Split an L-shape into non-overlapping rectangles

model. An L-shape starts as a 6-by-4 rectangle with a 2-by-2 corner removed from the upper right. What is its area?

1. Locate the missing corner before calculating. Its width is 2 units and its height is 2 units; it is not part of the covered shape.
2. Draw a vertical line where the missing corner begins. The left part is 4 by 4, while the lower-right part is 2 by 2.
3. Add the two non-overlapping areas: 4 × 4 + 2 × 2 = 20. Check that every remaining unit square belongs to one part.
4. Check another way: the full 6-by-4 rectangle has area 24, and the missing corner has area 4. Removing it also leaves 20 square units.

Worked example: 20 square units. Split the remaining shape into a 4-by-4 rectangle and a 2-by-2 rectangle: 16 + 4 = 20.

p1. An L-shape is a 5-by-4 rectangle with an upper-right 2-by-1 corner removed. Find its area in square units.

Answer: 18

Teacher response: The containing area is 20 and the missing area is 2.

p2. Two non-overlapping rectangles, 3 by 4 and 2 by 2, join along an edge. Find their total area in square units.

Answer: 16

Teacher response: Shared edges have no area; add the twelve and four covered squares.

p3. An 8-by-3 rectangle loses a 2-by-2 corner. Find the remaining area in square units.

Answer: 20

Teacher response: Subtract only the four missing squares from twenty-four.

e1. Draw a 7-by-5 rectangle with a 3-by-2 corner removed. Find the L-shape area and show a valid decomposition.

Answer: 29

Look for:

• Marks dimensions of each smaller rectangle rather than guessing them.
• Explains complete coverage with no overlap.

Teacher response: A 4-by-5 part and a 3-by-3 part give 20 + 9 = 29.

transfer. A learner draws two rectangles that overlap in one unit square and adds their areas. How should the count be repaired?

Answer: The overlap was counted twice. Either choose non-overlapping parts or subtract the area of the overlap once; explain which square was double-counted.

teach-1. 3 minutes: Trace the L-shape and identify the removed corner. Ask which squares are inside the actual shape.

teach-2. 4 minutes: Colour two non-overlapping rectangles and connect each multiplication to one coloured part.

teach-3. 5 minutes: Try the practice with both an addition method and a containing-rectangle check when possible.

teach-4. 3 minutes: Complete the independent check and explain how the decomposition avoids counting a square twice.

misconception. Use the full width and full height, so the L-shape has area 24.

Teacher response: Point to the four counted squares that are missing. The outer dimensions describe a containing rectangle; they do not show which part of that rectangle is actually covered.

support. Count the missing squares first, then colour each remaining row or rectangle.

extend. Draw a different dividing line and show that both decompositions cover exactly the same unit squares.