Representing the Commutative Property with an Array
Representing the Commutative Property with an Array is a Grade 3 math skill from Eureka Math using arrays to show that a × b = b × a. A 3 × 4 array (3 rows, 4 columns) and a 4 × 3 array (4 rows, 3 columns) have the same total number of objects—12. The visual equivalence of these rotated arrays makes the Commutative Property concrete. Third graders use this to understand that switching the order of factors does not change the product, which cuts the number of multiplication facts to memorize roughly in half.
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
What is the Commutative Property of Multiplication?
The Commutative Property states that changing the order of the factors does not change the product: a × b = b × a.
How does an array demonstrate the Commutative Property?
A 3 × 4 array (3 rows, 4 columns) has 12 total objects. Rotating it gives a 4 × 3 array (4 rows, 3 columns), which also has 12 objects. Same total, different arrangement.
Why is the Commutative Property useful for learning multiplication facts?
Because a × b = b × a, learning 3 × 7 = 21 automatically means you know 7 × 3 = 21. This roughly halves the number of independent facts to memorize.
What does it mean to rotate an array?
Rotating an array means turning it 90 degrees so rows become columns and columns become rows. A 5 × 2 array becomes a 2 × 5 array, but the total count stays the same.
In which textbook is Representing the Commutative Property with an Array taught?
This skill is taught in Eureka Math, Grade 3.