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.

Mental Calculations and Estimation of Products

Mental Calculations

Computative numbers and substitution/distributive property (numbers that are easy to calculate).

Compatible Numbers
Rearrangement of numbers.
                         1.         5 x 346 x 2 = 5 x 2 x 346 = 10 x 346 = 3460
              
                         2.         2 x 25 x 5 x 2 = 2 x 2 x 25 x 5 = 100 x 5 = 500
                                          
Substitutions

Distributive property is useful for mental calculations of products.

Example:

               14 x 102 = 14 x (100 + 2) = 1400 + 28 = 1428

                                            Distributive property

Equal Products

This method is based on the fact that the product of two numbers is unchanged when one of the numbers is divided by a given number and the other number is multiplied by the same number.

Example:

              12 x 52 =

                          12/2 x (52 x 2 ) = 

                                                   6 x 104 = 624 

Estimation of Products

Rounding

Products can be estimated by rounding one or both numbers. 
  
                         1.      71 x 58 » 70 x 60 = 4200

                               

                         2.      205 x 29 »  200 x 30 = 6000

                                          or » 210 x 30 = 6300

Compatible Numbers

Estimation of products combined with mental calculations become a useful tool when using compatible numbers. For example, to estimate 4 x 237 x 26, we might replace 26 by 25 and use a different ordering of the numbers.

                4 x 237 x 26 » 4 x 25 x 237 = 100 x 237 = 23,700

Front-End Estimation
Is related to rounding. keep the leading digit and everything else becomes a zero. For example, to estimate 62 x 83, keep the leading digits 60 x 80 so the estimated product is 4800.

Example:
               62 x 83 » 60 x 80 = 4800
Kids Olr  is a good website to collect information about math.             
Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers and from other sources.