Teaching Models

Multiply and Divide 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.

squares with 6 shaded

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.

Multiplication array

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 answer 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.
2 digit 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.

Division by a One-Digit Divisor
The division algorithm for whole numbers is based on the base-ten numeration system. Proficiency in division requires proficiency in multiplication and subtraction. In discussing division, the use of correct vocabulary is essential.

Example: Divide 59 by 8

59/8

Example: Divide 1,646 by 7.

test
Look at the thousands place.
Since 1 < 7, there will be no thousands digit
in the quotient. The first digit will be in the hundreds place.
Think: 7 × n ≤ 16 so 2 is the greatest possible value of n.
Multiply 2 × 7.
Subtract 16 − 14. Compare to make sure the remainder is less than the
divisor. If the remainder is greater, an error has been made.
test
Rename the remainder of 2 hundreds as 20 tens.
Bring down the 4 tens in the dividend and add to the 20 tens.
Think: 7 × n ≤ 24 so 3 is the greatest possible value of n.
Multiply 3 × 7.
Subtract 24 − 21. Compare to make sure the remainder is less than the divisor. If the remainder is greater, an error has been made.
test
Rename the remainder of 3 tens as 30 ones.
Bring down the 6 ones in the dividend and add to the 30 ones.
Think: 7 × n ≤ 36 so 5 is the greatest possible value of n.
Multiply 5 × 7.
Subtract 36 − 35. Compare to make sure the remainder is less than the divisor. If the remainder is greater, an error has been made.
 
To check division, multiply the quotient by the divisor and add the remainder. The result should be equal to the dividend.
test

Since division problems with zeros in the quotient cause some students difficulty, extra practice with such computations should be provided. Point out to students that when dividing money, the decimal point in the quotient is placed directly above the decimal point in the dividend.

Finding the Mean
One application of division is finding the mean, or average, of a set of numbers. The mean is found by adding the numbers and then dividing the sum by the number of addends. The mean tells what each number would be if the sum remained the same and the individual numbers were equal.

Example: Find the mean for 9, 19, 44, 8, 25, 4, 27, and 32.

Find the sum:
Divide the sum by the number of addends:
    9 + 19 + 44 + 8 + 25 + 4 + 27 + 32 = 168
   168 ÷ 8 = 21
The mean, or average, of the group is 21.

The mean is only one measure of central tendency of a set of data. Other measures of central tendency are discussed elsewhere.

Dividing by Multiples of 10
Patterns can be used when dividing by multiples of 10.

63 ÷ 7 = 9
630 ÷ 7 = 90
6,300 ÷ 7 = 900
63,000 ÷ 7 = 9,000
630,000 ÷ 7 = 90,000
 
630 ÷ 70 = 9
6,300 ÷ 700 = 9
63,000 ÷ 7,000 = 9
630,000 ÷ 70,000 = 9
    
 
Think: (63 ÷ 7) × 10
Think: (63 ÷ 7) × 100
Think: (63 ÷ 7) × 1,000
Think: (63 ÷ 7) × 10,000
 
Think: (63 ÷ 7) × (10 ÷ 10)
Think: (63 ÷ 7) × (100 ÷100)
Think: (63 ÷ 7) × (1,000 ÷ 1,000)
Think: (63 ÷ 7) × (10,000 ÷ 10,000)

Division by a Two-Digit Number
The division algorithm for dividing by two-digit divisors is the same as that for one-digit divisors. Students must develop skill in estimating the first digit in the dividend, including making adjustments when the first estimate is too great or too small.

Example: Divide 1,138 by 23.

test
Look at the dividend. Since 11 < 23, there will be no thousands or hundreds digit in the quotient. The first digit will be in the tens place.
Think: 23 × n ≤ 113 so 4 is the greatest possible value of n.
Multiply 23 × 4.
Subtract 113 − 92. Compare to make sure the remainder is less than the divisor. If the remainder is greater, an error has been made.
test
Rename the remainder of 21 tens as 210 ones. Bring down the 8 ones in the dividend and add to the 210 ones.
Think: 23 × n ≤ 218 so 9 is the greatest possible value of n.
Multiply 23 × 9.
Subtract 218 − 207. Compare to make sure the remainder is less than the divisor. If the remainder is greater, an error has been made.
 
To check a division problem, multiply the quotient by the divisor and add the remainder. The result should be equal to the dividend.
test

Since division problems with zeros in the quotient cause some students difficulty, extra practice with such computations should be provided.


Teaching Model 2.6:Use Multiplication Properties


Houghton Mifflin Math Grade 6