Part B (Bottom Block): \(3 \times 2 = 6\) sq units
Sum: \(2 + 6 = 8\) sq units
Problem 3 A B C
Decomposition Breakdown (3 Parts): Total Area = 14 sq units
Part A (Left Arm): \(2 \times 2 = 4\) sq units • Part C (Right Arm): \(2 \times 2 = 4\) sq units
Part B (Bottom Connector): \(3 \times 2 = 6\) sq units
Sum: \(4 + 6 + 4 = 14\) sq units
Problem 4 A B
Decomposition Breakdown: Total Area = 11 sq units
Part A (Full Left Column): \(1 \times 5 = 5\) sq units
Part B (Bottom-Right Block): \(3 \times 2 = 6\) sq units
Sum: \(5 + 6 = 11\) sq units (Or horizontal cut: \(1 \times 3 = 3\) plus \(4 \times 2 = 8 \rightarrow 11\))
Problem 5: Two Ways Verification Verified Area = 8 sq units
Way 1: Horizontal Cut
• Top Bar: \(3 \times 1 = 3\) sq units
• Right Column (below cut): \(1 \times 5 = 5\) sq units
Calculation: \(3 + 5 = \mathbf{8\text{ sq units}}\)
Way 2: Vertical Cut
• Left Arm (before cut): \(2 \times 1 = 2\) sq units
• Full Right Column: \(1 \times 6 = 6\) sq units
Calculation: \(2 + 6 = \mathbf{8\text{ sq units}}\)
Key Teaching Point: Because decomposing merely regroups the unit squares without adding or removing any squares, the sum of the partial areas remains constant regardless of cut orientation.
Composite Shapes Answer Key Page 2 of 3
Answer Key
Teacher Reference
Solutions for advanced rectilinear floor plans, geometric explanations, and open-ended design tasks.
Problem 1 A B C
Decomposition Breakdown (3 Parts): Total Area = 22 sq units
Part A (Top Arm): \(2 \times 2 = 4\) sq units
Part B (Central Horizontal Bar): \(7 \times 2 = 14\) sq units
Part C (Bottom Arm): \(2 \times 2 = 4\) sq units
Sum: \(4 + 14 + 4 = 22\) sq units
Problem 2 R1 R2 R3
Room Analysis:
3 Rooms Total Area = 17 sq units
Room 1 (Top-Left): \(2 \times 3 = 6\) sq units
Room 2 (Bottom-Left): \(2 \times 2 = 4\) sq units
Room 3 (Long Hallway): \(1 \times 7 = 7\) sq units
Total Floor Area: \(6 + 4 + 7 = 17\) sq units
Problem 3: Area Calculation & Conceptual Explanation Total Area = 9 sq units
Area Calculation:
• Top Bar: \(5 \times 1 = 5\) sq units
• Left Arm (below cut): \(1 \times 4 = 4\) sq units
Sum: \(5 + 4 = 9\) sq units
Exemplar Student Explanation:
"This shape is not a single rectangle—it is a non-convex polygon with 6 sides and an inner corner. Multiplying overall length by overall width (\(5 \times 5 = 25\)) would count a large empty section that isn't part of the shape. To find the real area, we must decompose it into simpler rectangles."
Problem 4: Three Decomposition Methods Target Area = 10 sq units
Way 1: Horizontal Cut
• Top piece: \(1 \times 2 = 2\)
• Bottom piece: \(4 \times 2 = 8\)
Total: \(2 + 8 = 10\)
Way 2: Vertical Cut
• Left column: \(1 \times 4 = 4\)
• Right block: \(3 \times 2 = 6\)
Total: \(4 + 6 = 10\)
Way 3: Subtraction
• Bounding box: \(4 \times 4 = 16\)
• Empty notch: \(3 \times 2 = 6\)
Total: \(16 - 6 = 10\)
Problem 5: Super Challenge Teacher Evaluation Criteria Exact Target: 15 sq units
Student designs will vary widely. Verify that the student's figure is rectilinear (all 90° corners on grid lines) and contains exactly 15 shaded unit squares.
Exemplar A (T-Shape):
Top crossbar \(5 \times 1 = 5\) sq units + vertical stem \(2 \times 5 = 10\) sq units. Total: \(5 + 10 = 15\) sq units.
Exemplar B (L-Shape):
Vertical wing \(3 \times 3 = 9\) sq units + horizontal wing \(2 \times 3 = 6\) sq units. Total: \(9 + 6 = 15\) sq units.
Composite Shapes Answer Key Page 3 of 3
