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Lattice algorithm for multiplication

Lattice algorithm for multiplication

 
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Objective:

To master the lattice algorithm.

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Tutorial

Overview

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

  • We do not pay attention to the values of the numbers as we work through the algorithm-only to the digits themselves,
  • 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
  • We have to take a little bit of time to set up the lattice before we can begin multiplying.

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 demonstrates the lattice algorithm with two two-digit numbers.

Three-digit by two-digit example

This video demonstrates the lattice algorithm with larger numbers.

Lattice v. standard algorithm smackdown

So you spend a lot of time setting up the lattice, right? It must take a lot longer than the standard algorithm, right?
Let's see.
I'll multiply two 10-digit numbers. Which will be completed first? And by how much?
(Note: While the video is sped up, it still takes two minutes...feel free to fast forward to the end!)

Questions and Answers

  • Answer 1
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    Doug Freese about 3 years ago

    History of the algorithm. The wikipedia entry sounds right, got any citable sources on history?
    History from WIkipedia: Lattice multiplication also known as sieve multiplication, shabakh, or the Hindu lattice is a method of multiplication that uses a lattice to multiply two numbers. It was described by Al-Khwarizmi in the 9th century[1], and brought to Europe by Fibonacci. Derivations of this method also appear in Matrakci Nasuh's Umdet-ul Hisab[2], which led to the discovery of Napier's bones. It is algorithimically the same as regular long multiplication, but it breaks the process into smaller steps, which some students find easier to use.[3]
    The connection with Napier's bones makes this method a natural to discuss with the bones in hand.
    What is the 'same' about column and lattice is that in both the partial products move along a diagonal. With column that diagonal comes from 'adding a zero' or shifting each partial product to the left, row by row. What I notice as the difference is that with column you carry both when you multiply and when you add, with lattice it's only when you add.

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      Christopher Danielson answered about 3 years ago

      I have no wisdom to add to the Wikipedia account of the history of this algorithm. I have heard it claimed that the algorithm fell out of favor when arithmetic books went to press, as it was too hard to typeset. I have no citable sources on this claim.

      Because this algorithm is taught in one of the newer math curricula (Everyday Math), people often assume that it is brand-new. But as you document, it is very, very old indeed.

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  • Answer 1
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    Doug Freese about 3 years ago

    About 5% of my High School students have been taught this method, so I've learned it too (they taught me). I've never been too sure how decimals are dealt with, nor do my students know. They seem to be taught this lattice method before decimals and it's not followed on to higher grades maybe? ANYWAY I would throw the decimals away, multiply like integers, then count the decimal places in the original multiplicands, add them up and adjust the decimal in the final product.

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      Author
      Christopher Danielson answered about 3 years ago

      Here's something to try...Do a multiplication problem where each number has a decimal point in it. Something such as 2.34x34.2

      The decimal point along the top goes above a vertical divider in the lattice. The decimal point on the side goes next to a horizontal divider. Multiply as usual, then follow the vertical divider down to where it meets the horizontal divider. The decimal point slides down this diagonal to fall in the correct place.

      Why does this work? I think about the decimal point as marking the spot between the units and the tenths places. When you follow the horizontal and vertical dividers, you are looking for where the unitsxunits multiplication is-that's going to be the units place in the product. And each place (as you observe in the other question) has its own diagonal, so you slide down the diagonal divider to stay between the units diagonal and the tenths diagonal. Voila!

      There's a bit more on place value in the lattice algorithm on my blog:
      http://tiny.cc/oj1h6

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