Apply Distance to Real-World Maps and Polygons

Lesson 2 of 2: 6.NS.C.8

In this lesson:

• Apply the distance rule to city grids, navigation, and elevation maps
• Find side lengths and perimeters of axis-aligned polygons

Learning Objectives for This Deck

By the end of this lesson, you will be able to:

4. Apply the plot-and-measure approach to real-world problems involving city grids, coordinate maps, navigation grids, and elevation profiles
5. Compute side lengths and perimeters of polygons on the coordinate plane when sides are horizontal or vertical

Recap: The Distance Rule from Lesson 1

• Two points sharing an x-coordinate → vertical segment → distance =
• Two points sharing a y-coordinate → horizontal segment → distance =
• First step always: identify which coordinate is shared

City Grids: Streets and Avenues

• Streets run east-west: (horizontal lines)
• Avenues run north-south: (vertical lines)
• Two places on the same avenue share an x-coordinate → vertical distance

Walking Distance: A Worked Example

School at ; library at ; post office at

Step 1: Check for a shared coordinate
Library x = post office x = ✓ (same avenue)

Step 2: Compute vertical distance

Ship Navigates Due North on a Grid

A ship moves from to .

Step 1: Shared coordinate?
Both x-values = ✓ (same north-south line)

Step 2: Distance traveled

"Due north" → x stays constant → vertical segment.

Elevation Map: Zero Is a Real Coordinate

Cable car: from to .

Step 1: Shared coordinate?
Both x-values = ✓ (both on the y-axis)

Step 2: Altitude gained

Sharing x = 0 is still sharing a coordinate.

When Neither Coordinate Is Shared

Diagonal Pairs: Stop and Name the Limit

Given: and

• Same x? — No
• Same y? — No
• Conclusion: diagonal → 8.G.B.8 (Grade 8)

and are not the distance.

Quick Check: Horizontal, Vertical, or Diagonal?

For each pair — which type of segment?

1. and
2. and
3. and

Check the shared coordinate before subtracting.

Quick Check: Three Pairs, Three Types

1. and : share → vertical, length
2. and : share → horizontal, length
3. and : no shared coordinate → diagonal

From Single Distances to Full Polygons

So far: one pair of points → one segment length.

Now: a polygon has multiple sides — each side is a segment.

• Apply the rule side by side
• Sum the side lengths → perimeter
• Works when every side is horizontal or vertical

Rectangle on the Coordinate Plane

Find Each Rectangle Side Length

Rectangle: , , ,

Side AB: shares →

Side BC: shares →

CD = AB = 8; DA = BC = 6 (opposite sides match).

Rectangle Perimeter: Add All Side Lengths

Using the shortcut:

Preview: area = sq units (6.G.A.1).

Right Triangle: Two Legs, One Diagonal

• Vertices: , ,
• Horizontal leg: shares → units
• Vertical leg: shares → units

Hypotenuse Is Diagonal: Out of Scope

• Two legs: units ✓
• Hypotenuse: diagonal → requires 8.G.B.8
• Full perimeter: not computable at Grade 6

Finding the two legs is the complete Grade 6 answer.

Soccer Field Perimeter from Coordinates

Corners: , , ,

Length (horizontal side, or ):

Width (vertical side, or ):

Perimeter: m

Quick Check: Rectangle from Coordinates

Rectangle vertices: , , ,

1. What is the length of the horizontal side?
2. What is the length of the vertical side?
3. What is the perimeter?

Apply the two-step habit for each side before advancing.

Rectangle Check: Side Lengths and Perimeter

Horizontal side (shares ):

Vertical side (shares ):

Perimeter: units

Both sides equal 5 — this rectangle is a square.

Lesson Summary: Two Chunks, One Tool

✓ Check the shared coordinate first — always
✓ Real-world grids: streets share y; avenues share x
✓ Diagonal pairs: name the limitation, don't compute
✓ Polygon perimeters: apply the rule side by side, then sum

Sharing counts — zero is a valid coordinate

Exit Task: Apply Everything You Know

Plot , , .

• Distance = ?
• Distance = ?
• Perimeter of = ?

Coming up: 6.G.A.1 (area), 6.G.A.3 (polygons), 8.G.B.8 (diagonal).