Grouping Partial Products from an Area Model
The four partial products from a 2×2 area model can be regrouped into two partial products by combining those associated with the ones digit and those associated with the tens digit of one factor, as taught in Grade 4 Eureka Math. For 26 × 35, the four products are 6×30, 6×5, 20×30, and 20×5. Regrouping: (6×30 + 6×5) = 6×35 = 210, and (20×30 + 20×5) = 20×35 = 700. Total: 910. This grouping bridges the full area model and the standard two-row multiplication algorithm.
Key Concepts
The four partial products from an area model can be grouped to form two partial products. This grouping combines the products related to the ones digit and the products related to the tens digit of one of the factors. For a problem like $26 \times 35$: $$ (6 \times 30 + 6 \times 5) + (20 \times 30 + 20 \times 5) = (6 \times 35) + (20 \times 35) $$.
Common Questions
How do you group partial products from a 4-section area model into 2 partial products?
Combine the two sections related to the ones digit of one factor, and the two sections related to the tens digit. Each combined group equals that digit times the entire other factor.
Why group four partial products into two?
Two partial products correspond directly to the two rows in the standard multiplication algorithm (ones row and tens row), making the transition from area model to algorithm clearer.
What are the two partial products for 26 × 35?
Group by digits of 26: (6 × 35) = 210 (ones row) and (20 × 35) = 700 (tens row). Total: 210 + 700 = 910.
How does grouping partial products connect to the standard algorithm?
In the standard algorithm, the first row (multiplying by ones digit) equals the ones partial product, and the second row (multiplying by tens digit, shifted left) equals the tens partial product.
What is the distributive property’s role here?
Grouping works because of the distributive property: (a+b)×c = a×c + b×c. The area model makes this visual; grouping shows how partial products can be combined in different ways.