Rotating Arrays and the Commutative Property
Rotating Arrays and the Commutative Property is a Grade 3 math skill from Eureka Math demonstrating a × b = b × a by physically rotating arrays. A 2 × 8 array (2 rows, 8 columns) rotated 90 degrees becomes an 8 × 2 array (8 rows, 2 columns). Both arrangements contain exactly 16 objects. The rotation makes the Commutative Property visible and tangible. Third graders use this to recognize related multiplication facts and reduce the number of new facts to learn.
Key Concepts
The commutative property of multiplication states that changing the order of the factors does not change the product. $$a \times b = b \times a$$.
Common Questions
How does rotating an array show the Commutative Property?
Rotating an array 90 degrees swaps its rows and columns. A 3 × 5 array becomes a 5 × 3 array. Both have 15 objects total, showing 3 × 5 = 5 × 3.
What happens to the total count when you rotate an array?
The total count stays the same. Rotating changes the visual arrangement—rows become columns—but does not add or remove any objects.
Why is the Commutative Property useful when learning multiplication facts?
Knowing 4 × 9 = 36 automatically tells you 9 × 4 = 36. This reduces the multiplication table to roughly half the number of unique facts to memorize.
Give an example of rotating an array.
A 6 × 4 array has 6 rows of 4 = 24 objects. Rotate it: now it is 4 × 6 with 4 rows of 6 = 24 objects. Same total, different dimensions.
In which textbook is Rotating Arrays and the Commutative Property taught?
This skill is taught in Eureka Math, Grade 3.