Math // Measurement

Area of compound shapes

Recognise area as additive; find areas of rectilinear figures by decomposing into non-overlapping rectangles and summing their areas

Ages
8 to 9
Type
Method
Builds on
1 topic
Leads to
1 topic
Standards
1 code

Quick check

If the learner is given an L-shaped room on squared paper, can they split it into two rectangles, find the area of each part, and add them to get the total area?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Decompose an L-shape into two rectangles, find each area, and add them
  • Find the area of a floor plan shaped like a T by splitting into rectangles
  • Solve: A room is L-shaped (3m×5m plus 2m×4m) — what is the total area?

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

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Area of compound shapes

Ages 8 to 9 // Method

Leads to

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Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Teaches children to find the area of compound rectilinear shapes (L- and T-shapes) by decomposing them into non-overlapping rectangles, finding each rectangle's area by multiplying, and summing. The two hard steps are partitioning cleanly (no gaps, no overlaps) and deriving a sub-rectangle's missing internal side from the given outer measurements. This rests on two prior ideas: area is additive, and rectangle area is found by multiplying rows by columns. For the worked example, a 3 m × 5 m rectangle gives 15 m² and a 2 m × 4 m rectangle gives 8 m², summing to 23 m². The sub-areas are ADDED (not multiplied) because area is additive.

Common misconceptions

  • Treats the figure as one big rectangle — multiplies the outer bounding dimensions as if the notch weren't there
  • Overlaps the two sub-rectangles and double-counts the shared corner
  • Cannot find a sub-rectangle's side when it isn't printed
  • Adds side lengths / a perimeter-like total instead of areas, or drops the square unit

Useful representations

  • An L/T-shape with dashed decomposition lines
  • A grid overlay for counting/checking sub-rectangles
  • The two sub-rectangles pulled apart and labelled
  • A figure with some sides blank, prompting missing-length reasoning

Teaching practices

  • Physically cut or draw the figure into rectangles before calculating, so the decomposition is concrete
  • Show that more than one valid decomposition gives the SAME total — the point of additivity
  • Teach how to derive missing internal lengths from the given outer lengths (e.g. 5 − 3 = 2)
  • Insist the two pieces are non-overlapping — shade them different colours and check no tile is both
  • Keep square units on every sub-area and the total

Activities to try

  1. 1

    Cut the L-shaped floor plan into rectangles

    Why it works: A rectilinear figure can be split into non-overlapping rectangles whose areas ADD to the whole; the pieces are summed (not multiplied) because area is additive, and any valid split gives the same total.

    1. Look at an L-shaped room on a grid with its outer measurements labelled.
    2. Draw a cut line to split the shape into two non-overlapping rectangles.
    3. Find each sub-rectangle's area by multiplying its sides (e.g. 3×5 = 15 m² and 2×4 = 8 m²).
    4. Add the two areas to get the total, keeping square-metre units (23 m²).
    5. Say why the two areas are ADDED rather than multiplied: each rectangle covers a separate part of the floor, and area is additive.
  2. 2

    Find the hidden internal side

    Why it works: A sub-rectangle's unlabelled side can be derived by reasoning about the outer lengths, not guessed or taken only from printed numbers.

    1. Look at a compound shape (which may be a separate illustrative figure, not the earlier floor plan) where one internal side is left blank and only the outer measurements are printed.
    2. Work out which outer measurements determine the missing internal length by subtracting; for example, 5 − 3 = 2 is a generic illustration of taking one printed outer length from another, not the derivation of a hidden side in the 23 m² floor plan (whose hidden sides come from 5 − 2 = 3 and 7 − 3 = 4).
    3. Mark the derived length on the figure.
    4. Complete the decomposition and compute the total area.
  3. 3

    Spot the whole-rectangle error

    Why it works: Multiplying the outer bounding dimensions — or multiplying all the printed numbers together — counts tiles the shape does not contain; the missing notch must be accounted for, and here we do that by decomposing the shape into rectangles before multiplying each piece.

    1. Examine a worked solution where someone multiplied the outer bounding dimensions (e.g. 5×7 = 35 m²) as if the notch weren't there, or multiplied every printed number together.
    2. Explain what that person forgot about the shape.
    3. Overlay or imagine a grid and count: the tiles in the missing corner are not there.
    4. Give the correct total by decomposing first (15 + 8 = 23 m²).

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
3.MD.7.d

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