Isometry Transformations
Objective To review transformations that produce another figure
while maintaining the same size and shape of the original figure.
w
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Plot, name, and label points in any of the
four quadrants of a coordinate grid. [Measurement and Reference Frames Goal 3]
• Identify congruent figures. [Geometry Goal 2]
• Practice and perform isometry
transformations with geometric figures. [Geometry Goal 3]
• Classify a rotation by the number of
degrees needed to produce a given image. [Geometry Goal 3]
Key Activities
Students review and perform isometry
transformations, including reflections,
translations, and rotations.
Ongoing Assessment:
Recognizing Student Achievement
Family
Letters
Assessment
Management
Ongoing Learning & Practice
Making a Circle Graph
with a Protractor
Math Journal 1, p. 182
Geometry Template/protractor compass
Students find fraction, decimal, and
percent equivalencies and calculate
degree measures of sectors to create
a circle graph.
Math Boxes 5 5
Math Journal 1, p. 183
inch ruler
Students practice and maintain skills
through Math Box problems.
Study Link 5 5
Math Masters, p. 160
Students practice and maintain skills
through Study Link activities.
Use an Exit Slip (Math Masters,
page 404). [Measurement and Reference Frames
Goal 3]
Key Vocabulary
isometry transformation transformation translation (slide) reflection (flip) rotation
(turn) image preimage line of reflection
Materials
Math Journal 1, pp. 178–181
Student Reference Book, pp. 180 and 181
Math Masters, p. 404
Study Link 54
transparent mirror tracing paper (optional)
Advance Preparation
For the optional Readiness activity in Part 3, set aside trapezoid pattern blocks.
Teacher’s Reference Manual, Grades 4–6 pp. 193, 196–199
356
Unit 5
Common
Core State
Standards
Geometry: Congruence, Constructions, and Parallel Lines
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Reviewing Degrees and Directions
of Rotation
Math Masters, p. 161
trapezoid pattern block (1 per student)
Students use a full-circle protractor to
practice rotating a figure about a point.
ENRICHMENT
Performing a Scaling Transformation
Math Masters, p. 162
ruler
Students perform size-change
transformations.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 131
Students add the terms translation, reflection,
and rotation to their Math Word Banks.
Mathematical Practices
SMP1, SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
6.NS.6, 6.NS.6b, 6.NS.6c, 6.NS.8
Mental Math and Reflexes
Math Message
Students find the signed number that is halfway between each of the two given
numbers. Suggestions:
Study Problem 1 and
complete Problem 2 on
journal page 178.
between 5 and -5 0
between -7 and -3 -5
between -3 and 2 -0.5
Study Link 5 4
Follow-Up
between 1.5 and -4.5 -1.5
Review answers.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 1, p. 178; Student Reference Book,
pp. 180 and 181)
Students worked with isometry transformations in Fifth Grade
Everyday Mathematics, but some of the vocabulary in this lesson
may be new. Provide students with multiple opportunities to read,
write, and say the vocabulary words. Whenever possible, relate
vocabulary to students’ experiences.
Briefly review each transformation (translation, reflection,
and rotation) and go over the answers to Problems 2a–c. Draw
students’ attention to the fact that each new figure—the image
(2)—is the same size and shape as the original figure—the
preimage (1). Point out that the distance between points also
remains unchanged.
Student Page
Date
Language Arts Link The word isometry comes from the
Greek words iso, meaning same, and metron, meaning
measure.
Time
LESSON
Isometry Transformations
5 5
䉬
Math Message
180 181
Translations (slides), reflections (flips), and rotations (turns) are basic
transformations that can be used to move a figure from one place to another
without changing its size or shape.
1.
Adjusting the Activity
Study each transformation shown below.
Translation (Slide)
Reflection (Flip)
y
y
5
5
4
4
3
Have students identify the following locations on a coordinate grid:
2
2
2
1
1
• the x-axis (or horizontal axis)
• the y-axis (or vertical axis)
Write several ordered number pairs on the board. Ask students to identify the
x- and y-coordinates for each ordered number pair.
1
2
3
4
5
x
3
1
⫺3
⫺4
⫺5
K I N E S T H E T I C
T A C T I L E
3
4
5
x
2
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
1
1
2
3
4
5
x
90⬚
counterclockwise
⫺2
⫺3
⫺4
⫺5
A rotation (turn) moves a
figure around a point.
A reflection (flip) of a figure
gives its mirror image over
a line.
Identify whether the preimage (1) and image (2) are related by a translation, a
reflection, or a rotation. Record your answer on the line below each coordinate grid.
y
A U D I T O R Y
1
2
⫺3
⫺5
4
2
1
⫺2
⫺4
5
3
2
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
⫺2
A translation (slide) moves
each point of a figure a certain
distance in the same direction.
2.
y
270⬚
clockwise
1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
• the origin (0,0)
Rotation (Turn)
V I S U A L
1
5
4
4
3
3
2
2
1
1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
2
a.
y
5
1
2
3
4
5
x
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
⫺2
⫺2
⫺3
⫺3
⫺4
⫺4
⫺5
⫺5
reflection
b.
y
5
4
3
2
2
1
2
1
3
4
5
x
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
1
2⫺2
3
4
5
x
⫺3
1
translation
2
1
⫺4
⫺5
c.
rotation
Math Journal 1, p. 178
Lesson 5 5
357
Student Page
Date
▶ Translating Geometric Figures
Time
LESSON
Translations
5 5
䉬
y
Example:
Translate quadrangle ABCD 6 units
to the right and 5 units up.
4
3
2
D'
1
Plot and label the vertices of the
image that would result from
the translation.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0
⫺1
A
D
B
1
180–181
A'
5
2
4
x
5
⫺2
⫺3
⫺4
C
⫺5
Plot and label the vertices of the image that would result from each translation.
1.
Translate triangle DEF 4 units
to the right and 3 units down.
2.
Translate pentagon LMNOP
0 units to the right and 7 units down.
y
y
D
5
5
4
4
3
E
F
⫺4
⫺3
⫺2
⫺1
0
⫺1
F'
N
L
P
2
1
⫺5
O
M
3
D'
2
1
2
E'
1
3
4
x
5
⫺5
⫺4
⫺3
⫺2
PROBLEM
PRO
P
RO
R
OBL
BLE
B
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VING
VI
VIN
ING
B'
C'
3
(Math Journal 1, p. 179)
⫺1
0
1
⫺2
⫺2
⫺3
⫺3
⫺4
⫺4
⫺5
⫺5
2
3
M'
⫺1
4
N'
L'
x
5
O'
P'
WHOLE-CLASS
ACTIVITY
When a figure is translated, each point on the preimage slides the
same distance in the same direction to create the image. If the
translation is done on a coordinate plane, the image of a point can
be found by adding translation numbers to the coordinates of the
point being translated. In the example on journal page 179, each
point of Figure ABCD is translated 6 units to the right (+6) and
5 units up (+5). Each point of the preimage has a corresponding
point in the image. Point out that to distinguish between the
image and the preimage, students should label an image point
with the same letter as the preimage point and the additional
symbol ('). For example, the image of A is A', read “A prime.”
Try This
3.
Square WXYZ has the following vertices:
W (⫺3,⫺2), X (⫺1,⫺2), Y (⫺1,⫺4), Z (⫺3,⫺4)
Ongoing Assessment:
Recognizing Student Achievement
Without graphing the preimage, list the vertices of image W⬘X⬘Y⬘Z⬘ resulting from
translating each vertex 3 units to the right and 2 units up.
W⬘ (
0 , 0
2 , 0
); X⬘ (
); Y⬘ (
2 ,⫺2 ); Z⬘ ( 0 ,⫺2 )
Math Journal 1, p. 179
Exit Slip
Use the example on journal page 179 to assess students’ ability to name
ordered number pairs in the third quadrant. Students are making adequate
progress if they can correctly name the vertices of preimage ABCD. Ask
students to record the number pairs on an Exit Slip (Math Masters, p. 404).
[Measurement and Reference Frames Goal 3]
List the coordinates of the vertices of ABCD and A'B'C'D'. Show
students how they can produce the image (A'B'C'D') from the
preimage (ABCD) by adding +6 units to each x-coordinate of a
vertex and +5 units to each y-coordinate.
A = (-3,-1)
B = (-2,-2)
C = (-3,-4)
D = (-4,-2)
Student Page
Date
Ask students to pay attention to the coordinates of the vertices of
preimages and images as they complete journal page 179.
Time
LESSON
Reflections
5 5
䉬
Reflect each figure over the indicated axis or line of reflection. Then plot and label the
vertices of the image that results from the reflection. Use a transparent mirror to check
your placement of each image.
Reflect triangle TAM over the x-axis.
2.
Reflect rectangle STUV over the y-axis.
y
y
5
M'
T'
⫺4
T
V
U
⫺3
⫺2
⫺1
A
M
S'
1
U'
V'
0
1
4
0
1
2
3
4
5
x
⫺5
⫺4
⫺3
⫺2
⫺1
2
⫺1
⫺1
⫺2
⫺2
⫺3
⫺3
⫺4
⫺4
⫺5
⫺5
Reflect triangle PQR over the y-axis.
4.
5
4
2
⫺3
⫺1
0
⫺1
E
G
F
R'
1
2
3
Q'
4
5
x
⫺5
⫺4
⫺3
⫺2
4
1
⫺1
0
⫺1
⫺2
⫺2
⫺3
⫺3
⫺4
⫺4
⫺5
⫺5
F'
1
G'
2
3
4
5
E'
D'
Unit 5
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
x
The reflection of a figure appears to be a reversal, or mirror image
of the preimage. Each point on the preimage is the same distance
from the line of reflection as the corresponding point on the
image. When reflecting geometric figures over an axis on the
coordinate grid, the axis is the line of reflection. The preimage
and image are on opposite sides of the axis. If the figure is
reflected across the x-axis, the x-coordinates stay the same and the
y-coordinates differ only by sign. If the figure is reflected across the
y-axis, the y-coordinates stay the same and the x-coordinates differ
only by sign.
Math Journal 1, p. 180
358
WHOLE-CLASS
ACTIVITY
x
2
R
⫺2
5
3
1
⫺4
D
P'
3
⫺5
4
y
5
Q
3
Reflect square DEFG over line m.
y
P
2
▶ Reflecting Geometric Figures
(Math Journal 1, p. 180)
3
1
T
3.
S
3
2
⫺5
T'
5
4
A'
180
m
1.
A' = (3,4)
B' = (4,3)
C' = (3,1)
D' = (2,3)
Geometry: Congruence, Constructions, and Parallel Lines
Student Page
Have students complete journal page 180. Bring the class
together to discuss answers. Ask a volunteer to read the preimage
coordinates and the image coordinates for Problem 1. Record them
on the board, listing corresponding coordinates next to each other.
Ask: What do you notice about the coordinates of the preimage and
the reflected image? The x-coordinates stayed the same and the
y-coordinates changed signs. Repeat this procedure for Problem 2.
The y-coordinates stayed the same and the x-coordinates changed
signs. Ask: Look at Problem 3. Describe a pattern you notice about
the coordinates of figures that are reflected across an axis. The
preimage and image of a figure reflected across the x-axis have the
same x-coordinates and their y-coordinates differ only by sign. The
preimage and image of a figure reflected across the y-axis have the
same y-coordinates and their x-coordinates differ only by sign.
Date
Time
LESSON
Rotations
5 5
䉬
Rotate each figure around the point in the direction given. Then plot and label the
vertices of the image that results from that rotation.
1.
Rotate triangle XYZ 180⬚ clockwise
about Point X.
y
y
5
5
4
3
Z'
2
4
B
⫺4
⫺3
⫺2
X'
⫺1
X
1
0
▶ Rotating Geometric Figures
2
3
4
5
x
⫺5
⫺4
⫺3
⫺1
⫺2
3.
⫺2
B'C
⫺1
⫺2
⫺3
⫺4
⫺5
⫺5
4.
⫺3
P'
M
⫺1
E
3
N
O
2
S' S
1
M'
0
x
5
4
3
1
⫺2
4
V'
5
4
N'
⫺4
3
C' D
y
E'
5
⫺5
E
2
Rotate triangle SEV 270⬚ clockwise
about (0,2).
y
O'
1
0
⫺4
Rotate trapezoid MNOP 90⬚
counterclockwise about point M.
D'
2
1
⫺1
Z
Y
3
⫺3
2
Ask students to look closely at Problem 2 and decide whether
the same image could be obtained by a horizontal translation of
the preimage. No. The rectangles have the same orientation in
relation to the x-axis, but the order of the vertices changes from the
preimage to the image. When a translation is performed, the order
of the vertices stays the same.
E'
Y'
1
⫺5
180
Rotate quadrangle BCDE 90⬚
counterclockwise about the origin.
2.
2P
1
3
4
5
x
⫺5
⫺4
⫺3
⫺2
⫺1
⫺1
⫺2
⫺2
⫺3
⫺3
⫺4
⫺4
⫺5
⫺5
V
1
0
⫺1
2
3
4
x
5
Math Journal 1, p. 181
WHOLE-CLASS
ACTIVITY
(Math Journal 1, pp. 178 and 181)
Adjusting the Activity
ELL
PROBLEM
PR
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IN
ING
Draw attention again to Problem 2c of the Math Message on
journal page 178. Point out that to rotate a figure, the following
are needed:
Have students set a transparent mirror
on the line of reflection to more clearly see the
reversal of the points on a reflected image.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
a specific point about which the figure is to rotate.
a number of degrees the figure is to rotate about that
specific point.
Student Page
Time
LESSON
Making a Circle Graph with a Protractor
5 5
䉬
1.
One way to convert a percent to the degree measure of a sector is to multiply
360⬚ by the decimal equivalent of the percent.
In Problem 2c, the preimage (1) is rotated 90° counterclockwise
about the point (0,-3) to produce the image (2). A clockwise
rotation of 270° of the preimage (1) will also produce the
same image.
55%
Complete the table below.
40%
0.4
Degree Measure
of Sector
0.4 ⴱ 360° ⫽
144⬚
90%
0.9
0.9 ⴱ 360⬚ ⫽ 324⬚
65%
0.65
0.65 ⴱ 360⬚ ⫽ 234⬚
5%
0.05
0.05 ⴱ 360⬚ ⫽ 18⬚
1%
0.01
0.01 ⴱ 360⬚ ⫽ 3.6⬚
Elective Courses
Taken by
Seventh Graders
Music
Art
2.
Make copies of journal page 181 from which students can cut out
each preimage and perform each rotation manually. Students can also trace the
preimage onto another sheet of paper and rotate the paper to help determine
the orientation of the image.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
The table below shows the elective courses taken by
a class of seventh graders. Complete the table. Then,
in the space to the right, use a protractor to make a
circle graph to display the information. Do not use the
Percent Circle. (Reminder: Use a fraction or a
decimal to find the degree measure of each sector.)
Write a title for the graph.
Course
Number
Fraction
of Students of Students
Photography
rs
Decimal
Equivalent
te
Percent
of Circle
Decimal
Equivalent
Percent
of Students
pu
Have students work in partnerships on journal page 181.
Adjusting the Activity
59 60
147
Example: What is the degree measure of a sector that is 55% of a circle?
55% of 360⬚ ⫽ 0.55 ⴱ 360⬚ ⫽ 198⬚
m
To support English language learners, discuss the meaning of
clockwise and counterclockwise.
Co
a specific direction of rotation (clockwise or counterclockwise).
Date
Degree
Measure
of Sector
Music
6
6
ᎏᎏ
30
0.2
20%
72⬚
Art
9
9
ᎏᎏ
30
0.3
30%
108⬚
Computers
10
10
ᎏᎏ
30
0.33
–
33 ᎏ13ᎏ%
120⬚
5
5
ᎏᎏ
30
–
16 ᎏ23ᎏ%
60⬚
Photography
0.16
Math Journal 1, p. 182
Lesson 5 5
359
Student Page
Date
Time
LESSON
Links to the Future
Math Boxes
55
䉬
Rectangle MNOP has sides parallel to the
axes. What are the coordinates of point O?
1.
2.
Solve mentally.
a.
20% of 50 ⫽
y
3
b. ᎏᎏ
8
M (⫺3,2)
N (3,2)
48
50
c.
x
0
d.
O
P (⫺3,⫺2)
of 48 ⫽
10
18
Offer an example of a transformation that is not an isometry transformation. In
Unit 8 of Sixth Grade Everyday Mathematics, students will study similar figures,
which are produced by scaling transformations. A scaling transformation
produces a figure that is the same shape as the original figure but not
necessarily the same size. The following shows a scaling (but not an
isometry) transformation:
⫽ 75% of 64
5
9
⫽ ᎏᎏ of 90
3 , ⫺2 )
Coordinates of point O: (
49 50
87
234
3. a.
b.
image
preimage
5
Draw a line segment that is 3 ᎏ1ᎏ
inches long.
6
By how many inches would you need to extend the line segment you drew
to make it 5 inches long?
11
1ᎏ16ᎏ inches
85 86
Multiply or divide.
4.
9
15
1
5
a.
45 ⴱ ᎏᎏ ⫽
b.
60 ⫼ 4 ⫽
7
9
c.
d.
5.
⫽ 56 ⴱ
1
ᎏᎏ
8
Find the value that makes each number
sentence true.
a.
32 ⫹ n ⫽ 52
n⫽
b.
y ⫺ 15 ⫽ 20
y⫽
c.
(2 ⴱ m) ⫹ 5 ⫽ 17
⫽ 108 ⫼ 12
m⫽
2 Ongoing Learning & Practice
20
35
6
88
242 243
▶ Making a Circle Graph
INDEPENDENT
ACTIVITY
with a Protractor
Math Journal 1, p. 183
(Math Journal 1, p. 182)
Students find fraction, decimal, and percent equivalencies;
calculate degree measures of sectors; and use a compass
and a protractor to create a circle graph to display elective
course information.
▶ Math Boxes 5 5
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 183)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 5-7. The skills in Problems 4 and 5
preview Unit 6 content.
Study Link Master
Name
Date
Time
Transforming Patterns
STUDY LINK
55
䉬
A pattern can be translated, reflected, or rotated to create
many different designs. Consider the pattern at the right.
180 181
The following examples show how the pattern can be
transformed to create different designs:
A
D
C
B
1.
Reflections
Rotations
Translations
Writing and Reasoning Have students write a response to
the following: Explain how to rewrite Problems 4b and 4d
as multiplication problems. Sample answer: The division
60 . Because
problem 60 ÷ 4 can be expressed as the fraction _
4
60 = 60 ∗ _
1 , 60 ÷ 4 and 60 ∗ _
1 are equivalent expressions.
_
4
4
4
108 or 108 ∗ _
1.
Similarly, 108 ÷ 12 can be expressed as _
12
12
Translate the pattern at the right across
2 grid squares. Then translate the
resulting pattern (the given pattern and
its translation) down 2 grid squares.
▶ Study Link 5 5
INDEPENDENT
ACTIVITY
(Math Masters, p. 160)
2.
Rotate the given pattern clockwise 90⬚
around point X. Repeat 2 more times.
X
Home Connection Students practice performing
transformations on a grid. If necessary, review the
examples at the top of page 160 with the class.
J
3.
Reflect the given pattern over line JK.
Reflect the resulting pattern (the given
pattern and its reflection) over line LM.
L
M
K
Practice
4.
26 ⫽
64
5.
35 ⫽
243
6.
70 ⫽
1
7.
43 ⫽
64
Math Masters, p. 160
360
Unit 5
Geometry: Congruence, Constructions, and Parallel Lines
Teaching Master
Name
3 Differentiation Options
READINESS
▶ Reviewing Degrees and
Date
LESSON
55
Time
Degrees and Directions of Rotation
When a figure is rotated, it is turned a certain number of degrees around a
particular point. A figure can be rotated clockwise or counterclockwise.
INDEPENDENT
ACTIVITY
5–15 Min
Position a trapezoid pattern block
on the center point of the angle
measurer as shown at the right. Then
rotate the pattern block as indicated
and trace it in its new position.
degrees
11 12 1
Example: Rotate 90° clockwise.
2
10
9
3
8
Directions of Rotation
4
7
6
5
(Math Masters, p. 161)
To provide experience using an angle measurer and rotating a
figure, have students complete Math Masters, page 161. Students
use an angle measurer with an embedded clock face to practice
rotating a trapezoid pattern block a given number of degrees in a
clockwise or counterclockwise direction. If necessary, provide
additional opportunities to rotate the trapezoid.
ENRICHMENT
▶ Performing a Scaling
INDEPENDENT
ACTIVITY
5–15 Min
For each problem below, rotate and then trace the pattern block in its new position.
1.
2.
Rotate 90° counterclockwise.
Rotate 270° clockwise.
degrees
degrees
11 12 1
11 12 1
10
2
9
10
4
7
6
2
9
3
8
3
8
5
4
7
6
5
Math Masters, p. 161
Transformation
(Math Masters, p. 162)
To deepen students’ understanding of transformations and to
introduce a size-change transformation, have students complete
Math Masters, page 162. Students follow steps to perform scaling
transformations to produce images that are twice and half the size
of preimages. Have students describe the relationships between
the original figure and the enlarged or reduced figure. Encourage
them to use the vocabulary they have developed in this unit.
ELL SUPPORT
▶ Building a Math Word Bank
INDEPENDENT
ACTIVITY
Teaching Master
Name
5–15 Min
(Differentiation Handbook, p. 131)
To provide language support for transformational geometry
terms, have students use the Word Bank template found on
Differentiation Handbook, page 131. Ask students to write the
terms translation, reflection, and rotation, draw pictures depicting
each term, and write other related words. See the Differentiation
Handbook for more information.
LESSON
55
䉬
Date
Time
Scaling Transformations
Some scaling transformations produce a figure that is the same shape as the original
figure but not necessarily the same size. Enlargements and reductions are types of
scaling transformations.
Enlargement: Follow the steps to draw a triangle D⬘E⬘F⬘ with angles that are congruent to
triangle DEF and sides that are twice as long as triangle DEF.
P
E
D
F
Step 1
Draw rays from P through each vertex. The first ray P៮៮៬
D has been drawn
for you.
Step 2
Measure the distance from point P to vertex D. Then locate the point on
P៮៮៬
D that is 2 times that distance. Label it D⬘.
Step 3
Use the same method from Step 2 to locate point F⬘ on P៮៮៬
F and
point E ⬘ on P៮៮៬
E.
Step 4
Connect points D⬘, E⬘, and F⬘.
Reduction: Change Steps 2 and 3 to draw a triangle D⬙E ⬙F⬙ with angles that are
congruent to triangle DEF and sides that are half as long as triangle DEF.
Math Masters, p. 162
Lesson 5 5
361