# Partial products

The partial products method is a method for multiplying multi-digit numbers. It is typically used as an introductory method to multiplying numbers larger than 10. Although it is generally less efficient, using the partial products method of multiplication usually precedes traditional long multiplication because it fosters an understanding of the concept of multiplication.

The partial products method breaks the factors in a multiplication problem down into its parts based on place value, allowing students to see what exactly is being multiplied rather than just following a step-by-step process, as they would with traditional long multiplication. This helps promote number sense rather than rote memorization, tends to be easier for students to remember, and is considered more reliable since students usually make fewer mistakes with the partial products method as compared to traditional long multiplication.

To perform the partial products method of multiplication:

- Write each factor in expanded form.
- Multiply each number in the expanded form of one factor by each of the numbers in the expanded form of the other factor
- Calculate the sum of all of the products that resulted from step 2. The sum of all the products is the product of the multiplication problem.

Example

Use the partial products method to find the product of 124 and 3.

- The expanded form of 124 is 100 + 20 + 4. 3 cannot be further expanded.
- 100 × 3 = 300 ; 20 × 3 = 60 ; 4 × 3 = 12
- 300 + 60 + 12 = 372

124 | |||

× | 3 | ||

12 | (3 × 4) ← partial products | ||

60 | (3 × 20) ← partial products | ||

+ | 300 | (3 × 100) ← partial products | |

372 | (the sum of the partial products) |

The partial products method is arguably simpler to perform than traditional long multiplication since using the place value of each number frequently involves multiplying values that end in 0, which often results in compatible numbers.

The partial products method also serves as an introduction to the distributive property. A student who understands how to use the partial product method will have a good foundation for understanding and using the distributive property, which is used throughout mathematics. Particularly in algebra, which is used in virtually all areas of mathematics, an understanding of the distributive property (as well as many other properties of numbers) is necessary, and will provide a student the means to solve algebraic equations.