Mathematics · Grade 3

Active learning ideas

Area of Rectilinear Figures

Active learning works for rectilinear area because students need to physically manipulate shapes to see how parts combine into a whole. Moving unit squares or blocks lets them experience the concept of area as filled space, not just a number to memorize. This hands-on approach builds lasting understanding beyond worksheets or formulas.

• Ontario Curriculum Expectations
• 3.MD.C.7.D

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Think-Pair-Share
• 35 min · Small Groups

Small Groups: Cut-and-Assemble Stations

Prepare station trays with grid paper rectilinear figures, scissors, and tape. Students cut shapes into rectangles, calculate each area, and reassemble to verify the total. Groups rotate stations, discussing one new decomposition per shape.

Explain how to decompose a complex shape into simpler rectangles.

Facilitation Tip

During Cut-and-Assemble Stations, circulate to ask each group: 'How did you decide where to cut? Does your total match the grid count?'

What to look for

Provide students with a worksheet showing 2-3 different rectilinear figures. Ask them to draw lines to decompose each figure into rectangles and write the area of each smaller rectangle. Then, they should write the total area of the figure.

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

Activity 02

• Think-Pair-Share
• 25 min · Pairs

Pairs: Block Building Challenge

Provide unit blocks or squares. Pairs construct L- or T-shaped figures, decompose by separating into rectangles, measure areas, and record sums. Partners swap builds to test alternative decompositions.

Analyze how the area of the parts relates to the area of the whole figure.

Facilitation Tip

For Block Building Challenge, remind pairs to trade one block at a time and recalculate the area after each move to see how small changes affect the total.

What to look for

Present a rectilinear figure to the class. Ask students: 'How could we break this shape into smaller rectangles?' Facilitate a discussion where students share different decomposition strategies. Ask: 'Does everyone's strategy give the same total area? Why or why not?'

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

Activity 03

• Think-Pair-Share
• 20 min · Whole Class

Whole Class: Projection Decomposition

Project large rectilinear figures on the board. Class suggests and votes on decomposition lines, calculates collective areas, and compares methods. Record top strategies on chart paper for reference.

Design a strategy to find the area of an L-shaped figure.

Facilitation Tip

In Projection Decomposition, pause after each student shares a strategy to ask: 'Who can restate this in their own words? Does this method change the total area?'

What to look for

Give each student an L-shaped figure drawn on grid paper. Ask them to: 1. Draw one way to decompose it into two rectangles. 2. Write the area of each rectangle. 3. Write the total area of the L-shape.

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

Activity 04

• Think-Pair-Share
• 30 min · Individual

Individual: Design and Decompose

Students draw original rectilinear figures on grid paper, decompose into three rectangles minimum, label areas, and sum totals. They explain their strategy in a short journal entry.

Explain how to decompose a complex shape into simpler rectangles.

Facilitation Tip

During Design and Decompose, model tracing around the outside of one rectangle before cutting to emphasize non-overlapping boundaries.

What to look for

Provide students with a worksheet showing 2-3 different rectilinear figures. Ask them to draw lines to decompose each figure into rectangles and write the area of each smaller rectangle. Then, they should write the total area of the figure.

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

A few notes on teaching this unit

Start with concrete tools like grid paper or interlocking cubes to ground the abstract formula. Avoid rushing to the formula length x width; let students discover why it works through repeated decomposition. Use language like 'filled space' instead of 'inside the shape' to reinforce area as coverage. Research shows students who physically manipulate shapes retain concepts longer than those who only compute.

Successful learning looks like students confidently breaking shapes into rectangles, calculating each piece’s area, and adding them without gaps or overlaps. They explain their process clearly and compare strategies with peers to confirm the total area matches. Missteps become visible through their actions, not just their answers.

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Watch Out for These Misconceptions

• During Cut-and-Assemble Stations, watch for students who measure the outer edges of the figure instead of counting the unit squares inside.

  Circulate with a blank grid and ask them to cover their shape with unit squares. Point out that the number of squares touching the edges is the perimeter, not the area.

• During Block Building Challenge, watch for pairs who assume all rectangles must be the same size to add up correctly.

  Hand them two different-sized blocks and ask: 'Does this still cover the whole shape? How does the total change if one piece is bigger?'

• During Projection Decomposition, watch for students who leave gaps or overlaps when drawing decomposition lines on the board.

  Hand them scissors and grid paper to cut out the pieces. Ask: 'Does this fit perfectly back together? What happens if you leave a gap here?'

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Methods used in this brief

• Think-Pair-Share