## Multiply Whole Numbers

Distributive Property
The Distributive Property states that when you multiply the sum of two or more addends by a factor, the product is the same as if you multiplied each addend by the factor and then added the partial products. The Distributive Property is illustrated below graphically, arithmetically, and algebraically. At this time, students do not need to know the algebraic explanation of the Distributive Property.

To find the total number of squares, you can multiply 3 × (2 + 6) or you can add (3 × 2) and (3 × 6).
arithmetically
3 × (2 + 6) = (3 × 2) + (3 × 6)
3 × 8 = 6 +18
24 = 24

algebraically
a × (b + c) = (a × b) + (a × c)

Multiplication by a One-Digit Number
One way to multiply a number by a one-digit number is to multiply the value of each digit by that one-digit number and then find the sum of the partial products. The traditional multiplication algorithm for multiplying by a one-digit number has the product written in place with necessary regrouping recorded above the number being multiplied.

Examples:

The above example shows how the Distributive Property applies to the multiplication algorithm. 546 × 7 = (500 × 7) + (40 × 7) + (6 × 7) Emphasize that the traditional algorithm starts by multiplying in the ones place and that makes recording and regrouping of the product easier. Students sometimes find it difficult to multiply numbers with internal zeros, such as 302 or 10,809, and may need extra practice with such examples.

To multiply money amounts by a one-digit number, students multiply as if the numbers were whole numbers. They place the decimal point so the answer is given in dollars and cents.

Multiplication Patterns
Patterns can be used when multiplying multiples of 10.

8 × 7 = 56
80 × 7 = 560
800 × 7 = 5,600
8,000 × 7 = 56,000

80 × 70 = 5,600
800 × 700 = 560,000

Think: (8 × 7) × 10
Think: (8 × 7) × 100
Think: (8 × 7) × 1,000

Think: (8 × 7) × (10 × 10)
Think: (8 × 7) × (100 × 100)

In each example, the number of zeros in the product is the same as the sum of the number of zeros in each factor.

Multiplication by a Two-Digit Number
To solve problems such as 392 × 50, students can use patterns and what they know about multiplication by 1-digit numbers to find (392 × 5) × 10.

The algorithm for multiplying by a two-digit number simply extends the algorithm for multiplying by a one-digit number. The Distributive Property can be used to show the multiplication.

Estimation can be used to check that the answer to a multiplication problem is reasonable. Students round each factor to a multiple of 10 that has only one nonzero digit. Then they use mental math to recall the basic fact product and patterns to determine the correct number of zeros in the estimate.

Teaching Model 3.6: Estimate Products