adding. The algorithms can be verified by decomposing the factors according to the values of their digits (in this case, 23=20+3 and 15=10+5) and using the distributive law in one of several ways:
A more compelling justification uses the area model of multiplication. If the sides of a 23×15 rectangle are subdivided as 20+3 and 10+5, then the area of the whole rectangle can be computed by summing the areas of the four smaller rectangles.
Note the correspondence between the areas of the four smaller rectangles and the partial products in Method 3. With more careful examination, it is possible to see the same four partial products residing in the four cells in Method 6. (The 2 in the upper left cell, for example, actually represents 200.) Methods 1,