Mathematics · Grade 6

Active learning ideas

Area of Composite Figures

Hands-on work with composite figures helps students move beyond memorized formulas by showing how irregular shapes relate to familiar ones. When students cut, rearrange, and measure, they build spatial reasoning and justify their own strategies, which leads to deeper understanding than abstract calculations alone.

• Ontario Curriculum Expectations
• 6.G.A.1

• 30–50 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 35 min · Pairs

Grid Paper Decomposition: Cut and Calculate

Provide composite shapes printed on grid paper. Students cut along lines to separate into rectangles and triangles, label each area, and sum totals. Pairs then swap shapes to verify calculations and discuss strategies.

Justify why any polygon can be decomposed into triangles to find its total area.

Facilitation Tip

During Grid Paper Decomposition, have students trace each piece they cut out in a different color to track which parts belong to which shape.

What to look for

Provide students with a worksheet showing 2-3 composite figures made of rectangles and triangles. Ask them to draw lines to decompose each figure and then calculate its total area, showing all steps.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 02

• Problem-Based Learning
• 40 min · Small Groups

Dimension Shift Challenge: Predict and Measure

Give groups a base composite figure. They change one length or width, sketch the new version, predict area change, then calculate to compare. Record patterns in a class chart.

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

Facilitation Tip

In Dimension Shift Challenge, ask students to record their predictions first before measuring to make their thinking visible.

What to look for

Give each student an image of a composite shape. Ask them to write down two different ways to decompose the shape and calculate its area using one of their methods. They should also state one dimension they could change and predict how it would affect the total area.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 03

• Problem-Based Learning
• 50 min · Small Groups

Habitat Design: Build and Budget

Students design a park or room habitat using composite shapes on grid paper, ensuring total area fits a budget limit. Calculate areas, justify decompositions, and present to class.

Analyze how changing one dimension of a composite shape affects its total area.

Facilitation Tip

Have students label each partial shape with its dimensions and formula during Habitat Design so peer reviews can focus on accuracy.

What to look for

Pose the question: 'Imagine you need to tile a floor that is shaped like a large rectangle with a smaller rectangular section removed from one corner. How would you find the exact area to purchase tiles?' Facilitate a discussion where students share strategies and justify their approaches.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 04

• Problem-Based Learning
• 30 min · Individual

Polygon Puzzle: Triangle Breakdown

Distribute irregular polygons. Students draw lines to divide into triangles, calculate areas multiple ways, and justify why totals match. Compare methods in whole-class share.

Justify why any polygon can be decomposed into triangles to find its total area.

Facilitation Tip

For Polygon Puzzle, require students to sketch diagonals lightly before cutting to ensure their shapes match the original polygon’s angles.

What to look for

Provide students with a worksheet showing 2-3 composite figures made of rectangles and triangles. Ask them to draw lines to decompose each figure and then calculate its total area, showing all steps.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teachers should let students struggle productively with decomposition before stepping in, because the cognitive work of visualizing lines and justifying cuts deepens spatial reasoning. Avoid showing the ‘right’ decomposition too soon, as this removes the chance for students to discover strategies themselves. Research shows that sketching before cutting strengthens mental models more than immediate hands-on work.

By the end of these activities, students will decompose composite shapes cleanly, calculate areas using appropriate formulas, and explain how their methods connect to the original figure. They will also recognize when overlap occurs and adjust their calculations to avoid double counting.

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Watch Out for These Misconceptions

• During Grid Paper Decomposition, watch for students who treat every partial shape as a rectangle and apply length times width.

  Have them compare their cut pieces to the original dimensions and ask, 'Does this triangle look like a rectangle?' before calculating.

• During Dimension Shift Challenge, watch for students who add overlapping regions twice when predicting area.

  Ask them to slide transparent overlays of their shapes to check for gaps or overlaps before recording final measurements.

• During Polygon Puzzle: Triangle Breakdown, watch for students who avoid triangles and only use rectangles.

  Prompt them to draw diagonals on a regular polygon and ask, 'How can you prove this shape is made of triangles?'

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Methods used in this brief

• Problem-Based Learning