Practice 1
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 and teaching response
18 · The containing area is 20 and the missing area is 2.
Split an L-shape into non-overlapping rectangles
Find the area of a composite shape by choosing parts that cover it once, then check the result with a second decomposition.
Get the student and teacher materialsPrepare the group
Learning goal: A learner can mark a valid decomposition and explain why adding the parts neither omits nor double-counts area.
You will need: Squared paper, two coloured pencils and unit-square counters if available.
Start with this question
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?
20 square units. Split the remaining shape into a 4-by-4 rectangle and a 2-by-2 rectangle: 16 + 4 = 20.
- 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.
- 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.
- Add the two non-overlapping areas: 4 × 4 + 2 × 2 = 20. Check that every remaining unit square belongs to one part.
- 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.
Run the 15-minute workshop
- 3 minutes. Trace the L-shape and identify the removed corner. Ask which squares are inside the actual shape.
- 4 minutes. Colour two non-overlapping rectangles and connect each multiplication to one coloured part.
- 5 minutes. Try the practice with both an addition method and a containing-rectangle check when possible.
- 3 minutes. Complete the independent check and explain how the decomposition avoids counting a square twice.
A useful error to discuss
A learner might say: Use the full width and full height, so the L-shape has area 24.
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.
Practise, then check independently
Use the first three tasks for guided practice. Try the final task without help. Record a drawing or equation as well as the answer.
Practice 2
Two non-overlapping rectangles, 3 by 4 and 2 by 2, join along an edge. Find their total area in square units.
Answer and teaching response
16 · Shared edges have no area; add the twelve and four covered squares.
Practice 3
An 8-by-3 rectangle loses a 2-by-2 corner. Find the remaining area in square units.
Answer and teaching response
20 · Subtract only the four missing squares from twenty-four.
Independent check
Draw a 7-by-5 rectangle with a 3-by-2 corner removed. Find the L-shape area and show a valid decomposition.
Answer and teaching response
29 · A 4-by-5 part and a 3-by-3 part give 20 + 9 = 29.
Transfer the idea
A learner draws two rectangles that overlap in one unit square and adds their areas. How should the count be repaired?
Discuss a possible response
The overlap was counted twice. Either choose non-overlapping parts or subtract the area of the overlap once; explain which square was double-counted.
Choose the next teaching step
Look for:
- Marks dimensions of each smaller rectangle rather than guessing them.
- Explains complete coverage with no overlap.
If the idea is not secure: Count the missing squares first, then colour each remaining row or rectangle.
If the learner is ready: Draw a different dividing line and show that both decompositions cover exactly the same unit squares.
Materials for your group
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