Obtaining answers to such multiplication questions as 2 x 17, 3 x 200, and even 4 x 592 can be accomplished relatively easily and quickly by using a "repeated addition" algorithm. For example, for 3 x 200, you can do: 200 + 200 + 200 = 600 However, once numbers get large or involve multiplication by 2-digit or greater multipliers, using 'repeated addition' to multiply is cumbersome. Ancient cultures (e.g. Egyptian, Mayan) realized this and figured out a way to do multiplication without using repeated addition. The method involved a mathematical principle now referred to as the distributive principle. Consider 5 x 12. Using the array (rows and columns) model for the 'groups of' meaning of multiplication. A 5 x 12 array has 5 rows with 12 in each row: You can cut up the 5 x 12 array in many ways. Here is one way. Two smaller arrays. The left one is a 5 x 8 and the right one is a 5 x 4. We can obtain an answer to 5 x 12 by doing: 5 x 8 + 5 x 4. Thus 5 x 12 = 40 + 20 = 60. When we cut up the array in the way shown above, the column count (12) was split into two parts. We can split the row count (5) as well. This results in a 2 x 12 array and a 3 x 12 array. This time the answer to 5 x 12 can be worked out by doing: 5 x 12 = 2 x 12 + 3 x 12 = 24 + 36 = 60 We can also cut up both the row count and the column count. This results in the four smaller arrays shown on the right. For this way of cutting up the 5 x 12 array, the answer to 5 x 12 can be worked out by doing: 5 x 12 = 3 x 5 + 3 x 7 + 2 x 5 + 2 x 7 5 x 12 = 15 + 21 + 10 + 14 5 x 12 = 60 All of the previous have been examples of the distributive principle. If we revisit the three previous ways of cutting up the 5 x 12 array, using brackets, the work would look as follows: 5 x 12 = 5 x (8 + 4) = 5 x 8 + 5 x 4 5 x 12 = (3 + 2) x 12 = 3 x 12 + 2 x 12 5 x 12 = (3 + 2) x (5 + 7) = 3 x 5 + 3 x 7 + 2 x 5 + 2 x 7 Notice that the third way above looks like the "infamous" FOIL method. For a 3 stages example of the distributive principle in action, refer to: Grade 4 Multiplication Algorithm Try these: Make an array for 6 x 13. Split both the number of rows and the number of columns. Obtain the answer to 6 x 13 by adding the subproducts. Also, show the thinking by using brackets. Now that you have examined the 3 stages example, do the following 1-digit x 3-digit multiplications, using the distributive multiplication algorithm: 7 x 291 4 x 850