This document discusses using the coordinate plane to solve problems involving distance, line segments, and geometric shapes. It includes the following key points: 1) Students solve problems finding distances between points on the coordinate plane using absolute value. The distance between two points is the sum of the absolute values of their x- and y-coordinates. 2) The length of a line segment can be determined by finding the distance between its endpoints. Side lengths of geometric shapes like rectangles can then be used to calculate perimeter and area. 3) Knowing one endpoint and the length of a line segment, the other endpoint can be found by counting units above or below for vertical segments, or left or right for horizontal segments.

1. Lesson 19: Problem-Solving and the CoordinatePlane
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
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Lesson 19: Problem-Solving and the Coordinate Plane
Student Outcomes
 Students solveproblems related to the distancebetween points that lieon the same horizontal or vertical line.
 Students use the coordinateplaneto graph points,linesegments and geometric shapes in the various
quadrants and then use the absolutevalueto find the related distances.
Lesson Notes
The grid provided in the Opening Exerciseis also used for Exercises 1–6 sinceeach exercise is sequential. Students
extend their knowledge about findingdistances between points on the coordinateplaneto the associated lengths of line
segments and sides of geometric figures.
Classwork
OpeningExercise (3 minutes)
Opening Exercise
In the coordinate plane, find thedistance betweenthe pointsusing absolute value.
The distancebetween thepoints is 𝟖 units. Thepoints havethesamefirst coordinates andtherefore lieon thesame
vertical line. |−𝟑| = 𝟑, and | 𝟓| = 𝟓, and the numbers lieon oppositesides of 𝟎 so their absolutevalues areadded
together; 𝟑 + 𝟓 = 𝟖. Wecan check our answer by just counting thenumber ofunits between thetwo points.
| 𝟓| = 𝟓
|−𝟑| = 𝟑

2. Lesson 19: Problem-Solving and the CoordinatePlane
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
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(−𝟓,−𝟑)
𝟖units
(𝟒,−𝟑)
(𝟒,𝟓)
.
..
(−𝟓, −𝟑)
(−𝟓, 𝟓)
(𝟒, −𝟑)
(𝟒, 𝟓)
Exercises1–2 (8 minutes): The Length of a Line Segmentisthe Distance BetweenitsEndpoints
Students relate the distancebetween two points lyingin different quadrants of the coordinateplaneto the length of a
linesegment with those endpoints. Students then use this relationship to graph a horizontal or vertical linesegment
usingdistanceto find the coordinates of endpoints.
Exercises
1. Locate and label (𝟒,𝟓)and (𝟒,– 𝟑). Draw the line segment
between theendpointsgiven on the coordinateplane. How long
isthe line segmentthat you drew? Explain.
The length ofthelinesegment is also 𝟖units. I found that the
distancebetween ( 𝟒,−𝟑)and (𝟒,𝟓)is 𝟖units, and becausethese
aretheendpoints ofthelinesegment, thelinesegmentbegins
and ends at thesepoints, so thedistancefrom end to end is 𝟖
units.
2. Draw a horizontal line segmentstarting at (𝟒,−𝟑)that has a
length of 𝟗units. What are the possiblecoordinatesofthe other
endpoint ofthe line segment? (There ismorethan oneanswer.)
(−𝟓,−𝟑) or (𝟏𝟑,−𝟑)
Which point do you choose to betheother endpoint ofthe
horizontal line segment? Explain how and why you chosethat point. Locate and label thepoint on thecoordinate
grid.
The other endpoint ofthehorizontal linesegment is (−𝟓,−𝟑); I chosethis point becausetheother option
(𝟏𝟑,– 𝟑)is located offofthegiven coordinategrid.
Note: Students may choosetheendpoint (𝟏𝟑,−𝟑)but they must changethenumber scaleofthe 𝒙-axis to do so.
Exercise 3 (5 minutes): ExtendingLengthsof Line Segmentsto Sidesof GeometricFigures
The two line segmentsthat you havejust drawn could beseenastwo sidesofa rectangle. Given this, the endpointsof
the two line segmentswouldbe threeofthe verticesofthisrectangle.
3. Find the coordinatesofthe fourth vertex ofthe rectangle. Explain how you find thecoordinatesofthe fourth vertex
using absolute value.
The fourth vertex is (−𝟓,𝟓). Theoppositesides ofa rectangleare
thesamelength, so thelength oftheverticalsidestarting at
(−𝟓,−𝟑) has to be 𝟖 units long. Also, thesidefrom (−𝟓,−𝟑)to
theremaining vertex is a verticalline, so theendpoints must have
thesamefirst coordinate. |−𝟑|= 𝟑, and 𝟖 − 𝟑 = 𝟓, so the
remaining vertex must befiveunits above the 𝒙-axis.
*Students can usea similar argumentusingthelengthofthe
horizontal sidestartingat ( 𝟒,𝟓),knowingit has to be 𝟗units long.
How doesthe fourth vertex that you found relateto eachofthe
consecutivevertices in eitherdirection? Explain.
The fourth vertex has thesame first coordinateas (−𝟓,−𝟑)
becausethey aretheendpoints ofa verticallinesegment. The
fourth vertex has thesamesecond coordinateas (𝟒,𝟓)sincethey
aretheendpoints ofa horizontal linesegment.
Draw the remaining sidesofthe rectangle.
MP.7

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
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..
(−𝟓, −𝟑)
(−𝟓, 𝟓)
(𝟒, −𝟑)
(𝟒, 𝟓)
X
Y
(−𝟔, 𝟑)
(−𝟔, −𝟏)
(𝟒, 𝟑)
(𝟒, −𝟏)
Exercises4–6 (6 minutes): Using Lengthsof Sidesof GeometricFigurestoSolve Problems
4. Using the verticesthat you havefound and the lengthsofthe linesegmentsbetweenthem,
find the perimeter oftherectangle.
𝟖+ 𝟗+ 𝟖 + 𝟗 = 𝟑𝟒; Theperimeter oftherectangleis 𝟑𝟒units.
5. Find the areaofthe rectangle.
𝟗 × 𝟖 = 𝟕𝟐; Thearea oftherectangleis 𝟕𝟐 𝒖𝒏𝒊𝒕𝒔𝟐
.
6. Draw a diagonal line segment through the rectanglewith
opposite verticesfor endpoints. What geometric figures
are formed by thisline segment? What are theareasof
each ofthese figures? Explain.
The diagonal linesegment cuts therectangleintotwo
right triangles. Theareas ofthetriangles are 𝟑𝟔 𝒖𝒏𝒊𝒕𝒔𝟐
each becausethetriangles each makeup halfofthe
rectangleand halfof 𝟕𝟐is 𝟑𝟔.
EXTENSION [Iftime allows]: Line theedgeofapieceofpaper
up to the diagonal in the rectangle. Mark thelength ofthe
diagonal on the edge ofthe paper. Align your marks
horizontally or vertically on the gridand estimatethelength of
the diagonal to the nearest integer. Use that estimationto
now estimate the perimeter ofthe triangles.
The length ofthediagonal is approximately 𝟏𝟐units,and theperimeter ofeach triangleis approximately 𝟐𝟗units.
Exercise 7 (8 minutes)
7. Construct a rectangleon thecoordinateplane that satisfieseach ofthe criterialisted below. Identify the coordinate
of each ofitsvertices.
 Each of the verticesliesin adifferent quadrant.
 Itssidesare either vertical or horizontal.
 The perimeter ofthe rectangle is28units.
Answers will vary. Theexample to the right shows a
rectanglewith sidelengths 𝟏𝟎and 𝟒units. The
coordinates ofthe rectangle’s vertices
are (−𝟔,𝟑),(𝟒,𝟑),(𝟒,−𝟏)and (−𝟔,−𝟏).
Using absolute value, show how the lengthsofthe
sidesofyour rectangle provide aperimeter of 𝟐𝟖units.
|−𝟔| = 𝟔, | 𝟒| = 𝟒, and 𝟔 + 𝟒 = 𝟏𝟎, so thewidth ofmy
rectangleis 𝟏𝟎units.
| 𝟑| = 𝟑, |−𝟏| = 𝟏, and 𝟑 + 𝟏 = 𝟒, so theheight ofmy
rectangle is 𝟒units.
𝟏𝟎+ 𝟒 + 𝟏𝟎+ 𝟒 = 𝟐𝟖; The perimeter ofmy rectangle
is 𝟐𝟖 units.
Scaffolding:
Students may need to review
and discuss theconcepts of
perimeter and area from
earlier grades.
MP.1

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
Closing(5 minutes)
 How do we determine the length of a horizontal linesegment whose endpoints liein different quadrants of
the coordinateplane?
 If the points are in different quadrants, then the 𝑥-coordinates lie on opposite sides of zero. The
distance between the 𝑥-coordinates can be found by adding the absolute values of the 𝑥-coordinates.
(The 𝑦-coordinates are the same and show that the points lie on a horizontal line.)
 If we know one endpoint of a vertical linesegment and the length of the linesegment, how do we find the
other endpoint of the linesegment? Is the process the same with a horizontal linesegment?
 If the line segment is vertical, then the other endpoint could be above or below the given endpoint. If
we know the length of the line segment then we can count up or down from the given endpoint to find
the other endpoint. We can check our answer using the absolute values of the 𝑦-coordinates.
Exit Ticket (10 minutes)
Lesson Summary
 The length ofaline segment onthe coordinateplanecan be determined by finding thedistancebetween its
endpoints.
 You can find the perimeter and areaoffiguressuch asrectanglesand righttrianglesby finding the lengthsof
the line segmentsthat make up their sides, and then using theappropriateformula.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
Name ___________________________________________________ Date____________________
Lesson 19: Problem-Solving and the Coordinate Plane
Exit Ticket
1. The coordinates of one endpoint of a linesegment are (−2, −7). The linesegment is 12 units long. Give three
possiblecoordinates of the linesegment’s other endpoint.
2. Graph a rectangle with area 12 units2,such that its vertices liein atleasttwo of the four quadrants in the coordinate
plane. State the lengths of each of the sides,and useabsolutevalueto show how you determined the lengths of the
sides.
X
Y

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19
X
Y
1 unit 1 unit
3 units
3 units
6 units
2 units
Exit Ticket Sample Solutions
1. The coordinatesofone endpoint ofaline segmentare (−𝟐,−𝟕). The linesegment is 𝟏𝟐unitslong. Give three
possible coordinatesofthe line segment’sother endpoint.
( 𝟏𝟎,−𝟕); (−𝟏𝟒,−𝟕); (−𝟐,𝟓); (−𝟐,−𝟏𝟗)
2. Graph arectangle with area 𝟏𝟐units2,such that itsverticeslie in at least two ofthe four quadrantsin the coordinate
plane. List the lengthsofthe sides, and use absolutevalue to show how you determined thelengthsofthe sides.
Answers will vary. Therectanglecan have
sidelengths of 𝟔and 𝟐or 𝟑and 𝟒. A
sampleis provided on thegridon the
right. 𝟔× 𝟐 = 𝟏𝟐
Problem Set Sample Solutions
Pleaseprovidestudents with three coordinategrids to usein completingthe ProblemSet.
1. One endpoint ofaline segment is (−𝟑,−𝟔). The length ofthe linesegment is 𝟕units. Find four pointsthat could
serve asthe other endpointofthe givenlinesegment.
(−𝟏𝟎,−𝟔); ( 𝟒,−𝟔); (−𝟑,𝟏); (−𝟑,−𝟏𝟑)
2. Two of the verticesofarectangle are ( 𝟏,−𝟔) and (−𝟖,−𝟔). Ifthe rectanglehasaperimeter of 𝟐𝟔units, what are
the coordinatesofitsother two vertices?
( 𝟏,−𝟐) and (−𝟖,−𝟐); or ( 𝟏,−𝟏𝟎) and (−𝟖,−𝟏𝟎).
3. A rectangle has aperimeterof 𝟐𝟖units, an areaof 𝟒𝟖square units, and sidesthat are eitherhorizontal or vertical.
If one vertex isthe point (−𝟓,−𝟕)and theoriginisin theinterior ofthe rectangle, findthe vertex ofthe rectangle
that isopposite (−𝟓,−𝟕).
(𝟏,𝟏)