Equivalent Volume Expressions
Equivalent Volume Expressions is a Grade 5 math skill from Illustrative Mathematics Chapter 1 (Finding Volume) that demonstrates how the same rectangular prism can be viewed from different orientations, each producing a different multiplication expression but always the same volume. Whether computing (l × w) × h or l × (w × h), the associative property ensures the total cubic units never change.
Key Concepts
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 $l$, width $w$, and height $h$, the volume $V$ is always the same. $$(l \times w) \times h = l \times (w \times h) = V$$.
Common Questions
Why can you calculate volume in different ways and get the same answer?
Because multiplication is associative: (l × w) × h = l × (w × h). Which two dimensions you multiply first doesn't matter — the total product (volume) is always the same. Each way corresponds to a different orientation of the prism.
What are the equivalent volume expressions for a prism with dimensions 3, 4, and 5?
(3×4)×5 = 12×5 = 60, 3×(4×5) = 3×20 = 60, and (3×5)×4 = 15×4 = 60. All three expressions give 60 cubic units, regardless of which face you treat as the base.
What chapter covers equivalent volume expressions in Illustrative Mathematics Grade 5?
Equivalent volume expressions is covered in Chapter 1 of Illustrative Mathematics Grade 5, titled Finding Volume.
What mathematical property makes equivalent volume expressions possible?
The associative property of multiplication allows changing the grouping of factors without changing the product. For volume: (l × w) × h = l × (w × h) = (l × h) × w, all giving the same cubic volume.
How does viewing a prism from different angles connect to equivalent expressions?
Each orientation defines a different face as the base. Viewing horizontal layers gives (l × w) × h. Viewing from the side gives l × (w × h). Each perspective gives a different expression, but since the prism hasn't changed, all products are equal.