f ). Use the trace and turn test if helpful. a. b.
c.
N
d.
R
23. Describe the reectional and rotational symmetry properties of each object. Use a plastic reector, paper folding, or GES when needed. a. Propeller
e.
b. Flag f.
c. Advertising logo
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25. Which uppercase letters of the alphabet have reectional symmetry?
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26. Which uppercase letters of the alphabet have rotational symmetry? 27. Which uppercase letters of the alphabet have point symmetry? 28. Use a compass and a straightedge to construct the image of point A under a rotation with center C and indicated by the angle a shown in the gure. (See Appendix D for construction techniques.)
A
C
B
O
A
a. Find the image of AB under the size transformation SC,3 . Then, nd the values of the ratios in (b)(d). CA CB AB b. c. d. CA CB AB e. Find the image of AB under the size transformation SC,0.5 . Then, nd the values of the ratios (f ) through (h). AB CA CB f. g. h. CA CB AB 35. Explain how to use the transformational denition of similarity to test whether each pair of gures is similar.
B
29. The pattern that follows is called a frieze pattern. Imagine that it continues indenitely in each direction, and tell as much about the symmetry of the pattern as you can.
A D E
C
30. If you place a plastic reector on line /, you form a word from the half-word shown. Explain why this is true, and create another half-word.
B
F (a) C G H A (b) F E D
31. Decide whether the pair of objects in each part of the gure are topologically equivalent. Support your answer.
(a)
B. Deepening Understanding
36. Describe the symmetry properties of the following geometric shapes or gures:
(b)
ISBN: 0-536-08809-8
32. Identify two physical objects that are topologically equivalent to each of the following: a. Ball b. Donut c. Sack with two handles 33. Classify the letters of the alphabet into groups of topologically equivalent gures. 34. Copy the following gure to nd the images and ratios specied and verify your answers:
(a) (b)
(d) (c)
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37. Make a chart showing how the isometries are alike and how they are different. Consider the following characteristics: a. Reverse orientation b. Points that map into themselves c. Mathematical notation d. Role of segments, angles, and lines 38. Figures that correspond under a translation, rotation, or reection are said to be translation congruent, rotation congruent, or reection congruent. Position two congruent gures on a piece of paper to satisfy each combination of congruences listed. Use tracing paper, mirrors, or GES to justify your choices. a. All three congruences b. Translation and rotation congruences only c. Translation and reection congruences only d. Rotation and reection congruences only e. Translation congruence only f. Rotation congruence only g. Reection congruence only h. None of the three congruences 39. Find the image of the quadrilateral when it is subjected to the glidereection TRS followed by M4 . Does it RS matter whether you translate rst and then reect, or reect rst and then translate? Justify your answer.
R F G H
41. What different types of symmetry, if any, do you see in these automobile manufacturers logos?
(a)
(b)
(c)
(d)
S E
42. Create three logos for an imaginary corporation as follows: a. One is to have rotational symmetry but no reectional symmetry. b. One is to have reectional symmetry but not rotational symmetry. c. One is to have both rotational and reectional symmetry. 43. Draw, if possible, a quadrilateral that has a. a line of symmetry but no rotational symmetry. b. rotational symmetry but no line of symmetry. c. both reectional and rotational symmetry.
40. Use tracing paper, geoboards, or GES to nd the images obtained from the following combinations of motions, where, for example, TAB*TCD indicates a translation from A to B followed by a translation from C to D.
C. Reasoning and Problem Solving
44. Copy the following gures on dot paper and nd the original shape if the shaded shape is its image under the given transformation:
A A B
P E r A B s D (a) TAB C
G F E
B C D M N (c) MMN
O
(b) RF,90
(a) RO,30 * RO,60
(b) Mr * Ms
C
A
B
(c) TAB * TBC
45. Give a type of triangle, if possible, that has a. no lines of symmetry. b. exactly one line of symmetry. c. exactly two lines of symmetry. d. exactly three lines of symmetry. e. more than three lines of symmetry. 46. Gina showed Maria the following GES sketch. Gina claimed that she created gure XYZ by using a
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combination of two motions applied to the original gure ABC. The dashed lines show the result of the rst motion. Describe the combination of motions that Gina could have used. Explain how XYZ could result from ABC in just one motion.
P
X A B Z C P Q Y
Copy the sketch and use this information to recreate gure W on it. Are P and W congruent? Explain. 51. The Recreation Room Construction Problem. A building contractor had instructions from the architect to make a rectangular recreation room larger by adding 4 feet to both the length and the width. Is the new room rectangle similar to that of the original room rectangle? Explain and support your conclusion. 52. The Mirror Problem. What is the smallest size of mirror, hung appropriately, that would allow Julie to see her total height? Explain with a diagram. 53. The Flower Garden Problem. A gardener made a model showing the layout of a planned ower garden that was quadrilateral-shaped with dimensions w, x, y, and z. If she builds the actual garden with dimensions 25w, 25x, 25y, and 25z, can she be sure the model and the actual garden are the same shape? Explain. 54. The Castle Court Problem. When the gatekeeper at a castle was asked how to get to the inner chamber, he said, After you enter, always stick to the right-hand wall. Use the topological transformation of the map of the inner court in the gure for this exercise to decide whether the gatekeeper is correct. In following this path, does the visitor go through all the halls of the inner court? If the gatekeeper had said stick to the lefthand wall, would the visitor have reached the inner chamber?
m n
47. Give a type of quadrilateral, if possible, that has a. no lines of symmetry. b. exactly one line of symmetry. c. exactly two lines of symmetry. d. exactly three lines of symmetry. e. exactly four lines of symmetry. 48. How can GES be used to show whether a transformation preserves distance? Create various gures to illustrate your techniques. 49. Matisto told his classmates that he obtained gure K from a gure A by using the following two GES transformations: TXY (A) = D and R (0,0),90(D) = K.
K
Y X
a. Copy the grid and use this information to locate gure A on it. Note that X is located at (0, - 1) and that Y is located at (2, 0). b. Is A congruent to K?
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50. Lana told her classmates that she got figure P from a gure W by using the size transformation S(0,0),2(W ) = P.
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55. Draw a topological transformation of a map of the main oor of a house with which you are familiar. Would following the gatekeepers instructions in Exercise 54 take you through each room of the house and return you to the front door? 56. The Crossword Puzzle Problem. Crossword puzzles are designed in such a way that, if C is the center of the square, the transformation R C,180, generates an image of the colored squares in the blank puzzle that is exactly the same as those in the original blank puzzle. Copy the following crossword puzzle template and color in at least 16 squares to design a crossword puzzle having that characteristic.
60. If you have access to LOGO computer software, try the following translation: Input numbers for the distance D and the direction angle A and check to see if the following program will translate the gure HALF-ARROW to a new position. Then create a new gure, and try the translation again.
TO SLIDE :A :D HALF-ARROW PU RT :A FD :D PD HALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD 10 LT 30 BK 25 END
61. If you have access to LOGO computer software, try the following rotation: Input numbers for the distance D from the center and the angle A telling how far to turn, and check to see if the following program will rotate the figure HALF-ARROW to a new position. Then create a new gure and try the rotation again.
TO TURN :D :A HALF-ARROW PU BK :D FD PD FD 2 BK 2 PU RT :A FD :D PD HALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD 10 LT 30 BK 25
57. The Box Pattern Problem. A manufacturer found that the machines in his factory could make boxes most efciently if the six-square box pattern had four squares in a row and had rotational symmetry. He asked a staff engineer to analyze all 35 possible box patterns and present a report on the patterns that had the needed characteristics. He asked the engineer to give reasons to assure him that all such patterns had been included. Prepare the engineers report for the manufacturer. 58. How would you classify different types of triangles and quadrilaterals according to their symmetry properties? Make a chart to show your classications. 59. How many other pentominos, like the one shown below, have one line of symmetry? Draw a picture of each pentomino.
62. If you have access to LOGO computer software, try the following ip: Input numbers for the distance D and the direction angle A, and check to see if the following program will ip the gure HALF-ARROW to a new position. Then create a new gure and try the ip again.
TO FLIP :A :D HALF-ARROW PU LT :A BK :D RT 90 PD FD 60 BK 120 FD 60 LT 90 PU BK :D RT 180 - :A PD REVERSEHALF-ARROW END TO HALF-ARROW FD 25RT 30 BK 10 FD0 1 LT 30 BK 25 END
TO REVERSEHALFARROW FD 25 LT 30 BK 10 FD 10 BK 30 END
D. Communicating and Connecting Ideas
63. Historical Pathways. In 1872, Felix Klein published Erlanger Programm, a paper that described a new view of geometry as the study of those properties, such as the commutative property of transformation combination, that are preserved by certain types of transformations. In your group, answer the following questions and devise a way to support your answer.
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a. b. c. d.
Does TAB*TPQ = TPQ*TAB? Does Mr*Ms = Ms*Mr? Does RA,180*RB,180 = RB,180*RA,180? Does RC,a*RC,b = RC,b*RC,a?
64. Making Connections. Write conjectures about the connections between the following: a. Reections and line symmetry b. Rotations and turn symmetry
Section
11.2
Geometric Patterns and Tessellations Polygons That Tessellate Combinations of Regular Polygons That Tessellate Other Ways to Generate Tessellations Tessellations with Irregular or Curved Sides
In this section, we explore tiling patterns that consist of geometric gures that t without gaps or overlaps to cover a plane. We use characteristics and properties of geometric gures to decide which polygon shapes can be used to tile a oor by themselves and which combinations of polygon shapes can be used to tile a oor. We also explore tiling with irregular gures.
Geometric Patterns
Essential Understandings for Section 11.2
Some shapes or combinations of shapes can be put together without overlapping to completely cover the plane. There are a nite number of ways one type of regular polygon or a pair of types of regular polygons can be put together to completely cover the plane.
Mini-Investigation 11.4 gives you an opportunity to look at the general idea of tiling a oor and which polygons can be used to do so.
Draw tilings by using tracing paper to convince someone that your solution is correct. Verify your solution in another way and describe tilings you have seen in real-world situations.
M I N I - I N V E S T I G A T I O N 11 . 4
Solving a Problem
Which of the tiles shown can a manufacturer advertise correctly as tiles that can be used by themselves to tile a rectangular oor? (Note: Partial tiles may be used at the sides to exactly ll the rectangle.)
60 90 108 120
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Equilateral triangle
Square
Regular pentagon
Regular hexagon
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(a) F I G U R E 11 .1 3
(b)
Interesting geometric patterns.
Source: (a) Weyl, Herman, Symmetry. Copyright 1952 Princeton University Press. Reprinted by permission of Princeton University Press; (b) Adapted from Branko, Grnbaum, and G. C. Shephard, Tilings and Patterns An Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission.
Geometric Patterns and Tessellations
From oor tiling to Escher art, the world is full of geometric patterns. The Greek wall pattern shown in Figure 11.13(a) is an example of a design repeating some basic element in a systematic manner, commonly known as a pattern. Another example is the interesting Mexican strip pattern shown in Figure 11.13(b). Note the translations in both patterns: the reectional symmetry in the top part of the pattern in Figure 11.13(a), and the rotational symmetry in Figure 11.13(b). We now dene a special type of pattern.
Denition of a Tessellation
A tessellation is a special type of pattern that consists of geometric gures that t without gaps or overlaps to cover the plane.
Figure 11.14 shows an example of a tessellation. Because this pattern involves the use of regular octagons and squares, we say that a combination of these gures will tessellate the plane or, more simply, will tessellate. Sometimes the word tiling is used to mean tessellation, and the word tile is used to mean tessellate. In Mini-Investigation 11.4, you may have found that an equilateral triangle will tessellate the plane. As indicated by the tessellations in Figure 11.15, the gures in a tessellation can be curved and do not need to be polygons.
F I G U R E 11 .14
Example of a tessellation.
Source: Adapted from Branko, Grnbaum, and G. C. Shepard, Tilings and PatternsAn Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission.
F I G U R E 11 .1 5
Curved tessellations.
Source: Adapted from Branko, Grnbaum, and G. C. Shepard, Tilings and PatternsAn Introduction, p. 5. Copyright 1987 by W. H. Freeman and Company. Used with permission.
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Connection to the PreK8 Classroom
Analyzing tessellations enhances students understanding of the various properties of different polygons, including angle measures, congruences, and symmetries. Creating tessellations provides students with an interesting context in which to apply their knowledge of translations, rotations, and reections.
Polygons That Tessellate
The process of nding which polygons will tessellate the plane is an example of what makes mathematics challenging and interesting. It also enables us to broaden our ideas about polygons and transformations by applying them in analyzing tessellations. Lets take a brief look at part of this process. In Mini-Investigation 11.4, you may have drawn the correct conclusion that an equilateral triangle, a square, and a regular hexagon can each be used to form a tessellation. Such a tessellation, made up of congruent regular polygons of one type, all meeting edge to edge and vertex to vertex, is called a regular tessellation. Now lets consider tessellations involving nonregular polygons. Because the sum of the measures of the angles of a triangle is 180 and the sum of the measures of the angles of a quadrilateral is 360, we can show that these types of gures can be made to t around their vertices to tessellate the plane. We conclude that any triangle or quadrilateral will tessellate the plane. A natural extension of that idea is to investigate whether pentagons and hexagons tessellate the plane. You may have found in Mini-Investigation 11.4 that a regular pentagon does not tessellate the plane but that a regular hexagon will. Some nonregular pentagons and hexagons, such as regions A and B in Figure 11.16, will also tessellate the plane. No convex polygon with more than six sides will tessellate.
B
F I G U R E 11 .1 6
ISBN: 0-536-08809-8
A
Nonregular pentagons and hexagons that tessellate.
Source: ODaffer/Clemens, Geometry: An Investigative Approach, Second Edition, Figure 4.15 from p. 95, 1992. Reprinted by permission of Pearson Education, Inc.
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In Mini-Investigation 11.5 you are asked to verify a generalization about which regular polygons will not tessellate the plane.
Write a convincing argument in support of your conclusion.
M I N I - I N V E S T I G A T I O N 11 . 5
Using Mathematical Reasoning
How could you use the fact that the measure of an interior angle of a regular hexagon is 120 to convince someone that no regular polygon with more than six sides will tessellate the plane?
Combinations of Regular Polygons That Tessellate
Piet Hein has stated, We will have to evolve problem-solvers galore, for each problem they solve creates ten problems more [Hein (1969), p. 32]. In this spirit, the question about whether combinations of regular polygons can tessellate the plane naturally arises from the question about single-polygon regular tessellations that we answered earlier. Figure 11.17 depicts two combinations of regular polygons that do tessellate the plane. The pattern of polygons in Figure 11.17(a) is a special kind of tessellation, dened as follows.
Denition of a Semiregular Tessellation
A tessellation formed by two or more regular polygons with the arrangement of polygons at each vertex the same is called a semiregular tessellation.
The tessellation in Figure 11.17(b) has two different arrangements at vertices and is not a semiregular tessellation. Because a semiregular tessellation is a special kind of tessellation, another natural question arises: How many different semiregular tessellations are there? This question is explored more fully in ODaffer and Clemens (1992), in which they established that there are only 21 ways to arrange regular polygons around a point, with no gaps or overlaps. Of these 21 ways, only 8 can be extended to form a semiregular tessellation. Thus we conclude that only eight semiregular tessellations are possible. These tessellations are shown in Figure 11.18.
44 333 44 333 333 44 (a) F I G U R E 11 .17 34 A 3 3B 4
34 4A 6
(b)
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Two types of tessellations involving the combination of regular polygons.
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(a)
(b)
(c)
(d)
(e) F I G U R E 11 .1 8
(f)
(g)
(h)
The eight possible semiregular tessellations.
The symbol 4, 8, 8 or 4, 82 can be used to describe tessellation (h) in Figure 11.18 because the arrangement around each vertex is square, octagon, octagon. The other tessellations can be symbolized in a similar manner. Thus we symbolize tessellations a), c), and f ) in Figure 11.18 as 34, 6; 32, 4, 3, 4; and 3, 122. The problem in Example 11.5 can be solved using ideas about semiregular tessellations. The problem-solving strategies make a model, draw a diagram, and use reasoning are helpful when solving the problem.
Example
11.5
Problem Solving: The Ancient Temple Floor
A oor restoration specialist was called upon to restore the oor tile in an ancient temple. He knew that the oor had been tiled only with equilateral triangles and regular hexagons and was a semiregular tessellation. The only remaining section of original tile is shown. Can the specialist use this information to restore the oor to its original appearance?
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Working Toward a Solution
Understand the problem What does the situation involve? What has to be determined? What are the key data and conditions? What are some assumptions? Develop a plan What strategies might be useful? Are there any subproblems? Should the answer be estimated or calculated? What method of calculation should be used? Implement the plan How should the strategies be used? Restoring a tiled oor in an ancient temple What the original tiling looked like The only remaining section of tile is pictured. Only equilateral triangles and hexagons were used in the semiregular tiling. Assume that the tiling must be formed by extending the available section. Use a model; draw a diagram; use reasoning Deciding what arrangements will t around a point No estimation or calculation is needed. Not applicable Use a model with cutout triangles and hexagons, or a diagram to extend the arrangementfor example, the following:
Then, use reasoning to decide whether this arrangement is a semiregular tessellation. What is the answer? Look back Is the interpretation correct? Is the calculation correct? Is the answer reasonable? Is there another way to solve the problem? (Continue this process in Your Turn to look for other tessellations and solve the problem.) Yes. The diagram meets the conditions. Not applicable Look for all possibilities and decide. We could nd the number of degrees in the angles of the polygons and use these measures to look for different ways to arrange them around a point.
YOUR TURN
Practice: Complete the solution to the example problem. Reect: What if the example problem didnt specify that the oor tiling was a semiregular tessellation? Would that change the solution? Explain.
Other Ways to Generate Tessellations
Using a Kaleidoscope. A three-mirror kaleidoscope provides a model that uses reections to create tessellations of the plane. To make a three-mirror kaleidoscope, you can fasten three mirrors of the same size together to form the vertical faces of an equilateral triangular prism. When you place an equilateral triangle piece of construction paper with a pattern on it in the base of the kaleidoscope and peer over the edge, as in Figure 11.19, an interesting geometric pattern appears. If we investigate the semiregular tessellations further, we nd that only those with lines of symmetry that form a superimposed tessellation of equilateral triangles
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S E C T I O N 11 . 2
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F I G U R E 11 .1 9
Viewing tessellations through a three-mirror kaleidoscope.
Source: Photo: Farnsworth Kaleidoscope Images.
can be produced with a three-mirror kaleidoscope. The other tessellations can be produced with other types of kaleidoscopes, based on their lines of symmetry. Using Computer Software. The following LOGO procedures will create a portion of the tessellation of hexagons shown in the next gure:
TO HEXAGON :SIDE REPEAT 6 [FD :SIDE RIGHT END 60]
TO TESSHEX PENUP BACK 80 LEFT 90 PENDOWN REPEAT 4 [HEXAGON 30 RIGHT 120 FD 30 LEFT 60 HEXAGON 30 FD 30 LEFT 60] END
Similar procedures can be created easily to produce a portion of a regular tessellation of triangles or squares. In fact, writing LOGO procedures to produce the semiregular tessellations is both possible and interesting. Exercise 46 at the end of this section asks you to do so.
Tessellations with Irregular or Curved Sides
The Moors, who occupied Spain from 711 until 1492, were forbidden by their religion to draw living objects. To compensate, they mastered creative design, as indicated by the drawings of two of the simpler designs, shown in Figure 11.20, with which they decorated the walls of the Alhambra in Granada. M. C. Escher, a Dutch artist whose drawings, including the one shown in Figure 11.21, have long been a special source of enjoyment for people interested in mathematics, made the following remarks about the Moors inuence on his art:
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This is the richest source of inspiration I have ever struck: nor has it yet dried up. [A] surface can be regularly divided into, or lled up with, similar-shaped gures (congruent) which are contiguous to one another, without leaving any open spaces. The Moors were past masters of this. [Escher (1960), p. 11]
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F I G U R E 11 . 2 0
Drawings from the Alhambra.
Source: M. C. Eschers Drawings from the Alhambra. 2006 The M. C. Escher CompanyHolland. All rights reserved. www.mcescher.com
F I G U R E 11 . 2 1
A classic drawing by M. C. Escher.
Source: M. C. Eschers Day and Night. 2006 The M. C. Escher CompanyHolland. All rights reserved. www.mcescher.com
Interestingly, techniques are readily accessible for creating curved tessellation art. The basic approach is to begin with a shape that will tessellate and alter it, often using transformations, to produce the effect wanted. In the following subsections, we describe some ways to do so. Replacing Polygon Edges with Curved Lines. The basic gure for the tessellation in Figure 11.22 was created by starting at the midpoint of each edge of the basic quadrilateral, cutting out a piece, rotating it 180 about the midpoint, and taping it back on the original gurecalled a cut and turn method. A similar method uses only tessellating polygons with congruent adjacent sides to make a basic gure for a curved tessellation. As shown in Figure 11.23, it involves starting with a tessellating polygon, cutting out a piece that contains a complete side, rotating it about a corner, and taping it along the adjacent side. Another way to replace each edge of a square or rectangle with a curve is shown in Figure 11.24. In this procedure, the basic gure of the tessellation was created by cutting out a piece that includes a side of the basic rectangle and sliding the piece so that the side
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Tape on here.
Cut out this piece of the quadrilateral. Turn piece about this point.
F I G U R E 11 . 2 2
The cut and turn method.
Rotate around this point Cut out
Cut out Rotate around this point (b) Cut out, rotate, and tape
(a) Start with a square, draw cutout lines. F I G U R E 11 . 2 3
(c) Decorate the figure
The cut and turn method with special polygons.
Rectangle
Slide piece to left
Slide piece up
Tessellating shape
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F I G U R E 11 . 2 4
The cut and slide method.
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Design
Decorated design F I G U R E 11 . 2 6
The interior design method.
on the cutout piece coincides with its opposite sidecalled a cut and slide method. Notice how the transformation we have called a translation is involved in this procedure. Making a Design in the Polygon Interior. An appropriate design drawn in the interior of a polygon can produce interesting curved tessellations, as shown in Figure 11.26. When creating such a tessellation, it is helpful to begin with a rough sketch of the tessellation you want to produce and work backward to develop the interior design. Example 11.6 demonstrates the creation of a curved tessellation and then gives you an opportunity to use one of the methods just described.
Example
11.6
Making a Curved Tessellation
Use one of the methods described to make a curved tessellation from a tessellation of equilateral triangles.
SOLUTION
Figure
Decorated figure YOUR TURN
Practice: Use a method not used in the example problem to make a curved tessellation from a tessellation of equilateral triangles.
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Reect: What type of curved tessellation can you make with one method you could have used that you cant make with another?
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
Analyzing a Textbook Page: Tessellations from Transformations
Connection to the PreK8 Classroom
Experiences with tessellations enable elementary school students to integrate art with mathematics, as illustrated in Figure 11.25. They provide an interesting environment in which students can apply what they have learned about polygons and their angles. Geometric design fosters creative thinking and helps students further develop their spatial visualization abilities.
What idea about
tessellations of polygons is this activity based upon?
Use the rst example to
explain why this procedure works to produce a gure that tessellates the plane.
What transformation is
used in carrying out this procedure?
F I G U R E 11 . 2 5
Excerpt from a sixth-grade mathematics textbook
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Source: Scott ForesmanAddison Wesley Mathematics, Grade 6, p. 516. 2004 Pearson Education. Reprinted with permission.
685
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Problems and Exercises
for Section 11.2
A. Reinforcing Concepts and Practicing Skills
1. Describe or draw an arrangement of polygons that has a characteristic that prohibits it from being a tessellation. 2. Describe or draw an arrangement of polygons that has a characteristic, different from the one selected in Exercise 1, that prohibits it from being a tessellation. 3. How many equilateral triangles will t about a point when tessellating the plane? How do you know? 4. How many rectangles will t about a point when tessellating the plane? How do you know? 5. How many quadrilaterals will t about a point when tessellating the plane? How do you know? 6. Use the idea that all parallelograms tessellate to show that the following triangle tessellates the plane:
16. Give an example of a regular polygon, other than a pentagon, that wont tessellate the plane. 17. Could you tile a oor by using only pentagons such as the ones shown? Use tracing paper and show enough of the tessellation to convince someone of your conclusion.
7. How many different types of regular tessellations are there? Describe them. 8. Give an example of a tessellation made of squares that is not a regular tessellation. 9. Give an example of a tessellation made of equilateral triangles that is not a regular tessellation. 10. Why does a regular pentagon not tessellate the plane? 11. Are the curved tessellations in Figure 11.15 regular tessellations? Explain. 12. Will a rhombus tessellate the plane? How do you know? 13. Will the nonconvex quadrilateral shown in the next gure tessellate the plane? Devise a way to convince someone that your answer is correct.
(a)
(b)
18. Why wont a regular octagon tessellate the plane by itself? Describe a combination of a regular octagon and another regular polygon that will tessellate the plane. 19. Use symbols to represent semiregular tessellations (b), (d), and (g) in Figure 11.18. 20. How many different semiregular tessellations are there? 21. Why is the tessellation shown below not a regular tessellation?
22. Why is the tessellation shown not a regular tessellation? 14. Why is it impossible for a regular polygon with more than six sides to tessellate the plane? 15. If this pattern were placed in a three-mirror kaleidoscope, what tessellation do you think would be produced? Describe the tessellation as accurately as possible, and draw a picture to explain your thinking.
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23. Why is the tessellation shown not a semiregular tessellation?
tessellate the plane. Draw an example of each of these types of tessellating gures. 31. Make a curved tessellation from one of the following geometric gures: a. A tessellation of equilateral triangles b. A tessellation of squares c. A tessellation of general quadrilaterals 32. A vertex gure is formed by connecting the midpoints of the edges that form the vertex of a tessellation, as shown in the following gure:
24. Why is the tessellation shown not a semiregular tessellation? A vertex gures tessellation is created by drawing all the vertex gures of a tessellation. Draw the vertex gures tessellation for the tessellation of regular hexagons. Describe this tessellation. 33. Adapt the LOGO procedure on p. 681 to produce a portion of a tessellation of a. squares. b. equilateral triangles. 25. Draw the arrangement of regular polygons around a vertex described by 6, 6, 3, 3. Explain why it cant be extended to form a semiregular tessellation of the plane. 26. What type(s) of symmetry, if any, is present in the completed tessellation suggested in Figure 11.22? Describe the symmetry as accurately as possible. 27. What type(s) of symmetry, if any, is present in the completed tessellation suggested in Figure 11.24? Describe the symmetry as accurately as possible. Figures for Exercise 28
B. Deepening Understanding
28. Trace each tessellation shown in the gure for this exercise, recognizing that it actually covers the entire plane. In how many ways can you slide, turn, and ip each tessellation to match the original tessellation? Devise ways to describe these slides, turns, and ips. 29. The dual of a tessellation is a tessellation created by connecting the centers of neighboring polygons in the tessellation. a. Draw the dual of the tessellation of regular hexagons. b. Draw the dual of the tessellation of equilateral triangles. c. Describe any relationships that you see between the results of parts (a) and (b). 30. Every pentagon with a pair of parallel sides and every hexagon with three pairs of equal parallel sides will
(a)
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(b)
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34. Write a LOGO procedure that will produce a gure consisting of an equilateral triangle and a square with a common side. Such a conguration might be used as a part of a procedure to produce the semiregular tessellation 33, 42.
40.
C. Reasoning and Problem Solving
35. Of the 12 types of pentomino, nd at least 2 that will tessellate the plane. Show a portion of the tessellation. 36. Of the 35 hexominos, nd at least 2 that will tessellate the plane and show a portion of the tessellation. 37. A heptiamond is a gure formed by seven connected, nonoverlapping, and congruent equilateral triangles, each touching others only along a complete side. Of the 24 heptiamonds, all but 1 will tessellate the plane. Choose a heptiamond and show how to tessellate the plane with it. 38. Which tessellations should the following design make when placed in a three-mirror kaleidoscope? Explain.
41.
42.
A demi-regular tessellation is a tessellation of regular polygons that has exactly two or three different polygon arrangements about its vertices. There are 9 demiregular tessellations with exactly two vertex arrangements, and 5 that have exactly three vertex arrangementsa total of 14 demiregular tessellations in all. Four demiregular tessellations are shown in exercises 3942. List all their vertex arrangements. 39.
43. Is the following tessellation a semiregular or a demiregular tessellation? Explain why.
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44. This tessellation of pentagons was made by rst drawing a tessellation of hexagons and then dividing each hexagon into ve congruent pentagons. Trace the tessellation of hexagons that was used, and use observation to give some of the characteristics of the hexagon.
48. The Kaleidoscope Problem. A geometer made a kaleidoscope out of three mirrors put together to form a 454590 right triangle. He asserted that he could put a pattern in the kaleidoscope that would produce a semiregular tessellation. His colleague disagreed. Who was correct, the geometer or his colleague? Justify your conclusion. 49. The famous tessellation shown was created by Johannes Kepler. Describe the types of gures that make up the tessellation.
45. The following are instructions showing how to create a tessellation of nonregular hexagons. Use this method to create a tessellation of nonregular hexagons that have opposite sides parallel and congruent. Describe your technique. (See gure below.) 46. How would you instruct the LOGO turtle to have it produce semiregular tessellation (b) in Figure 11.18? a. Write a paragraph describing the instructions that you would give the turtle. b. Use GES or a computer drawing program to create a tessellation. What functions of the technology did you nd most useful in completing your tessellation? 47. The Roman Bath House Tiling Problem. A oor restoration specialist was called upon to restore the tile oor in a Roman bath house. She knew that the oor had been tiled only with equilateral triangles and squares and that it was a semiregular tessellation. The only remaining pieces of the original tile are shown.
D. Communicating and Connecting Ideas
50. The following pattern shows what appears to be a tessellation of squares, regular pentagons, regular hexagons, regular heptagons, and regular octagons. How would you convince someone that it is a fake tessellation of regular polygons?
Can you help the specialist by drawing a diagram of what the original tiled oor looked like? Is there more than one possibility? Explain. Figure for Exercise 45
M M
ISBN: 0-536-08809-8
Start with any parallelogram
Create two new sides on each of a pair of opposite sides.
Form a tessellation by sliding, and rotating 180 about the midpoint, M, of a side.
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51. Discuss what is communicated to you by a tessellation such as the one shown in Exercise 49 or by an Escher design such as the one shown in Figure 11.21. How do you feel when you see it? What message, if any, does it convey to you about mathematics? 52. Historical Pathways. The following pattern is a marble design found in a thirteenth-century Roman church:
the vertical and horizontal lines are squares on the hypotenuse of the right triangle. How does this relate to the puzzle gure in Figure 10.34(b), p. 623?
a. Is it a tessellation? Why or why not? b. Describe at least two lines of symmetry of the design. 53. Making Connections. Illustrate the connection between tessellations and the Pythagorean Theorem by explaining how the dissection of the following tessellation by the vertical and horizontal lines proves the Pythagorean Theorem. Assume that the squares that make up the tessellation are the squares on the legs of a right triangle and that the squares formed by
Section
11.3
Special Polygons
Golden Triangles and Rectangles Star-Shaped Polygons
Star Polygons
In this section, we examine some special topics that involve polygons. We explore and generalize some properties of the pentagram, the golden rectangle, and star polygons.
Essential Understandings for Section 11.3
Polygons shaped like stars can be uniquely described by their sides and angles. Special triangles and rectangles can be described uniquely by the ratios of pairs of their sides.
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Mini-Investigation 11.6 gives you an opportunity to make some interesting historical connections.
Talk about why the ancient Greeks, who chose the pentagram as a sacred symbol because of its special beauty, called ^BFG a golden triangle.
M I N I - I N V E S T I G A T I O N 11 . 6
Making a Connection
How many different pairs of segments can you nd in the following pentagram with a ratio of lengths equal to the golden ratio, F 1.618?
B x x F x G x
Estimation Application
Golden Shape The golden rectangle, in which the ratio of its length to its width is approximately 1.618, is purported to be the most aesthetically pleasing of all rectangular shapes. Choose two rectangular shapes in your environment that you think are the closest to golden rectangles, and estimate their golden ratios. Measure to see how good your estimates were.
A 1 B A
C
J I
H
E
D
Golden Triangles and Rectangles
In the pentagram shown in Mini-Investigation 11.6, an isosceles triangle that forms one of the points of the star, such as ^BFG, is called a golden triangle because the ratio of its longer side to its shorter side is the golden ratio, f = (1 + 15)> 2 L 1.618. We know that the measure of an interior angle of the pentagram, such as ABC, is 108, so we can conclude from the equations 2x + a = 108, 4x + a = 180, and b = 2x, that x = a = 36 and b = 72. Using this information, try to identify 9 other golden triangles in the pentagram. The golden triangle is interesting but hasnt received as much attention as the often-discussed and used golden rectangle. Mathematicians, artists, architects, and others have long considered the golden rectangle an especially pleasing formin nature, in buildings, and in works of art. To construct a golden rectangle, we start with a unit square, as in Figure 11.27(a), and nd the midpoint M of one side. Next, we consider MB. Then, as shown in Figure 11.27(b), we use M as the center and MB as the radius and strike an arc on the extension of DC at E. Finally, we construct EF DE and complete the golden rectangle AFED. To verify that the ratio of the length to the width of rectangle AFED is the golden ratio, and that AFED is a golden rectangle, we use the Pythagorean Theorem and show that d = 15 . It then follows that DE = 1 + 15 = (1 +2 15) . 2 2 2 This length of DE is approximately (1 + 22.236) , or 1.618, and AFED is a golden rectangle. Note that 1.618 is an approximation of the golden ratio, referred to on p. 563.
1
D
M (a) 1 B
C
A
F
1
d
1
D
1 2
M
1 2 (b)
C d
E
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F I G U R E 11 . 2 7
Construction of a golden rectangle.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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V1
Star Polygons
Using Circles to Produce Star Polygons. The pentagram shown in MiniInvestigation 11.6 leads to consideration of another interesting type of polygon, called a star polygon. Star polygons are frequently used in advertising logos, artistic designs, quilts, and other decorative situations. The star polygon shown in Figure 11.28 is constructed from ve equally spaced points on a circle. Beginning at V1 , we go in one direction and draw a segment to every second point, returning to V1 . The segments drawn to form the star polygon are V1V3 , V3V5 , V5V2 , V2V4 , and V4V1 . Because the star polygon shown in Figure 11.28 was constructed from ve points with segments connecting every second point, it is denoted {5}. This star polygon is a nonsimple polygon having ve ver2 tices with angles of equal measure and ve congruent sides and is called a regular star polygon.
V5
V2
V4
V3
F I G U R E 11 . 2 8
A regular star polygon.
Roman Soumar/CORBIS
Mini-Investigation 11.7 lets you explore the properties of star polygons and make some generalizations.
Talk about which are regular polygons and which are the same star polygonand why. Technology Extension: If you have access to LOGO, use it to construct the star polygons with eight points and formulate possible generalizations.
M I N I - I N V E S T I G A T I O N 11 . 7
Finding a Pattern
What possible relationships can you discover by constructing the following star polygons? a. E 8 F 4 b. E 8 F 6 c. E 6 F 2 d. E 6 F 7
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Mini-Investigation 11.7 and the analysis of the star polygons with eight points suggest the following generalizations.
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Generalizations About Star Polygons
n = the number of equally spaced points on the circle. d = the dth point to which segments are drawn.
n E n F is the same as the star polygon E n - d F . d n The gure produced for E n F and E n - 1 F is not a star polygon, but is a regular n-gon. 1
The star polygon
The n-sided star polygon are relatively prime.
E n F exists if and only if d Z 1, d Z n - 1, and n and d d
If fraction a is equivalent to a lowest terms fraction n , where E n F is a star polygon, b d d then E a F , sometimes called an improper representation of a star polygon, b represents an n-sided star polygon.
To illustrate the generalizations for star polygons, note, from the rst bulleted statement, that E 5 F is the same star polygon as E 5 F , even though they are 2 3 produced differently. From the second bulleted statement, we see that E 5 F and E 5 F 1 4 are each regular pentagons. In the third and fourth bulleted statements, we see that E 7 F ts all the requirements and represents a seven-sided star polygon, and that 3 E 14 F is an improper (taken from the improper fraction terminology) representa6 tion of a seven-sided star polygon. In subsequent discussion, we will omit the improper designation, and simply agree that symbols such as E 7 F and E 14 F repre3 6 sent the same star polygon. In Mini-Investigation 11.7, you may have discovered some other interesting results when you used the procedure suggested by the notation E n F . The following generald izes one possible discovery and gives some examples. To interpret E a F write the fraction a in lowest terms, say n , and determine what b b d gure E n F produces. E a F produces the same gure, but in a different way. So E a F can d b b be a star polygon, a regular polygon, and in the special case of E 2 F , a line segment. 1 For example, using E 16 F , we see that 16 is equivalent to 8 , and since E 8 F is an 6 6 3 3 eight-sided star polygon, E 16 F is also a representation of an eight-sided star 6 polygon. As a second example, consider E 9 F and E 9 F . 9 is equivalent to 3 , and 9 is 6 3 6 2 3 equivalent to 3 . Since E 3 F and E 3 F are triangles, we might say that E 9 F and E 9 F also 1 1 2 6 3 represent triangles. Finally, since 4 is equivalent to 2 , and E 2 F produces a line 2 1 1 segment, E 4 F also represents a line segment. 2 The following Web site has a very useful little device that produces star polygons. You insert the values for n and d, and it automatically draws the star polygon: http://www.cut-the-knot.org/Generalization/PolyStar.shtml In Exercise 40 at the end of this section, you study star polygons further. In Example 11.7, we apply some of the ideas about star polygons.
Example
11.7
Identifying Star Polygons
Describe the star polygons having seven vertices and verify that all of them have been identied.
ISBN: 0-536-08809-8
SOLUTION
There are two star polygons with seven vertices, namely,
E 7 F and E 7 F , as follows: 2 3
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Note that E 7 F and E 7 F form regular polygons, not star polygons; E 7 F and E 7 F are 1 6 5 4 the same star polygons as those shown here. So there are only two star polygons with seven vertices.
YOUR TURN
7 2
7 3
Practice: Describe the star polygons with nine vertices and verify that you have identied all of them. Reect: Explain how the process for producing star polygon E 7 F is different 2 from the process for producing the star polygon E 7 F even though the nal pictures 5 are the same.
Dent angle
30
Star-Shaped Polygons
Point angle
(a) 360 n 120 60
(b) F I G U R E 11 . 2 9
Examples of star-shaped polygons.
If we stop after only considering star polygons that can be produced by sequentially connecting points on a circle with line segments, we miss a lot of very interesting star-shaped gures. For example, a quiltmaker might want to use star-shaped gures like those shown in Figure 11.29. To compare these six-pointed gures with six-pointed star polygons, we observe that E 6 F and E 6 F produce regular hexagons, E 6 F and E 6 F produce triangles, and E 6 F 1 5 2 4 3 produces a straight line. Thus the polygons shown in Figure 11.29 are not star polygons, nor can they be produced by connecting points on a circle with line segments and then erasing the interior parts of the segments. However, we can use the term star-shaped polygon to describe a nonconvex symmetric gure like the one shown in Figure 11.29(a) or (b) that isnt a star polygon. Star-shaped polygons have n star-tip points, 2n congruent sides, n congruent point angles with measure a, and n congruent dent angles b such that b = A 360 B + a, or a = b - A 360 B . Exercise n n 48 at the end of this section asks you to verify this relationship. A six-pointed, starshaped polygon with point angle 30 is denoted by 630 . Figure 11.30 compares a star polygon and star-shaped polygon. It illustrates the importance of precise denitions, which are required to differentiate two different but closely related ideas. Although the basic shape of the ve-pointed stars shown are the same here, that isnt always the case when an n-pointed star polygon and an n-pointed star-shaped polygon are compared. This information about the relationship between dent angles and point angles is useful in constructing star-shaped polygons that meet our specications. For example, if you want to produce a star-shaped polygon with a specic point angle for a special quilt, you can quickly calculate the measure of the dent angle for that gure. Conversely, if you want to make a star-shaped polygon with a specic dent angle for tessellation or other
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5 (a) Star polygon 2 Nonsimple, nonconvex 5 vertices 5 sides 5 point angles F I G U R E 11 . 3 0
(b) Star-shaped polygon 5 36 Simple, nonconvex 10 vertices 10 sides 5 point angles
Comparison of a star polygon and a star-shaped polygon.
design purposes, you can quickly calculate the measure of the point angle for that gure. Example 11.8 demonstrates how to construct a desired star-shaped polygon.
Example
11.8
Problem Solving: The Star Design
An artist wants to make a painting that includes a ve-pointed star-shaped polygon with a fairly thin point angle of 18. How could the artist accurately construct the star-shaped polygon 518 ?
SOLUTION
The measure of the point angle is 18, so the dent angle is construct the star-shaped polygon as follows:
A 360 B + 18 = 90. We 5
a. Plot ve equally spaced points on a circle. b. Connect two of the points, A and B, and construct 45 angles at those two points. Then produce the 90 angle at point D. c. Use a compass to nd the other dent angle points.
B 45 A D 45 45 D B
A
45
YOUR TURN
Practice: Construct a six-pointed star-shaped polygon with a point angle of 30.
ISBN: 0-536-08809-8
Reect: If you want a dent angle of 120, how will you perform the construction to nd the dent angle vertex D?
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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The results in Example 11.9 allow us to investigate how star-shaped polygons can be used in tessellations. The problem-solving strategies draw a picture and use logical reasoning are helpful here.
Example
11.9
Problem Solving: The Quilt
A quiltmaker wanted to make a quilt from six-pointed star-shaped polygons and either squares or equilateral triangles. Is this combination possible? If so, what would the quilt pattern look like?
WORKING TOWARD A SOLUTION
The quiltmaker needs to decide which type of six-pointed star-shaped polygon to use. Drawing a picture and reasoning indicate that the dent angles of the starshaped polygon must be either 90 (to t a square), as shown in part (a) of the next gure or 120 (to t two equilateral triangles), as shown in part (b). Then the quiltmaker can calculate the measure of the point angles, if needed.
120
(a)
(b)
The following quilt pattern meets the conditions of the problem for the starshaped polygon shown in part (a) of the preceding gure. The measure of the point angle is 30.
YOUR TURN
Practice: Draw the quilt pattern for the polygon shown in part (b) of the gure.
ISBN: 0-536-08809-8
Reect: Why cant you use a dent angle of 60, which would hold a single equilateral triangle, to produce a third quilt pattern solution to the problem?
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Connection to the PreK8 Classroom
Making geometric designs has benets for elementary school students. It provides an opportunity to learn and extend ideas of geometry. It also provides a path to success for students who may not be as successful in other areas of mathematics. It promotes active involvement and is an excellent activity for integrating mathematics and art.
Problems and Exercises
for Section 11.3
10. How are the processes for constructing star polygons E 5 F and E 5 F alike? How are they different? What can 2 3 you conclude about the resulting star polygons? 11. What is the number of degrees in the point angle of star polygon a. E 5 F ? b. E 8 F ? c. E 9 F ? 2 3 2 For Exercises 1216, tell if the gure is a star polygon, a polygon, or other, and how many sides it has. 12. E 9 F 13. E 8 F 3 3
A. Reinforcing Concepts and Practicing Skills
1. Which of the following is shaped most like a golden rectangle? Explain. a. a 5 * 7 photo b. a 3 * 5 card c. a 20 * 35 picture frame 2. Find an object or part of the place where you live that is close to being a golden rectangle. Measure it to the nearest centimeter and check. What is the ratio of the length to the width? 3. Perform the construction of a golden rectangle described in Figure 11.27. Then, measure to the nearest millimeter and calculate the ratio AF> FE. By how much does the ratio you calculated differ from the golden ratio, which is approximately equal to 1.618? 4. What are the dening characteristics of a golden triangle? 5. Name two pairs of segments in the following pentagram for which the ratio of their measures is the golden ratio.
Q
E 15 F 9 16. E 10 F 5
14.
15.
E 12 F 5
17. Give an example to illustrate that the star polygons E n F 1 n and E n - 1 F , where n is the number of equally spaced points on the circle, are regular polygons. For Exercises 1820, nd the number of degrees in the point angle of each gure. 18. E 5 F 19. E 8 F 20. E 6 F 2 2 2 21. How are star polygons and star-shaped polygons alike? How are they different? 22. How does the number of sides of the star polygon E 9 F 4 differ from the number of sides of the star-shaped polygon 920 ? For Exercises 2326, calculate the measures of the point angles of the star-shaped polygons shown. 23.
P H
D
E
R F
G T S
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8. Why is the polygon represented by E 8 F not a star 6 polygon? 9. Describe the polygons produced by interpretation of the following symbols: a. E 8 F b. E 8 F 4 6 c.
6. Describe the measures of the angles of a golden triangle. 7. How many sides does the star polygon E 5 F have? 2
110
E6F 2
d.
E6F 7
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698 24.
C H A P T E R 11
EXTENDING GEOMETRY
29.
140
70
25. 30.
75 40
26. 31. A wallpaper designer wants to use a star-shaped polygon with eight points and a point angle of 15. What should be the dent angle of this polygon?
40
B. Deepening Understanding
32. A graphics designer wants to use a star-shaped polygon with 10 points and a point angle of 12. What should be the dent angle of this polygon? 33. A quiltmaker thought that a six-pointed star-shaped polygon with a point angle of 30 would go together with squares to make a quilt. Was she correct? Explain. 34. How many star polygons are there with 12 sides? Give symbols for these polygons and verify your answer. 35. How many star polygons are there with 10 sides? Give symbols for these polygons and draw a picture of each. 36. Use GES and inductive reasoning to make a generalization about the sum of the measures of all the point angles in any nonsymmetric ve-pointed star. 37. Use GES and try several different examples to check the generalization about the relationship between the point angle and the dent angle in a star-shaped polygon. 38. For the pentagram shown in the gure in MiniInvestigation 11.6, it has been said that the ratio of the length of the side of the large outer pentagon to the length of the side of the smaller inner pentagon is f2. Do you agree? Support your decision. 39. Construct a star-shaped polygon with eight points and a point angle measure of 45.
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For Exercises 2730, calculate the measures of the dent angles of the star-shaped polygons shown. 27.
20
28.
10
C. Reasoning and Problem Solving
40. The number of numbers less than n and relatively prime to n is found by using the formula
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 3
S P E C I A L P O LY G O N S
699
nc1 - a
1 1 b d * c1 - a b d , p1 p2
Figure for Exercise 44
where p1 , p2 , are prime factors of n. Show how to use this formula to help nd the number of star polygons there are with 36 sides. 41. A student who liked algebra claimed that the following was a true statement: If the fraction n is equivalent to d n d n n-d n - d the gure produced for E d F is a regular n gon. Give two examples to illustrate the truth of n-d the statement or a counterexample to show that it is false. 42. Describe three different six-pointed star-shaped polygons. If the area of the smallest is 1, estimate the areas of the others. How might you check your estimates? 43. Use the gure shown and the ideas about the sum of the exterior angles of a polygon to prove that what you discovered in Exercise 36 is true.
(a)
q
g p f e j t d a
b h c i
r
s (b)
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44. Analyze the tessellations shown on the right. i. What regular polygons are used? ii. What are the measures of the point and dent angles of the star-shaped polygons that are used? iii. Are all vertices surrounded alike? If not, what different vertex arrangements are there? 45. The Wallpaper Design Problem. A wallpaper designer wants to use the tessellation shown in the gure as the base for some decorative wallpaper. To make the tessellation, he needs to make accurate templates for the 9-gon and the star-shaped polygon. What directions (angle measures, side lengths, and the like) would you give him for making the templates? 46. If the ratio of a to b in the square shown in the gure for this exercise (see p. 700) is the golden ratio, prove that the shaded rectangle is a golden rectangle. (Hint: Use properties of similar triangles.) 47. The ancient Greeks found that the proportion l (l + w) holds only for the length and width of a w= l golden rectangle. If the width in the proportion is 1, show that the length is (1 +2 15) .
Figure for Exercise 45
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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Figure for Exercise 46
a b
b a
51. Historical Pathways. The problem of constructing tessellations that utilized star-shaped polygons challenged early Islamic artists. The following design is adapted from Islamic art found in a Russian mosque. Describe the different geometric gures used in this tessellation. If you assume that the ve-pointed star is a pentagram star, draw some conclusions about the angle measures of the other gures.
a b
b
a
48. How would you use the following partial picture of a star-shaped polygon inscribed in a regular n-sided polygon to verify that b = (360) + a, where b is the measure n of the dent angle and a is the measure of the point angle?
360 n x
52. Making Connections. Show a connection between geometry and art by making and coloring a tessellation that uses the following polygons:
x
x
x {418} Regular pentagon
53. Making Connections. Connect algebra and geometry by nding an algebraic expression for the ratio of the length to width of golden rectangle AHCD. Do not use approximate values for square roots.
D. Communicating and Connecting Ideas
49. Discuss how you would produce an accurate drawing of a six-pointed star-shaped polygon with a point angle of 60. Describe your procedure in detail. 50. Work in a small group and devise a procedure to test whether people really do think that the golden rectangle is the most aesthetically pleasing rectangle. Write a paragraph explaining your procedure and the results.
A
B
H
1
1
D
1 2
M
1 2
F
C
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S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
701
Section
11.4
The Regular Polyhedra Prisms and Pyramids Cylinders, Cones, and Spheres Symmetry in Three Dimensions Visualizing Three-Dimensional Figures
In this section, we classify and dene various types of three-dimensional gures. We also explore properties of these gures, such as symmetry, and relationships involving them. Finally, we describe and analyze different ways to view three-dimensional gures.
Three-Dimensional Figures
Essential Understandings for Section 11.4
All polyhedra can be described completely by their faces, edges, and vertices. There is more than one way to classify most solids. Polyhedra can have reectional or rotational symmetry. Solid gures can be viewed from different perspectives.
The Regular Polyhedra
Vertex Face
Edge F I G U R E 11 . 31
Parts of a polyhedron.
A polyhedron is a collection of polygons (triangles, rectangles, pentagons, hexagons, and the like) joined to enclose a region of space. A polyhedron has faces, edges, and vertices, as shown in Figure 11.31. The prex poly- means many or several and the sufx -hedron indicates surfaces or faces. The ve polyhedra shown in Figure 11.32 can be modeled from cardboard, Styrofoam, or connected drinking straws and are called regular polyhedra. The ancient Greeks used prexes that indicated the number of faces to name these polyhedra. Tetrameans four, octa- means eight, dodeca- means twelve, and icosa- means twenty. Because hexa- means six, a cube may be called a hexahedron. The regular polyhedra shown in Figure 11.32 are also called the Platonic solids because Plato associated earth, air, re, water, and creative energy with these ve solids and used them in his description of the universe. These regular polyhedra might be thought of as basic threedimensional building blocks of geometry because they are the only polyhedra that have edges of equal length and the arrangement of polygons at all vertices the same. A nonregular polyhedron with two different vertex arrangements is shown in Figure 11.33.
A
B Regular tetrahedron
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Regular hexahedron or cube
Regular octahedron
Regular dodecahedron
Regular icosahedron Three triangles surround vertex A. Four triangles surround vertex B. F I G U R E 11 . 3 3
F I G U R E 11 . 3 2
Regular polyhedra.
A nonregular polyhedron.
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F I G U R E 11 . 3 4
Geometric forms in the radiolarian skeleton.
Source: Darcy W. Thompson, On Growth and Form. In J. T. Bonner (ed.), abridged edition, Cambridge University Press, England, 1971, p. 168. Reprinted with permission of Cambridge University Press.
Evolving scientic theories of the universe long ago replaced Platos description, but nature hasnt ignored the Platonic solids. Skeletons of tiny sea creatures shown in Figure 11.34 and called radiolarians, made of silica and measuring only a fraction of a millimeter in diameter, have the form of the octahedron, icosahedron, and dodecahedron. Also, several mineral crystals and some viruses take the form of these and other polyhedra. Mini-Investigation 11.8 gives you an opportunity to explore further the regular polyhedra.
Write an equation that expresses the relationships you found.
M I N I - I N V E S T I G A T I O N 11 . 8
Finding a Pattern
When you complete the following table and look for a pattern, what relationships among vertices, faces, and edges do you nd?
Polyhedron V F V F E
Tetrahedron Cube Octahedron Dodecahedron Icosahedron
? ? 6 20 12
? ? 8 12 20
? ? 14 32 32
? ? 12 30 30
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The formula you discovered in Mini-Investigation 11.8 is called Eulers formula, named after the Swiss mathematician, Leonhard Euler (17071783). Example 11.10 involves the use of this formula.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
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Example
11.10
Using Eulers Formula
a. Is the polyhedron shown in the next gure a regular polyhedron? Why or why not? b. Does Eulers formula hold for it? Explain.
A
SOLUTION
B
It isnt a regular polyhedron because, although the faces might all be congruent equilateral triangles, the arrangements of polygons at all vertices arent the same: Five sides meet at vertex A, and four sides meet at vertex B. b. In the polyhedron, V = 7, F = 10, and E = 15; 7 + 10 = 15 + 2, so Eulers formula, V + F = E + 2, holds.
YOUR TURN
a.
Practice: Is the second polyhedron shown to the left a regular polyhedron? Why or why not? Does Eulers formula hold for it? Explain. Reect: Do you think that Eulers formula holds for all polyhedra? Explain.
Many different polyhedra can be created simply by slicing off parts of one of the ve regular polyhedra. Figure 11.35 shows the results of such slicing. The idea of
(a) Truncated cube
(b) Truncated octahedron
(c) Cube octahedron
F I G U R E 11 . 3 5
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(a) Slicing off the corners of a cube can change the original square faces to regular octagons; (b) slicing off the corners of a regular octahedron can change the original triangular faces to regular hexagons; (c) slicing off the corners of a cube or an octahedron at the midpoint of each edge changes each of these polyhedra to a cube octahedron.
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slicing or taking a cross section of a solid has many practical applications. For example, X-ray tomography is the use of computer graphic techniques to create a three-dimensional object solely from data about planar slices of the object. Contour maps, temperature analyses of materials, and biological analyses are other important uses of slicing. Example 11.11 involves analyzing the polyhedron that results when the faces of a regular polyhedron are sliced in a certain way.
Example
11.11
Analyzing a Sliced Polyhedron
Describe the number and type of faces of the polyhedron formed when every corner of a tetrahedron is sliced as shown to the left. Show that Eulers formula holds.
SOLUTION
The polyhedron has 4 hexagonal and 4 triangular faces. It has 12 vertices, 8 faces, and 18 edges. As 12 + 8 = 18 + 2, Eulers formula, V + F = E + 2, holds.
YOUR TURN
Practice: Complete the example problem by slicing off the corners of a cube. Reect: Do these examples prove that Eulers formula holds for any polyhedron formed by cutting off the corners of another polyhedron?
Prisms and Pyramids
A prism is a polyhedron with a pair of congruent faces, called bases, that lie in parallel planes. The vertices of the bases are joined to form the parallelogram-shaped lateral faces of the prism. Adjacent lateral faces share a common edge called a lateral edge. An altitude of a prism is a segment that is perpendicular to both bases with endpoints in the planes of the bases.
base lateral edge lateral face base Prism
altitude
As shown in Figure 11.36, a prism is named by the shape of its bases. If the lateral edges of a prism are perpendicular to its bases, the prism is a right prism. If the lateral edges are not perpendicular to the bases, the prism is an oblique prism. A pyramid is a polyhedron formed by connecting the vertices of a polygon to a point not in the plane of the polygon. The polygon is called the base of the pyramid, and the point is called the vertex of the pyramid. Since the vertex is also the highest point of the pyramid, relative to its base, it is also called the apex of the pyramid. The remaining faces are triangles and are called the lateral faces of the pyramid. The segment from the vertex perpendicular to the base is called the altitude of the pyramid.
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S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
705
(a) Right square, or rectangular, prism F I G U R E 11 . 3 6
(b) Right triangular prism
(c) Right pentagonal prism
(d) Oblique square prism
Prisms.
vertex
(a) Triangular pyramid F I G U R E 11 . 3 7
(b) Square, or rectangular, pyramid
(c) Pentagonal pyramid
(d) Oblique square pyramid
lateral face slant height altitude base Right Regular Pyramid
Pyramids.
As shown in Figure 11.37, a pyramid is named by the shape of its base. When the base of a pyramid is a regular polygon, the lateral faces are isosceles triangles, and the altitude is perpendicular to the base at its center, the pyramid is called a right regular pyramid. In such a pyramid, the height of an isosceles triangular lateral face is called the slant height. In contrast to the pyramid shown in the cartoon on p. 706, the pyramids built by the ancient Egyptians were spectacular. Mini-Investigation 11.9 suggests that you broaden your perspective on the applicability on Eulers formula.
M I N I - I N V E S T I G A T I O N 11 . 9
Draw a square pyramid and show that Eulers formula holds for it.
Using Mathematical Reasoning
Do you think that Eulers formula holds for prisms and pyramids? Copy and complete a chart like the following to help you investigate this question and support your conclusion.
Name of Polyhedron Number of Number of Number of Base Edges Vertices V Faces F Number of V F E 2? Edges E (Yes or No)
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Triangular prism Square prism n-gon prism Triangular pyramid Square pyramid n-gon pyramid
3 ? n 3 ? n
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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C H A P T E R 11
EXTENDING GEOMETRY
The New Yorker Collection 1963 Robert Weber from cartoonbank.com. All Rights Reserved.
Cylinders, Cones, and Spheres
If you imagine prisms, pyramids, and polyhedra with thousands of faces, these solid gures come very close to the solids with curved surfaces shown in Figure 11.38. In the cylinder and cone, the shaded circles are called bases. The radius, r, of the cylinder or cone is the radius of its base. The height, h, of a cylinder is the perpendicular distance from one base to the other. The line through the centers of the bases of a cylinder is called the axis of the cylinder. The height, h, of the cone is the perpendicular distance from the vertex, or apex, V, to the base. The line through the center of the base and the vertex is called the axis of the cone. Point O is the center of the sphere, and r is the radius of the sphere. From ice cream cones to cans to architecture
r
V
h
h
O
r
r (a) Circular cylinder F I G U R E 11 . 3 8
r (b) Circular cone (c) Sphere
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Cylinders, cones, and spheres.
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S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
707
to the earth and moon, the everyday importance of these building blocks of geometry with curved surfaces is apparent. Along with prisms, pyramids, and other polyhedra, cylinders, cones, and spheres are used to model real objects. Example 11.12 asks you to compare the characteristics of different three-dimensional gures.
Example
11.12
Analyzing Cylinders, Cones, and Spheres
a. How are a cylinder and a prism alike? b. How are a cylinder and a prism different?
SOLUTION
They both have pairs of congruent, parallel bases. Each has a dimension called height. They are both three-dimensional gures. b. The bases of a prism are polygons; the bases of a cylinder are not polygons. A prism has lateral faces that are polygons; a cylinder has a lateral surface that is not made up of polygons.
YOUR TURN
a.
Practice: How are a cone and a pyramid alike? How are they different? Reect: Which polyhedra are most like a sphere? Support your answer.
Symmetry in Three Dimensions
In Section 10.1, we discussed the symmetry of two-dimensional gures. Symmetry is also an important characteristic of some three-dimensional gures. The apple shown in Figure 11.39, in its ideal form, has reectional symmetry. The vertical slice that separates the apple into two symmetric parts goes through a plane of symmetry of the apple. You might think of this plane as a mirror. If half the apple were held against the mirror, it together with its reection would appear to be a whole apple.
ISBN: 0-536-08809-8
F I G U R E 11 . 3 9
Reectional and rotational symmetry in an ideal apple.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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(a) Plane of reflectional symmetry, parallel to the faces F I G U R E 11 . 4 0
(b) Plane of reflectional symmetry, containing two edges
(c) Axis of rotational symmetry, through the center of opposite faces
(d) Axis of rotational symmetry, through midpoints of pairs of opposite edges
(e) Axis of rotational symmetry, through opposite vertices
Symmetry of a cube.
Slicing an ideal apple horizontally provides an example of rotational symmetry. The part of the apple showing a star can be rotated 72 about an axis through its center to appear to be in the same position. Thus we say that the piece of apple has 72 rotational symmetry. The apple piece can be rotated through 72 ve times before returning to its original position. Thus we say that the apple piece has rotational symmetry of order 5. The axis through the center represents the axis of rotational symmetry. A cube has several planes of reectional symmetry. One plane of symmetry is parallel to a pair of faces of the cube and intersects edges at only one point, as shown in Figure 11.40(a). Because a cube has three different pairs of faces, there are three of this type of plane of symmetry. Another plane of symmetry contains a pair of edges of the cube, as shown in Figure 11.40(b). As there are six different pairs of edges, there are six of this type of plane of symmetry. A cube also has several axes of rotational symmetry. One type of axis goes through the center of opposite faces, as shown in Figure 11.40(c); there are three of this type of axis. Another type of axis goes through the midpoints of a pair of opposite edges of the cube, as shown in Figure 11.40(d); there are six of this type of axis. A third type of axis goes through opposite vertices of the cube, as shown in Figure 11.40(e); there are four pairs of opposite vertices, so there are four of this type of axis. Visualizing reectional and rotational symmetry properties of the cube is a starting point for analyzing symmetry properties of other solid gures in order to solve practical problems. Example 11.13 shows how to solve a practical problem by applying threedimensional symmetry ideas. The problem-solving strategies draw a diagram or make a model (with drinking straws) or both are helpful when solving this problem.
Example
11.13
Problem Solving: Planning a Sculpture
An artist wants to build a sculpture that includes a tetrahedral object with rods welded to its exterior. She decided to weld the rods so that they would look like extensions of all the axes of symmetry of the tetrahedron. What instructions would help her decide where to weld the rods?
WORKING TOWARD A SOLUTION
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One type of axis of rotational symmetry passes through a vertex and the center of a face opposite this vertex, as shown on the left, indicating where some rods should be welded as shown on the facing page.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
709
YOUR TURN
Practice: a. Give additional instructions to tell the artist where rods should be welded. b. Suppose that the artist wanted to weld sheets of metal to show extensions of all planes of symmetry of the tetrahedron. Use the gure on the left to help give her instructions about where to weld them. Reect: A plane of symmetry of a tetrahedron contains an edge and a midpoint of an opposite edge. How many different such planes are there? Explain.
Visualizing Three-Dimensional Figures
One component of spatial sense is representing a three-dimensional gure with a twodimensional picture. There are different types of two-dimensional representations. Visualizing Polyhedra from Their Patterns. A pattern or planar net for a polyhedron is an arrangement of polygons that can be folded to form the polyhedron. For example, each of the patterns shown in Figure 11.41 can be folded to make a cube. They are 3 of 35 possible hexominos, which are made from six squares that are always connected by at least one common side.
F I G U R E 11 . 4 1
Hexomino patterns for a cube.
In Exercise 48 at the end of this section, you are asked to look at pictures of hexominos and visualize how they can be folded to make a cube. Example 11.14 shows how to visualize other polyhedra from patterns.
Example
11.14
Visualizing Polyhedra from Patterns
What polyhedron could be made with the pattern shown on the left?
SOLUTION
If we visualize folding the pattern, we see that a tetrahedron could be formed.
YOUR TURN
Practice: What polyhedra could be made with these patterns?
ISBN: 0-536-08809-8
(a)
(b)
Reect: Do you think that other patterns can be folded to make a tetrahedron? Explain, using an example if possible.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
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C H A P T E R 11
EXTENDING GEOMETRY
Top End
Side Object
F I G U R E 11 . 4 2
End, top, and side views of an object.
Visualizing Three-Dimensional Figures from Different Views. Architects, engineers, and others often need to represent a three-dimensional real-world object on a two-dimensional piece of paper. To give a total picture of the object as they visualize it, they turn it to show an end view, a side view, and a top view and draw a picture of each view. Figure 11.42 illustrates this approach. Computer programs called computer-aided design (CAD) are now used to create such drawings. Some of these CAD programs can receive a three-dimensional image and create a two-dimensional drawing from it. Also, similar technology can receive a twodimensional drawing and create a three-dimensional image from it. These types of CAD programs are valuable aids to those involved in areas such as architecture, building, and manufacturing. In Example 11.15, we look at an object from three different viewpoints.
Example
11.15
Drawing End, Side, and Top Views
Draw the end view, side view, and top views of the iron casting shown.
SOLUTION
End view
Side view
Top view
YOUR TURN
Practice: Draw the end, top, and side view of the object on the left.
ISBN: 0-536-08809-8
Reect: Can you draw a gure accurately after seeing only the side view and the end view? Explain.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
Analyzing a Textbook Page: Different Views of Three-Dimensional Figures
Connection to the PreK8 Classroom
To develop space perception skills, students need to learn to look at actual twodimensional objects, or drawings of objects on a page, and visualize what the object would look like from different points of view. Study the page from a middle school mathematics textbook shown in Figure 11.43 and answer the questions.
Do you need to see all
three views (front, side, and top) in order to determine the original gure? Why or why not?
Can two gures have the
same front, side, and top views? Explain.
When would these views of
a three-dimensional object be useful?
F I G U R E 11 . 4 3
Excerpt from a middle school mathematics textbook.
ISBN: 0-536-08809-8
Source: Page 603 from Scott ForesmanAddison Wesley Middle School Math, Course 1 by R. I. Charles, J. A. Dossey, S. J. Leinwand, C. J. Seeley, and C. B. Vonder Embse. 1998 by Pearson Education, Inc. publishing as Prentice Hall. Used by permission.
711
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EXTENDING GEOMETRY
Visualizing Three-Dimensional Figures and Their Shadows. In the transformations discussed in Section 11.1, the dimensions of an object did not change. Transforming a three-dimensional gure into a two-dimensional shadow involves the use of shadow geometry. In such transformations, straightness is preserved, but length may change, as Trixie discovers with her sunbeam.
HI & LOIS, King Features Syndicate. Example 11.16 asks you to visualize the different shadows that could be made with a given three-dimensional object.
Example
11.16
Connecting Shadows to Three-Dimensional Objects
What shadow gures can you make using a cylinder with a circular base?
SOLUTION
Some possible answers are as follows: a. A circle, with a light source on the line through the center of the circle. b. A rectangle, with a light source on a line perpendicular to and bisecting the segment joining the centers of the bases. c. A nonrectangular parallelogram, if the light source creates a beam that is bisecting but not perpendicular to the segment joining the centers of the bases.
YOUR TURN
Practice: What are some shadow gures you can make with a circular cone? Reect: What shapes of shadows can you not make with a cylinder? Why?
Visualizing Three-Dimensional Figures from Perspective Drawings. From daily observation, you have probably noticed that an object nearer to you appears larger than an object of the same size that is farther away. Not until late in the thirteenth century, however, did artists, many of whom were mathematicians, begin to explore this idea of perspective and develop ways to attain a two-dimensional representation of the actual appearance of three-dimensional objects. One technique for depicting three dimensions on a at surface is to base the orientation of the picture on a vanishing point. In a drawing in which the front surface of the object or scene is in a plane parallel to the plane containing the picture, lines that move away from and should appear parallel to the viewer are drawn to meet at
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
713
(a) Draw the front (b) Choose a vanishing of the box parallel point and use a ruler to the plane of to connect the the paper. vertices of the box to the vanishing point. F I G U R E 11 . 4 4
(c) Draw the back of (d) Erase the lines the box with edges that are no longer parallel to the necessary. corresponding edges of the front of the box.
A rectangular box drawn with one-point perspective.
the vanishing point. Such representations are called perspective drawings and are illustrated in Figure 11.44. Example 11.17 gives you an opportunity to practice using a vanishing point to make a perspective drawing.
Example
11.17
Using a Vanishing Point to Make a Perspective Drawing
Use the technique shown in Figure 11.44 to make a perspective drawing of a hexagonal prism.
SOLUTION
(a) Draw the hexagonal base parallel to the plane of the paper.
(b) Choose a vanishing point and connect it to each vertex of the hexagon.
(c) Draw the other base of the prism with appropriate parallel edges.
(d) Erase unnecessary lines.
YOUR TURN
Practice: Draw the prism from a different perspective by putting the vanishing point in a different place on the paper. Reect:
F I G U R E 11 . 4 5
ISBN: 0-536-08809-8
How would you use this technique to draw a hexagonal pyramid?
A cube represented by a perspective drawing on isometric dot paper.
Another technique for creating perspective drawings is the use of isometric dot paper, as shown in Figure 11.45. With isometric dot paper, the front surface of the object is not parallel to the plane containing the picture.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
714
C H A P T E R 11
EXTENDING GEOMETRY
Problems and Exercises
for Section 11.4
10. A truncated cube 11. A truncated rectangular prism 12. Show that Eulers formula holds for the truncated octahedron and cube octahedron in Figure 11.35(b) and (c). 13. Describe the different planes of reectional symmetry for each of the gures shown in Exercise 7. Assume that the prism in (a) is a right prism with equilateral triangular bases and that the pyramids in (b) and (c) are right regular pyramids. 14. Describe the different axes of rotational symmetry for each of the gures shown in Exercise 7. Assume that the triangular faces of gure (a) are equilateral triangles and the nontriangular face of gure (c) is a square. 15. One type of axis of rotational symmetry and one type of plane of reectional symmetry are shown in the following square prism. How many different planes of symmetry and different axes of rotational symmetry does the square prism have?
A. Reinforcing Concepts and Practicing Skills
1. a. What is the difference between a regular tetrahedron and a nonregular tetrahedron? b. Describe or sketch a nonregular tetrahedron. 2. Complete Exercise 1 for an octahedron. 3. Name some real-world objects that can be modeled by the following geometric gures: a. a cube b. a rectangular prism other than a cube c. a triangular prism d. a pyramid e. a cylinder f. a cone g. a sphere 4. Name or describe some other hexahedrons besides a cube. 5. a. How many pairs of bases does a right rectangular prism have? b. Is there any other prism with more than one pair of faces that can be identied as bases? 6. Make a sketch of the following: a. a circular cylinder with radius of approximately 3 cm and height of approximately 20 cm b. a circular cylinder with radius of approximately 10 cm and height of approximately 3 cm c. a circular cone with radius of approximately 3 cm and height of approximately 20 cm d. a circular cone with radius of approximately 10 cm and height of approximately 3 cm 7. Show that Eulers formula holds for each of the following polyhedra:
(a)
(b)
(c)
Copy and complete the following table:
Polyhedron F V F V E
a. b. c.
? ? ?
? ? ?
? ? ?
? ? ?
Show that Eulers formula holds for the polyhedra described in Exercises 811. 8. A right pentagonal prism 9. A hexagonal pyramid
16. a. Describe the different types of planes of symmetry and axes of symmetry of a regular octahedron. b. How many planes and axes of symmetry does a regular octahedron have? c. Make a sketch and compare the symmetry properties of a regular octahedron and a cube. 17. Draw or describe a pyramid that has ve planes of symmetry. 18. Describe the characteristics that a pyramid must have for it to have an axis of rotational symmetry. 19. Describe the characteristics that a prism must have for it to have an axis of rotational symmetry. 20. A regular tetrahedron has two different types of axes of rotational symmetry and one type of plane of reectional symmetry, as shown in (a)(c) in the gure for this exercise on p. 715. Copy and complete the table (also on p. 715), giving the number of axes of rotational symmetry of each order and the number of different planes of symmetry for a regular tetrahedron. 21. Which of the patterns shown on p. 715 can be folded to make an open-top box? 22. A pentomino is a gure formed by ve connected, nonoverlapping congruent squares, each touching others only along a complete side. Of the 12 different
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
715
pentominos, how many can be folded to make an open-top box? 23. a. Draw a pattern for a rectangular prism that is not a cube. b. Draw another pattern for the same rectangular prism. 24. a. Draw a pattern for a rectangular pyramid. b. Draw another pattern for the same rectangular pyramid. 25. Each solid gure shown in the gure below is sliced by a plane. Identify the cross sections formed.
26. Sue built a gure with ve cubes. The top and side views of the gure look as follows:
Top view
Side view
Draw or describe the gure she made. 27. Draw or describe all the gures you could make by joining ve cubes that would have the following top view:
Table for Exercise 20
Number of Axes of Rotational Symmetry
Polyhedron
Regular tetrahedron
Order 2
Order 3
Order 4
Order 5
Order 6
Total
Number of Planes of Symmetry
Figure for Exercise 20
(a) Axis of rotational symmetry through a vertex and the center of the opposite face
(b) Axis of rotational symmetry through a pair of opposite edges
(c) Plane of rotational symmetry intersecting an edge and the midpoint of the edge opposite that edge
Figure for Exercise 21
Figure for Exercise 25
(a)
(b)
(a)
ISBN: 0-536-08809-8
(b)
(c)
(d)
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
716
C H A P T E R 11
EXTENDING GEOMETRY
28. Draw the end, side, and top views of the following object:
29. Describe at least four different shapes of shadows that you can make with a cube. 30. Make a perspective drawing of an isosceles triangular pyramid. Draw it so that all edges appear, using dotted lines if necessary. 31. Use isometric dot paper to draw a perspective representation of a shape that could be made with three cubes.
b. Keeping the front face of the box in the same position, move the vanishing point closer to the upper left corner of your paper and make a perspective drawing. c. Repeat with the vanishing point near the upper right corner of the paper, then near the lower right corner, and nally near the lower left corner. d. Compare the drawings and make a conjecture about the effect that the position of the vanishing point has on the perspective. 39. How could you show a cube from two different perspectives on the isometric dot paper shown in the gure below?
B. Deepening Understanding
32. What is the resulting polyhedron if a. the center points of the faces of an octahedron are connected? b. the center points of the faces of a tetrahedron are connected? 33. How can you form a tetrahedron by connecting selected vertices of a cube? Explain how you know that the edges of this tetrahedron are congruent. 34. Draw and describe one of the six identical polyhedra that are formed when a point at the center of a cube is connected to each of the vertices of the cube. 35. Why are the regular polyhedra good models for dice? 36. Compare the patterns in Exercises 23(a) and 24(a). a. How are they alike? b. How are they different? c. How does each illustrate the denition of prism and of pyramid? 37. An antiprism is like a prism except that the lateral faces are triangles and the bases have been rotated. a. Show that Eulers formula holds for the following square antiprism:
C. Reasoning and Problem Solving
40. Answer the following questions to determine why Eulers formula continues to hold for the polyhedron formed by cutting corners off of an octahedron, as in Figure 11.35(b): a. For an octahedron, V = ____, F = ____, and E = _____. b. When you slice off one corner of the octahedron, you (gain or lose) ____ vertices, (gain or lose) ____ faces, and (gain or lose) ____ edges. c. Therefore, the total change in V is ____; the total change in F is ____; and the total change in V + F is ____. d. The total change in E is ____. e. What does the comparison of the total change in V + F to the total change in E tell you? 41. Use the reasoning from the previous exercise to determine why Eulers formula would hold for the polyhedra formed by slicing off the corners of any prism. 42. Show how to slice a cylindrical piece of cheese into eight congruent pieces with three slices. 43. What do you think the new exposed surfaces will look like when the following cube is separated into two parts by a slice that goes through the marked midpoints of its edges? Justify your
How many faces, edges, and vertices would a triangular antiprism have? Does Eulers formula hold? 38. a. Place a vanishing point in about the middle of your paper to make a perspective drawing of a rectangular box.
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
S E C T I O N 11 . 4
THREE-DIMENSIONAL FIGURES
717
44. The Toy Blocks Problem. A toy factory manager has eight hundred 8-inch cubic blocks that have been painted red. She wants to produce small blocks by cutting each 8-inch cube into sixty-four 2-inch blocks. To utilize her machines as efciently as possible, she wants to make the fewest number of cuts possible. She needs unpainted blocks to package in a block set and blocks painted on exactly two faces to package in another set. Prepare a report for the manager indicating a. the fewest cuts needed to produce 64 cubes from one block. b. the total number of unpainted blocks and blocks painted on two faces that can be produced. Justify your conclusions in the report. 45. The Polyhedra Sculpture Problem. A sculptor found the following table of information about the dihedral angles (angles formed by adjacent faces) of the regular polyhedra. He knows that polyhedra that will t all the way around an edge with no gaps can be used to make a sculpture he is planning.
Regular polyhedron
Cube Tetrahedron Octahedron Dodecahedron Icosahedron
a. Which deltahedra are also regular polyhedra? b. Draw a picture of a deltahedron with six faces that is not a regular polyhedron. 48. Can every hexomino be folded to make a cube? Explain. 49. Could the following two gures describe actual objects? Explain your conclusions.
(a)
(b)
50. The Dollar Bill Problem. A magician claimed that he could make a tetrahedron out of a $1 bill. He began to fold the bill along the solid lines as shown. Do you believe that he could really do it? Explain your conclusion.
Degree Measure of Dihedral Angle
90 7032 10928 11634 13811 Midpoint
a. Use the table to determine which regular polyhedra could be used in his sculpture. b. Could he use two tetrahedra and two octahedra in combination for his sculpture? Why or why not? 46. Construct a tetrahedron from a sealed envelope, as follows: a. Find point C so that ^ ABC is equilateral. b. Make a cut DE parallel to AB. Discard the other part of the envelope. c. Fold AC and BC in both directions. d. Pinch the envelope to join D and E. e. Tape the opening to make a tetahedron.
A D
51. Without counting all the faces, edges, and vertices, how would you convince someone that Eulers formula still holds for the gure that remains when one corner of a cube is cut off? Make a sketch.
D. Communicating and Connecting Ideas
52. In a small group, discuss differences, if any, between mathematical and everyday meanings of each of the following terms: a. Sphere b. Cube c. Pyramid d. Prism Why is it important to have precise meanings for terms in mathematics? 53. Historical Pathways. The Greek philosopher Plato (430347 B.C.), in his book Timaeus, associated the cube with earth, the tetrahedron with re, the octahedron with air, and the icosahedron with water. He associated the dodecahedron with what was used to create the universe. To develop this cosmology, Plato used the relationships between these polyhedra. Follow the instructions given in order to view two of those relationships. a. Draw a cube as follows: Draw square ADHE in front and a partially dashed square BCGF in back of it.
C
ISBN: 0-536-08809-8
B
E
47. A deltahedron is a polyhedron with equilateral triangle faces.
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
718
C H A P T E R 11
EXTENDING GEOMETRY
B A D
C
F E H
G
Join vertices A and B, D and C, E and F, and H and G.
b. Now use a colored pencil to join all midpoints of adjacent faces of the cube. What polyhedron is formed? c. Now draw another cube and use a colored pencil to join vertices B, D, E, and G with all possible segments. What polyhedron is formed? 54. Making Connections. Put a familiar object in a bag and have another person visualize it only by feeling, using names of polyhedra or other geometric ideas to describe it. Discuss the value of appropriate mathematical language in doing this activity.
Chapter Summary
Reections on the Big Ideas and Essential Understandings: Questions and Answers
S E C T I O N 11 .1
What are some different types of transformations? (pp. 654661) An object can be mapped onto any other gure congruent to it through the use of one or more of three types of transformations called isometries: translations, rotations, and reections. How can we use congruence transformations, or isometries, to describe symmetry? ( pp. 661664) When a gure can be traced and folded so that one half exactly coincides with the other half, it has reectional symmetry about the line of symmetry. When it can be turned less than 360 about a xed point called a center of rotational symmetry to t back on itself, it has rotational symmetry. When a gure ts back on itself after a halfturn, it has point symmetry. What kind of transformations can change a gures size? (pp. 665668) A size transformation enlarges or shrinks a gure along lines determined by a point called the center of the size transformation and according to a multiplier called the scale factor. A combination of one or more isometries and a size transformation can map a gure onto any other similar gure. What kind of transformations can change both a gures size and shape? (pp. 668669) Topological transformations preserve the connectedness of curves and surfaces but can change both size and shape. What is meant by geometric patterns and tessellations? (pp. 676677) A design repeating some basic element in a systematic manner is called a pattern. A tessellation is a special type of pattern that consists of
geometric gures that t without gaps or overlaps to cover the plane. Which polygons will tessellate the plane by themselves? (pp. 677678) Equilateral triangles, squares, and regular hexagons are the only regular polygons that tessellate by themselves. Every triangle and quadrilateral will tessellate. Several nonregular pentagons and hexagons will tessellate. No convex polygon with more than six sides will tessellate, but some nonconvex polygons with more than six sides will tessellate. Can combinations of regular polygons tessellate the plane? (pp. 678680) Many different tessellations can be formed with a combination of regular polygons. Of interest are semiregular tessellations, formed by two or more polygons with the arrangement of polygons at each vertex the same. There are eight different semiregular tessellations of the plane. What other interesting ways are there to generate tessellations? (pp. 680681) A three-mirror kaleidoscope or computer software such as GES or LOGO can be used to generate tessellations using translations, rotations, and reections. How can tessellations with irregular or curved sides be formed? (pp. 681685) Techniques include replacing the edges of a tessellating polygon with curved lines (using rotation in the cut and turn or translation in the cut and slide methods) or making a design in the interior of a tessellating polygon. What are some characteristics of golden triangles and rectangles? (p. 691) In both a golden triangle and a golden rectangle, the ratio of its longer side to its shorter side is the golden ratio, f = (1 +2 15) L 1.618.
S E C T I O N 11 . 2
S E C T I O N 11 . 3
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
CHAPTER SUMMARY
719
What are some generalizations that can be made about star polygons? (pp. 692694) If n = the number of equally spaced points on the circle and d = the dth point that segments are drawn to, (a) the star polygon n {n} is the same as the star polygon {n - d}; (b) the polyd n n gons {1} and {n - 1} are regular polygons; and (c) the n-sided star polygon {n} exists only if d Z 1, d Z n - 1, d and n and d are relatively prime. What are some characteristics of star-shaped polygons? (pp. 694697) Star-shaped polygons have n startip points, 2n congruent sides, n congruent point angles with measure a, and n congruent dent angles b such that b = (360) + a. They differ from star polygons in n that they are simple polygons, whereas star polygons are nonsimple.
S E C T I O N 11 . 4
What are regular polyhedra? (pp. 701704) A polyhedron is a collection of polygons joined to enclose a region of space, forming a gure with faces, edges, and vertices. A regular polyhedrons faces are congruent regular polygons, and the arrangement of these polygons is the same at all vertices of the polyhedron. The ve regular polyhedra include the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron. What are some other important polyhedra? (pp. 704705) Prisms and pyramids are important types of polyhedra. A prism has two congruent, parallel bases
joined by quadrilateral lateral faces. A pyramid has one base and triangular lateral faces joined at a common vertex called the apex. What important three-dimensional gures are there other than polyhedra? (pp. 706707) Cylinders, cones, and spheres are three-dimensional gures with curved surfaces, rather than polygonal faces as in polyhedra. Cylinders are like prisms, in that they have two congruent, parallel bases. Cones are like pyramids, in that they have one base and an apex. A sphere might be considered like a regular polyhedron with a very large number of faces. How can we apply the idea of symmetry to threedimensional gures? (pp. 707709) A three-dimensional gure that can be sliced into two congruent parts along a plane of symmetry is said to have reectional symmetry. A three-dimensional gure that can be rotated less than 360 about an axis of rotational symmetry until it matches itself is said to have rotational symmetry. How can three-dimensional gures be visualized? (pp. 709713) Three-dimensional gures can be represented in two dimensions in a variety of ways. A pattern, or planar net, can be used to show an arrangement of polygons that can be folded to form a polyhedron. A three-dimensional object can be turned to show only its end view, its side view, or its top view. Shadows can also produce two-dimensional representations of threedimensional objects. Finally, perspective drawings based on a vanishing point can be created to represent threedimensional gures.
Terms, Concepts, and Generalizations
S E C T I O N 11 .1
ISBN: 0-536-08809-8
Image (p. 655) Slide arrow (p. 655) Directed segment (p. 655) Translation (p. 655) Rotation (p. 656) Reection (p. 657) Glidereection (p. 660) Congruence transformation (p. 660) Isometry (p. 660) Congruence (p. 661) Reectional symmetry (p. 661) Line of symmetry (p. 661) n rotational symmetry (p. 663) Center of rotational symmetry (p. 664) Point symmetry (p. 664) Size transformation (p. 665) Center of size transformation (p. 665) Scale factor (p. 665)
Similarity (p. 666) Topological transformation (p. 668) Topologically equivalent (p. 668)
S E C T I O N 11 . 2
Pattern (p. 676) Tessellation (p. 676) Regular tessellation (p. 677) Semiregular tessellation (p. 678)
S E C T I O N 11 . 3
Golden triangle (p. 691) Pentagram (p. 691) Golden rectangle (p. 691) Star polygon (p. 692) Regular star polygon (p. 692) Star-shaped polygon (p. 694)
S E C T I O N 11 . 4
Polyhedron (p. 701) Regular polyhedra (p. 701)
Eulers formula (p. 702) Prism (p. 704) Right prism (p. 704) Oblique prism (p. 704) Pyramid (p. 704) Right regular pyramid (p. 705) Cylinder (p. 706) Cone (p. 706) Sphere (p. 706) Reectional symmetry (p. 707) Plane of symmetry (p. 707) Rotational symmetry (p. 708) Axis of rotational symmetry (p. 708) Pattern or planar net (p. 709) Shadow geometry (p. 712) Vanishing point (p. 712) Perspective drawings (p. 713)
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
720
C H A P T E R 11
EXTENDING GEOMETRY
Chapter Review
Concepts and Skills
1. Describe the ve regular polyhedra and explain how they are named. 2. Give the dimensions of a picture frame that is a golden rectangle. 3. Show that Eulers formula holds for a square pyramid. 4. Draw two patterns of squares that can be used to form cubes. 5. Draw the front, side, and top views of the following object: 15. Consider the following gure:
a. If it is drawn on a balloon that can be stretched, shrunk, and twisted, draw two different shapes that could be produced. b. If it is made of wire, draw two different shapes that its shadow could take.
Reasoning and Problem Solving
6. Draw a cube in perspective. 7. Draw a tetrahedron in perspective. 8. Draw a sketch of the star polygon {8}. Give another sym3 bold for this same star polygon. Is it a regular polygon? 9. An eight-point star-shaped polygon has a point angle of 36. What is the measure of its dent angle? 10. Describe the symmetry properties of the following gures: 16. If you marked the midpoints of the edges of a cube and sliced off all its corners through the midpoints of its edges, how many and what type of faces would the truncated gure have? 17. Do you believe the following generalization about the measure of a point angle of a star polygon is true? The measure of a point angle of the star polygon E n F is d (|n - 2d|180) . Support your belief in writing, giving n evidence. 18. How do the axes of rotational symmetry of an octahedron compare to the axes of rotational symmetry of a cube?
(a)
(b)
11. How many planes and axes of rotational symmetry does a right rectangular prism have? 12. Name three regular polygons that will tessellate the plane. 13. Draw a small portion of a semiregular tessellation. a. Tell why it is semiregular. b. What is true about a tessellation that utilizes a combination of regular polygons but isnt semiregular? 14. Use tracing paper to nd the image of the quadrilateral shown for the following transformations:
B E D A C
a. TAB 4 c. MAB e. TAB*RB,180
b. R A,90 d. SE,2
19. The Fancy Quilt Problem. Stacy wants to make a quilt that uses a 12-pointed star-shaped polygon that tessellates with equilateral triangles. To make the star, she needs to know the measure in degrees of the point angles and the dent angles of the star-shaped polygon. If possible, supply this information and sketch the quilt pattern. If not, explain what additional information you need in order to do so.
ISBN: 0-536-08809-8
Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.
CHAPTER REVIEW
721
20. a. How are congruence transformations and size transformations alike? b. How are they different? 21. Draw two topologically equivalent gures that look as different as you can make them. Justify why they are topologically equivalent. 22. Measuring Trees Problem. Jared wants an estimate of the height of some of the trees near his lake house. Use ideas of size transformations to design a plan for him to use his height to estimate accurately a trees height without directly measuring the tree. 23. The Open-Top Box Pattern Problem. A manufacturer found that the machines in his factory could make open-top boxes most efciently if the vesquare box pattern had no more than two squares in a row and had reectional symmetry. He could also manufacture the pattern more easily if it would tessellate the plane. Assume that you are the engineer given
the task of analyzing all 12 possible open-top box patterns, nd a pattern or patterns that meet the conditions, and verify that you have found the correct pattern or patterns.
Alternative Assessment
24. Use paper cutouts or GES to make a tessellation based on the following: a. Slide images b. Turn images c. Flip images Write a paragraph for each one, describing your procedure. 25. Choose ve different types of familiar rectangular objects that you estimate to be in the shape of a golden rectangle and analyze them to see how close they are to being a golden rectangle. Devise a technique to rate the closeness of rectangles to golden rectangles.
ISBN: 0-536-08809-8 Mathematics for Elementary School Teachers, Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc.

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