G6 m3-c-lesson 19-t

Lesson 19: Problem-Solving and the Coordinate PlaneDate: 4/13/23 174

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 19

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 8units. 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 0so 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

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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.

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. 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 ) 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 thex-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 8units 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.

*Students can use a similar argument using the length of the

horizontal side starting at (4,5 ) , knowing it has to be 9 units

(4 ,5)

(4 ,−3)

8 units

(−5 ,−3)

.

.

.

(4 ,5)

(4 ,−3)

(−5 ,5)

(−5 ,−3)

. .

.





long.

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

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.

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. Find the area of the rectangle.

9×8=72; The area of the rectangle is 72unit s2.

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.

The diagonal line segment cuts the rectangle into two

right triangles. The areas of the triangles are 36units2 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.

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.

Each of the vertices lies in a different quadrant.

Its sides are either vertical or horizontal.

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

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

(4 ,5)

(4 ,−3)

(−5 ,5)

(−5 ,−3)

. .

.

(4 ,−1)

(4 ,3)

(−6 ,−1)

(−6 ,3)

Y

X

MP.7

MP.1





coordinates of the rectangle’s vertices are and (−6 ,−1).

Using absolute value, show how the lengths of the sides of your rectangle provide a perimeter of 28 units.

|−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.

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 thex-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.

Exit Ticket (10 minutes)

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.





Name ___________________________________________________ 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





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

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 28units, an area of 48square 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).

2 units

6 units

3 units

3 units

1 unit1 unit

Y

X





(1,1)


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G6 m3-c-lesson 19-t