Form II: Arithmetic: Advanced Multiplication with Napier's Bones
I have posted about using materials such as tiles to calculate answers to multiplication problems. As problems increase in size, the use of these materials becomes impractical. Traditionally an abstract system of long multiplication the distributive process has been taught.
The lattice method came out of a calculating aid made by John Napier called "Napier's Rods."
To use them, you take out the rods that have at the top the first number you are working with.
"Research has shown, however, that the lattice system of multiplication allows students to compute multi-digit multiplication problems in significantly less time and with greater accuracy than is possible using the distributive method. "- A Comparison of Two Methods of Teaching Multidigit Multiplication [University of Tennessee, Frank George Hughes]
The lattice method came out of a calculating aid made by John Napier called "Napier's Rods."
"John Napier was a Scottish nobleman who loved mathematics. He invented logarithms, worked in spherical trigonometry and designed "Napier's rods," a mechanical calculating aid...
These were an assortment of rods marked off with numbers. When these rods were arranged correctly, they could be used for multiplication and division...They were a sort of movable multiplication table -an early type of slide rule, which is what people used before pocket calculators. Because they were made of bone or ivory strips they were sometimes called "Napier's Bones."
-Mathematicians Are People, Too, Reimer & Reimer
To use them, you take out the rods that have at the top the first number you are working with.
For example if your problem is 298 x 7, you take out the 2 rod, the 9 rod and the 8 rod and lay them down on the table. 
You can see, if you go down to the 7th row of this set (because 7 is the second number you are working with), on the top of the slashes is 165. Write that down. These are your tens. Below the slashes are the numbers 436. Since these are your ones, you will write them under the 165, but you will indent out one column, to make the number above start in the tens column. Add these two numbers together, and you will get 2086, which is the answer to 298 x 7.
Let's try another problem; 31 x 24. This system is sometimes called the "Lattice System" because when you get to working with larger numbers, it becomes easier to make a grouping of boxes. You make a graph with the number of boxes across as there is in your first numeral; in this case two. You make the number of boxes down as you have numerals in your second numeral; in this case also two. 
So you have a graph with two boxes going across and two boxes going down for this problem.
Now put diagonal lines going from one corner of each box diagonally to the other corner. I extend my diagonal to make it clear where the answers go. The drawing of the boxes may seem complicated, but it is easy once you have done it a few times and kids find it easy to do on any blank piece of paper. 
Next, write the digits of your problem along the sides of your boxes. Now, get out your rods for the first two digits of your problem; in this case the 3 rod and the 1 rod. Go down the number of boxes according to the numerals along the side of your graph; in this case, 2 first. Copy down the boxes just as they are on the rods, the numerals above and below the slashes will correspond to the boxes and slashes you have in your graph; in this case 0/6 and 0/2.
Continue this way with the next digit; in this case, 4. 
Now you have numerals all around your boxes. Ignore the numerals along the top and right sides now, and add only the numbers within the boxes on the diagonal. Starting at the bottom corner the one diagonal triangle box has the numeral 4, so I write 4 down below it. The next row of diagonal triangle boxes contain the numerals 2,0 and 2, which if you add them together equal 4, so I write 4 below them. The next diagonal row of numerals are 1,6 and 0, which equals 7, so I write 7 below it. The corner diagonal box contains 0, which I chose to just leave out since it won't affect the answer, but you could have your students write a 0 there just to be consistent and get in the habit of always writing down the numerals so as not to forget any by accident. It depends on how old they are and their understanding of the concepts.
The answer to this problem, reading the numerals from right to left is 744.
This method can be used for as large a problem as you want. Just make sure you draw the correct number of boxes according to the numerals that are in your problem. Here is the problem 123 x 12. We got out our 1, 2 and 3 rods 
and made our graph 3 boxes by 2 boxes and wrote the numerals around the top and right hand edges. We then copied what the first (1) and second (2) boxes of the bones had in them. We then added the numerals on a diagonal, getting the correct answer; 1,476.
You have to use a slightly different method of adding up the numbers if you get numbers that have carrying in them. For this problem, for example, 123 x 67, you will get the numerals from left to right, 7 1,14,11 and 7. If you get numerals over 9, you must carry them over into column addition. You write down the numeral(s) and then add the number of 0's after it that corresponds to the number of columns after it. Since 1 is in the first row, it gets no 0's after it. 14 is in the next row, and it has only 1 row after it, so you write down 14 with 1 0 behind it, or 140. The next row has 11 in it (pardon the odd looking numerals; my son made a mistake in his addition the first time and got 10, but then changed it to an 11) so you write 11 with two 0's behind it for the two columns, or 1100. The last row has a 7 in it and it has three 0's behind it for the three rows, or 7000. Add these numbers together and you will get 8,241; the correct answer. This is something that is complicated to explain but easy to use once you understand how it goes. All a child needs to be able to do is add two digit numbers.
You can instead go to Mathwire and print and cut out these Napier's Bones. They can be used on their own or cut out and glued to large craft sticks.
You can see, if you go down to the 7th row of this set (because 7 is the second number you are working with), on the top of the slashes is 165. Write that down. These are your tens. Below the slashes are the numbers 436. Since these are your ones, you will write them under the 165, but you will indent out one column, to make the number above start in the tens column. Add these two numbers together, and you will get 2086, which is the answer to 298 x 7.
Let's try another problem; 31 x 24. This system is sometimes called the "Lattice System" because when you get to working with larger numbers, it becomes easier to make a grouping of boxes. You make a graph with the number of boxes across as there is in your first numeral; in this case two. You make the number of boxes down as you have numerals in your second numeral; in this case also two. So you have a graph with two boxes going across and two boxes going down for this problem.Now put diagonal lines going from one corner of each box diagonally to the other corner. I extend my diagonal to make it clear where the answers go. The drawing of the boxes may seem complicated, but it is easy once you have done it a few times and kids find it easy to do on any blank piece of paper. Next, write the digits of your problem along the sides of your boxes. Now, get out your rods for the first two digits of your problem; in this case the 3 rod and the 1 rod. Go down the number of boxes according to the numerals along the side of your graph; in this case, 2 first. Copy down the boxes just as they are on the rods, the numerals above and below the slashes will correspond to the boxes and slashes you have in your graph; in this case 0/6 and 0/2. Continue this way with the next digit; in this case, 4. Now you have numerals all around your boxes. Ignore the numerals along the top and right sides now, and add only the numbers within the boxes on the diagonal. Starting at the bottom corner the one diagonal triangle box has the numeral 4, so I write 4 down below it. The next row of diagonal triangle boxes contain the numerals 2,0 and 2, which if you add them together equal 4, so I write 4 below them. The next diagonal row of numerals are 1,6 and 0, which equals 7, so I write 7 below it. The corner diagonal box contains 0, which I chose to just leave out since it won't affect the answer, but you could have your students write a 0 there just to be consistent and get in the habit of always writing down the numerals so as not to forget any by accident. It depends on how old they are and their understanding of the concepts.The answer to this problem, reading the numerals from right to left is 744.
This method can be used for as large a problem as you want. Just make sure you draw the correct number of boxes according to the numerals that are in your problem. Here is the problem 123 x 12. We got out our 1, 2 and 3 rods and made our graph 3 boxes by 2 boxes and wrote the numerals around the top and right hand edges. We then copied what the first (1) and second (2) boxes of the bones had in them. We then added the numerals on a diagonal, getting the correct answer; 1,476.You have to use a slightly different method of adding up the numbers if you get numbers that have carrying in them. For this problem, for example, 123 x 67, you will get the numerals from left to right, 7 1,14,11 and 7. If you get numerals over 9, you must carry them over into column addition. You write down the numeral(s) and then add the number of 0's after it that corresponds to the number of columns after it. Since 1 is in the first row, it gets no 0's after it. 14 is in the next row, and it has only 1 row after it, so you write down 14 with 1 0 behind it, or 140. The next row has 11 in it (pardon the odd looking numerals; my son made a mistake in his addition the first time and got 10, but then changed it to an 11) so you write 11 with two 0's behind it for the two columns, or 1100. The last row has a 7 in it and it has three 0's behind it for the three rows, or 7000. Add these numbers together and you will get 8,241; the correct answer. This is something that is complicated to explain but easy to use once you understand how it goes. All a child needs to be able to do is add two digit numbers.
"The lattice method produces the same kind of understanding as the distributive method but is easier to teach, faster to use, and less prone to error. " -Mathematics...A Way of Thinking, Robert Baratta-Lorton
If you would like to make some Napier Rods, just get 9 wide craft sticks (like tongue depressors) and divide them into 9 fairly even sections. I just eyed mine; I did not measure them. If you are using these with children younger than 3rd or 4th grade, I would divide them into 10 sections and use the top section to put the number of the rod on the top. My Kindergartner can use these, but sometimes has difficulty reading the top number to identify which rod he is using. I ended up putting the number of the rod on the back for him. He chooses the rods by the numbers on the back and then turns them over to use them. My severely dyslexic son has trouble sometimes counting down the blocks correctly. Perhaps using different colors for the block divisions than for the numbers would help with this. (It would also be possible to make a thin column down one side to mark the rows with numbers.) Divide each of these sections with a diagonal line and copy the numbers above. The numbers are just the traditional multiplication tables. I used an Ultra-fine point Sharpie to write with, but it did bleed into the wood a little, making the numerals fuzzy. I am not sure if there would be a better writing instrument to use for this.
"Students who have difficulty reasoning abstractly with numbers are frequently unable to grasp the numerical logic behind either the distributive or the lattice approach to long multiplication. Knowledge of why an abstract system of producing answers works is not as important as the knowledge that it does wok. For this reason, answers to the initial problems students work using a lattice method are checked against the materials, usually chips on the chip trading boards."
source:
- Mathematics; a Way of Thinking, Bob Baratta-Lorton
