Shapes and Space: Geometry and Area · Weeks 19-27

The Concept of Area

Understanding area as an attribute of plane figures and measuring area by counting unit squares.

About This Topic

Area is a fundamental geometric concept that describes the amount of space inside a flat shape. In third grade, students transition from measuring length to measuring 'coverage' using unit squares, as defined in CCSS.Math.Content.3.MD.C.5 and 6. They learn that area is measured in square units and that these units must be packed together without gaps or overlaps. This concept is the precursor to understanding multiplication as a tool for finding area.

Students begin by physically tiling shapes with square blocks, which helps them see area as an additive property. They also learn to decompose irregular shapes into smaller rectangles to find the total area. This topic comes alive when students can physically tile large areas of the classroom floor or use 'area robots' to map out different spaces, making the abstract concept of 'square units' concrete.

Learning Objectives

• Calculate the area of a plane figure by counting unit squares.
• Compare the area of two different plane figures by counting the unit squares within each.
• Explain how breaking a larger rectangle into smaller rectangles affects the calculation of total area.
• Justify why uniform unit squares without gaps or overlaps are necessary for accurate area measurement.
• Differentiate between the concepts of area and perimeter for a given shape.

Before You Start

Introduction to Length and Measurement

Why: Students need to understand the concept of linear measurement and units of length before they can grasp the concept of measuring area using square units.

Identifying Basic 2D Shapes

Why: Familiarity with shapes like squares and rectangles is essential for understanding how to measure their area.

Key Vocabulary

Area	The amount of two-dimensional space a flat shape covers. It is measured in square units.
Unit Square	A square with sides of length one unit. It is used to measure area.
Square Unit	A unit of measurement for area, such as a square inch or a square centimeter. It represents the area of one unit square.
Tiling	Covering a surface or plane figure completely with unit squares without any gaps or overlaps.

Active Learning Ideas

Inquiry Circle: Tiling the Territory

Give groups various 'irregular' shapes drawn on large grid paper. Students must use physical square tiles to cover the shape perfectly and then count the tiles to determine the area, ensuring no gaps are left.

⏱ 30 min·Small Groups

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Simulation Game: The Area Architects

Students are 'hired' to design a floor plan for a small house using a specific number of square units. They must work in pairs to arrange their 'rooms' (rectangles) on a grid and calculate the total area of the house.

⏱ 40 min·Pairs

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Think-Pair-Share: Gap or Overlap?

Show two examples of 'bad' area measurement, one with gaps between tiles and one with overlapping tiles. Students discuss with a partner why these methods give an incorrect area and how to fix them.

⏱ 15 min·Pairs

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Real-World Connections

Carpenters and flooring installers calculate the area of rooms to determine how much carpet, tile, or wood flooring is needed. They measure in square feet or square meters to ensure they purchase the correct amount of material for a project.

Graphic designers and artists use the concept of area when arranging elements on a page or canvas. They consider the space each element occupies to create a balanced and visually appealing composition.

Watch Out for These Misconceptions

Common Misconception

Students often confuse area (inside space) with perimeter (outside boundary).

What to Teach Instead

Use a piece of string for perimeter and square tiles for area. Having students physically 'fence' a shape with string and then 'fill' it with tiles in a small group setting makes the distinction clear.

Common Misconception

Students may count the grid lines instead of the square spaces when finding area.

What to Teach Instead

Encourage students to place a physical counter or a 'dot' inside each square as they count. Peer-checking each other's 'tiled' shapes helps catch this error early.

Assessment Ideas

Exit Ticket

Provide students with a drawing of a rectangle made of 1-inch grid lines. Ask them to write the area of the rectangle in square inches. Then, ask them to explain in one sentence why it's important that the squares are all the same size.

Quick Check

Show students two irregular shapes made of unit squares, one larger than the other. Ask: 'Which shape has a larger area? How do you know?' Observe student responses to gauge their understanding of area as coverage.

Discussion Prompt

Present a large rectangle divided into two smaller rectangles. Ask: 'How can we find the total area of the large rectangle? Can we find the area of each small rectangle first and then add them? Why or why not?' Facilitate a discussion about decomposing shapes.

Frequently Asked Questions

What is a 'unit square'?

A unit square is a square with side lengths of 1. It is the standard building block used to measure area. In 3rd grade, we use square inches, square centimeters, or non-standard units like 'tile units.'

How do you find the area of an irregular shape?

You can find the area by breaking the irregular shape into smaller rectangles, finding the area of each rectangle, and then adding those areas together.

How can active learning help students understand area?

Active learning through 'Tiling the Territory' or 'Area Architects' allows students to see area as a physical covering. When they have to physically fit tiles into a space, they internalize the rule about 'no gaps or overlaps.' This tactile experience is much more effective than just counting squares on a worksheet, as it builds a spatial understanding of how area is constructed.

Why is area taught before the formula L x W?

Counting squares builds the conceptual understanding of what area actually is. If students jump straight to the formula, they often confuse it with perimeter or don't understand why they are multiplying.