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Determine Distance on the Coordinate Plane

Find horizontal and vertical distances between points on the coordinate plane using absolute value.

6.NOS.C.9 · The Number System
Level
Guided practice with vocabulary support

🟠 Level 0 — Extra Support

Sentence starters: “First, I…” · “The answer is… because…” · “I know this because…”
W

Warm-Up

2 questions
Warm-Up 1
What is the distance between the points (2, 3) and (2, −4) on the coordinate plane?
Vocabulary: distance is how far apart two points are (always positive). When two points share the same x-coordinate, find the distance by subtracting the y-coordinates and taking the absolute value.
Formula reference: |3 − (−4)| = |3 + 4| = 7
x y 1 2 3 -1 1 2 3 -1 -2 (2, 3) (2, -4)
✓Correct! The points share x = 2, so distance = |3 − (−4)| = |3 + 4| = 7 units.
✗Not quite. Both points have x = 2, so find the vertical distance: |3 − (−4)| = |7| = 7 units (answer C).
Warm-Up 2
True or False: To find the distance between two points on the same horizontal line, you subtract the x-coordinates and take the absolute value.
Vocabulary: A horizontal line goes left-right. Points on the same horizontal line share the same y-coordinate.
Absolute value makes any number positive, so distance is always positive.
✓Correct! On a horizontal line the y-coordinates are the same, so you subtract the x-coordinates and use absolute value to get the distance.
✗Actually, the statement is true. Points on the same horizontal line share a y-coordinate, so distance = |x₂ − x₁|.
P

Practice

5 questions
Practice 1
Find the distance between each pair of points.
Strategy: Check whether the points share the same x-coordinate (vertical distance) or the same y-coordinate (horizontal distance). Then subtract and use absolute value.
Formula: distance = |coordinate₁ − coordinate₂|
a. (5, 1) to (5, −3) = units
b. (−2, 4) to (6, 4) = units
c. (0, −5) to (0, 3) = units

Sentence frame: "The distance is ___ units because |___ − ___| = ___."

✓All correct! a. |1 − (−3)| = 4, b. |−2 − 6| = 8, c. |−5 − 3| = 8.
✗Some answers need another look. Remember: subtract the coordinates that differ and take the absolute value. a = 4, b = 8, c = 8.
Practice 2
What is the distance between the two points shown on the grid?
Hint: Both points are on the same horizontal line (same y-coordinate). Count the spaces between them, or use |x₂ − x₁|.
-3 -2 -1 1 2 3 4 2 1 -1 -2 (-3, 2) (4, 2)
Explain your reasoning:
✓Correct! Both points have y = 2. Distance = |−3 − 4| = |−7| = 7 units.
✗The points share y = 2, so find the horizontal distance: |−3 − 4| = |−7| = 7 units (answer C).
Practice 3
Match each pair of points to the correct distance. Drag the distance to the matching pair.
Strategy: For each pair, check if the x- or y-coordinates match. Then use absolute value to find the distance.
Remember: distance is always positive.
10 units
5 units
9 units
7 units
(1, 3) and (1, −2)
(−4, 0) and (3, 0)
(2, −5) and (2, 4)
(−6, 1) and (4, 1)
✓All correct! |3 − (−2)| = 5, |−4 − 3| = 7, |−5 − 4| = 9, |−6 − 4| = 10.
✗Some matches are wrong. Find the shared coordinate, subtract the other coordinates, and take the absolute value. Try again!
Practice 4
True or False: The distance between (−3, 1) and (−3, −5) is −6 units.
Key idea: Distance is always positive (or zero). You can never have a negative distance. Use absolute value: |1 − (−5)| = ?
Explain your reasoning:

Sentence frame: "The statement is ___ because distance is always ___. The actual distance is |___ − ___| = ___ units."

✓Correct! Distance is always positive. |1 − (−5)| = |6| = 6 units, not −6.
✗The statement is false. Distance is always positive. |1 − (−5)| = 6 units, not −6 units.
Practice 5
A rectangular park has two corners at (−3, 2) and (5, 2). How long is this side of the park?
Real-world connection: The two corners are on the same horizontal line (y = 2). Find the horizontal distance by subtracting the x-coordinates and using absolute value.
Explain your reasoning:

Sentence frame: "The side is ___ units long because |___ − ___| = ___."

✓Correct! Both points share y = 2. Distance = |−3 − 5| = |−8| = 8 units.
✗The y-coordinates match (y = 2), so subtract the x-coordinates: |−3 − 5| = |−8| = 8 units (answer C).
★

Challenge

1 question
Challenge
A map has a school at (−4, 3), a library at (−4, −2), and a store at (3, −2). Find the total walking distance going school → library → store. Show your work.
Break it into two parts:
1. School to Library: same x-coordinate (−4). Find the vertical distance: |3 − (−2)| = ?
2. Library to Store: same y-coordinate (−2). Find the horizontal distance: |−4 − 3| = ?
3. Add the two distances together.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 3 2 1 -1 -2 -3 School (-4, 3) Library (-4, -2) Store (3, -2)
School to Library (show work):
Library to Store (show work):
Total walking distance = units

Sentence frame: "From school to library the distance is ___ units because |___ − ___| = ___. From library to store the distance is ___ units because |___ − ___| = ___. The total is ___ + ___ = ___ units."

✓Correct! School to Library = |3 − (−2)| = 5 units. Library to Store = |−4 − 3| = 7 units. Total = 5 + 7 = 12 units.
✗The total is 12 units. School → Library: |3 − (−2)| = 5. Library → Store: |−4 − 3| = 7. Total: 5 + 7 = 12.