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### Course: Arithmetic (all content)>Unit 3

Lesson 6: Multiplication: place value and area models

# Why lattice multiplication works

Sal explains why lattice multiplication works. Created by Sal Khan.

## Want to join the conversation?

• Why did they name this "lattice" multiplication?
• Because the box looks like an old-fashioned fence, also called a lattice.
• Does lattice multiplication work with negative numbers?
• Yes, I believe it would. When doing the lattice problem I would disregard any negative signs. When you arrive at your lattice "answer" I would then follow the rules around multiplying negative numbers.

If only one of the numbers that you are multiplying is negative, then the answer will be a negative number. If both numbers that you are multiplying are negative, then the answer will be a positive amount since the negative signs cancel out.
• I just noticed that the sum of the number of the diagonals are always the rows you have plus the columns. So extending this pattern I get to the following, since I have 9999 x 99 witch is 4 digits + 2 digits, the result will always be less than 1.000.000.
Another example is say I have 999999 x 999 witch is 6 + 3 digits = 9 that denotes that the result will always be less than 1.000.000.000, 9 zeros and 1.
• Good observation! Yes, the sum of the total number of digits being multiplied together is always going to be the maximum number of digits in the answer.

Why do I say maximum and not exactly? Let's look at some examples:
3x2=6
3 (one digit) multiplied by 2 (one digit) equals 6 (one digit).
9x9=81
9 (one digit) multiplied by 9 (one digit) equals 81 (two digits).
10x9=90
10 (two digits) multiplied by 9 (one digit) equals 90 (two digits)
99x9=891
99 (two digits) multiplied by 9 (one digit) equals 891 (three digits)
84x11=924
84 (two digits) multiplied by 11 (two digits) equals 924 (three digits)
99x99=9801
99 (two digits) multiplied by 99 (two digits) equals 9801 (four digits)

If you continue like this, you will see that the number of digits in your answer will always be somewhere between the number of digits in the larger number (the minimum number of digits you will end up with) and the total number of digits being multiplied (the maximum number of digits you will end up with).
• This is cool I never have learned this in high school or college. my question is When did this type of multiplication came out or was discovered?
• Actually, it looks as if the origin of lattice multiplication dates back further, and to outside Europe.
"Lattice multiplication, also known as sieve multiplication or the jalousia (gelosia)
method, dates back to 10th century India and was introduced into Europe by Fibonacci in the 14th century (Carroll & Porter, 1998)."
SOURCE: An Introduction to Various Multiplication Strategies by Lynn West, University of Nebraska-Lincoln, July, 2011
CITING THIS REFERENCE:
Carroll, W. M., & Porter, D. (1998). Alternative algorithms for whole-number operations. In The
National Council of Teachers of Mathematics, The teaching and learning of algorithms
in school mathematics (pp. 106-114). Reston, VA: The National Council of Teachers of
Mathematics, Inc.
• Can you use lattice with negative numbers?
• Of course. Just do it the regular way, and then decide whether it should be positive or negative.
• can you do this with division?
• No, the lattice method only works for multiplication.
• Does this work with decimals or percents
• Yes, it works with decimals. Ignore the decimals during the initial multiplication. Once multiplication is completed, go back and add up the number decimal places, then move the decimal in the answer this number to the left (similar to this: https://www.youtube.com/watch?feature=player_embedded&v=JEHejQphIYc#t=100s).
• so which is better the traditional method or the lattice method
• It depends. Lattice is harder to set up but is easier to do. I like traditional, though.
• Hey everyone, this does work for decimal numbers, and you don't even have to ignore the decimal point!

Write the numbers and draw the lattice as you usually would, but put the decimal point of the numbers being multiplied right on the row and column lines that separate the place value columns.

Do everything like you normally would, but once you're done, look at the row and column lines "coming out of" the decimal points and go to where they intersect on the lattice. Follow the diagonal line coming out of that intersection point down, and that is where your decimal point is gonna go.

I hope that made sense, sorry, it's a little difficult to explain with just text.