Nys common core mathematics curriculum



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reference to each axis.


(πŸ‘, πŸ’) and (βˆ’πŸ‘, πŸ’)
(πŸ‘, πŸ’) and (πŸ‘, βˆ’πŸ’)
(πŸ‘, πŸ’) and (βˆ’πŸ‘, βˆ’πŸ’)
Similarities of
Coordinates


Same π’š-coordinates
The 𝒙-coordinates have the
same absolute value.
Same 𝒙-coordinates
The π’š-coordinates have the
same absolute value.
The 𝒙-coordinates have the
same absolute value.
The π’š-coordinates have the
same absolute value.
Differences of
Coordinates


The 𝒙-coordinates are
opposite numbers.
The π’š-coordinates are
opposite numbers.
Both the 𝒙- and
π’š-coordinates are opposite
numbers.
Similarities in
Location


Both points are πŸ’ units
above the 𝒙-axis and πŸ‘ units
away from the π’š-axis.
Both points are πŸ‘ units to the
right of the π’š-axis and πŸ’
units away from the 𝒙-axis.
Both points are πŸ‘ units from
the π’š-axis and πŸ’ units from
the 𝒙-axis.
Differences in
Location


One point is πŸ‘ units to the
right of the π’š-axis; the other
is πŸ‘ units to the left of the
π’š-axis.
One point is πŸ’ units above
the 𝒙-axis; the other is πŸ’
units below.
One point is πŸ‘ units right of
the π’š-axis; the other is πŸ‘
units left. One point is πŸ’
units above the 𝒙-axis; the
other is πŸ’ units below.
Relationship Between
Coordinates and
Location on the Plane


The 𝒙-coordinates are
opposite numbers, so the
points lie on opposite sides
of the π’š-axis. Because
opposites have the same
absolute value, both points
lie the same distance from
the π’š-axis. The points lie the
same distance above the
𝒙-axis, so the points are
symmetric about the π’š-axis.
A reflection across the π’š-axis
takes one point to the other.
The π’š-coordinates are
opposite numbers, so the
points lie on opposite sides
of the 𝒙-axis. Because
opposites have the same
absolute value, both points
lie the same distance from
the 𝒙-axis. The points lie the
same distance right of the
π’š-axis, so the points are
symmetric about the 𝒙-axis.
A reflection across the 𝒙-axis
takes one point to the other.
The points have opposite
numbers for 𝒙- and π’š-
coordinates, so the points
must lie on opposite sides of
each axis. Because the
numbers are opposites and
opposites have the same
absolute values, each point
must be the same distance
from each axis. A reflection
across one axis followed by ab iibreflection across the other
axis takes one point to the
other.



6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
160
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Exercises 1–2 (5 minutes)

Exercises
In each column, write the coordinates of the points that are related to the given point by the criteria listed in the first
column of the table. Point 𝑺(πŸ“, πŸ‘) has been reflected over the 𝒙- and π’š-axes for you as a guide, and its images are shown
on the coordinate plane. Use the coordinate grid to help you locate each point and its corresponding coordinates.
Given Point
𝑺(πŸ“, πŸ‘)
(βˆ’πŸ, πŸ’)
(πŸ‘, βˆ’πŸ)
(βˆ’πŸ, βˆ’πŸ“)

The given point
is reflected
across the
𝒙-axis.
𝑴(πŸ“, βˆ’πŸ‘)
(βˆ’πŸ, βˆ’πŸ’)
(πŸ‘, 𝟐)
(βˆ’πŸ, πŸ“)
The given point
is reflected
across the
π’š-axis.
𝑳(βˆ’πŸ“, πŸ‘)
(𝟐, πŸ’)
(βˆ’πŸ‘, βˆ’πŸ)
(𝟏, βˆ’πŸ“)
The given point
is reflected first
across the
𝒙-axis and then
across the
π’š-axis.
𝑨(βˆ’πŸ“, βˆ’πŸ‘) (𝟐, βˆ’πŸ’)
(βˆ’πŸ‘, 𝟐)
(𝟏, πŸ“)
The given point
is reflected first
across the
π’š-axis and then
across the
𝒙-axis.
𝑨(βˆ’πŸ“, βˆ’πŸ‘) (𝟐, βˆ’πŸ’)
(βˆ’πŸ‘, 𝟐)
(𝟏, πŸ“)

1.
When the coordinates of two points are (𝒙, π’š) and (βˆ’π’™, π’š), what line of symmetry do the points share Explain.
They share the π’š-axis because the π’š-coordinates are the same and the 𝒙-coordinates are opposites, which means the
points will be the same distance from the π’š-axis but on opposite sides.

2.
When the coordinates of two points are (𝒙, π’š) and (𝒙, βˆ’π’š), what line of symmetry do the points share Explain.
They share the 𝒙-axis because the 𝒙-coordinates are the same and the π’š-coordinates are opposites, which means the
points will be the same distance from the 𝒙-axis but on opposite sides.
Example 2 (8 minutes Navigating the Coordinate Plane Using Reflections
Have students use a pencil eraser or finger to navigate the coordinate plane given verbal prompts. Then, circulate the room during the example to assess students understanding and provide assistance as needed.
 Begin at (7, 2). Move 3 units down, and then reflect over the axis. Where are you
οƒΊ
(βˆ’7, βˆ’1)
S
M
L
A
𝒙
π’š
Scaffolding:
Project each prompt so that visual learners can follow along with the steps.


6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
161
This work is derived from Eureka Math β„’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License Begin at (4, βˆ’5). Reflect over the axis, and then move 7 units down and then to the right 2 units. Where are you
οƒΊ
(6, βˆ’2)
 Begin at (βˆ’3, 0). Reflect over the axis, and then move 6 units to the right. Move up two units, and then reflect over the axis again. Where are you?
οƒΊ
(3, βˆ’2)
 Begin at (βˆ’2, 8). Decrease the coordinate by 6 units. Reflect over the axis, and then move down 3 units. Where are you
οƒΊ
(2, βˆ’1)
 Begin at (5, βˆ’1). Reflect over the axis, and then reflect over the axis. Where are you
οƒΊ
(βˆ’5, 1)

Examples 2–3: Navigating the Coordinate Plane
Example 3 (7 minutes Describing How to Navigate the Coordinate Plane
Given a starting point and an ending point, students describe a sequence of directions using at least one reflection about an axis to navigate from the starting point to the ending point. Once students have found a sequence, have them find another sequence while classmates finish the task.
 Begin at (9, βˆ’3), and end at (βˆ’4, βˆ’3). Use exactly one reflection.
οƒΊ
Possible answer Reflect over the 𝑦-axis, and then move 5 units to the right.
 Begin at (0, 0), and end at (5, βˆ’1). Use exactly one reflection.
οƒΊ
Possible answer Move 5 units right, 1 unit up, and then reflect over the π‘₯-axis.
 Begin at (0, 0), and end at (βˆ’1, βˆ’6). Use exactly two reflections.
οƒΊ
Possible answer Move right 1 unit, reflect over the 𝑦-axis, up 6 units, and then reflect over the π‘₯-axis.




6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
162
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Closing (4 minutes)
 When the coordinates of two points differ only by one sign, such as (βˆ’8, 2) and (8, 2), what do the similarities and differences in the coordinates tell us about their relative locations on the plane
οƒΊ
The 𝑦-coordinates are the same for both points, which means the points are on the same horizontal
line. The π‘₯-coordinates differ because they are opposites, which means the points are symmetric across
the 𝑦-axis.
 What is the relationship between (5, 1) and (5, βˆ’1)? Given one point, how can you locate the other
οƒΊ
If you start at either point and reflect over the π‘₯-axis, you will end at the other point.
Exit Ticket (4 minutes)


6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
163
This work is derived from Eureka Math β„’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 16: Symmetry in the Coordinate Plane

Exit Ticket

1. How are the ordered pairs (4, 9) and (4, βˆ’9) similar, and how are they different Are the two points related by a reflection over an axis in the coordinate plane If so, indicate which axis is the line of symmetry between the points. If they are not related by a reflection over an axis in the coordinate plane, explain how you know.
2. Given the point (βˆ’5, 2), write the coordinates of a point that is related by a reflection over the π‘₯- or axis. Specify which axis is the line of symmetry.




6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
164
This work is derived from Eureka Math β„’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License, βˆ’πŸ‘)
𝑩(βˆ’πŸ“, βˆ’πŸ‘)
π‘ͺ(βˆ’πŸ“, πŸ‘)
𝑫(πŸ“, πŸ‘)
Exit Ticket Sample Solutions

1.
How are the ordered pairs (πŸ’, πŸ—) and (πŸ’, βˆ’πŸ—) similar, and how are they different Are the two points related by ab breflection over an axis in the coordinate plane If so, indicate which axis is the line of symmetry between the points.
If they are not related by a reflection over an axis in the coordinate plane, explain how you know.
The 𝒙-coordinates are the same, but the π’š-coordinates are opposites, meaning they are the same distance from zero
on the 𝒙-axis and the same distance but on opposite sides of zero on the π’š-axis. Reflecting about the 𝒙-axis
interchanges these two points.

2.
Given the point (βˆ’πŸ“, 𝟐), write the coordinates of a point that is related by a reflection over the 𝒙- orb b-axis. Specify
which axis is the line of symmetry.
Using the 𝒙-axis as a line of symmetry, (βˆ’πŸ“, βˆ’πŸ); using the π’š-axis as a line of symmetry, (πŸ“, 𝟐)


Problem Set Sample Solutions

1.
Locate a point in Quadrant IV of the coordinate plane. Label the point 𝑨, and write its ordered pair next to it.
Answers will vary Quadrant IV (πŸ“, βˆ’πŸ‘)

a.
Reflect point 𝑨 over an axis so that its image is in Quadrant
III. Label the image 𝑩, and write its ordered pair next to it.
Which axis did you reflect over What is the only
difference in the ordered pairs of points 𝑨 and 𝑩?
𝑩(βˆ’πŸ“, βˆ’πŸ‘); reflected over the π’š-axis
The ordered pairs differ only by the sign of their
𝒙-coordinates: 𝑨(πŸ“, βˆ’πŸ‘) and 𝑩(βˆ’πŸ“, βˆ’πŸ‘).

b.
Reflect point 𝑩 over an axis so that its image is in Quadrant
II. Label the image π‘ͺ, and write its ordered pair next to it.
Which axis did you reflect over What is the only
difference in the ordered pairs of points 𝑩 and π‘ͺ? How
does the ordered pair of point π‘ͺ relate to the ordered pair
of point 𝑨?
π‘ͺ(βˆ’πŸ“, πŸ‘); reflected over the 𝒙-axis
The ordered pairs differ only by the signs of their π’š-coordinates: 𝑩(βˆ’πŸ“, βˆ’πŸ‘) and π‘ͺ(βˆ’πŸ“, πŸ‘).
The ordered pair for point π‘ͺ differs from the ordered pair for point 𝑨 by the signs of both coordinates
𝑨(πŸ“, βˆ’πŸ‘) and π‘ͺ(βˆ’πŸ“, πŸ‘).

c.
Reflect point π‘ͺ over an axis so that its image is in Quadrant I. Label the image 𝑫, and write its ordered pair
next to it. Which axis did you reflect over How does the ordered pair for point 𝑫 compare to the ordered
pair for point π‘ͺ? How does the ordered pair for point 𝑫 compare to points 𝑨 and 𝑩?
𝑫(πŸ“, πŸ‘); reflected over the π’š-axis again
Point 𝑫 differs from point π‘ͺ by only the sign of its 𝒙-coordinate: 𝑫(πŸ“, πŸ‘) and π‘ͺ(βˆ’πŸ“, πŸ‘).
Point 𝑫 differs from point 𝑩 by the signs of both coordinates 𝑫(πŸ“, πŸ‘) and 𝑩(βˆ’πŸ“, βˆ’πŸ‘).
Point 𝑫 differs from point 𝑨 by only the sign of the π’š-coordinate: 𝑫(πŸ“, πŸ‘) and 𝑨(πŸ“, βˆ’πŸ‘).


6β€’3
NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16
Lesson 16:
Symmetry in the Coordinate Plane
165
This work is derived from Eureka Math β„’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
2.
Bobbie listened to her teacher’s directions and navigated from the point (βˆ’πŸ, 𝟎) to (πŸ“, βˆ’πŸ‘). She knows that she has
the correct answer, but she forgot part of the teacher’s directions. Her teacher’s directions included the following
β€œMove πŸ• units down, reflect about the ? -axis, move up πŸ’ units, and then move right πŸ’ units
Help Bobbie determine the missing axis in the directions, and explain your answer.
The missing line is a reflection over the π’š-axis. The first line would move the location to (βˆ’πŸ, βˆ’πŸ•). A reflection over
the π’š-axis would move the location to (𝟏, βˆ’πŸ•) in Quadrant IV, which is πŸ’ units left and πŸ’ units down from the end
point (πŸ“, βˆ’πŸ‘).

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