Lesson 7 Reference Sheet CCSS.6.G.A.3

Cartographer's Rule: Distance on the Grid

Visual Grid Map

Fundamental Law

The Axis-Parallel Rule

Calculate distances between points sharing the same x-coordinate (vertical) or same y-coordinate (horizontal).

Different Signs: Cross Axis

Rule: Add Absolute Values (|a| + |b|)

Same Signs: Same Quadrant

Rule: Subtract Absolute Values (|a| - |b|)

Step-by-Step Navigator

1
Identify Shared Coordinate
Find the coordinate that is identical in both pairs. Ignore it!
2
Check the Signs
Are the changing coordinates opposite (cross axis) or same sign?
3
Calculate
Take absolute values. ADD if signs are opposite; SUBTRACT if same.

Absolute Value: | -5 | = 5 and | 5 | = 5. Distance is always positive!

Worked Proofs

Case 1: Different Quadrants (Points A & B)

Points: A(-4, 3) & B(2, 3)
• Examine x-values: -4 and 2 (Diff Signs)
• Action: ADD Absolute Values
• Calculation: | -4 | + | 2 | = 4 + 2 = 6 units.

Case 2: Same Quadrant (Points C & D)

Points: C(3, -1) & D(3, -4)
• Examine y-values: -1 and -4 (Same Signs)
• Action: SUBTRACT Absolute Values
• Calculation: | -4 | - | -1 | = 4 - 1 = 3 units.

"Count grid steps to easily verify your math!"

Lesson 8 Reference Sheet CCSS.6.G.A.3

Plotting & Planning: Drawing Polygons

Constructed Shape on Grid

Drafting Rule

Building the Polygon

Coordinate points act as vertices (corners). Connecting them in sequence forms the closed perimeter of your polygon!

Key Checklist:

• Vertices: Plotted at coordinates (x, y)

• Side Align: Same coordinate = perfect flat line

• Closed Path: Connect last vertex to the first

Mapping Procedure

1
Plot Coordinates
Start at (0, 0). Move left/right on the x-axis, then up/down the y-axis. Mark with a dot.
2
Connect in Sequence
Draw neat straight lines from point to point in the specific alphabetical order (A → B → C → D → A).
3
Classify the Shape
Measure vertical & horizontal side lengths to identify properties (equal sides, parallel sides).

Alignment Check: Points sharing a coordinate are aligned! A(-3, 3) & B(4, 3) share y = 3, forming a perfect horizontal line.

Finding a Missing Vertex

The Problem: Three vertices of rectangle EFGH are plotted at E(-2, 4), F(5, 4), and G(5, -1). Find missing vertex H.

1. Vertical Align: Since G and F share x = 5, side FG is vertical. Side EH must also be vertical!

2. Horizontal Align: Since E and F share y = 4, side EF is horizontal. Side GH must also be horizontal!

3. Solve for H:
• H must share x with E → x = -2.
• H must share y with G → y = -1.
• Therefore, H = (-2, -1).

"Opposite corners share matching x and y coordinates!"

Lesson 9 Reference Sheet CCSS.6.G.A.3

Boundary & Space: Perimeter & Area

Sized Polygon Visuals

Measurement Rule

Boundary vs. Capacity

Perimeter (P) measures the linear boundary path and is summed. Area (A) measures inside 2D space in square units (u²).

Rectangle: A = base × height

Triangle: A = ½ × base × height

Perimeter: P = Sum of all sides

Area Decomposition

For complex shapes, use **decomposition** to split them into simple parts!

1
Decompose Shape
Draw horizontal or vertical lines to partition the shape into rectangles and right triangles.
2
Calculate Area of Parts
Find base and height of each simple shape, then calculate individual areas.
3
Sum Up Total Area
Add the calculated areas of all parts together to find the total area.

Perpendicular Sides: Base and height must meet at a perfect right angle (90°)!

Proof Walkthroughs (ABC)

Calculating Perimeter (P)

Triangle sides from coordinates:
• Side AC (vertical): From y=1 to y=5 → 4 units
• Side AB (horizontal): From x=-3 to x=3 → 6 units
• Side BC (diagonal): Labeled 7.2 units
• Perimeter Calculation:
P = 4 + 6 + 7.2 = 17.2 units

Calculating Area (A)

Triangle dimensions on coordinate plane:
• base (b) = 6 u | height (h) = 4 u
• Formula: A = ½ × b × h
• Area Calculation:
A = ½ × 6 × 4 = 12 units²

"Base and height must be perpendicular!"

Lesson 10 Reference Sheet CCSS.6.G.A.3

The Grid in Action: Real-World Coordinates

Scale Layout Map

🏫 S: (-3,3)

🌳 P: (3,3)

📚 L: (3,-2)

Scaling Rule

Converting Map to World

On real-world maps, 1 grid unit corresponds to a defined real-world scale (e.g. 50 yards). Multiply coordinates-based distance by this scale factor!

Scale Conversion Equation:

Real Dist = Grid Units × Scale Unit

Map Scale Key: 1 Unit = 50 Yards

Scale Master Guide

1
Measure on the Grid
Find grid distance by counting blocks or using coordinate values.
2
Convert the Dimensions
For real-world distances, multiply grid length directly by the scale factor.
3
Calculate Area Carefully
Convert first! Always convert base and height to real units before calculating area.

Area Scaling Trap: If map area is 12 u² and scale is 1 u = 5m, real area is not 12 × 5. It is 12 × (5 × 5) = 300 m². Always scale sides first!

Real Map Workouts

Example 1: Road Distance (S to P)

• Coordinates: S(-3,3) to P(3,3).
• Grid units: | -3 | + | 3 | = 6 units.
• Scale Key: 1 unit = 50 yards.
• Real Distance Calculation:
6 units × 50 yards = 300 yards

Example 2: School Lot Area (Rectangle)

Let a park plot on map have dimensions:
• Grid Base = 4u | Grid Height = 3u
• Scale Key: 1 unit = 20 feet
• Step 1: Convert dimensions to feet:
Base: 4 × 20 = 80 ft | Height: 3 × 20 = 60 ft
• Step 2: Multiply real dimensions:
Real Area = 80 ft × 60 ft = 4,800 ft²

"Always scale individual sides first to avoid area errors!"

Grid Geometry Cartography

Grid Cartographer Reference Sheets

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