#partial products

How many meatballs do you see?

How many do you see now?

See how arrays are so powerful? An array is a systematic arrangement of objects. This model is as important to multiplication and division as the number line model is to addition and subtraction. The visual representation of rows and columns helps students as they develop their proportional reasoning. Like the part/whole box for addition and subtraction, the array identifies the parts (factors) and the whole (total area of product) and can be used to demonstrate and prove student strategies. (Number Talks, Helping Children Build Mental Math and Computation Strategies) As student move on beyond the basic multiplication and division facts, the array can model and prove partial products which is also called the distributive property. Students build arrays with color tiles, base-ten materials, or on grid paper. Look at the array below. How many different ways can you find the answer to 8 x 25?

Here are just a few different ways that can be proved by cutting or folding the array. Halving and Doubling: (4 x 25) x 2 Four 25s equals 100 (think money) and double it you get 200.

Partial Products: (8 x 20) + (8 x 5) Eight times 20 is 160 and 8 times 5 is 40. 160 + 40 = 200

Using Multiples of 10: (8 x 10) + (8 x 10) + (8 x 5) 80 + 80 + 40 equals 200

Here is an example of how base-ten materials can be used to build a 24 x 36 array. How would you count this?