G6 m3-c-lesson 16-t

Lesson 16: Symmetry in the CoordinatePlane
Date: 2/10/15 153
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 16
Lesson 16: Symmetry in the Coordinate Plane
Student Outcomes
 Students understand that two numbers are said to differ only by signs if they areopposite of each other.
 Students recognize that when two ordered pairs differ only by sign of one or both of the coordinates,then the
locations of the points are related by reflections acrossoneor both axes.
Classwork
OpeningExercise (3 minutes)
Opening Exercise
Give an example oftwo oppositenumbersand describe wherethe numberslieon thenumber line. How are opposite
numberssimilar and how are they different?
Example 1 (14 minutes): ExtendingOpposite Numbersto the Coordinate Plane
Students locateand label points whoseordered pairs differ only by the sign of one or both coordinates. Together,
students and their teacher examine the relationshipsof the points on the coordinateplane, and express these
relationshipsin a graphicorganizer.
 Locate and label the points (3,4) and (−3,4).
 Record observations in the firstcolumn of the graphic organizer.
The firstcolumn of the graphic organizer is teacher-led so that students can pay particularattention to the absolute
values of coordinates and the general locations of the correspondingpoints with regard to each axis. Followingthis lead,
columns 2 and 3 are more student-led.
 Locate and label the point (3, −4).
 Record observations in the second column of the graphic organizer.
 Locate and label the point (−3, −4).
 Record observations in the third column of the graphic organizer.
MP.8

Date: 2/10/15 154
Extending Opposite Numbers to the Coordinates of
Points on the Coordinate Plane
Locate and label your pointson thecoordinate planeto the right. For each given pair of
points in the table below,record yourobservationsand conjecturesin theappropriatecell.
Pay attention to the absolute valuesof the coordinates and wherethepointsliein
reference toeach axis.
( 𝟑,𝟒)and (−𝟑,𝟒) ( 𝟑,𝟒)and ( 𝟑,−𝟒) ( 𝟑,𝟒)and (−𝟑,−𝟒)
Similaritiesof Coordinates Same 𝒚-coordinates.
The 𝒙-coordinates havethesame
absolutevalue.
Same 𝒙-coordinates.
The 𝒚-coordinates havethesame
absolutevalue.
The 𝒙-coordinates havethesame
absolutevalue.
The 𝒚-coordinates havethesame
absolutevalue.
Differencesof Coordinates The 𝒙-coordinates areopposite
numbers.
The 𝒚-coordinates areopposite
numbers.
Both the 𝒙- and 𝒚-coordinates are
opposite numbers.
Similaritiesin Location Both points are 𝟒units abovethe
𝒙-axis; and 𝟑units away from the
𝒚-axis.
Both points are 𝟑units to theright
ofthe 𝒚-axis; and 𝟒units away
from the 𝒙-axis.
Both points are 𝟑units from the
𝒚-axis; and 𝟒 units from the
𝒙-axis.
Differences in Location Onepoint is 𝟑units to theright of
the 𝒚-axis; theother is 𝟑units to
theleft of the 𝒚-axis.
Onepoint is 𝟒units above the
𝒙-axis; theother is 𝟒units below.
Onepoint is 𝟑units right ofthe
𝒙-axis; theother is 𝟑units left. One
point is 𝟒units abovethe 𝒚-axis;
theother is 𝟒units below.
Relationship between
Coordinatesand Location
on the Plane
The 𝒙-coordinates areopposite
numbers so thepoints lieon
oppositesides ofthe 𝒚-axis.
Becauseopposites havethesame
absolutevalue, both points liethe
samedistancefrom the 𝒚-axis.
The points liethesamedistance
abovethe 𝒙-axis, so thepoints are
symmetric about the 𝒚-axis. A
reflection across the 𝒚-axis takes
onepoint to theother.
The 𝒚-coordinates areopposite
numbers so thepoints lieon
oppositesides ofthe 𝒙-axis.
Becauseopposites havethesame
absolutevalue, both points liethe
samedistancefrom the 𝒙-axis.
The points liethesamedistance
right ofthe 𝒚-axis, so thepoints
aresymmetric about the 𝒙-axis. A
reflection across the 𝒙-axis takes
The points haveoppositenumbers
for 𝒙- and 𝒚-coordinates, so the
points must lieon oppositesides of
each axis. Becausethenumbers are
opposites, and opposites havethe
sameabsolutevalues each point
must bethesamedistancefrom
each axis. A reflection across one
axis followed by theother will take

Date: 2/10/15 155
Exercise (5 minutes)
Exercise
In each column, write thecoordinatesofthe pointsthat are related to the given point by thecriterialisted in thefirst
column ofthe table. Point 𝑺(𝟓,𝟑)hasbeen reflected over the 𝒙-and 𝒚-axesfor you asa guide and itsimagesare shown
on the coordinateplane. Use the coordinategrid to help you locate eachpointand itscorresponding coordinates.
Given Point: 𝑺(𝟓,𝟑) (– 𝟐,𝟒) (𝟑,– 𝟐) (–𝟏, – 𝟓)
Reflected
acrossthe
𝒙-axis. 𝑴 (𝟓,– 𝟑) (– 𝟐,– 𝟒) (𝟑,𝟐) (–𝟏, 𝟓)
Reflected
acrossthe
𝒚-axis. 𝑳 (– 𝟓,𝟑) (𝟐,𝟒) (– 𝟑,– 𝟐) (𝟏,– 𝟓)
Reflected first
acrossthe
𝒙-axis then
across the
𝒚-axis.
𝑨 (– 𝟓,– 𝟑) (𝟐,– 𝟒) (–𝟑, 𝟐) (𝟏,𝟓)
Reflected first
acrossthe
𝒚-axis then
across the
𝒙-axis.
𝑨 (– 𝟓,– 𝟑) (𝟐,– 𝟒) (–𝟑, 𝟐) (𝟏,𝟓)
a. When the coordinatesoftwo pointsare (𝒙,𝒚) and (−𝒙, 𝒚),what lineofsymmetry do the pointsshare?
Explain.
They sharethe 𝒚-axis, becausethe 𝒚-coordinates arethesameand the 𝒙-coordinates areopposites, which
means thepoints will bethesamedistancefrom the 𝒚-axis, but on oppositesides.
b. When the coordinatesoftwo pointsare (𝒙, 𝒚)and (𝒙, −𝒚), what line ofsymmetry do thepointsshare?
Explain.
They sharethe 𝒙-axis, becausethe 𝒙-coordinates arethesameand the 𝒚-coordinates areopposites, which
means thepoints will bethesamedistancefrom the 𝒙-axis buton oppositesides.
Example 2 (8 minutes): Navigating the Coordinate Plane usingReflections
Have students use a pencil eraser or their finger to navigatethe coordinateplanegiven verbal prompts. Then circulate
the room during the example to assess students’ understandingand provideassistanceas needed.
 Begin at (7, 2). Move 3 units down, then reflect over the 𝑦-axis. Where areyou?
 (−7, −1)
 Begin at (4, −5). Reflect over the 𝑥-axis,then move 7 units down, then to the right2 units. Where are you?
 (6, −2)
S. .
.. M
L
A
x
y

Date: 2/10/15 156
 Begin at (−3, 0). Reflect over the 𝑥-axis then move 6 units to the right. Move up two units,then reflect over
the 𝑥-axis again. Whereare you?
 (3, −2)
 Begin at (−2, 8). Decrease the 𝑦-coordinateby 6. Reflect over the 𝑦-axis,then move down 3 units. Where are
you?
 (2, −1)
 Begin at (5, −1). Reflect over the 𝑥-axis,then reflect over the 𝑦-axis. Where are you?
 (−5, 1)
Example 3 (7 minutes): DescribingHow to Navigate the Coordinate Plane
Given a startingpointand an ending point, students describea sequence of directions usingatleas tone reflection about
an axis to navigate from the startingpointto the ending point. Once students have found a sequence, have them find
another sequence whiletheir classmates finish thetask.
 Begin at (9, −3) and end at (−4,−3). Use exactly one reflection.
 Possible Answer: Reflect over the 𝑦-axis then move 5 units to the right.
 Begin at (0, 0) and end at (5, −1). Use exactly one reflection.
 Possible Answer: Move 5 units right, 1 unit up, then reflect over the 𝑥-axis.
 Begin at (0, 0) and end at (−1, −6). Use exactly two reflections.
 Possible Answer: Move right 1 unit, reflect over the 𝑦-axis, up 6 units, then reflect over the 𝑥-axis.
Closing(4 minutes)
 When the coordinates of two points differ only by one sign, such as (−8, 2) and (8, 2), what do the similarities
and differences in the coordinates tell us about their relativelocations on the plane?
 What is the relationship between (5, 1) and (5, −1)? Given one point, how can you locatethe other?
Exit Ticket (4 minutes)

Date: 2/10/15 157
Name ___________________________________________________ Date____________________
Lesson 16: Symmetry in the Coordinate Plane
Exit Ticket
1. How are the ordered pairs (4, 9) and (4, −9) similar,and how arethey different? Are the two points related by a
reflection over an axis in the coordinateplane? If so, indicatewhich axis isthe lineof symmetry between the points.
If they arenot related by a reflection over an axis in the coordinateplane,explain how you know.
2. Given the point (−5, 2), write the coordinates of a point that is related by a reflection over the 𝑥- or 𝑦-axis. Specify
which axis is thelineof symmetry.

Date: 2/10/15 158
𝑨(𝟓,−𝟑)𝑩(−𝟓,−𝟑)
𝑪(−𝟓,𝟑) 𝑫(𝟓,𝟑)
Exit Ticket Sample Solutions
1. How are the orderedpairs (𝟒,𝟗)and (𝟒,−𝟗) similar and how are they different? Arethetwo pointsrelated by a
reflection over an axisin the coordinate plane? Ifso, indicate which axisisthe lineofsymmetry between the points.
If they are not related by areflectionover an axisin thecoordinateplane, explain how you know?
The 𝒙-coordinates arethesame, but the 𝒚-coordinates areopposites, meaning they arethesamedistancefrom zero
on the 𝒙-axis, and thesamedistancebutoppositesides ofzero on the 𝒚-axis. Reflectingabout the 𝒙-axis
interchanges thesetwo points.
2. Given the point (−𝟓,𝟐), write thecoordinatesofapoint that isrelatedby areflectionover the 𝒙-or 𝒚-axis. Specify
which axisisthe line ofsymmetry.
Using the 𝒙-axis as a lineofsymmetry, (−𝟓,−𝟐); using the 𝒚-axis as a lineofsymmetry, (𝟓,𝟐).
Problem Set Sample Solutions
1. Locate apoint in QuadrantIV ofthe coordinateplane. Label the point A and write itsorderedpair nextto it.
Answers will vary; Quadrant IV (𝟓,−𝟑);
a. Reflect point 𝑨 over an axisso that itsimage isin
Quadrant III. Label the image 𝑩 and write itsordered
pair next to it. Which axisdid you reflect over? What
isthe only differencein theordered pairsofpoints 𝑨
and 𝑩?
𝑩(−𝟓,−𝟑); Reflected over the 𝒚-axis.
The ordered pairs differ only by thesign oftheir 𝒙-
coordinates: 𝑨(𝟓,−𝟑)and 𝑩(−𝟓,−𝟑).
b. Reflect point 𝑩 over an axisso that itsimage isin
Quadrant II. Label the image 𝑪 and write itsordered
pair next to it. Which axisdid you reflect over? What
isthe only differencein theordered pairsofpoints 𝑩
and 𝑪? How doesthe ordered pair ofpointC relate to
the ordered pair ofpoint 𝑨?
𝑪(−𝟓,𝟑); Reflected over the 𝒙-axis.
The ordered pairs differ only by thesign oftheir 𝒚-coordinates: 𝑩(−𝟓,−𝟑)and 𝑪(−𝟓,𝟑).
The ordered pair for pointC differs from theordered pair for point A by thesigns ofboth coordinates:
𝑨 (𝟓,−𝟑)and 𝑪(−𝟓,𝟑).
c. Reflect point 𝑪 over an axisso that itsimage isin Quadrant I. Label the image 𝑫 and write itsorderedpair
next to it. Which axisdid you reflect over? How doesthe ordered pair for point 𝑫 compare to the ordered
pair for point 𝑪? How doesthe ordered pair for point 𝑫 compare to points 𝑨 and 𝑩?
𝑫( 𝟓,𝟑); Reflected over the 𝒚-axis again.
Point 𝑫 differs from point 𝑪 by only thesign of its 𝒙-coordinate: 𝑫( 𝟓,𝟑) and 𝑪(−𝟓,𝟑).
Point 𝑫 differs from point 𝑩 by thesigns ofboth coordinates: 𝑫( 𝟓,𝟑)and 𝑩(−𝟓,−𝟑).
Point 𝑫 differs from point 𝑨 by only thesign of the 𝒚-coordinate: 𝑫(𝟓,𝟑)and 𝑨( 𝟓,−𝟑).

Date: 2/10/15 159
2. Bobbie listened to her teacher’sdirectionsand navigated from thepoint (−𝟏,𝟎)to ( 𝟓,−𝟑). She knowsthat she has
the correct answerbut, sheforgot part ofthe teacher’sdirections. Her teacher’sdirectionsincluded thefollowing:
“Move 𝟕 units down, reflect about the ? -axis, moveup 𝟒units, then moveright 𝟒units.”
Help Bobbie determine themissing axisin thedirections, and explain your answer.
The missing lineis a reflectionover the 𝒚-axis. The first linewould movethelocation to (−𝟏,−𝟕). A reflection over
the 𝒚-axis would movethelocation to(𝟏,−𝟕) in Quadrant IV, which is 𝟒units left and 𝟒units down from theend
point (𝟓,−𝟑).

G6 m3-c-lesson 16-t

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