1. Rational Numbers & Coordinate Plane

2. Ordered Pairs

3. Learning Intention
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

4. Success Criteria
-Students use ordered pairs to name points in a grid and to locate points on a map.
-Students identify the first number in an ordered pair as the first coordinate and the second number as the second coordinate.

5. Opening Exercise How did you find your seat in the classroom?
Did the order of the numbers matter? Why or why not?

6. Example #1 The first number of an ordered pair is called the
First coordinate.
The second number of an ordered pair is called the
Second coordinate.

7. Example #1 Everyone stand. Raise your hand if you are at ( ).
Why are you at that ordered pair?

8. Example #2
Describe how the ordered pair is being used in your scenario.
Indicate what defines the first coordinate and what defines the second coordinate in your scenario.

9. Example #2
Scenario 1: The seats in a college football stadium are arranged into 210 sections, with 144 seats in each section. Your ticket to the game indicates the location of your seat using the ordered pair of numbers (123,37). Describe the meaning of each number in the ordered pair and how you would use them to find your seat.

10. Example #2
Scenario 2: Airline pilots use measurements of longitude and latitude to determine their location and to find airports around the world. Longitude is measured as 0–180ᵒ east or 0–180ᵒ west of a line stretching from the North Pole to the South Pole through Greenwich, England, called the prime meridian. Latitude is measured as 0–90ᵒ north or 0–90ᵒ south of the earth’s equator. A pilot has the ordered pair (90ᵒ west, 30ᵒ north). What does each number in the ordered pair describe? How would the pilot locate the airport on a map? Would there be any confusion if a pilot were given the ordered pair (90°,30°)? Explain.

11. Example #2
Scenario 3: Each room in a school building is named by an ordered pair of numbers that indicates the number of the floor on which the room lies, followed by the sequential number of the room on the floor from the main staircase. A new student at the school is trying to get to science class in room 4–13. Describe to the student what each number means and how she should use the number to find her classroom. Suppose there are classrooms below the main floor. How might these rooms be described?

12. Practice
Name the letter from the grid below that corresponds with each ordered pair of numbers below.

13. Practice
List the ordered pair of numbers that corresponds with each letter from the grid below.

14. Closing time
Why does order matter when using ordered pairs of numbers?
Alayna says the order in which the values are given in an ordered pair does not always matter. Who can give an example of when the order does matter and an example of when the order does not matter.
Who can explain how to locate points when pairs of integers are used?

15. Independent Practice Answers

16. Independent Practice Answers

17. Success Criteria
Can you use ordered pairs to name points in a grid and to locate points on a map?
Can you identify the first number in an ordered pair as the first coordinate and the second number as the second coordinate?

19. Locating Ordered Pairs on the Coordinate Plane

20. Learning Intention
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

21. Success Criteria
-Students extend your understanding of the coordinate plane to include all four quadrants and recognize that the axes of the coordinate plane divide the plane into four regions called quadrants .
-Students identify the origin and locate points other than the origin, which lie on an axis.
-Students locate points in the coordinate plane that correspond to given ordered pairs of integers and other rational numbers.

22. Opening Exercise
With a partner, walk around the room and read the posters.
As a pair, write down at least 1 thing on each poster that you know about this vocabulary term from 5th grade or before. When you have written at least 1 thing on each poster, sit back down.

23. Opening Exercise
Lets discuss. Who can share some things that they wrote on the posters?

24. Who can explain what an X-AXIS should look like?
Example #1
Who can explain what an X-AXIS should look like?

25. Example #1
The point below represents zero on the number line. Draw a number line to the right starting at zero.

26. Example #1
Use a straightedge to extend the 𝑥-axis to the left of zero to represent the real number line horizontally, and complete the number line using the same scale as on the right side of zero.

27. Who can explain what an Y-AXIS should look like?
Example #1
Who can explain what an Y-AXIS should look like?

28. Example #1
Use a straightedge to make the Y-axis to draw a vertical line, and complete the numbers above and below zero.

29. Example #2
All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?

30. Example #2
To describe locations of points in the coordinate plane, we use _________of numbers. Order is important, so on the coordinate plane, we use the form (__, __). The first coordinate represents the point’s location from zero on the ___ -axis, and the second coordinate represents the point’s location from zero on the ___-axis.

31. Practice 1. Use the coordinate plane to answer parts (a)–(c).
a. Graph at least five points on the 𝑥-axis, and label their coordinates.
b. What do the coordinates of your points have in common?
c. What must be true about any point that lies on the 𝑥-axis? Explain.

32. Practice 2. Use the coordinate plane to answer parts (a)–(c).
a. Graph at least five points on the y-axis, and label their coordinates.
b. What do the coordinates of your points have in common?
c. What must be true about any point that lies on the y-axis? Explain.

33. Practice
3. If the origin is the only point with 0 for both coordinates, what must be true about the origin?

34. Example #3
Examine the four regions of the coordinate plane cut by the intersecting axes.

35. Example #3
The 𝑥- and 𝑦-axes divide the coordinate plane into regions called quadrants. Why are the regions called quadrants?

36. Example #3
Which of the four regions did you work with most in Grade 5, and why was it the only region you used?

37. Example #3
The four quadrants are numbered one through four using roman numerals. The upper-right quadrant is Quadrant I, and the remaining quadrants are numbered moving counterclockwise from Quadrant I: Quadrant II, Quadrant III, and Quadrant IV. What was the first axis that we extended in Example 1, and what did it reveal?

38. Practice a. (7,2) b. (3,−4) c. (1,−5) d. (−3,8) e. (−2,−1)
Locate and label each point described by the ordered pairs below. Indicate which of the quadrants the points lie in.
a. (7,2)
b. (3,−4)
c. (1,−5)
d. (−3,8)
e. (−2,−1)

39. Practice
Write the coordinates of at least one other point in each of the four quadrants.
a. Quadrant I b. Quadrant II
c. Quadrant III d. Quadrant IV

40. Closing Time
If a point lies on an axis, what must be true about its coordinates? Specifically, what is true for a point that lies on the 𝑥-axis? The 𝑦-axis?

41. Closing Time
What do you know about the location of a point on the coordinate plane if:
Both coordinates are positive?
Only one coordinate is positive?
Both coordinates are negative?
One coordinate is zero?
Both coordinates are zero?

42. Independent Practice Answers

43. Independent Practice Answers

44. Independent Practice Answers

45. Success Criteria
Can you extend your understanding of the coordinate plane to include all four quadrants and recognize that the axes of the coordinate plane divide the plane into four regions called quadrants ?
Can you identify the origin and locate points other than the origin, which lie on an axis?
Can you locate points in the coordinate plane that correspond to given ordered pairs of integers and other rational numbers?

47. Symmetry in the Coordinate Plane

48. Learning Intention
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

49. Success Criteria
-Students understand that two numbers are said to differ only by signs if they are opposites of each other.
-Students recognize that when two ordered pairs differ only by the sign of one or both of the coordinates, then the locations of the points are related by reflections across one or both axes.

50. Opening Exercise
Who can give an example of two opposite numbers, and describe where the numbers lie on the number line.
How are opposite numbers similar, and how are they different?

51. Example #1
Locate and label your points on the coordinate plane to the right. For each given pair of points in the table below, record your observations and conjectures in the appropriate cell. Pay attention to the absolute values of the coordinates and where the points lie in reference to each axis.

52. Example #1

53. Example #1

54. Practice
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 𝑆(5,3) has been reflected over the 𝑥- and 𝑦-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.

55. Practice

56. Practice

57. Practice
When the coordinates of two points are (𝑥,𝑦) and (−𝑥,𝑦), what line of symmetry do the points share? Explain.

58. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (7,2). Move 3 units down, and then reflect over the 𝑦-axis. Where are you?

59. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (7,2). Move 3 units down, and then reflect over the 𝑦-axis. Where are you?
(-7, -1)

60. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (4,−5). Reflect over the 𝑥-axis, and then move 7 units down and then to the right 2 units. Where are you?

61. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (4,−5). Reflect over the 𝑥-axis, and then move 7 units down and then to the right 2 units. Where are you?
(6, -2)

62. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (−3,0). Reflect over the 𝑥-axis, and then move 6 units to the right. Move up two units, and then reflect over the 𝑥-axis again. Where are you?

63. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (−3,0). Reflect over the 𝑥-axis, and then move 6 units to the right. Move up two units, and then reflect over the 𝑥-axis again. Where are you?
(3, -2)

64. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (−2,8). Decrease the 𝑦-coordinate by 6 units. Reflect over the 𝑦-axis, and then move down 3 units. Where are you?

65. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (−2,8). Decrease the 𝑦-coordinate by 6 units. Reflect over the 𝑦-axis, and then move down 3 units. Where are you?
(2, -1)

66. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (5,−1). Reflect over the 𝑥-axis, and then reflect over the 𝑦-axis. Where are you?

67. Practice
Use your pencil eraser or finger to navigate the coordinate plane given verbal prompts.
Begin at (5,−1). Reflect over the 𝑥-axis, and then reflect over the 𝑦-axis. Where are you?
(-5, 1)

68. Closing Time
When the coordinates of two points differ only by one sign, such as (−8,2) and (8,2), what do the similarities and differences in the coordinates tell us about their relative locations on the plane?

69. Closing Time
What is the relationship between (5,1) and (5,−1)? Given one point, how can you locate the other?

70. Independent Practice Answers

71. Independent Practice Answers

72. Independent Practice Answers

73. Success Criteria
Do you understand that two numbers are said to differ only by signs if they are opposites of each other?
Can you recognize that when two ordered pairs differ only by the sign of one or both of the coordinates, then the locations of the points are related by reflections across one or both axes?

75. Drawing the Coordinate Plane & Points on the Plane

76. Learning Intention
Use coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

77. Success Criteria
-Students draw a coordinate plane on graph paper in two steps: (1) Draw and label the horizontal and vertical axes; (2) Mark the number scale on each axis.
-Given some points as ordered pairs, students make reasonable choices for scales on both axes and locate and label the points on graph paper.

78. Opening Exercise
Draw all necessary components of the coordinate plane on the blank 20 × 20 grid provided below, placing the origin at the center of the grid and letting each grid line represent 1 unit.

79. Opening Exercise
What should every coordinate plane have on it?

80. Example #1 Locate and label the points (3,2),(8,4),(−3,8),
(−2,−9),(0,6),(−1,−2),
(10,−2) on the grid.

81. Example #2
Draw a coordinate plane on the grid, and then locate and label the following points:
(−4,20),(−3,35),
(1,−35),(6,10),(9,−40).

82. Example #3
Draw a coordinate plane on the grid, and then locate and label the following points:
(0.1,4),(0.5,7),
(−0.7,-5), (−0.4,3), (0.8,1)

83. Why is it important to label the axes when setting up a coordinate plane?
Why shouldn’t you draw and label the entire coordinate grid before looking at the points to be graphed?
Closing Time

84. Independent Practice Answers

85. Independent Practice Answers

86. Success Criteria
Did you successfully draw a coordinate plane on graph paper in two steps: (1) Draw and label the horizontal and vertical axes; (2) Mark the number scale on each axis.
-Given some points as ordered pairs, did you make reasonable choices for scales on both axes and locate and label the points on graph paper?

88. Distance on the Coordinate Plane

89. Learning Intention
Use coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

90. Success Criteria
Students compute the length of horizontal and vertical line segments with integer coordinates for end points in the coordinate plane by counting the number of units between end points and using absolute value.

91. Opening Exercise
Four friends are touring on motorcycles. They come to an intersection of two roads; the road they are on continues straight, and the other is perpendicular to it. The sign at the intersection shows the distances to several towns. Draw a map/diagram of the roads, and use it and the information on the sign to answer the following questions:

92. What is the distance between Albertsville and Dewey Falls?
Opening Exercise
What is the distance between Albertsville and Dewey Falls?

93. What is the distance between Blossville and Cheyenne?
Opening Exercise
What is the distance between Blossville and Cheyenne?

94. Opening Exercise
On the coordinate plane, what represents the intersection of the two roads?

95. Example #1 Consider the points (−4,0) and (5,0).
What do the ordered pairs have in common, and what does that mean about their location in the coordinate plane?

96. Example #1 Consider the points (−4,0) and (5,0).
How did we find the distance between two numbers on the number line?

97. Example #1 Consider the points (−4,0) and (5,0).
Use the same method to find the distance between (−4,0) and (5,0).

98. Example #2
Consider the line segment with end points (0,−6) and (0,−11).
What do the ordered pairs of the end points have in common, and what does that mean about the line segment’s location in the coordinate plane?

99. Example #2
Consider the line segment with end points (0,−6) and (0,−11).
Find the length of the line segment described by finding the distance between its end points (0,−6) and (0,−11).

100. Example #3
Consider the line segment with end points (−3,3) and (−3,−5).
What do the end points, which are represented by the ordered pairs, have in common? What does that tell us about the location of the line segment on the coordinate plane?

101. Example #3
Consider the line segment with end points (−3,3) and (−3,−5).
Find the length of the line segment by finding the distance between its end points.

102. Practice
Find the lengths of the line segments whose end points are given below. Show how you determined that the line segments are horizontal or vertical.
a. (−3,4) and (−3,9)
b. (2,−2) and (−8,−2)
c. (−6,−6) and (−6,1)
d. (−9,4) and (−4,4)
e. (0,−11) and (0,8)

103. Closing Time
Why is it possible for us to find the length of a horizontal or vertical line segment even if it’s not on the 𝑥- or 𝑦-axis?
Can you think of a real-world situation where this might be useful?

104. Independent Practice Answers

105. Independent Practice Answers

106. Success Criteria
Can you compute the length of horizontal and vertical line segments with integer coordinates for end points in the coordinate plane by counting the number of units between end points and using absolute value?

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