Learn on PengiIllustrative Mathematics, Grade 5Chapter 2: Fractions as Quotients and Fraction Multiplication

Lesson 9: Area Situations

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students apply fraction multiplication to solve area situations involving rectangles with fractional side lengths. Students practice finding the area of a rectangle by multiplying a fraction by a fraction or a whole number by a fraction, connecting the area formula to their understanding of fractions as quotients. The lesson builds fluency with fraction multiplication in real-world geometric contexts.

Section 1

Measuring Area with Square Units

Property

The area of a shape is the total number of non-overlapping unit squares that cover it completely.
This measurement is expressed in square units.

Examples

Section 2

Unit Fractions

Property

The fractional unit is the name for the equal parts a whole is divided into (e.g., thirds, fourths).
A unit fraction is one of these equal parts, written as 1d\frac{1}{d}, where dd is the total number of equal parts.

Examples

Section 3

Area with a Unit Fraction Side

Property

To find the area of a rectangle with a whole number side length ww and a unit fraction side length 1b\frac{1}{b}, you can model it by partitioning a rectangle of area ww into bb equal parts. The area of one of those parts represents the area of the rectangle.

A=w×1bA = w \times \frac{1}{b}

Section 4

Area from Fractional Parts

Property

The total area of a shaded region is the number of identical fractional parts multiplied by the area of one part. If there are nn parts, each with an area of 1d\frac{1}{d} square units, the total area is A=n×1d=ndA = n \times \frac{1}{d} = \frac{n}{d} square units.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Measuring Area with Square Units

Property

The area of a shape is the total number of non-overlapping unit squares that cover it completely.
This measurement is expressed in square units.

Examples

Section 2

Unit Fractions

Property

The fractional unit is the name for the equal parts a whole is divided into (e.g., thirds, fourths).
A unit fraction is one of these equal parts, written as 1d\frac{1}{d}, where dd is the total number of equal parts.

Examples

Section 3

Area with a Unit Fraction Side

Property

To find the area of a rectangle with a whole number side length ww and a unit fraction side length 1b\frac{1}{b}, you can model it by partitioning a rectangle of area ww into bb equal parts. The area of one of those parts represents the area of the rectangle.

A=w×1bA = w \times \frac{1}{b}

Section 4

Area from Fractional Parts

Property

The total area of a shaded region is the number of identical fractional parts multiplied by the area of one part. If there are nn parts, each with an area of 1d\frac{1}{d} square units, the total area is A=n×1d=ndA = n \times \frac{1}{d} = \frac{n}{d} square units.

Examples