Area of Composite Figures

Active learning helps students grasp the concept of area for composite figures because they must physically manipulate shapes and visualize decompositions. Breaking down complex problems into simpler parts is a skill that improves with hands-on practice. This topic benefits from collaborative work where students articulate their reasoning and learn from peers.

• 6th Grade
• Mathematics
• 4 activities
• 20 min–50 min

Learning Objectives

1. Calculate the area of composite figures by decomposing them into rectangles and triangles.
2. Analyze different strategies for decomposing a complex polygon into simpler shapes.
3. Compare the results of area calculations using multiple decomposition methods for the same figure.
4. Explain the process of finding the area of a composite figure using addition and subtraction of areas.
5. Design a method to find the area of an irregular shape by approximating it with simpler polygons.

Inquiry Circle

⏱ 50 min·Small Groups

Inquiry Circle: Floor Plan Challenge

Each group receives an irregular floor plan sketch (L-shape, T-shape, or U-shape) and must decompose it at least two different ways. They calculate the total area using each decomposition and confirm both methods give the same result, then present their strategies to the class.

Prepare & details

Analyze how any polygon can be broken down into triangles and rectangles.

Facilitation Tip: During the Floor Plan Challenge, circulate and ask groups to explain how they chose their decomposition lines.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Add or Subtract?

Present two composite figures: one where students add areas and one where a shape has been removed (e.g., a rectangle with a triangular corner cut out). Pairs decide for each which strategy is more efficient and justify their reasoning before sharing with the class.

Prepare & details

Design a strategy to find the area of an irregular shape.

Facilitation Tip: In the Think-Pair-Share, provide a figure where both addition and subtraction are possible so students see the value of both approaches.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Gallery Walk

⏱ 40 min·Individual

Gallery Walk: Strategy Comparison

Post five composite figures around the room, each already solved using one decomposition method. Students must find and draw a different valid decomposition for each figure and verify that both methods give the same area.

Prepare & details

Justify the process of decomposing a complex figure to calculate its area.

Facilitation Tip: For the Gallery Walk, ask students to compare strategies by writing one sentence on each poster about what they learned.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Simulation Game

⏱ 45 min·Individual

Simulation Game: Design Your Own Floor Plan

Students design an irregular polygon floor plan for an imaginary room on grid paper, add labeled measurements, and find its area by decomposing. They write a brief explanation of their decomposition strategy that another student could follow independently.

Prepare & details

Analyze how any polygon can be broken down into triangles and rectangles.

Facilitation Tip: During the Design Your Own Floor Plan, remind students to label all dimensions and include at least two different shapes in their composite figure.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Teaching This Topic

Start with simple composite figures made of two shapes and gradually increase complexity. Model decomposition by thinking aloud as you decide where to draw lines. Avoid moving too quickly to advanced figures before students are comfortable with basic decompositions. Research shows that students benefit from seeing multiple solutions to the same problem, so present figures that can be solved using different methods.

What to Expect

Students will confidently decompose composite figures into familiar shapes, choose appropriate area formulas, and combine results accurately. They will also recognize when subtraction is more efficient than addition. Group discussions should include clear explanations of strategies and justifications for chosen methods.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Floor Plan Challenge, watch for students who draw decomposition lines that overlap or leave gaps.

What to Teach Instead

Ask students to use colored pencils to shade each sub-region a different color and check that no area is double-shaded before calculating. Have them present their shaded sketches to the group before moving to calculations.

Common Misconception

During Think-Pair-Share: Add or Subtract?, watch for students who always use addition even when subtraction is more efficient.

What to Teach Instead

Provide figures where subtraction is clearly the better method, such as a rectangle with a corner cut out. After students share their solutions, show both addition and subtraction methods for the same figure to demonstrate equivalence and build confidence in subtraction.

Assessment Ideas

Quick Check

After Collaborative Investigation: Floor Plan Challenge, collect one decomposition sketch and calculation from each group to check for overlapping regions and accurate area formulas.

Exit Ticket

During Think-Pair-Share: Add or Subtract?, collect student responses that include two decomposition methods and calculations for one method. Review these to assess their ability to choose and justify an efficient strategy.

Discussion Prompt

After Gallery Walk: Strategy Comparison, facilitate a class discussion about which strategies students found most effective. Ask them to explain why a particular method worked better for certain figures.

Extensions & Scaffolding

• Challenge students to design a composite figure that requires both addition and subtraction to find the total area.
• For students who struggle, provide figures with pre-drawn decomposition lines and ask them to calculate areas using the given shapes.
• Ask students to create a composite figure using only triangles and then calculate its area in two different ways, explaining which method they prefer.

Key Vocabulary

Composite Figure	A shape made up of two or more simpler geometric shapes, such as rectangles and triangles.
Decomposition	The process of breaking down a complex shape into smaller, simpler shapes whose areas are easier to calculate.
Area	The amount of two-dimensional space a shape occupies, measured in square units.
Polygon	A closed shape made of straight line segments.