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

Area of Rectangles by Tiling

Students calculate the area of rectangles by tiling with unit squares and relating it to multiplication.

• Ontario Curriculum Expectations
• 3.MD.C.6
• 3.MD.C.7.A

About This Topic

Area of rectangles by tiling teaches students to measure surface space by covering shapes with unit squares. They place tiles edge-to-edge inside rectangles without gaps or overlaps, then count the tiles for the area. This process shows that area equals the number of unit squares, directly linking to multiplication as rows along the length times columns along the width.

This topic fits within Ontario's Grade 3 geometry and spatial systems unit, reinforcing multiplication facts from earlier terms. Students explore how side lengths determine area, compare rectangles with the same area but different dimensions, and derive the formula A = length x width. Real-life connections include calculating floor space or fabric needs.

Active learning benefits this topic greatly because students handle physical or virtual tiles to construct and measure rectangles. They test ideas like building rectangles for a given area, discuss findings in pairs, and verify through peer review. These experiences build spatial reasoning, correct misconceptions on the spot, and solidify the multiplication relationship through concrete manipulation.

Key Questions

1. Explain how tiling a rectangle with unit squares helps us understand its area.
2. Analyze the relationship between the side lengths of a rectangle and its area.
3. Construct a formula for finding the area of any rectangle.

Learning Objectives

• Calculate the area of a rectangle by counting unit squares arranged in rows and columns.
• Compare the areas of different rectangles by tiling them with unit squares.
• Explain the relationship between the number of unit squares along the length and width of a rectangle and its total area.
• Construct a formula for the area of a rectangle using the measurements of its sides.
• Identify rectangles with equal areas but different dimensions through tiling and calculation.

Before You Start

Introduction to Multiplication

Why: Students need to understand the concept of multiplication as repeated addition or as arrays to grasp how it relates to rows and columns of unit squares.

Identifying Rectangles

Why: Students must be able to recognize the properties of a rectangle (four sides, four right angles, opposite sides equal) to apply tiling and area concepts accurately.

Key Vocabulary

Unit Square	A square with sides that are one unit long, used to measure area. It has an area of 1 square unit.
Tiling	Covering a surface or shape completely with unit squares without any gaps or overlaps. This process is used to measure area.
Area	The amount of two-dimensional space a shape covers. It is measured in square units.
Square Unit	A standard unit for measuring area, such as a square centimeter or a square inch. It represents the area of a unit square.

Watch Out for These Misconceptions

Common Misconception

Area measures the distance around the rectangle, like perimeter.

What to Teach Instead

Tiling focuses on covering the inside space with unit squares, distinct from edge length. Hands-on activities where students tile and trace perimeters side-by-side help them see the difference through direct comparison and group discussion.

Common Misconception

Tiles can overlap or leave small gaps when finding area.

What to Teach Instead

Complete coverage without gaps or overlaps defines accurate tiling. Station rotations enforce rules with peer checks, allowing students to revise and discuss why precise placement matches multiplication results.

Common Misconception

Only square shapes have area; rectangles do not follow the same rules.

What to Teach Instead

All rectangles tile with unit squares based on side lengths. Building varied rectangles in pairs reveals the consistent length x width pattern, building confidence through shared constructions and explanations.

Active Learning Ideas

Stations Rotation: Tiling Challenges

Prepare four stations with grid paper rectangles of varying sizes and square tiles. Students tile each rectangle, record the area, and calculate length x width to check. Groups rotate every 10 minutes, then share one insight as a class.

45 min·Small Groups

Pairs Build: Target Area Rectangles

Give pairs a target area number and square tiles. They build as many different rectangles as possible that tile to that area, measure side lengths, and list multiplication sentences. Pairs swap builds to verify areas.

30 min·Pairs

Whole Class: Rectangle Design Contest

Students design a rectangle on grid paper for a playground or garden with a given area. They tile to confirm, label dimensions and multiplication fact, then vote on the most creative design that matches.

35 min·Whole Class

Individual: Virtual Tiling Exploration

Using online grid tools or apps, students create rectangles, tile digitally, and record areas with side lengths. They experiment with changing one side and predict area changes before tiling.

25 min·Individual

Real-World Connections

• Interior designers use the concept of area to determine how much carpet or tile is needed for a room floor. They measure the length and width of the space to calculate the total square footage required.
• Construction workers calculate the area of walls to estimate the amount of paint or wallpaper needed for a building project. This ensures they purchase the correct quantity of materials.

Assessment Ideas

Quick Check

Provide students with grid paper and ask them to draw a rectangle with an area of 12 square units. Then, ask them to write the dimensions (length and width) of their rectangle and explain how they know the area is 12.

Discussion Prompt

Present students with two different rectangles, each with an area of 16 square units but different dimensions (e.g., 4x4 and 2x8). Ask: 'How can two rectangles have the same area but look different? Use your unit squares to show your thinking.'

Exit Ticket

Give each student a card showing a rectangle tiled with unit squares. Ask them to: 1. Count the unit squares to find the area. 2. Write the multiplication sentence that represents the area (length x width = area). 3. Write one sentence explaining why this multiplication works.

Frequently Asked Questions

How do you introduce area of rectangles by tiling in grade 3?

Start with familiar objects like desks or books, have students estimate then tile with unit squares to find actual area. Guide them to notice rows and columns, leading to multiplication. Use grid paper for precision and connect to daily spaces like classroom rugs for relevance. Follow with guided practice on larger rectangles.

What is the link between tiling rectangles and multiplication?

Tiling shows area as the total unit squares, grouped into rows matching the length and columns matching the width. For a 4 by 3 rectangle, four rows of three tiles each equal 12, or 4 x 3 = 12. Students see multiplication as repeated addition in a visual array, strengthening both skills.

How can active learning help students understand area by tiling?

Active approaches like manipulating tiles to build rectangles make abstract area concrete and multisensory. Students discover patterns through trial, collaborate to verify measurements, and explain reasoning to peers. This reduces errors from misconceptions, boosts engagement, and deepens the side lengths to multiplication connection over passive worksheets.

What real-world examples work for rectangle area tiling?

Use scenarios like tiling a classroom floor with square tiles, planning a garden plot, or covering a table with square napkins. Students measure actual spaces with grid paper proxies, tile models, and compute areas. These contexts show practical value, encourage problem-solving, and link math to design decisions in homes or communities.