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5.2
Coordinate Rules for Translations
Goals
• In this problem, you will look for patterns that
relate each flag to the flag next to it. As you
work, ask yourself whether there is a rule that
relates a point to its translated image for a
given translation.
• Specify coordinate rules for translations in the
direction of the x-axis, the y-axis, and the line
y=x
• Recognize that a transformation of the form
(x, y) S (x + a, y + b) is a translation of point
(x, y) a units in the x direction and b units in the
y direction
In this problem, students examine pairs of
points and images for various translations. The
objective is for students to look for regular
features of algebraic representations of
translations so that they can both produce rules
for translations and interpret such rules.
Coordinate rules for translations are important
throughout mathematics; developing informal
knowledge through experience gives students
valuable time to build their intuitive
understanding.
The context is making designs with translation
symmetry. Students have to read the magnitude
and the direction from the images of the flags
under different translations. Have grid paper
available for students to make the patterns. Stress
that students should be looking for patterns in
what happens under the translations.
Some students may be ready to express the
relationships in Questions A and B by writing
coordinates that include a variable that represents
the length of the translation; others will describe
the relationships in words. Question C specifically
asks for a rule, so you may wait until that part to
ask students to capture their ideas in rules that
use variables to represent the x and y values of a
general point and its image point.
Have students work in pairs on the problem.
Explore 5.2
If students are having difficulty, focus their
attention on the pairs of coordinates between the
original and the first image. Have them organize
their work so that it is easy to observe patterns in
how the coordinates of the vertices change under
the translation.
In Question C, students get a chance to
experiment with translations where the possible
directions of translation are specified, but the
magnitude (distance) for the translations are not.
As students work on this part, if a group is
struggling, ask:
Suggested Questions
• When you translate a figure along the line
y = x, how is the change in the x-coordinate
related to the change in the y-coordinate?
(When the x-coordinate increases by 1, the
y-coordinate increases by 1.)
• What is the slope of the line y = x? (1)
• How does the slope of the line relate to
translating a figure along that line? (The slope
tells us that the y-coordinate would change by
1 whenever the x-coordinate changes by 1.)
Summarize 5.2
Call on students to share their answers for each
part of the problem.
Direct students’ attention to the two designs
shown in parts A and B.
Suggested Questions
Suggested Questions
• What type of symmetry do you see in these
designs? (translation symmetry)
translation symmetry?
• Give me a few rules showing what might
happen to a point (x, y) under a translation
parallel to the x-axis. [Students might suggest
such rules as (x, y) S (x + 2, y),
(x, y) S (x - 4, y),
and (x, y) S (x + 0.5, y).]
Investigation 5
Connecting Transformations to Coordinates
5
• How can you tell that each design exhibits
I N V E S T I G AT I O N
Launch 5.2
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• What do the rules for translations parallel to
the x-axis have in common? (The rules all add
or subtract a number from the x-coordinate,
but leave the y-coordinate alone.)
• Can you write a “super rule” to handle all of
You might sketch an example of a figure being
transformed according to this rule. The square
shown below increases in size in the x direction,
which means this rule does not describe a
translation.
these cases? (Help students to see that a
general rule is (x, y) S (x + a, y), where a
gives the length and direction of the change
parallel to the x-axis. If a is positive, the slide
is to the right. If a is negative, the slide is to
the left. )
y
D
A
x
C
• Now give me a few rules showing what might
B
happen to a point (x, y) under a translation
parallel to the y-axis. (Students might suggest
such rules as (x, y) S (x, y + 1),
(x, y) S (x, y - 4), and (x, y) S (x, y + 0.5).)
• What do the rules for translations parallel to
the y-axis have in common? [These rules are
of the form (x, y) S (x, y + a), where a gives
the length and direction of the change
parallel to the y-axis.]
y
D’
A’
x
C’
B’
• Now give me a few rules showing what might
happen to a point (x, y) under a translation
parallel to the line y = x. (Students might
suggest such rules as (x, y) S (x + 1, y + 1),
(x, y) S (x - 4, y - 4), and
(x, y) S(x + 0.5, x + 0.5).)
• What do the rules for translations parallel to
the line y = x have in common? [These rules
are of the form (x, y) S (x + a, y + a),
where a gives the length and direction of the
change in the x direction, which is identical to
the change in the y direction.]
• If you are given a rule, how can you recognize
whether it describes a translation? (It would
show that something is added to or
subtracted from x or y or both.)
• What about the rule (x, y) S (2x, y)? Does it
describe a translation? (No, the figure seems
to be stretched.)
118 Kaleidoscopes, Hubcaps, and Mirrors
Check for Understanding
• Which of the following rules represent
translations?
1. (x, y) S (x, y - 10)
1
1
2. (x, y) S (2x, y + 2)
3. (x, y) S (-2 + x, y)
4. (x, y) S (x + 3, y - 5)
(Answer: 1, 3, and 4 represent translations.
Number 2 does not.)
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At a Glance
5.2
Coordinate Rules for Translations
PACING 1 day
Mathematical Goals
• Specify coordinate rules for translations in the direction of the x-axis, the
y-axis, and the line y = x
• Recognize that a transformation of the form (x, y) S (x + a, y + b) is a
translation of point (x, y) a units in the x direction and b units in the
y direction
Launch
Direct students’ attention to the two designs shown in Questions A and B.
• What type of symmetry do you see in these designs?
• How can you tell that each design exhibits translation symmetry?
• In this problem, you will look for patterns that relate each flag to the flag
next to it. As you work, ask yourself whether there is a rule that relates a
point to its translated image for a given translation.
Have students work in pairs on the problem.
Materials
• Transparency 5.2A, B
• Grid paper
• Mirrors or transparent
reflecting tools
• Rulers, angle rulers,
or protractors
• Tracing paper
Explore
Materials
If students are struggling with Question C, ask
• When you translate a figure along the line y = x, how is the change in
• Labsheets 5.3A, B,
and C
the x-coordinate related to the change in the y-coordinate?
• What is the slope of the line y = x?
• How does the slope of the line relate to translating a figure along that line?
Summarize
• What might happen to a point (x, y) under a translation parallel to the
x-axis?
Materials
• Student notebooks
• What do all rules for translations parallel to the x-axis have in common?
• What might happen to a point (x, y) under a translation parallel to the
y-axis?
• What do all rules for translations parallel to the y-axis have in common?
• What might happen to a point (x, y) under a translation parallel to the
line y = x?
• What do the rules for translations parallel to the line y = x have in
common?
• If you are given a rule, how can you recognize whether it describes a
translation?
• Does the rule (x, y) S (2x, y) describe a translation?
See the extended Summarize for a Check for Understanding.
Investigation 5
Connecting Transformations to Coordinates
119
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ACE Assignment Guide
for Problem 5.2
Core 4, 20, 21
Adapted For suggestions about adapting ACE
exercises, see the CMP Special Needs Handbook.
Answers to Problem 5.2
A. 1. flag 2–Draw:
Line [(-3, -4), (-3, 2)]
Line [(-3, 2), (-2, 1)]
Line [(-2, 1), (-3, 0)]
flag 3–Draw:
Line [(-1, -4), (-1, 2)]
Line [(-1, 2), (0, 1)]
Line [(0, 1), (-1, 0)]
flag 4–Draw:
Line [(1, -4), (1, 2)]
Line [(1, 2), (2, 1)]
Line [(2, 1), (1, 0)]
flag 5–Draw:
Line [(3, -4), (3, 2)]
Line [(3, 2), (4, 1)]
Line [(4, 1), (3, 0)]
2. To find the coordinates of the flag to the
right of a given flag, add 2 to each
x-coordinate and keep the y-coordinates
the same.
3. To find the coordinates of the flag to the
left of a given flag, subtract 2 from each
x-coordinate and keep the y-coordinates
the same.
B. 1. flag 1–Draw:
Line [(-4, -4), (-4, 0)]
Line [(-4, 0), (-3, -1)]
Line [(-3, -1), (-4, -2)]
flag 2–Draw:
Line [(-2, -3), (-2, 1)]
Line [(-2, 1), (-1, 0)]
Line [(-1, 0), (-2, -1)]
flag 3–Draw:
Line [(0, -2), (0, 2)]
Line [(0, 2), (1, 1)]
Line [(1, 1), (0, 0)]
120 Kaleidoscopes, Hubcaps, and Mirrors
flag 4–Draw:
Line [(2, -1), (2, 3)]
Line [(2, 3), (3, 2)]
Line [(3, 2), (2, 1)]
flag 5–Draw:
Line [(4, 0), (4, 4)]
Line [(4, 4), (5, 3)]
Line [(5, 3), (4, 2)]
2. To find the coordinates of the flag to the
right of a given flag, add 2 to each
x-coordinate and 1 to each y-coordinate.
3. To find the coordinates of the flag to the
left of a given flag, subtract 2 from each
x-coordinate and 1 from each
y-coordinate.
C. 1. For sliding the figure along a line parallel
to the x-axis: The x-coordinate of the figure
to the right of a given figure is x plus the
length of the translation; the y-coordinate
does not change.
For sliding the figure along a line parallel
to the y-axis: The x-coordinate of the figure
above a given figure does not change; the
y-coordinate is y plus the length of the
translation.
For sliding the figure along a line parallel
to the line y = x: The same values are
added to both x and y coordinates. (Note:
The distance of the translation can be
found from these changes by using the
Pythagorean Theorem.)
2. a. (x, y) S (x + b, y)
b. (x, y) S (x, y + b)
c. (x, y) S (x + b, y + b)
D. 1. A translation that moves the point 2 to
the right and 3 down.
2. This is a stretch in the x direction, not a
translation.
3. This moves the figure 1 unit in the
x-direction, but stretches it in the
y-direction. It is not a translation.
4. This is a translation left 2 in the
x direction and up 1 in the y direction.
5. Answered above.