the product of a multiplication problem is empty product.
the product of a multiplication problem is empty product.
what is the product of a multiplication problem?
empty product
they showed how the matrix chain multiplication problem can be transformed (or reduced) into the problem of triangulation of a regular polygon. the polygon is oriented such that there is a horizontal bottom side, called the base, which represents the final result. the other "n" sides of the polygon, in the clockwise direction, represent the matrices. the vertices on each end of a side are the dimensions of the matrix represented by that side. with "n" matrices in the multiplication chain there are "n"−1 binary operations and "c" ways of placing parentheses, where "c" is the ("n"−1)-th catalan number. the algorithm exploits that there are also "c" possible triangulations of a polygon with "n"+1 sides.
matrix chain multiplication
the whole-part model can also be used to solve problems involving multiplication or division. a multiplication problem could be presented as follows:









































the student could solve this multiplication problem by drawing one bar to represent the unknown answer, and subdivide that bar into four equal parts, with each part representing $30. based on the drawn model, the student could then visualize this problem as providing a solution of $120.









































unlike the whole-part model, a comparison model involves comparing two bars of unequal lengths. it can be used to solve a subtraction problem such as the following:
singapore math