1. Lesson 18: Distanceon theCoordinate Plane
Date: 2/23/15 169
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 18
Albertsville  8 mi
Blossville  3 mi
Cheyenne  12 mi
Dewey Falls  6 mi
Lesson 18: Distance on the Coordinate Plane
Student Outcomes
 Students compute the length of horizontal and vertical linesegments with integer coordinates for endpoints in
the coordinateplaneby counting the number of units between end points and usingabsolutevalue.
Classwork
OpeningExercise (5 minutes)
Opening Exercise
Four friendsare touring on motorcycles. They cometo an intersection oftwo roads;
the road they are on continues straight, and theother isperpendicular to it. The sign
at the intersection shows thedistancesto several towns. Draw a map/diagram ofthe
roads and use it and the information on thesign to answer thefollowing questions:
What isthe distance between Albertsville and Dewey Falls?
Albertsvilleis 𝟖miles to theleft and Dewey Falls is 𝟔 miles to theright. Sincethe
towns arein oppositedirections from theintersection, their distances must be
combined. By addition, 𝟖 + 𝟔 = 𝟏𝟒, so thedistancebetween Albertsvilleand Dewey
Falls is 𝟏𝟒miles.
What isthe distance between Blossville and Cheyenne?
Blossvilleand Cheyenneareboth straight aheadfrom theintersection inthedirection that they aregoing. Sincethey are
on thesamesideoftheintersection, Blossvilleis on theway to Cheyenneso thedistanceto Cheyenneincludes the 𝟑 miles
to Blossville. To find thedistancefrom Blossvilleto Cheyenne, I haveto subtract, 𝟏𝟐− 𝟑 = 𝟗. So thedistancefrom
Blossvilleto Cheyenneis 𝟗miles.
On the coordinate plane, what represents the intersection ofthe two roads?
The intersection is represented by theorigin.
Example 1 (6 minutes): The Distance BetweenPointson an Axis
Students find the distancebetween points on the 𝑥-axis by findingthe distancebetween numbers on the number line.
They find the absolutevalues of the 𝑥-coordinates and add or subtracttheir absolutevalues to determine the distance
between the points.
Example 1: The Distance Between Pointson an Axis
What isthe distance between (−𝟒,𝟎)and (𝟓,𝟎)?

2. Lesson 18: Distanceon theCoordinate Plane
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 18
What do the orderedpairshave in common and what doesthat mean about their location in thecoordinateplane?
Both oftheir 𝒚-coordinates arezero so each point lies on the 𝒙-axis, thehorizontal number line.
How did we find the distancebetween two numberson thenumber line?
Wecalculated theabsolute values ofthenumbers,whichtold us how far thenumbers werefrom zero. Ifthenumbers
were located on oppositesides ofzero, then weadded theirabsolutevalues together. Ifthenumbers werelocated onthe
samesideof zero, then we subtracted their absolutevalues.
Use the same method to find thedistance between (−𝟒,𝟎)and (𝟓,𝟎).
|−𝟒| = 𝟒and | 𝟓| = 𝟓. The numbers areon oppositesides ofzero, so theabsolutevalues get combined, so 𝟒 + 𝟓 = 𝟗.
The distancebetween (−𝟒,𝟎)and (𝟓,𝟎)is 𝟗units.
Example 2 (5 minutes): The Length of a Line Segmenton an Axis
Students find the length of a linesegment that lies on the 𝑦-axis by findingthe distancebetween its endpoints.
Example 2: The Length ofaLine Segmenton an Axis
What isthe length of the linesegment with endpoints (𝟎,−𝟔)and (𝟎,−𝟏𝟏)?
What do the orderedpairs ofthe endpoints have in commonand what doesthat mean about theline segment’s location
in the coordinateplane?
The 𝒙-coordinates ofboth endpoints arezero so the points lieon the 𝒚-axis, thevertical number line. Ifits endpoints lie
on a vertical number line, thenthelinesegment itselfmust also lieon thevertical line.
Find the length ofthe linesegment describedby finding thedistance between its endpoints (𝟎,−𝟔)and (𝟎,−𝟏𝟏).
|−𝟔| = 𝟔and |−𝟏𝟏|= 𝟏𝟏. The numbers areon thesamesideofzero which means thelonger distancecontains the
shorter distance, so theabsolutevalues need to besubtracted. 𝟏𝟏− 𝟔 = 𝟓. Thedistancebetween (𝟎,−𝟔)and
(𝟎,−𝟏𝟏)is 𝟓 units, so thelength ofthelinesegment withendpoints (𝟎,−𝟔)and(𝟎,−𝟏𝟏)is 𝟓units.
Example 3 (10 minutes): Lengthof a Horizontal or Vertical Line Segmentthat Does Not Lie on an Axis
Students find the length of a vertical linesegment which does not lieon the 𝑦-axis by findingthe distancebetween its
endpoints.
Example 3: Length ofaHorizontal or Vertical LineSegment that DoesNot Lieon an Axis
Find the length ofthe linesegment withendpoints (−𝟑,𝟑)and (−𝟑,−𝟓).
What do the endpoints, which are represented by the ordered pairs, have incommon? What
doesthat tell us about the location ofthe line segment on thecoordinate plane?
Both endpoints have 𝒙-coordinates of −𝟑so thepoints lieon the vertical linethatintersects the 𝒙-axis at −𝟑. This means
that theendpoints ofthelinesegment,and thus thelinesegment, lieon a vertical line.
Find the length ofthe linesegment by finding the distancebetween itsendpoints.
The endpoints areon thesamevertical line, so weonly need to find thedistancebetween 𝟑and −𝟓on thenumber line.
| 𝟑| = 𝟑and |−𝟓| = 𝟓, and thenumbers areon oppositesides ofzero so thevalues must be added; 𝟑 + 𝟓 = 𝟖. So the
distancebetween (−𝟑,𝟑)and (−𝟑,−𝟓)is 8 units.
Scaffolding:
 Students may need to
draw an auxiliary line
through the endpoints to
help visualizea horizontal
or vertical number line.MP.7

3. Lesson 18: Distanceon theCoordinate Plane
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 18
Exercise 1 (10 minutes)
Students calculatethe distancebetween pairs of points usingabsolutevalues.
Exercise 1
1. Find the lengthsof the line segmentswhose endpointsare given below. Explain how you determined that the line
segmentsare horizontal or vertical.
a. (−𝟑,𝟒)and (−𝟑,𝟗)
Both endpoints have 𝒙-coordinates of −𝟑, so thepoints lie on a vertical line that passes through −𝟑 onthe
𝒙-axis. | 𝟒| = 𝟒 and | 𝟗| = 𝟗, and thenumbers areon thesamesideofzero. By subtraction, 𝟗− 𝟒 = 𝟓, so the
length ofthelinesegment with endpoints (−𝟑,𝟒)and (−𝟑,𝟗) is 𝟓units.
b. (𝟐,−𝟐)and (−𝟖,−𝟐)
Both endpoints have 𝒚-coordinates of −𝟐, so thepoints lie on a horizontal line thatpasses through −𝟐on the
𝒚-axis. | 𝟐| = 𝟐 and |−𝟖|= 𝟖, and thenumbers areon opposite sides ofzero, so the absolute values must be
added. By addition 𝟖+ 𝟐 = 𝟏𝟎, so thelengthofthelinesegment with endpoints (𝟐,−𝟐)and (−𝟖,−𝟐)is
𝟏𝟎units.
c. (−𝟔,−𝟔)and (−𝟔,𝟏)
Both endpoints have 𝒙-coordinates of −𝟔, so thepoints lie on a vertical line. |−𝟔| = 𝟔and | 𝟏| = 𝟏, and the
numbers areon opposite sides ofzero, so theabsolute values must be added. By addition 𝟔 + 𝟏 = 𝟕, so the
length ofthelinesegment with endpoints (−𝟔,−𝟔)and (−𝟔,𝟏)is 𝟕 units.
d. (−𝟗,𝟒)and (−𝟒,𝟒)
Both endpoints have 𝒚-coordinates of 𝟒, so thepoints lie on a horizontal line. |−𝟗| = 𝟗and |−𝟒| = 𝟒, and
thenumbers areon thesamesideofzero. By subtraction 𝟗− 𝟒 = 𝟓, so thelength ofthelinesegment with
endpoints (−𝟗,𝟒)and (−𝟒,𝟒)is 5 units.
e. (𝟎,−𝟏𝟏)and (𝟎,𝟖)
Both endpoints have 𝒙-coordinates of 𝟎, so thepoints lie on the 𝒚-axis. |−𝟏𝟏|= 𝟏𝟏and | 𝟖| = 𝟖, and the
numbers are on oppositesides ofzero, so their absolute values must be added. By addition 𝟏𝟏+ 𝟖 = 𝟏𝟗, so
the length ofthelinesegment with endpoints (𝟎,−𝟏𝟏)and (𝟎,𝟖)is 𝟏𝟗units.
Closing(3 minutes)
 Why can we find the length of a horizontal or vertical linesegment even if it’s not on the 𝑥- or 𝑦-axis?
 Can you think of a real-world situation wherethis might be useful?
Exit Ticket (6 minutes)
Lesson Summary
To find the distance betweenpointsthat lieon thesame horizontal line or on the same vertical line, wecan use the
same strategy that we used to find the distance betweenpointson the number line.

4. Lesson 18: Distanceon theCoordinate Plane
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 18
Name ___________________________________________________ Date____________________
Lesson 18: Distance on the Coordinate Plane
Exit Ticket
Determine whether each given pair of endpoints lies on the samehorizontal or vertical line. If so,find the length of the
linesegment that joins the pair of points. If not, explain how you know the points arenot on the same horizontal or
vertical line.
a. (0, −2) and (0,9)
b. (11,4) and (2,11)
c. (3, −8) and (3, −1)
d. (−4, −4) and (5, −4)

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 18
Exit Ticket Sample Solutions
Determinewhethereach given pair ofendpoints lies on the same horizontal or verticalline. Ifso, find the length ofthe
line segment that joinsthepair ofpoints. Ifnot, explain how you know the pointsare noton thesame horizontal or
vertical line.
a. (𝟎,−𝟐) and (𝟎,𝟗)
The endpoints bothhave 𝒙-coordinates of 𝟎so they both lieon the 𝒚-axis whichis a verticalline. They lieon
oppositesides ofzero so their absolutevalues haveto becombined to get thetotaldistance. |−𝟐| = 𝟐and | 𝟗| = 𝟗,
so by addition, 𝟐 + 𝟗 = 𝟏𝟏. Thelength ofthelinesegment withendpoints (𝟎,−𝟐)and(𝟎,𝟗) is 𝟏𝟏units.
b. (𝟏𝟏,𝟒)and (𝟐,𝟏𝟏)
The points do not lieon thesamehorizontalor verticallinebecausethey do notsharea common 𝒙-or 𝒚-coordinate.
c. (𝟑,−𝟖) and (𝟑,−𝟏)
The endpoints bothhave 𝒙-coordinates of 𝟑, so thepoints lie on a vertical line that passes through 𝟑 on the 𝒙-axis.
The 𝒚-coordinates numbers lieon thesameside ofzero. Thedistancebetween thepoints is determinedby
subtracting their absolutevalues, |−𝟖| = 𝟖and |−𝟏| = 𝟏. So by subtraction, 𝟖 − 𝟏 = 𝟕. Thelength oftheline
segment with endpoints (𝟑,−𝟖)and(𝟑,−𝟏)is 𝟕 units.
d. (−𝟒,−𝟒) and (𝟓,−𝟒)
The endpoints havethesame 𝒚-coordinateof −𝟒, so they lie on a horizontal line that passes through −𝟒on the 𝒚-
axis. The numbers lieon oppositesides ofzero on thenumber line, so their absolutevalues must be addedto obtain
thetotal distance, |−𝟒| = 𝟒and | 𝟓| =5. So by addition, 𝟒 + 𝟓 = 𝟗. Thelength ofthelinesegment with endpoints
(−𝟒,−𝟒)and (𝟓,−𝟒)is 𝟗units.
Problem Set Sample Solutions
1. Find the length ofthe linesegment withendpoints ( 𝟕,𝟐)and (−𝟒,𝟐), and explainhow you arrived at your solution.
𝟏𝟏units. Both points havethesame 𝒚-coordinate, so I knew they wereon thesamehorizontal line. I found the
distancebetween the 𝒙-coordinates by counting thenumber ofunits on a horizontalnumber linefrom −𝟒to zero,
and then from zero to 𝟕, and 𝟒 + 𝟕 = 𝟏𝟏.
2. Sarah and Jamal were learning partnersin math classand wereworking independently. They each startedat the
point (−𝟐,𝟓) and moved 𝟑unitsvertically in the plane. Each student arrived at adifferent endpoint. How isthis
possible? Explain and list the two differentendpoints.
It is possiblebecauseSarah could havecounted up and Jamalcould havecounted down, or vice-versa. Moving 𝟑
units in either direction vertically would generatethefollowingpossibleendpoints: (−𝟐,𝟖) or (−𝟐,𝟐)
3. The length ofaline segment is 𝟏𝟑units. One endpoint ofthe line segment is (−𝟑,𝟕). Find four pointsthat could be
the other endpointsofthe linesegment.
(−𝟑,𝟐𝟎),(−𝟑,−𝟔),(−𝟏𝟔,𝟕) or (𝟏𝟎,𝟕)