**reference to each axis. **** **** **(π, π) ** and **(βπ, π) ** **(π, π) ** and **(π, βπ) ** **(π, π) ** and **(βπ, βπ) ** ****Similarities of ****Coordinates **** **** ****Same **π **-coordinates ****The **π **-coordinates have the ****same absolute value. ****Same **π **-coordinates ****The **π **-coordinates have the ****same absolute value. ****The **π **-coordinates have the ****same absolute value. ****The **π **-coordinates have the ****same absolute value. ****Differences of ****Coordinates **** **** ****The **π **-coordinates are ****opposite numbers. ****The **π **-coordinates are ****opposite numbers. ****Both the **π **- and **π **-coordinates are opposite ****numbers. ****Similarities in ****Location **** **** ****Both points are **π ** units ****above the **π **-axis and **π ** units ****away from the **π **-axis. ****Both points are **π ** units to the ****right of the** π **-axis and **π ** ****units away from the **π **-axis. ****Both points are **π ** units from ****the **π **-axis and **π **units from ****the **π **-axis. ****Differences in ****Location **** **** ****One point is **π ** units to the ****right of the **π **-axis; the other ****is **π ** units to the left of the **π **-axis. ****One point is **π ** units above ****the **π **-axis; the other is **π ** ****units below. ****One point is **π ** units right of ****the **π **-axis; the other is **π ** ****units left. One point is **π ** ****units above the **π **-axis; the ****other is **π ** units below. ****Relationship Between ****Coordinates and ****Location on the Plane **** **** ****The **π **-coordinates are ****opposite numbers, so the ****points lie on opposite sides ****of the** π **-axis. Because ****opposites have the same ****absolute value, both points ****lie the same distance from ****the **π **-axis. The points lie the ****same distance above the **π **-axis, so the points are ****symmetric about the **π **-axis. ****A reflection across the **π **-axis ****takes one point to the other. ****The **π **-coordinates are ****opposite numbers, so the ****points lie on opposite sides ****of the **π **-axis. Because ****opposites have the same ****absolute value, both points ****lie the same distance from ****the** π **-axis. The points lie the ****same distance right of the** π **-axis, so the points are ****symmetric about the** π **-axis. ****A reflection across the **π **-axis ****takes one point to the other. ****The points have opposite ****numbers for **π **- and** π **-****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 **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. **Exercises 1β2 (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 **πΊ(π, π) ** ****has been reflected over the **π **- and **π **-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 **πΊ(π, π) (βπ, π) (π, βπ) (βπ, βπ) ** ****The given point ****is reflected ****across the **π **-axis. **π΄(π, βπ) (βπ, βπ) (π, π) (βπ, π) **The given point ****is reflected ****across the **π **-axis. **π³(βπ, π) (π, π) (βπ, βπ) (π, βπ) **The given point ****is reflected first ****across the **π **-axis and then ****across the **π **-axis. **π¨(βπ, βπ) ** ** (π, βπ) (βπ, π) (π, π) **The given point ****is reflected first ****across the **π **-axis and then ****across the **π **-axis. **π¨(βπ, βπ) ** ** (π, βπ) (βπ, π) (π, π) ** ****1. ****When the coordinates of two points are **(π, π) **and **(βπ, π) **, what line of symmetry do the points share Explain. ****They share the **π **-axis because the **π **-coordinates are the same and the **π **-coordinates are opposites, which means the ****points will be the same distance from the **π **-axis but on opposite sides. **** ****2. ****When the coordinates of two points are** (π, π) **and **(π, βπ) **, what line of symmetry do the points share Explain. ****They share the **π **-axis because the** π **-coordinates are the same and the **π **-coordinates are opposites, which means the ****points will be the same distance from the **π **-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. ο§ Begin at (7, 2). Move 3 units down, and then reflect over the axis. Where are you οΊ (β7, β1) **S ****M ****L ****A **π ** **π ** ***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, β5). Reflect over the axis, and then move 7 units down and then to the right 2 units. Where are you οΊ (6, β2) ο§ Begin at (β3, 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? ** ** οΊ (3, β2) ο§ Begin at (β2, 8). Decrease the coordinate by 6 units. Reflect over the axis, and then move down 3 units. Where are you οΊ (2, β1) ο§ Begin at (5, β1). Reflect over the axis, and then reflect over the axis. Where are you οΊ (β5, 1) ** ****Examples 2β3: 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. ο§ Begin at (9, β3), and end at (β4, β3). Use exactly one reflection. οΊ *Possible answer Reflect over the *π¦ *-axis, and then move *5 *units to the right. *ο§ Begin at (0, 0), and end at (5, β1). Use exactly one reflection. οΊ *Possible answer Move *5 * units right, *1 * unit up, and then reflect over the *π₯ *-axis. *ο§ Begin at (0, 0), and end at (β1, β6). Use exactly two reflections. οΊ *Possible answer Move right *1 * unit, reflect over the *π¦ *-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 ** 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.
**Closing (4 minutes) ** ο§ When the coordinates of two points differ only by one sign, such as (β8, 2) and (8, 2), what do the similarities and differences in the coordinates tell us about their relative locations on the plane οΊ
*The *π¦*-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 *π¦*-axis. * ο§ What is the relationship between (5, 1) and (5, β1)? Given one point, how can you locate the other οΊ
*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, β9) 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 (β5, 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.
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**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, βπ) π©(βπ, βπ) πͺ(βπ, π) ** **π«(π, π) **Exit Ticket Sample Solutions **** ****1. ****How are the ordered pairs **(π, π) ** and **(π, βπ) ** 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 **π **-coordinates are the same, but the **π **-coordinates are opposites, meaning they are the same distance from zero ****on the **π **-axis and the same distance but on opposite sides of zero on the** π **-axis. Reflecting about the **π **-axis ****interchanges these two points. **** ****2. ****Given the point **(βπ, π) **, write the coordinates of a point that is related by a reflection over the **π **- orb b-axis. Specify ****which axis is the line of symmetry. ****Using the **π **-axis as a line of symmetry,** (βπ, βπ) **; using the **π **-axis as a line of symmetry, **(π, π) ** **** **** ****Problem Set Sample Solutions **** ****1. ****Locate a point in Quadrant IV of the coordinate plane. Label the point **π¨ **, ****and write its ordered pair next to it. ****Answers will vary Quadrant IV **(π, βπ) ** **** ****a. ****Reflect point **π¨ ** over an axis so that its image is in Quadrant ****III. Label the image **π© **,*** *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 **π¨ ** ****and **π© **? **π©(βπ, βπ) **; reflected over the **π **-axis ****The ordered pairs differ only by the sign of their **π **-coordinates: **π¨(π, βπ) ** and **π©(βπ, βπ) **. **** ****b. ****Reflect point **π© ** ****over an axis so that its image is in Quadrant ****II. Label the image **πͺ **,*** *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 **π© ** ****and **πͺ **? How ****does the ordered pair of point **πͺ ** ****relate to the ordered pair ****of point **π¨ **? **πͺ(βπ, π) **; reflected over the **π **-axis ****The ordered pairs differ only by the signs of their** π **-coordinates: **π©(βπ, βπ) ** and** πͺ(βπ, π) **. ****The ordered pair for point **πͺ ** differs from the ordered pair for point **π¨ ** by the signs of both coordinates **π¨(π, βπ) ** and** πͺ(βπ, π) **. **** ****c. ****Reflect point **πͺ ** ****over an axis so*** *that its image is in Quadrant I. Label the image π« **, and write its ordered pair ****next to it. Which axis did you reflect over How does the ordered pair for point **π« ** ****compare to the ordered ****pair for point **πͺ **?*** * How does the ordered pair for point π« ** compare to points **π¨ ** ****and **π© **? **π«(π, π) **; reflected over the **π **-axis again ****Point **π« ** differs from point **πͺ ** by only the sign of its **π **-coordinate: **π«(π, π) **and** πͺ(βπ, π) **. ****Point **π« ** differs from point **π© ** by the signs of both coordinates **π«(π, π) ** and **π©(βπ, βπ) **. ****Point **π« ** differs from point **π¨ ** by only the sign of the **π **-coordinate: **π«(π, π) ** and **π¨(π, βπ) **. **
**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 teacherβs directions and navigated from the point **(βπ, π) ** to **(π, βπ) **. She knows that she has ****the correct answer, but she forgot part of the teacherβs directions. Her teacherβs directions included the following ****βMove **π ** units down, reflect about the ? -axis, move up **π ** units, and then move right **π ** units ****Help Bobbie determine the missing axis in the directions, and explain your answer. ****The missing line is a reflection over the **π **-axis. The first line would move the location to **(βπ, βπ) **. A reflection over ****the **π **-axis would move the location to **(π, βπ) ** in Quadrant IV, which is **π ** units left and **π ** units down from the end ****point **(π, βπ) **. **
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