Models and Properties of Products

Multiplication of Whole Numbers

For any whole numbers r and s, the product or r and s is the sum with s occurring r times. This is written as

                                    r x s = s + s + s + . . . + s

                                                     r times

If r ¹ 0 and s ¹ 0, r and s are called factors.

Models for Multiplication

Repeated Addition

Three groups of 6 to illustrate 6 + 6 + 6.

Rectangular Array

Shows the figures pushed together to form a 3 x 6 rectangle.

Tree Diagram
A tree diagram is useful for certain types of multiplication. Here is a learning about making tree diagrams video: Tree Diagram Video

Partial Products
When a two-digit number is multiplied by a two-digit number, there are four partial products.
Example:
                                17
                             x 13
           Partial          21   (3 x 7)
         products         30   (3 x 10)                             
                                70   (7 x 10)
                              100   (10 x 10)
                              221

                                          Number Properties

Closure Property of Multiplication

For any whole numbers a and b, a x b is a unique whole number  (the product of any two whole numbers is also a whole number).

               

                      (whole #) (whole #) = whole #

                      (even #) (even #) = even #

                      (odd #) (odd #) = odd #

Identity Property for Multiplication

For any whole number b, 1 x b = b x 1 = b, and 1 is a unique identity for multiplication (when number 1 is multiplied by another number, it leaves the identity of the number unchanged).
Example:

              1 x 7 = 7                   14 x 1 = 14                  1 x 0 = 0

Commutative Property for Multiplication

For any whole numbers a and b, a x b = b x a (any product of two numbers may be interchanged (commuted) without affecting the product.

Example:

                347 x 26 = 26 x 347

Associative Property for Multiplication

For any whole numbers a, b, and c, a x (b x c) = (a x b) x c (in any product of  three numbers, the middle number may be associated with and multiplied by either of the two end numbers.

Example:

                   2 x (3 x 10) = (2 x 3) x 10 = 60

Distributive Property for Multiplication over Addition

For any whole numbers a, b, and c, a x (b + c) = a x b + a + c.

Example:
                 7 x 8 = 7 x (7 + 1) = 49 + 7 = 56
                                Distributive property

Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers.