Learn on PengiIllustrative Mathematics, Grade 5Chapter 1: Finding Volume

Lesson 3: Side Lengths and Volume Expressions

In this Grade 5 Illustrative Mathematics lesson, students learn to describe rectangular prisms using the terms length, width, and height, and calculate volume by multiplying these three side lengths together. They connect this formula to the earlier method of multiplying the area of the base by the height, reinforcing that any face of a prism can serve as the base. The lesson builds toward standard 5.MD.C.5.b as students move from counting unit cubes to writing and applying volume expressions.

Section 1

Area of a Prism's Base

Property

The number of unit cubes in a single layer of a rectangular prism is the product of the layer's length and width. This quantity is also known as the area of the base.

Cubes per layer=length×width \text{Cubes per layer} = \text{length} \times \text{width}

Section 2

Base Area Times Height

Property

The volume (VV) of a rectangular prism is the area of its base (BB) multiplied by its height (hh). Since the base is a rectangle, its area is found by multiplying its length (ll) and width (ww).

V=BhV = B \cdot h
V=(lw)hV = (l \cdot w) \cdot h

Section 3

Equivalent Volume Expressions

Property

The volume of a rectangular prism can be expressed in different ways depending on which face is considered the base. For a prism with length ll, width ww, and height hh, the volume VV is always the same.

(l×w)×h=l×(w×h)=V(l \times w) \times h = l \times (w \times h) = V

Examples

Book overview

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Chapter 1: Finding Volume

  1. Lesson 1

    Lesson 1: Introduction to Volume and Measurement

  2. Lesson 2

    Lesson 2: Calculate Volume Using Layers

  3. Lesson 3Current

    Lesson 3: Side Lengths and Volume Expressions

  4. Lesson 4

    Lesson 4: Cubic Units of Measure

  5. Lesson 5

    Lesson 5: Composite Figures: Volume of Combined Prisms

  6. Lesson 6

    Lesson 6: Apply Volume Knowledge

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Area of a Prism's Base

Property

The number of unit cubes in a single layer of a rectangular prism is the product of the layer's length and width. This quantity is also known as the area of the base.

Cubes per layer=length×width \text{Cubes per layer} = \text{length} \times \text{width}

Section 2

Base Area Times Height

Property

The volume (VV) of a rectangular prism is the area of its base (BB) multiplied by its height (hh). Since the base is a rectangle, its area is found by multiplying its length (ll) and width (ww).

V=BhV = B \cdot h
V=(lw)hV = (l \cdot w) \cdot h

Section 3

Equivalent Volume Expressions

Property

The volume of a rectangular prism can be expressed in different ways depending on which face is considered the base. For a prism with length ll, width ww, and height hh, the volume VV is always the same.

(l×w)×h=l×(w×h)=V(l \times w) \times h = l \times (w \times h) = V

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 1: Finding Volume

  1. Lesson 1

    Lesson 1: Introduction to Volume and Measurement

  2. Lesson 2

    Lesson 2: Calculate Volume Using Layers

  3. Lesson 3Current

    Lesson 3: Side Lengths and Volume Expressions

  4. Lesson 4

    Lesson 4: Cubic Units of Measure

  5. Lesson 5

    Lesson 5: Composite Figures: Volume of Combined Prisms

  6. Lesson 6

    Lesson 6: Apply Volume Knowledge