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Partial products algorithm

Partial products algorithm

Author: Christopher Danielson

To master the partial products algorithm.

This packet is intended to be an introduction to the partial products multiplication algorithm. This algorithm is commonly taught in elementary schools, but many adults are unfamiliar with it.

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The partial products algorithm is used for multiplying multi-digit numbers. Some important features of the algorithm include:

  • We pay attention to the values of the numbers as we work through the algorithm (not just to the digits),
  • We do all of the multiplications first, followed by the additions (in contrast to the standard algorithm), meaning...
  • There is no carrying in the multiplication phase of the algorithm, and
  • The algorithm becomes unwieldy for large numbers of digits (more than, say 3 or 4 digits in each factor).

In this packet, I will solve two problems:

  1. A 2-digit by 2-digit problem, and
  2. A 3-digit by 2-digit problem

Two-digit by two-digit example

This video introduces the algorithm, including the basic thought processes behind it.

Three-digit by two-digit example

This video demonstrates that the number of partial products grows as the number of digits in the problem grows.

Comparing partial products to the standard algorithm

This video demonstrates relationships between the partial products algorithm and the standard algorithm for multi-digit multiplication.

So what?

So why does anybody teach the partial products algorithm? Here are several answers...

  1. When children are first learning multidigit algorithms, they are also still firming up their understanding of place value. It makes sense to have them work with algorithms that help to build their place value understanding, rather than with algorithms that ignore place value.
  2. When people are doing mental arithmetic, they actually do something much more like the partial products algorithm than like the standard algorithm. Most people find it very difficult to keep track of carried digits when computing in their heads. So the partial products algorithm more closely matches the ways people think about numbers (although it is less efficient for numbers with lots of non-zero digits).
  3. The connection between the partial products algorithm and the distributive property of multiplication over addition is more clear than with the standard algorithm, and so...
  4. The connection between the partial products algorithm and the ways people work with algebraic expressions is closer than with the standard algorithm.