🗺 What is 6.NS.C.8?
The Big Idea — navigating all four quadrants and measuring distance
📌 Standard
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Use coordinates and absolute value to find distances between points with the same first or second coordinate.
Distance = |x₁ − x₂| (same y) or |y₁ − y₂| (same x)
📐 The coordinate plane has 4 quadrants, an x-axis, a y-axis, and an origin (0,0). Every point is an ordered pair (x, y).
📏 Distance rule: Points on the same vertical line → subtract y-values. Points on the same horizontal line → subtract x-values. Always take absolute value!
⚠️ 6.NS.C.8 only requires distance for points sharing an x or y coordinate (horizontal/vertical lines). No diagonal distance (Pythagorean theorem is Grade 8).
📘 QUADRANT MAP
| Quadrant | Signs | Example |
|---|---|---|
| I (upper right) | (+, +) | (3, 4) |
| II (upper left) | (−, +) | (−3, 4) |
| III (lower left) | (−, −) | (−3, −4) |
| IV (lower right) | (+, −) | (3, −4) |
📏 Distance via Absolute Value — The Core Idea
Why absolute value? Because distance is always positive — direction doesn't matter.
Same x-coordinate → Vertical Distance
Points A(−3, 4) and B(−3, −2) share x = −3. They're on the same vertical line.
d = |4 − (−2)| = |4 + 2| = |6| = 6
Same y-coordinate → Horizontal Distance
Points C(2, 5) and D(7, 5) share y = 5. They're on the same horizontal line.
d = |2 − 7| = |−5| = 5
✍️ TWR Frame — Describe the Distance (EN/ES)
Points and share the same -coordinate, so I subtract their values: | − | = units.
🇪🇸 Los puntos ___ y ___ comparten la misma coordenada ___. La distancia es |___ − ___| = ___ unidades.
📖 Key Vocabulary
Academic language for 6.NS.C.8 — with Spanish cognates
Coordinate Plane
koh-OR-din-it playn
A flat surface formed by two perpendicular number lines (x-axis and y-axis) that intersect at the origin.
Has 4 quadrants, infinite points
🇪🇸 plano de coordenadas
Ordered Pair
OR-derd pair
Two numbers (x, y) that identify a point's location — x first (horizontal), then y (vertical).
(-3, 4) → left 3, up 4
🇪🇸 par ordenado
Quadrant
KWOD-rent
One of the four regions of the coordinate plane created by the x and y axes. Numbered I–IV counterclockwise.
QI: (+,+) QII: (−,+)
🇪🇸 cuadrante (cognate!)
Absolute Value
AB-suh-loot VAL-yoo
The distance from zero on a number line. Always non-negative. Written |n|.
|−6| = 6, |6| = 6, |0| = 0
🇪🇸 valor absoluto (cognate!)
Origin
OR-ih-jin
The center point (0, 0) where the x and y axes intersect on the coordinate plane.
Every axis starts here
🇪🇸 origen (cognate!)
x-axis / y-axis
eks AK-sis
x-axis = horizontal number line. y-axis = vertical number line. Both pass through the origin.
x: left/right · y: up/down
🇪🇸 eje x / eje y
🧠 Morphology: "Coordinate" from Latin coordinare = "arrange in order." "Quadrant" from Latin quadrans = "one quarter." "Absolute" from Latin absolutus = "set free" (from sign/direction).
💡 Cognate Alert for ELLs: coordinate / coordenada, quadrant / cuadrante, absolute / absoluto, origin / origen, axis / eje — many math terms share roots in Spanish and English!
🪜 How to Find Distance — Step by Step
Two methods: same x-coordinate (vertical) and same y-coordinate (horizontal)
1
Identify the two points
Write both ordered pairs clearly. Label them Point A and Point B.
A(−3, 4) and B(−3, −2)
2
Check: do they share an x-coord or y-coord?
Same x-value → vertical distance (subtract y's). Same y-value → horizontal distance (subtract x's).
Both have x = −3 → vertical → subtract y-values
3
Subtract the coordinates that differ
Take one value minus the other. It doesn't matter which order — absolute value fixes it.
4 − (−2) = 4 + 2 = 6
4
Apply absolute value
Distance is always positive — wrap the result in | |. This handles cases where you subtract in the "wrong" order.
|4 − (−2)| = |6| = 6 ✓
5
Label your answer with units
State the distance with units: "6 units." In real-world problems, use the actual unit (miles, feet, blocks).
Distance = 6 units
📘 Three Worked Examples
Vertical distance · Horizontal distance · Real-world context
📍 VERTICAL DISTANCE (same x)
Find the distance between P(2, 7) and Q(2, −3).
→
Same x-coordinate (x = 2) → vertical → subtract y-values
→
|7 − (−3)| = |7 + 3| = |10| = 10
→
Check: P is at y = 7 (above) and Q at y = −3 (below) → 7 units up + 3 units down = 10 ✓
Distance = 10 units
📍 HORIZONTAL DISTANCE (same y)
Find the distance between M(−5, 3) and N(2, 3).
→
Same y-coordinate (y = 3) → horizontal → subtract x-values
→
|−5 − 2| = |−7| = 7
→
Or: |2 − (−5)| = |7| = 7 — same result!
Distance = 7 units
🏙️ REAL-WORLD (city grid)
On a city map, a library is at (−4, 2) and a park is at (3, 2). One unit = 1 block. How many blocks apart?
→
Same y-coordinate (y = 2) → horizontal distance
→
|−4 − 3| = |−7| = 7
7 blocks apart
✍️ TWR — Distance Explanation Frame
Points and share the same -coordinate (), so I found the distance. I subtracted: | − | = units.
🎯 Interactive Coordinate Grid
Click to place points — practice plotting all four quadrants
Click the grid to plot a point
Plot two points with the same x or y coordinate to calculate distance automatically.
🗺
Score
0 / 7
Progress
✏️ Practice Problems
Type your answer · Click Check · Hints available
Problem 1
Plot
Point A is at (−3, 4) and Point B is at (−3, −2). Both share x = −3. What is the distance between them?
units
Same x-coordinate → subtract y-values. |4 − (−2)| = ?
Problem 2
Distance
Find the distance between M(−5, 3) and N(2, 3).
units
Same y-coordinate (y=3) → subtract x-values. |−5 − 2| = ?
Problem 3
Distance
Find the distance between P(0, 6) and Q(0, −4).
units
Same x-coordinate (x=0 — on the y-axis!) → |6 − (−4)| = ?
Problem 4
Real World
A soccer field on a coordinate grid has two corners at (−6, −2) and (6, −2). Each unit = 1 yard. How long is that side of the field?
yards
Same y-coordinate (y=−2) → |−6 − 6| = ? yards
Problem 5
Distance
Find the distance between R(4, −5) and S(4, 8).
units
Same x-coordinate (x=4) → |−5 − 8| = ? (remember: |−13| = 13)
Problem 6
Real World
A rectangle has two vertices at (−4, 3) and (5, 3), and two more at (−4, −2) and (5, −2). What is the perimeter of the rectangle?
units
Width: |(−4) − 5| = 9. Height: |3 − (−2)| = 5. Perimeter = 2×9 + 2×5 = ?
Problem 7
Extend
Point J is at (−7, y) and Point K is at (−7, 4). The distance between them is 9 units. If J is below K, what is the y-coordinate of Point J?
y-value
|4 − y| = 9. If J is below K, y is less than 4. So 4 − y = 9 → y = 4 − 9 = ?
🏙️ City Map Challenge
Solve real-world distance problems on a coordinate city grid. Each unit = 1 city block.
🗺 The City Grid
Positive x → East · Negative x → West · Positive y → North · Negative y → South
Origin (0,0) = City Hall
Origin (0,0) = City Hall
🏫 Question 1: The school is at (−4, 2) and the library is at (3, 2). How many blocks is the walk from school to library?
blocks
🏥 Question 2: The hospital is at (5, −3) and the fire station is at (5, 6). How many blocks apart are they?
blocks
🏪 Question 3: Store A is at (−6, −1) and Store B is at (2, −1). A delivery truck drives from Store A to Store B. How far does it travel?
blocks
🏟️ Question 4 (Challenge): A park has corners at (−3, 4), (−3, −2), (4, 4), and (4, −2). What is the perimeter of the park in blocks?
blocks
✍️ TWR — Real-World Distance Frame
The and are both at , so I found the horizontal distance. | − | = blocks.
🔁 Flash Review Cards
Tap to reveal the distance. Use the formula: |coord₁ − coord₂|
(2, 4) and (2, −3)
Same x → tap ↩
7 units
|4−(−3)| = 7
(−5, 1) and (4, 1)
Same y → tap ↩
9 units
|−5−4| = 9
(0, −6) and (0, 5)
Same x → tap ↩
11 units
|−6−5| = 11
(−8, 3) and (1, 3)
Same y → tap ↩
9 units
|−8−1| = 9
(6, 7) and (6, −4)
Same x → tap ↩
11 units
|7−(−4)| = 11
(−3, −5) and (9, −5)
Same y → tap ↩
12 units
|−3−9| = 12
(−2, 8) and (−2, −7)
Same x → tap ↩
15 units
|8−(−7)| = 15
(−4, 0) and (6, 0)
On x-axis → tap ↩
10 units
|−4−6| = 10
⚠️ Common Mistakes — Watch Out!
Real student errors and corrections
❌
Mistake #1 — Reversing x and y when plotting
Error: Plotting (3, −4) by going down 3 and right 4 instead of right 3 and down 4.
Fix: Ordered pair is always (x, y) — x is horizontal (left/right) FIRST, then y is vertical (up/down).
Fix: Ordered pair is always (x, y) — x is horizontal (left/right) FIRST, then y is vertical (up/down).
❌
Mistake #2 — Subtracting the wrong pair
Error: For same x-coordinate, student subtracts x-values instead of y-values.
Fix: Same x → they're on a vertical line → subtract the values that DIFFER (the y-values).
Fix: Same x → they're on a vertical line → subtract the values that DIFFER (the y-values).
❌
Mistake #3 — Forgetting to use absolute value
Error: Getting a negative distance. e.g., −5 − 2 = −7, then writing "distance = −7 units."
Fix: Distance is always positive. |−7| = 7. Always take the absolute value after subtracting.
Fix: Distance is always positive. |−7| = 7. Always take the absolute value after subtracting.
❌
Mistake #4 — Subtracting across quadrants incorrectly with negatives
Error: |4 − (−2)| → student computes as |4 − 2| = 2 instead of |4 + 2| = 6.
Fix: Subtracting a negative = adding its positive. 4 − (−2) = 4 + 2 = 6.
Fix: Subtracting a negative = adding its positive. 4 − (−2) = 4 + 2 = 6.
❌
Mistake #5 — Confusing quadrant numbers
Error: Thinking Quadrant I is upper left or numbering them clockwise.
Fix: Start Q I in the upper RIGHT (+,+). Go counterclockwise: II upper left (−,+), III lower left (−,−), IV lower right (+,−).
Fix: Start Q I in the upper RIGHT (+,+). Go counterclockwise: II upper left (−,+), III lower left (−,−), IV lower right (+,−).
🎯 Self-Check: After finding distance — is it positive? ✓ Does it make sense on the grid (can you count the squares)? ✓ Did you subtract the coordinates that CHANGE (not the ones that match)? ✓