Rectangle Multiplication
This virtual manipulative allows the user to visualize the product of two factors in three different ways: click on any of the options below the array to choose Grouping, Common, or Lattice.
Grouping. The array shows the product of two numbers, with height equal to one factor and the width equal to the other factor. The colored array may be considered as groups in two ways (horizontal or vertical), as displayed in the text at the right, and the result of the multiplication is also shown. To change either of the factors, use the sliders at the left and below the array.
Common. For the product of twodigit numbers, the Common Algorithm shows the result of multiplying by the units digit and by the tens digit. The array is colorcoded so that multiplying by the units digit is shown in red and the corresponding portion of the rectangular array appears in red. Similarly, multiplying by the tens digit is shown in blue text and in the blue portion of the colored array. The final product is the sum of the red and blue portions, and the total is shown in black text. You may change either of the factors by using the sliders.
Lattice. The Lattice Algorithm uses the distributive property of twodigit multiplication. For example,
22 x 15  =  (20 + 2)(10 + 5) 
=  (20 x 10) + (20 x 5) + (2 x 10) + (2 x 5) 

=  200 + 100 + 20 + 10  
=  330 
The array is colorcoded to match each of the terms above, so that 20 x 10 is blue, 20 x 5 is red, 2 x 10 is green, and 2 x 5 is yellow.
The process of using a lattice to compute the product is shown at the right, again colorcoded. The factors are written at the top and right of the lattice, and the product of each pair of digits in the distributive process is written in the corresponding block. Then each diagonal (upperright to lowerleft) is summed, with carries as needed.
Experiment with some examples by hand and then test yourself by adjusting the sliders to match your examples and to check your use of the Lattice Algorithm for your products.