Mathematics · Grade 3 · Geometry and Spatial Systems · Term 2

Area of Rectilinear Figures

Students find the area of rectilinear figures by decomposing them into non-overlapping rectangles.

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

About This Topic

Rectilinear figures feature sides that meet at right angles, creating shapes such as L's, T's, or U's. Grade 3 students calculate their area by decomposing them into non-overlapping rectangles, finding each rectangle's area through unit squares or length times width, then adding the results. This approach confirms that the whole area's measure equals the sum of the parts, aligning with Ontario's 3.MD.C.7.D standard.

Positioned in the Geometry and Spatial Systems unit for Term 2, this topic prompts students to explain decomposition steps, analyze part-whole relationships, and devise strategies for figures like L-shapes. It strengthens spatial reasoning, justification of methods, and problem-solving flexibility. Real-world links to floor plans or fenced yards show practical value, while building on prior rectangle area knowledge.

Active learning excels with this topic because students handle tangible shapes. Cutting grid paper figures, assembling with blocks, or sketching decompositions makes abstract ideas concrete. Collaborative sharing of strategies reveals multiple valid paths, corrects errors through peer feedback, and fosters confidence in tackling irregular forms.

Key Questions

1. Explain how to decompose a complex shape into simpler rectangles.
2. Analyze how the area of the parts relates to the area of the whole figure.
3. Design a strategy to find the area of an L-shaped figure.

Learning Objectives

• Calculate the area of rectilinear figures by decomposing them into non-overlapping rectangles.
• Explain the strategy used to decompose a given rectilinear figure into rectangles.
• Analyze how the sum of the areas of individual rectangles relates to the total area of the rectilinear figure.
• Design a method to find the area of an L-shaped figure by breaking it into two rectangles.
• Compare different strategies for decomposing a rectilinear figure and justify the chosen method.

Before You Start

Area of Rectangles

Why: Students need to know how to find the area of a single rectangle using unit squares or the formula length x width before they can find the area of composite shapes.

Identifying Right Angles

Why: Understanding that rectilinear figures are made up of right angles is fundamental to decomposing them into rectangles.

Key Vocabulary

Rectilinear figure	A shape whose sides are all either horizontal or vertical, meeting at right angles. Examples include L-shapes, T-shapes, and U-shapes.
Decomposition	The process of breaking down a complex shape into smaller, simpler shapes, such as rectangles, that do not overlap.
Non-overlapping	Shapes that do not share any area. When decomposing a figure, the smaller shapes fit together perfectly without any part of one shape covering another.
Area	The amount of two-dimensional space a shape covers, measured in square units.

Watch Out for These Misconceptions

Common Misconception

Area equals the perimeter of the figure.

What to Teach Instead

Students often confuse boundary length with interior coverage. Hands-on grid covering shows area as filled space, not edges. Group comparisons of perimeter and area calculations clarify the difference through visible unit square counts.

Common Misconception

Decompositions must use identical rectangles only.

What to Teach Instead

Varied rectangle sizes work if non-overlapping and complete. Building with mixed blocks lets students experiment, see valid sums match, and discuss why flexibility aids complex shapes during pair shares.

Common Misconception

Small gaps or overlaps in decomposition do not affect area.

What to Teach Instead

Gaps undermeasure, overlaps overmeasure the whole. Physical cutting and reassembling reveals exact fits, with peer reviews catching errors and reinforcing complete coverage in station rotations.

Active Learning Ideas

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.

35 min·Small Groups

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.

25 min·Pairs

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.

20 min·Whole Class

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.

30 min·Individual

Real-World Connections

• Architects and interior designers use the concept of area to calculate the square footage of rooms or entire buildings. They decompose complex floor plans into simpler rectangular sections to determine material needs for flooring or paint.
• Construction workers might measure and calculate the area of irregularly shaped yards to determine the amount of sod or fencing required. They would break down the yard into rectangular sections for easier measurement.

Assessment Ideas

Quick Check

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.

Discussion Prompt

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?'

Exit Ticket

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.

Frequently Asked Questions

How do I introduce decomposing rectilinear figures?

Start with familiar L-shapes on grid paper, model one decomposition by drawing lines to form rectangles, calculate areas, and sum. Invite students to try a second way on the same shape. Use visuals like room corners to connect to life, ensuring all see the part-whole logic before independent practice. (62 words)

What materials work best for rectilinear area activities?

Grid paper, unit squares, blocks, scissors, and tape provide hands-on options. Digital tools like grid apps suit extensions. These allow cutting, building, and measuring, making decomposition visible and reducing abstraction. Stock trays for stations keep prep simple and reuse easy across lessons. (58 words)

How can active learning help students master rectilinear areas?

Active methods like cutting shapes or block building let students manipulate figures, testing decompositions kinesthetically. Pair discussions expose varied strategies, building flexibility and error correction. Whole-class projections encourage collective justification, turning passive calculation into dynamic spatial exploration that retains better than worksheets alone. (60 words)

What strategies help with L-shaped figures?

Guide students to draw horizontal or vertical lines from indent corners to form two rectangles. Calculate each as length times width, add results. Practice multiple paths, like splitting the long arm first, to show equivalent totals. Real-world ties, such as playground layouts, motivate strategy design and part-whole analysis. (59 words)