G6 m3-c-lesson 19-t

Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

6•3

Lesson 19: Problem-Solving and the Coordinate Plane
Student Outcomes


Students solve problems related to the distance between points that lie on the same horizontal or vertical line.



Students use the coordinate plane to graph points, line segments and geometric shapes in the various
quadrants and then use the absolute value to find the related distances.

Lesson Notes
The grid provided in the Opening Exercise is also used for Exercises 1–6 since each exercise is sequential. Students
extend their knowledge about finding distances between points on the coordinate plane to the associated lengths of line
segments and sides of geometric figures.

Classwork
Opening Exercise (3 minutes)
Opening Exercise
In the coordinate plane, find the distance between the points using absolute value.
The distance between the points is 𝟖 units. The points have the same first coordinates and therefore lie on the same
vertical line. |−𝟑| = 𝟑, and |𝟓| = 𝟓, and the numbers lie on opposite sides of 𝟎 so their absolute values are added
together; 𝟑 + 𝟓 = 𝟖. We can check our answer by just counting the number of units between the two points.

.
|𝟓| = 𝟓

|−𝟑| = 𝟑

.

Lesson 19:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org

Problem-Solving and the Coordinate Plane
3/4/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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Exercises 1–2 (8 minutes): The Length of a Line Segment is the Distance Between its Endpoints
Students relate the distance between two points lying in different quadrants of the coordinate plane to the length of a
line segment with those endpoints. Students then use this relationship to graph a horizontal or vertical line segment
using distance to find the coordinates of endpoints.
Exercises
1.

Locate and label (𝟒, 𝟓) and (𝟒, – 𝟑). Draw the line segment
between the endpoints given on the coordinate plane. How long
is the line segment that you drew? Explain.

(𝟒, 𝟓)
.

The length of the line segment is also 𝟖 units. I found that the
distance between (𝟒, −𝟑) and (𝟒, 𝟓) is 𝟖 units, and because these
are the endpoints of the line segment, the line segment begins
and ends at these points, so the distance from end to end is 𝟖
units.
2.

𝟖 units

.

.

Draw a horizontal line segment starting at (𝟒, −𝟑) that has a
length of 𝟗 units. What are the possible coordinates of the other
endpoint of the line segment? (There is more than one answer.)

(−𝟓, −𝟑)

(𝟒, −𝟑)

(−𝟓, −𝟑) or (𝟏𝟑, −𝟑)
Which point do you choose to be the other endpoint of the
horizontal line segment? Explain how and why you chose that point. Locate and label the point on the coordinate
grid.
The other endpoint of the horizontal line segment is (−𝟓, −𝟑); I chose this point because the other option
(𝟏𝟑, – 𝟑) is located off of the given coordinate grid.
Note: Students may choose the endpoint (𝟏𝟑, −𝟑) but they must change the number scale of the 𝒙-axis to do so.

Exercise 3 (5 minutes): Extending Lengths of Line Segments to Sides of Geometric Figures
The two line segments that you have just drawn could be seen as two sides of a rectangle. Given this, the endpoints of
the two line segments would be three of the vertices of this rectangle.
3.

Find the coordinates of the fourth vertex of the rectangle. Explain how you find the coordinates of the fourth vertex
using absolute value.
The fourth vertex is (−𝟓, 𝟓). The opposite sides of a rectangle are
the same length, so the length of the vertical side starting at
(−𝟓, −𝟑) has to be 𝟖 units long. Also, the side from (−𝟓, −𝟑) to
the remaining vertex is a vertical line, so the endpoints must have
the same first coordinate. |−𝟑| = 𝟑, and 𝟖 − 𝟑 = 𝟓, so the
remaining vertex must be five units above the 𝒙-axis.

.

(−𝟓, 𝟓)

.

(𝟒, 𝟓)

*Students can use a similar argument using the length of the
horizontal side starting at (𝟒, 𝟓), knowing it has to be 𝟗 units long.

.
How does the fourth vertex that you found relate to each of the
consecutive vertices in either direction? Explain.

MP.7

(−𝟓, −𝟑)

.
(𝟒, −𝟑)

The fourth vertex has the same first coordinate as (−𝟓, −𝟑)
because they are the endpoints of a vertical line segment. The
fourth vertex has the same second coordinate as (𝟒, 𝟓) since they
are the endpoints of a horizontal line segment.
Draw the remaining sides of the rectangle.

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Date:

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Exercises 4–6 (6 minutes): Using Lengths of Sides of Geometric Figures to Solve Problems
4.

Using the vertices that you have found and the lengths of the line segments between them,
find the perimeter of the rectangle.
𝟖 + 𝟗 + 𝟖 + 𝟗 = 𝟑𝟒; The perimeter of the rectangle is 𝟑𝟒 units.

5.

Scaffolding:
Students may need to review
and discuss the concepts of
perimeter and area from
earlier grades.

Find the area of the rectangle.
𝟗 × 𝟖 = 𝟕𝟐; The area of the rectangle is 𝟕𝟐 𝒖𝒏𝒊𝒕𝒔 𝟐.

6.

Draw a diagonal line segment through the rectangle with
opposite vertices for endpoints. What geometric figures
are formed by this line segment? What are the areas of
each of these figures? Explain.

(−𝟓, 𝟓)

(𝟒, 𝟓)
.

The diagonal line segment cuts the rectangle into two
right triangles. The areas of the triangles are 𝟑𝟔 𝒖𝒏𝒊𝒕𝒔 𝟐
each because the triangles each make up half of the
rectangle and half of 𝟕𝟐 is 𝟑𝟔.

.
EXTENSION [If time allows]: Line the edge of a piece of paper
up to the diagonal in the rectangle. Mark the length of the
diagonal on the edge of the paper. Align your marks
horizontally or vertically on the grid and estimate the length of
the diagonal to the nearest integer. Use that estimation to
now estimate the perimeter of the triangles.

.
(𝟒, −𝟑)

(−𝟓, −𝟑)

The length of the diagonal is approximately 𝟏𝟐 units, and the perimeter of each triangle is approximately 𝟐𝟗 units.

Exercise 7 (8 minutes)
7.

Construct a rectangle on the coordinate plane that satisfies each of the criteria listed below. Identify the coordinate
of each of its vertices.


Its sides are either vertical or horizontal.



MP.1

Each of the vertices lies in a different quadrant.



The perimeter of the rectangle is 28 units.

Answers will vary. The example to the right shows a
rectangle with side lengths 𝟏𝟎 and 𝟒 units. The
coordinates of the rectangle’s vertices
are (−𝟔, 𝟑), (𝟒, 𝟑), (𝟒, −𝟏) and (−𝟔, −𝟏).
Using absolute value, show how the lengths of the
sides of your rectangle provide a perimeter of 𝟐𝟖 units.

Y

(−𝟔, 𝟑)

(𝟒, 𝟑)

X
(−𝟔, −𝟏)

(𝟒, −𝟏)

|−𝟔| = 𝟔, |𝟒| = 𝟒, and 𝟔 + 𝟒 = 𝟏𝟎, so the width of my
rectangle is 𝟏𝟎 units.
|𝟑| = 𝟑, |−𝟏| = 𝟏, and 𝟑 + 𝟏 = 𝟒, so the height of my
rectangle is 𝟒 units.
𝟏𝟎 + 𝟒 + 𝟏𝟎 + 𝟒 = 𝟐𝟖; The perimeter of my rectangle
is 𝟐𝟖 units.

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Closing (5 minutes)


How do we determine the length of a horizontal line segment whose endpoints lie in different quadrants of
the coordinate plane?




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 line segment and the length of the line segment, how do we find the
other endpoint of the line segment? Is the process the same with a horizontal line segment?


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.

Lesson Summary


The length of a line segment on the coordinate plane can be determined by finding the distance between its
endpoints.



You can find the perimeter and area of figures such as rectangles and right triangles by finding the lengths of
the line segments that make up their sides, and then using the appropriate formula.

Exit Ticket (10 minutes)

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Date:

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Name ___________________________________________________

6•3

Date____________________

Lesson 19: Problem-Solving and the Coordinate Plane
Exit Ticket
1.

The coordinates of one endpoint of a line segment are (−2, −7). The line segment is 12 units long. Give three
possible coordinates of the line segment’s other endpoint.

2.

Graph a rectangle with area 12 units2, such that its vertices lie in at least two of the four quadrants in the coordinate
plane. State the lengths of each of the sides, and use absolute value to show how you determined the lengths of the
sides.

Y

X

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Exit Ticket Sample Solutions
1.

The coordinates of one endpoint of a line segment are (−𝟐, −𝟕). The line segment is 𝟏𝟐 units long. Give three
possible coordinates of the line segment’s other endpoint.
(𝟏𝟎, −𝟕); (−𝟏𝟒, −𝟕); (−𝟐, 𝟓); (−𝟐, −𝟏𝟗)

2.

Graph a rectangle with area 𝟏𝟐 units2, such that its vertices lie in at least two of the four quadrants in the coordinate
plane. List the lengths of the sides, and use absolute value to show how you determined the lengths of the sides.
Answers will vary. The rectangle can have
side lengths of 𝟔 and 𝟐 or 𝟑 and 𝟒. A
sample is provided on the grid on the
right. 𝟔 × 𝟐 = 𝟏𝟐

Y
2 units
1 unit 1 unit

3 units

6 units

X

3 units

Problem Set Sample Solutions
Please provide students with three coordinate grids to use in completing the Problem Set.

1.

One endpoint of a line segment is (−𝟑, −𝟔). The length of the line segment is 𝟕 units. Find four points that could
serve as the other endpoint of the given line segment.
(−𝟏𝟎, −𝟔); (𝟒, −𝟔); (−𝟑, 𝟏); (−𝟑, −𝟏𝟑)

2.

Two of the vertices of a rectangle are (𝟏, −𝟔) and (−𝟖, −𝟔). If the rectangle has a perimeter of 𝟐𝟔 units, what are
the coordinates of its other two vertices?
(𝟏, −𝟐) and (−𝟖, −𝟐); or (𝟏, −𝟏𝟎) and (−𝟖, −𝟏𝟎).

3.

A rectangle has a perimeter of 𝟐𝟖 units, an area of 𝟒𝟖 square units, and sides that are either horizontal or vertical.
If one vertex is the point (−𝟓, −𝟕) and the origin is in the interior of the rectangle, find the vertex of the rectangle
that is opposite (−𝟓, −𝟕).
(𝟏, 𝟏)

Lesson 19:
Date:

3/4/14

179

G6 m3-c-lesson 19-t

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