Multiplying with Partial Products
Multiplying with partial products breaks a two-digit multiplication into two simpler multiplications—one by the ones digit and one by the tens digit—then adds the results. In Grade 6 Saxon Math Course 1 (Chapter 1: Number, Operations, and Algebra), students compute 36 × 15 by first finding 36 × 5 = 180 (the ones partial product), then 36 × 10 = 360 (the tens partial product), and adding 180 + 360 = 540. The tens partial product is always a multiple of 10. This method illuminates the distributive property: 36 × 15 = 36 × (10 + 5) = 360 + 180 = 540, and serves as a conceptual bridge to the standard multiplication algorithm.
Key Concepts
Property When multiplying a number by a two digit number, you can multiply by the ones digit first, then by the tens digit, and add the two partial products together to get the final product. $$ \begin{array}{rl} 36 & \\ \times 15 & \\ \hline 180 & \text{partial product } (36 \times 5) \\ 360 & \text{partial product } (36 \times 10) \\ \hline 540 & \text{product } (15 \times 36) \end{array} $$.
Examples To find $42 \times 23$, calculate the partial products: $(42 \times 3) + (42 \times 20) = 126 + 840 = 966$.
Calculating $205 \times 31$ becomes $(205 \times 1) + (205 \times 30)$, which is $205 + 6150 = 6355$.
Common Questions
What is the partial products method for multiplication?
Multiply the first number by the ones digit of the second, then by the tens digit (which is a multiple of 10), then add the two partial products.
Use partial products to compute 36 × 15.
36 × 5 = 180 (ones partial product). 36 × 10 = 360 (tens partial product). 180 + 360 = 540.
Use partial products to compute 47 × 23.
47 × 3 = 141. 47 × 20 = 940. 141 + 940 = 1,081.
How does partial products connect to the distributive property?
36 × 15 = 36 × (10 + 5) = (36 × 10) + (36 × 5) = 360 + 180 = 540. Each partial product corresponds to one term in the distribution.
Why is the tens partial product always a multiple of 10?
Because you are multiplying by the tens digit, which represents 10, 20, 30, etc. Multiplying any number by a multiple of 10 results in a number ending in zero.