Problem
4.1
Topic 4: Translations in the Coordinate Plane
for use after
Shapes and Designs
(
Investigation 4)
A
transformation
is the change in the size, shape, or position of a figure.
A
translation
is a transformation in which each point of a figure moves the
same distance and in the same direction. This design contains many such
figures.
A. Copy ∆
ABC
and translate it to ∆
A
’
B
’
C
’ using the steps below.
1. From
A
, count down 2 units and to the left 3 units. Label the new
point
A
’(ay-prime).
2. Find and label points
B’
and
C’
by counting down 2 units and left
3 units.
3. Draw ∆
A
’
B
’
C
’.
B. Draw a line from
A
to
A
’, from
B
to
B
’, and from
C
to
C
’.
1. Compare the length of the three lines.
2. Compare the direction of the three lines.
3. Explain why ∆
A
’
B
’
C
’ is a translation of ∆
ABC
.
2
4
6
y
O
2
46
x
A
BC
Exercises
1. For each of the directions below, copy the graph and translate
∆
RST.
Label the image ∆
R’S’T’
.
a. Translate ∆
RST
up 2 units.
b. Translate ∆
RST
to the right 2 units.
c. Translate ∆
RST
to the left 2 units and down 4 units.
d. Translate ∆
RST
to the right 1 unit and down 1 unit.
e. Translate ∆
RST
to the left 2 units and up 1 unit.
2. Danielle drew the figures at the right to represent a
translation.
a. Describe the translation of point
E
to point
E
’.
b. Name the coordinates of an unlabeled point on the
bottom figure, then give the coordinates of the
translated image of that point.
c. Jeremy says that Danielle only plotted 6 points to do
the translation, so that means only 6 points on the
original figure were translated. Do you agree with
Jeremy?
3. Chee wrote this rule to describe the translation of ?
ABC
to ∆
A’B’C’
:
(
x
,
y
)
(
x
? 1,
y
4)
a. How does Chee’s rule use coordinates to translate a figure?
b. Chee drew ∆
KLM
with vertices at
K
(2, 7),
L
(4, 6), and
M
(3, 4). He
then followed his own rule to draw ∆
K’L’M’
. Draw both of these
triangles.
2
2
4
6
8
y
2468
x
EK
FG
JH
E
?
K
?
F
?
G
?
J
?
H
?
O
2
4
6
y
O
2
46
x
S
R
T
Topic 4: Translations in the Coordinate
Plane
Guided Instruction
Mathematical Goals
• Identify the translations used to move a polygon from one location to
another in the coordinate plane.
• Explain how translations affect the location of a polygon in the
coordinate plane.
Vocabulary
•
transformation
•
translation
Materials
•
Labsheets 4. 1,
4ACE Exercise 1
At a Glance
Among the three main types of transformations your students will study—
translations, rotations, and reflections—translations should be the easiest
for the students to grasp. Students will find translations relatively easy to
model by tracing the outline of a flat shape on graph paper before and after
sliding the shape from one location to another. Have students work in pairs
for this type of modeling. In those activities where students copy a shape
and then draw its prescribed translation, tracing paper can be used to
confirm that the original shape and the translated image are congruent.
In the translation problems and exercises, the points being used for the
translation are all vertices. You should briefly review the definition of a
vertex as the point in a polygon where two sides meet.
Although the translations in this lesson are shown as occurring only in
the first quadrant, more advanced students should be encouraged to
translate shapes between any two locations in the coordinate plane.
Before Problem 4.1:
•
Describe some of the figures in the design at the top of the page that are
repeated as you move from left to right.
(various descriptions of
triangles and a square.)
During Problem 4.1, A:
•
Why do you think it is a good idea to name the translated triangle with
A’, B’, and C’ instead of just using other letters altogether?
(It makes it
easy to match up the original points with the translated points.)
After Problem 4.1:
•
How can you prove that the direction from A to A’ is the same as the
direction from B to B’ and C to C’?
(They all form a diagonal of a
2 unit ? 3 unit rectangle.)
•
How could you make sure that A’B’C’ is the same size and shape as
ABC?
(Trace one triangle onto tracing paper and see if the shapes
match; or measure all the sides and the angles.)
•
What changes when you translate a figure?
(Its position.)
You will find additional work on transformations in the grade 8 unit
Kaleidoscopes, Hubcaps, and Mirrors.
D
D
PACING
1 day
ACE Assignment Guide
for Topic 4
Core
1–3
Answers to Topic 4
Problem 4.1
A.
B. 1.
The lines are the same length.
2.
The lines are in the same direction.
3.
Every point on
ABC
moved the same
distance and in the same direction.
Exercises
1. a.
b.
c.
d.
e.
2. a.
To the right 2 units then up 2 units.
b.
(2, 1); (4, 3)
c.
Answers may vary. Sample: Every single
point on the original figure was translated.
The number of points translated is infinite.
3. a.
It says that every point on
ABC
moves 1
unit to the right, then 4 units down.
b.
46
28
2
4
6
8
0
0
y
x
M
K
L
K
?
L
?
M
?
D
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6
8
0
0
y
x
R
S
T
R
?
S
?
T
?
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8
0
0
y
x
R
S
R
?
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S
?
T
?
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y
x
R
S
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R
?
S
?
T
?
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0
y
x
R
S
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R
?
S
?
T
?
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28
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0
0
y
x
R
S
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R
?
S
?
T
?
D
2
46
2
4
6
O
y
x
A
BC
A
?
B
?
C
?
Name
Date
Class
Labsheet 4.1
Topic 4
Translations in the Coordinate Plane
A.
2
4
6
y
O
246
x
A
BC
© Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
© Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
Name
Date
Class
Labsheet 4ACE Exercise 1
Topic 4
a.
b.
c.
d.
e.
2
4
6
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246
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