Partial Product Algorithm for Multiplication


This website is designed for students learning how to multiply multi-digit numbers together. It will be helpful in learning an additional algorithm for multiplication called thepartial product algorithm for multiplication. Students will learn the step-by-step process for multiplying by using partial products. The product would be exactly the same if you used the traditional algorithm; the student will simply learn another way to multiply multi-digit numbers together.

Latest update to this document: 10 November 2004




Partial Product Multiplication
The first step in partial product multiplcation is to line up the numbers you are multiplying vertically. (It also helps to have the number with the fewest digits on the bottom.)
Multiply the two digits in the one's column. In this example you would multiply the 2 in 72 by the 4 in 164. 2 X 4 = 8; write an 8 in the one's column, just as you would in the traditional multiplication algorithm.
In the traditional algorithm for multiplication, you would now multiply 2 X 6 = 12 and put the two in the ten's place and carry the one. In the partial products algorithm, you still multiply 2 X 6, but you need to look at the digit in the ten's place in 164 as 6 tens (60). So using partial product multiplication you multiply 2 X 60 = 120; write 120 beneath the 8 and align the numbers so you can add them together later.
Keeping in mind the last step you just performed, look at the digit in the hundred's place. In this example 164 has a 1 in the hundred's place, so 1 hundred (100). Using partial product multiplication, multiply 2 X 100 = 200; write 200 beneath the 8 and 120 and remember to align the numbers to add them together at a later step.
You have now multiplied the 2 in 72 by all the digits in 164, and just as like traditional algorithm, you now need to multiply the digits by the 7. The key to partial product multiplication is to remember to look at the 7 in 72 as 7 tens (70), so every time you are multiplying using the 7 you are really multiplying by 70. In this example you multiply 70 by the digit in the one's place of the other number; 70 X 4 = 280. Write 280 the in the same way you wrote the other numbers keeping in mind that you will need to add them together later.
Remember that you are multiplying by 70 and not 7, and look at the digit in the ten's place of 164. There are 6 tens (60), so 70 X 60 = 4,200. Write 4,200 under all of the other numbers you have written (in the same way so you can add them together afterwards).
Now you need to multiply 70 by the digit in the hundreds place of the other number. Remember to look at the digit in the hundred's place and use it as that many hundreds (1 hundred in this example). 70 X 100 = 7,000; write 7,000 under the 4,200 so you can find the sum of all the numbers you have just written.
In this step of using partial products to find the answer to a multi-digit multiplcation problem, make sure your numbers are aligned to add them up without and confusion. You should not have any trouble with finding the sum if you were careful in placing the numbers so they could be added together in the first place. Make sure all the digits in the one's, ten's, hundred's and thousand's columns line up. Add the + and = sign, and you are ready for the next step.
This is now a problem of adding multi-digit numbers together. Add the digits in the one's column, and write the answer below (in this example write 8 in the one's place of the answer).
Now add the digits in the ten's column. In this example the answer is 10, so you put the 0 in the ten's place of the answer and will carry the 1 to the digits in the hundred's place.
Add all the digits in hundred's column, and write the sum in the hundred's place of the answer. In this example the sum is 8, so write an 8 in the hundred's place of the answer.
Add the digits in the thousand's column, and write the sum in the thousand's place of the answer. In this example the sum is 11, so write an 11 in thousand's place of the answer (actually you will put a 1 in the thousand's place and a 1 in the ten thousand's place).


You have now completed a problem using partial product multiplcation. You used the same concepts as the traditional algorithm for multiplying multi-digit numbers together. By using the partial products algorithm, you actually get more of an understanding as to why you get the answer you do when using the traditional algorithm. As long as you have have a decent understanding of place value, using partial product multiplcation should come fairly easy (with practice of course).


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J. J. Kelto: jebowerm@nmu.edu