Finding Products with Two Partial Products
Students learn to find the product of two 2-digit numbers by breaking one factor into its ones and tens components and calculating two partial products that are then added together, as taught in Illustrative Mathematics Grade 5, Chapter 4: Wrapping Up Multiplication and Division with Multi-Digit Numbers. For example, 47 x 36 = (47 x 6) + (47 x 30) = 282 + 1,410 = 1,692.
Key Concepts
You can find the total product by calculating two partial products. First, multiply the top number by the ones digit of the bottom number. Then, multiply the top number by the tens part of the bottom number and add the results. $$N \times (10a + b) = (N \times b) + (N \times 10a)$$.
Common Questions
What is the partial products method?
The partial products method breaks one factor by place value (ones and tens), multiplies the other factor by each part, and adds the results; it applies the distributive property to organize multiplication.
How do you use two partial products?
To find 47 x 36: multiply 47 x 6 = 282 (ones partial product), then 47 x 30 = 1,410 (tens partial product), and add 282 + 1,410 = 1,692.
Why is the partial products method useful?
The partial products method organizes multiplication into manageable steps using only two calculations, is less error-prone than long multiplication for many students, and clearly shows the distributive property in action.
How is this related to the distributive property?
The partial products method is an application of the distributive property: N x (10a + b) = (N x 10a) + (N x b), breaking one factor into its component parts to simplify the calculation.
When would you use partial products versus standard algorithm?
Partial products is useful for understanding why multiplication works and for students who find the standard algorithm confusing; both give the same answer but partial products shows the mathematical structure more clearly.