Distributive Law

The purpose of the worksheet is to use geometry for illustrating the concept of the distributive law that relates to the operations of multiplication and addition. With parameters a, b, and c, building a rectangle ABCD and two smaller rectangles G and H have the same width of a, and their lengths are (b + c), b, and c in turn. In addition, the widths and the lengths of three rectangles are, in turn, parallel each other. Point G is on a line segment AE, and H is on CF. When changing parameters, the sizes of the sides of the rectangles change; thus leading to the increase or decrease of their areas. The area of the rectangle ABCD can be computed in two different ways, as the product of the width and the length a(b+c), or as the sum of the areas ab and ac of the two rectangles G and H. The two different ways implies the distributive law for products of sums a(b+c) = ab+ac. The student expects to see what will happen for various values of t. When changing values of t, the rectangles G and H change their positions. • When t = 0, the rectangles G and H overlap and fit the rectangle ABCD, and these two rectangles lie next to each other in the rectangle ABCD. • When t = 1, the rectangle G has the point G overlapping the point E of the segment AE. At the same time, the rectangle H has the point H overlapping the point F of the segment DF. • When t between 0 and 1, the rectangle G moves between the rectangle ABCD and the point E, parallel to the rectangle ABCD, and has the point G moving on the segment AE. Similarly, the rectangle H moves between the rectangle ABCD and the point F, parallel to the rectangle ABCD, and has the point H moving on the segment DF.