Explore · composite area
A rectangular garden is 15 m by 12 m. A 8 m by 2 m rectangular corner is removed. Find the remaining area.
Great Inheritance Mathematics Term A · Independent learning
Area of Complex Figures
Learning target: Decompose or subtract rectangles and triangles to calculate compound area.
Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.
Opening task
Lesson-specific prerequisite checkpoint
Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.
Find the area of an 8 m by 5 m rectangle with a 2 m by 3 m rectangle removed.
Understand the idea
Decompose or subtract rectangles and triangles to calculate compound area.
A complex polygon can be partitioned into nonoverlapping rectangles and triangles, or enclosed in a larger shape from which a cutout is removed. Check that every region is counted once and only once. Label all areas in square units.
Vocabulary focus: unit conversion, perpendicular height, square units. Make a small labeled model, work through the arithmetic, and explain the result in words.
Watch for: Double-counting overlapping pieces.
Model to study: Labeled subregion polygons with area arithmetic.
Worked examples
Worked examples 1 · composite area
A rectangular garden is 13 m by 13 m. A 6 m by 6 m rectangular corner is removed. Find the remaining area.
Solution: 133 m²
Why: Outer rectangle: 13×13=169 m². Removed corner: 6×6=36 m². Subtract disjoint parts: 169−36=133 m².
Worked examples 2 · composite area
A rectangular garden is 10 m by 13 m. A 7 m by 10 m rectangular corner is removed. Find the remaining area.
Solution: 60 m²
Why: Outer rectangle: 10×13=130 m². Removed corner: 7×10=70 m². Subtract disjoint parts: 130−70=60 m².
Worked examples 3 · composite area
A rectangular garden is 14 m by 13 m. A 5 m by 10 m rectangular corner is removed. Find the remaining area.
Solution: 132 m²
Why: Outer rectangle: 14×13=182 m². Removed corner: 5×10=50 m². Subtract disjoint parts: 182−50=132 m².
Check understanding
Check understanding 1 · composite area
A rectangular garden is 11 m by 9 m. A 3 m by 6 m rectangular corner is removed. Find the remaining area.
Check understanding 2 · composite area
A rectangular garden is 12 m by 12 m. A 9 m by 2 m rectangular corner is removed. Find the remaining area.
Guided practice
Guided practice 1 · composite area
A rectangular garden is 12 m by 13 m. A 9 m by 5 m rectangular corner is removed. Find the remaining area.
Guided practice 2 · composite area
A rectangular garden is 11 m by 13 m. A 5 m by 6 m rectangular corner is removed. Find the remaining area.
Guided practice 3 · composite area
A rectangular garden is 10 m by 13 m. A 7 m by 4 m rectangular corner is removed. Find the remaining area.
Independent practice
Independent practice 1 · composite area
A rectangular garden is 15 m by 10 m. A 3 m by 3 m rectangular corner is removed. Find the remaining area.
Independent practice 2 · composite area
A rectangular garden is 11 m by 13 m. A 2 m by 8 m rectangular corner is removed. Find the remaining area.
Independent practice 3 · composite area
A rectangular garden is 13 m by 10 m. A 3 m by 3 m rectangular corner is removed. Find the remaining area.
Independent practice 4 · composite area
A rectangular garden is 15 m by 13 m. A 5 m by 4 m rectangular corner is removed. Find the remaining area.
Independent practice 5 · composite area
A rectangular garden is 14 m by 13 m. A 3 m by 4 m rectangular corner is removed. Find the remaining area.
Challenge and error analysis
Challenge and error analysis 1 · worked-error diagnosis
Error-analysis notebook: Learner notebook (LESSON): “The missing corner ADDS area: 12×11+3×6=150 m².” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: A rectangular garden is 12 m by 11 m. A 3 m by 6 m rectangular corner is removed. Find the remaining area.
Challenge and error analysis 2 · reasoning transfer
Reasoning and transfer: Solve, then explain a different way to check your answer (inverse operation, estimate, model, or units as appropriate). A rectangular garden is 11 m by 10 m. A 8 m by 6 m rectangular corner is removed. Find the remaining area.
Targeted mastery: model, calculate, explain
These additional original examples address specific 5A mastery requirements. Compare the visual and the reasoning before trying the independent questions.
Targeted 1 · targeted mastery: geometry
A 8 m by 6 m rectangle has a triangular corner removed: base 4 m, perpendicular height 3 m. Find area.
Solution: 42 m²
Why: Outer area 8×6=48 m²; removed right triangle 1/2×4×3=6 m²; remaining 42 m².
Targeted 2 · targeted mastery: geometry
A 12 m by 8 m rectangle is extended by a nonoverlapping right triangle attached along 6 m of one side; perpendicular outward height 4 m. Area?
Solution: 108 m²
Why: Rectangle 12×8=96 m²; added triangle 1/2×6×4=12 m²; total 108 m².
Targeted 3 · targeted mastery: geometry
A 9 m by 5 m rectangle loses a right triangular corner with legs 3 m and 4 m. What remains?
Targeted 4 · targeted mastery: geometry
A 10 m by 7 m rectangle has a triangular corner of base 2 m, perpendicular height 5 m removed. Find area.
Exit check
Exit check 1 · composite area
A rectangular garden is 15 m by 10 m. A 5 m by 3 m rectangular corner is removed. Find the remaining area.
Exit check 2 · composite area
A rectangular garden is 11 m by 7 m. A 4 m by 3 m rectangular corner is removed. Find the remaining area.
Target for independent mastery: all exit-check problems correct plus a justified explanation. A instructor may reteach the concept if calculations are correct without meaningful reasoning.