Representing Multiplication with an Array
Representing Multiplication with an Array is a Grade 4 math skill that uses the distributive property to multiply two two-digit numbers by decomposing each into tens and ones and summing four partial products. For 23 x 45, the array splits into four sections: 20 x 40 = 800, 20 x 5 = 100, 3 x 40 = 120, and 3 x 5 = 15, giving a total of 1,035. Covered in Chapter 16: Multiplication of Two-Digit by Two-Digit Numbers in Eureka Math Grade 4, this visual method makes the distributive property concrete and prepares students for the standard multiplication algorithm.
Key Concepts
To multiply two two digit numbers, we can decompose each number into tens and ones and use the distributive property. This can be visualized with an array, where the total area is the sum of four smaller areas, known as partial products. For factors $(a+b)$ and $(c+d)$: $$(a + b) \times (c + d) = (a \times c) + (a \times d) + (b \times c) + (b \times d)$$.
Common Questions
What is an array model for two-digit multiplication?
An array model represents a multiplication problem as a rectangle divided into four smaller rectangles, one for each combination of place value parts. For 23 x 45, the array shows 20 x 40, 20 x 5, 3 x 40, and 3 x 5 as four partial products that sum to the total.
How do I use the distributive property to multiply 23 x 45?
Decompose: 23 = 20 + 3 and 45 = 40 + 5. Then multiply each pair: (20 + 3) x (40 + 5) = (20 x 40) + (20 x 5) + (3 x 40) + (3 x 5) = 800 + 100 + 120 + 15 = 1,035.
What are partial products in multiplication?
Partial products are the individual results of multiplying each part of one factor by each part of the other. For two-digit multiplication using decomposition, there are four partial products. Adding all partial products gives the final product.
How does the array model connect to the standard multiplication algorithm?
The standard algorithm organizes the same four partial products vertically, stacking and adding them. Understanding the array model first helps students see why the algorithm works and why each row is shifted one place to the left when multiplying by tens.
Why do Grade 4 students learn multiplication with arrays?
Arrays make the distributive property visible, helping students understand why multiplying two-digit numbers works the way it does. This conceptual foundation prevents common algorithm errors and builds multiplicative reasoning for algebra in later grades.
What chapter uses arrays for two-digit multiplication in Eureka Math Grade 4?
Chapter 16: Multiplication of Two-Digit by Two-Digit Numbers in Eureka Math Grade 4 develops two-digit multiplication using arrays and area models as visual scaffolds before introducing the standard algorithm.