Download Multiplying Whole Numbers

Section 1.4
PRE-ACTIVITY
PREPARATION
Multiplying Whole Numbers
Martha and Tom have taken on a 36-month automobile loan for $12,000 with fixed
monthly payments of $364. What will their payments total?
Their latest home improvement project is to tile the floor of their remodeled
kitchen. How many square feet of tiles do they need to cover the 12 feet by 16 feet
rectangular space?
In each case, they will multiply whole numbers.
Competence with multiplying whole numbers is practical in everyday situations
such as these. Furthermore, with an eye to your future classes, the material in
math intensive courses such as computer information systems, science, health, and
business will be easier to learn when you understand and are proficient in the basic
operations, multiplication being one of them. Multiplying “by hand” rather than by
using a calculator will help you review and retain your multiplication skills.
LEARNING OBJECTIVE
Master the multiplication of whole numbers.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
carrying
factor
operation
multiplicand
place digit
multiplier
place value
partial product
validate
power of ten
value
product
rectangular array
65
Chapter 1 — Whole Numbers
66
BUILDING MATHEMATICAL LANGUAGE
The operation of multiplication might be considered in either of two ways—
•
as a shortcut to repeated addition of the same number
Example: six boxes of pencils with eight pencils in each box for a total of “six times eight,” or fortyeight pencils.
8 pencils
+
8 pencils
+
8 pencils
+
8 pencils
+
8 pencils
+
8 pencils
= 6 × 8 pencils = 48 pencils
OR
•
as a way to compute the number of items in a rectangular array, an orderly arrangement in rows
and columns
8 columns (8 desks per row)
Example: six rows of desks in a classroom with eight desks in each row for
a total of 6 rows × 8 desks per row, or
48 desks
6 rows
The numbers you multiply together are called factors.
The name for the factor being multiplied is the multiplicand, and the factor by which it is multiplied is
the multiplier.
The answer is called the product.
67
Section 1.4 — Multiplying Whole Numbers
“6 times 8,” “multiply 8 by 6,” or “find the product of 6 and 8,” might be written symbolically in any of
the following ways;
multiplier multiplicand
6×8
Horizontally
8
6•8
or
×6
Vertically
6 (8)
48
multiplicand
multiplier
product
(6) (8)
6 * 8 (on many calculators)
The table below presents the basic multiplication facts from 1 × 1 through 9 × 9. The box where two factors
intersect gives their product. As you will use these facts repeatedly, you must know them confidently for
proficiency, speed, and accuracy when multiplying larger numbers.
factor
factor
×
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10
12
14
16
18
3
3
6
9
12
15
18
21
24
27
4
4
8
12
16
20
24
28
32
36
5
5
10
15
20
25
30
35
40
45
6
6
12
18
24
30
36
42
48
54
7
7
14
21
28
35
42
49
56
63
8
8
16
24
32
40
48
56
64
72
9
9
18
27
36
45
54
63
72
81
The numbers 10, 100, 1000, 10,000, and so on are called the powers of ten. You already know them
as the place values of the decimal (base 10) number system (see Section 1.1). Recall that they are
generated on the place value chart by multipling each place value by 10 as you move to the left; that is,
10 × 10 = 100, 10 × 100 = 1000, 10 × 1000 = 10,000 and so on.
See the Methodologies section for the shortcut to use when a multiplication factor is a power of ten.
Chapter 1 — Whole Numbers
68
Following are the Mathematical Properties of Multiplication. A simple example is given for each.
Mathematical Properties of Multiplication
Commutative Property of Multiplication
Two numbers can be multiplied in either order without affecting their product.
Example:
4 × 5 = 5 × 4 = 20
Identity Property of Multiplication
The product of any number and one (1) is that number.
Example:
5×1=5
1×5=5
Multiplication Property of Zero
The product of any number and zero (0) is zero (0).
Example:
7×0=0
0×7=0
Associative Property of Multiplication
When finding the product of three numbers, the numbers can be grouped in
different ways without affecting their product.
6
×4=2×
}
}
Example: (2 × 3) × 4 = 2 × (3 × 4)
12
= 24
The final property to consider includes both multiplication and addition. The Distributive Property of
Multiplication over Addition is the basis for the traditional methodology that you will use to multiply
larger numbers.
Distributive Property of Multiplication over Addition
Multiplying a factor by the sum of two or more numbers yields the same
answer as multiplying the factor by each of the numbers and then adding their
respective products.
Example: 7(4 + 5) = 7 × 4 + 7 × 5
7(9)
=
28
63 = 63
+
35
(4 + 5)7 = 4 × 7 + 5 × 7
(9)7
=
28
63 =
63
+
35
The methodology for multiplying larger whole numbers will use the term partial product. A partial
product is the result of multiplying a whole number by the value of a place digit in the multiplier.
69
Section 1.4 — Multiplying Whole Numbers
METHODOLOGIES
The first methodology to understand is the process of multiplying any whole number by a single-digit multiplier.
Its carrying notation is similar to what you used in addition.
Multiplying any Whole Number by a Single-digit Whole Number
►
►
Example 1:
Multiply: 5496 × 7
Example 2:
Multiply: 3918 × 6
Steps in the Methodology
Step 1
Set up the
problem.
Try It!
Example 1
Write the problem vertically, right aligning
the ones place digits. For ease of calculation,
use the single-digit number as the multiplier
(bottom number).
5496
× 7
???
Why can you do this?
Step 2
Multiply each
digit in the top
number.
Multiply each digit in the top number by the
multiplier, beginning with the ones place and
working left.
Use carry notation, as necessary.
????
3 6 4
5496
× 7
384 7 2
How and why do you do this?
Step 3
Present the
answer.
Present your answer.
Step 4
Validate your answer by using the opposite
Validate your
answer.
operation—division.
Divide your answer by the single-digit
multiplier to get the other original factor.
(Use the Methodology for Long Division.
See Section 1.5)
38,472
5496 9
7 38 472
)
−35
2 1
34
−2 8
67
−63
42
−4 2
0
Example 2
Chapter 1 — Whole Numbers
70
???
Why can you do Step 1?
The Commutative Property of Multiplication permits you to choose which factor is to be the multiplier and
which to be the multiplicand without affecting the answer.
????
How and why do you do Step 2?
In Example 1, if you think of multiplication as repeated addition, then you are adding 5496 seven times:
+
5496
5496
5496
5496
5496
5496
5496
To add each column separately, you would keep in mind place values
and add seven 6’s in the ones column, then seven 9’s, seven 4’s, and
seven 5’s, also taking into account the addition “carries.”
Because multiplication is a welcome shortcut to repeated addition, instead of adding the repeated digits in each
column, you can compute 7 × 6, 7 × 9, 7 × 4, and 7 × 5, always keeping in mind the place values of the columns.
The carrying process for multiplication comes from an understanding of the value of the place digits.
4
5496
× 7
2
64
5496
× 7
72
3 64
5496
× 7
472
3 64
5496
× 7
38472
Begin by multiplying 7 times the ones place digit of the top number (6).
7 × 6 = 42, meaning 4 tens and 2 ones.
Write the 2 in the ones place of the answer and carry the 4 (tens) to the tens column.
Multiply 7 times the tens place digit of the top number (9). 7 × 9 tens = 63 tens;
63 tens plus the 4 tens carried over = 67 tens (670)
Write 7 in the tens place and carry the 6 (hundreds).
Multiply 7 times the hundreds place digit of the top number (4).
7 × 4 hundreds = 28 hundreds
28 hundreds plus the 6 hundreds carried over = 34 hundreds (3400)
Write 4 in the hundreds place and carry the 3 (thousands).
Finally, multiply 7 times the thousands place digit (5). 7 × 5 thousands = 35 thousands
35 thousands plus the 3 thousands carried over = 38 thousands (38,000)
Write 8 in the thousands place and 3 in the ten-thousands place.
71
Section 1.4 — Multiplying Whole Numbers
Multiplying any Whole Number by a Two-or-more Digit Whole Number
►
►
Example 1:
Multiply: 2859 × 374
Example 2:
Multiply: 4387 × 162
Steps in the Methodology
Step 1
Set up the
problem.
Write the problem vertically, right aligning the
ones place digits. For ease of calculation, use
the number with fewer digits as the multiplier
(bottom number).
Try It!
Example 1
Example 2
2859
×374
???
Why can you do this?
Special
Case:
More than two factors to multiply
(see page 76, Model 5)
Shortcut #1 Multiplying by a power of ten
(see page 75, Model 3)
Step 2
Find the
first partial
product.
Multiply the top number by the ones place digit
of the multiplier (bottom number), carrying
when necessary, to determine the first partial
product.
Right align it with the ones place.
???
Why do you do this?
Step 3
Determine the next partial product.
Find the
second
partial
product.
Multiply the top number by the tens place digit
of the multiplier, again using “carry” notation
as necessary.
Hold the ones place with a zero and align your
answer with the tens place digit.
???
Why do you do this?
3 2 3
2 85 9
×374
11436
first partial
product
(4 × 2859)
6 46
3 2 3
2 85 9
×374
1 14 3 6
200130
second partial
product
(70 × 2859)
continued on the next page
Chapter 1 — Whole Numbers
72
Steps in the Methodology
Step 4
Find the
next partial
product.
Example 1
Determine the next partial product if the
multiplier has more than two digits.
2 1 2
6 46
3 2 3
2 85 9
×374
Multiply the top number by the next higher
place digit of the multiplier.
1 14 3 6
2 0 013 0
857700
Align the answer below that place digit, and fill
in the appropriate number of zeros to hold the
remaining places in this partial product.
third partial
product
(see Why do you do Step 3?)
Special Zero (0) as a digit in the multiplier
Case: (bottom number) (see page 74, Model 2)
Step 5
Find the
remaining
partial
products.
Step 6
Add the
partial
products.
Repeat Step 4 until you have used each digit in
the multiplier to find its corresponding partial
product and you have properly aligned each of
them according to place values.
(300 × 2859)
partial products
completed
2 1 2
6 46
3 2 3
Add the partial products to get the final answer.
2 85 9
× 374
Shortcut #2
One or both factors end in a string
of zeros (see page 75, Model 4)
1
11 43 6
20 0 13 0
+8 5 7 7 0 0
10 6 9 2 6 6
Step 7
Present the
answer.
Present your answer.
Step 8
Validate your answer by using the opposite
operation—division.
Validate
your answer.
Divide your answer by the factor you chose to
be the multiplier to get the other factor in the
original problem.
(Use the Methodology for Long Division.
See Section 1.5)
1,069,266
6 3
3 2
5 3
1
9
2 859
37 4 0110 6 9 266
)
−7 4 8
1
2 1 1
32 12
−2 9 9 2
1
1 1 1
2 2 06
−1 8 70
336 6
−336 6
0
Example 2
73
Section 1.4 — Multiplying Whole Numbers
???
Why can you do Step 1?
The Commutative Property of Multiplication allows you to choose which factor is to be the multiplier and
which to be the multiplicand without affecting the answer.
???
Why do you do Step 2?
This methodology is based on the Distributive Property of Multiplication over Addition.
Think of 374 × 2859 as (300 + 70 + 4) × 2859.
By the Commutative Property of Addition, this is the same as (4 + 70 + 300) × 2859.
second partial
product
}
first partial
product
}
}
The Distributive Property says you can multiply 2859 by each of the addends and add their respective
products to compute the answer; that is, compute 4 × 2859 + 70 × 2859 + 300 × 2859.
third partial
product
In Step 2, you compute the first partial product with the ones digit of the multiplier, using the previous
Methodology for Multiplying a Whole Number by a Single-digit Whole Number.
???
Why do you do Step 3?
Step 3 in the methodology tells you to multiply the tens digit in the multiplier (7) by 2859. This yields the
digits 20013.
However, this is just a convenient notation for multiplying the value of the tens digit in the multiplier times
the top number, that is, 70 × 2859, to get the second partial product 200,130. In order to align the digits of the
partial product correctly and according to their place values, you use a final zero to hold the ones place.
As you move to the third partial product and so on (if necessary), you use zeros to hold the last places in the
partial products to align them correctly, according to their place values.
Chapter 1 — Whole Numbers
74
MODELS
Model 1
Solve: 4035 × 27
2 3
4035
× 27
Step 1
4 0 35
× 27
28245
Step 2
Note: Be sure to apply the Multiplication Property of Zero correctly.
7×0=0
0+2=2
1
2 3
4 0 35
× 27
28245
80700
Step 3
first partial product (7 × 4035)
second partial product (20 × 4035)
Steps 4 & 5: partial products
complete
108945
Step 6
sum of the partial products
Step 7
Answer: 108,945
Step 8
Validate:
3
2
2
4035 9
2 7 108945
)
−108
94
−81
135
−135
0
Model 2
Special Case: Zero as a Digit in the Multiplier (bottom number)
Solve: 897 × 603
Step 1 897
×603
5 4
2 2
2 2
Step 2
897
×603
Step 3 & 8 9 7
Step 4
×60 3
2691
2691
0000
1
5 3 82 00
Step 5
not needed
Step 6
sum of partial products
Step 7
Answer: 540,891
Step 8
Validate:
2
2
2
5 4 08 91
)
4 13 1
8 97 9
6 0 3 5 4 08 9 1
− 4 82 4
5 84 9
−5 4 2 7
4221
−4 2 21
0
Partial products:
First, multiplication by 3
Second, multiplication by 0
Third, multiplication by 600
When the digit in the multiplier
is zero, multiply through by
zero to help with alignment.
75
Section 1.4 — Multiplying Whole Numbers
Model 3
Shortcut: Multiplying by a Power of Ten
Shortcut: To determine the product of a whole number and a power of ten, simply
attach the total number of zeros in the power of ten to the whole number.
A
►
70 × 10 = 700
536 × 100 =53,600
70 × 100 = 7000
536 × 100,000 = 53,600,000
70 × 100,000 = 7,000,000
Model 4
A
►
B
►
536 × 10 = 5360
Shortcut: One or Both Factors End in a String of Zeros
Solve: 42,000 × 500
Step 1
42, 000
× 500
Steps 2-6
1
4 2, 000
× 500
00000
000000
21000000
21000000
Step 7
Step 8
Answer: 21,000,000
42000 9
Validate: 500 21000000
−2000
1000
−1000
000
0
)
Shortcut: When one or both factors end in zeros, multiply the digits before the string
of zeros in the factors and attach the total number of ending zeros to their product.
42,000 × 500
Multiply 42 and 5 and attach five zeros to their product.
1
42
×5
21000000
Answer: 21,000,000
Chapter 1 — Whole Numbers
76
►
B
32,000 × 109,000
Use the shortcut:
2
1
10 9
×32
218
+3270
Answer: 3,488,000,000
Model 5
3488 000000
109000 9
Validate: 32000 3488000000
−32000
288000
−288000
0000
1
)
Special Case: More than Two Factors to Multiply
Multipy: 21 × 54 × 39
The Methodologies for Multiplying Whole Numbers are for two
factors. Apply the Commutative and Associative Properties of
Multiplication and choose any two factors to multiply first.
54
×21
1 54
10 80
Steps 1-6
1134
Then multiply the first product by the next factor.
Continue this process until all factors have been used.
11
13 3
Steps 1-6
113 4
× 39
10206
+34020
44226
Validate by successive divisions.
Following the Methodology for Long Division, use
the multipliers as the divisors, in reverse order.
Step 7
Answer: 44,226
Step 8
Validate:
1 13 4
3 9 4 4 2 26
−3 9
3
2
)
3 1
4 1
54 9
21 1 1 3 4
)
0 1
−1 0 5
84
−8 4
52
−3 9
0
2 1
132
−1 1 7
1 56
−1 5 6
0
Section 1.4 — Multiplying Whole Numbers
77
How Estimation Can Help
One way to easily estimate the product of whole numbers is to first round each factor to its largest place value
and then multiply the rounded factors. This allows you to use the shortcut for multiplying factors ending in
zeros (see Model 4) and to quickly do the calculation in your head. To determine if your answer is reasonable,
your estimate will give you an idea of how large your answer ought to be in terms of place value as well as an
approximation of its largest one or two place digits.
Look again at Example 1 in the Methodology for Multiplying a Whole Number by a Two-or-more Digit
Whole Number:
2859 × 374
Estimate:
3 0 0 0 × 4 0 0 = 1200000 = 1,200,000
This estimate will be greater than the actual product because both factors were rounded up, but you do know
from the estimate that your answer ought to be reasonably close to one million.
The answer to Example 1 (1,069,266) is, in fact, reasonably near the estimate.
Perhaps it may have been just as easy for you to round each factor to its hundreds place and calculate
2900 × 400 in your head (29 × 4 with four zeros attached, or 1,160,000)—a bit closer to the actual product.
Go back and estimate the answer to Example 2 in the same Methodology. Was your answer reasonably close
to your estimate?
Chapter 1 — Whole Numbers
78
ADDRESSING COMMON ERRORS
Issue
Misaligning
partial
products—not
using the
appropriate
zeros as
placeholders
Misaligning
place value
columns when
multiplying by
zero (0)
Incorrect
Process
28
×46
168
16
+112
112
2
280
439
×209
3951
395
1
+87
878
7
78
Resolution
The second partial
product multiplier is
the value of the tens
place digit (in this
case 40, not 4). Insert
a zero (0) in the ones
place to keep the
place value columns
correctly aligned.
Show multiplication
by zero as a partial
product to keep the
columns aligned
correctly.
12731
Correct
Process
Validation
4
1
28 9
4 6 1288
3
4
28
×46
−9 2
168
+1120
368
−3 68
1288
0
8
2
3
1
3 8
439
×209
2 4 1
2
4082
× 63
12646
2646
+26492
64920
277566
Any number times
zero (0) is zero (0).
Add the “carry digit”
to the zero (0)
product.
Never multiply the
“carry digit.”
4399
9 17 51
−8 3 6
)
209
1
8 1
7 10 1
8 15
−6 2 7
3 951
0000
1
+ 8 7800
1 8 81
−1 8 81
91751
Making partial
product errors
by multiplying
the “carry digit”
when zero (0)
is a digit in the
number
)
0
40 8 2 9
6 3 25716 6
−252
2 4 1
2
2
1
4082
× 63
)
1
51
−0
12 246
+244920
516
−5 04
257166
126
−12 6
0
Using poor
notation for
carrying when
adding partial
products
6 4
4 3
8 75
×96
9
1
11
1
5250
250
7875
87
7
75
9 300 0
Align the “carry digit”
with the proper
column and be
consistent and clear
in your notation for
this addition step.
6 4
4 3
8 75
×96
11
5250
1
7875 0
84000
3
4
4
96
)
8 759
7
13
1
8 4 0 00
−7 6 8
6 11 1
7 2 0
−6 7 2
480
−4 8 0
0
Section 1.4 — Multiplying Whole Numbers
79
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with multiplying whole numbers
the steps in the multiplication process
the mathematical property that gives you the flexibility to choose which number to be the multiplicand
and which to be the multiplier
the role of partial products in multiplying two numbers, each with more than one digit
the validation of multiplication by division
Section 1.4
ACTIVITY
Multiplying Whole Numbers
PERFORMANCE CRITERIA
• Multiplying any two whole numbers
– neatness of presentation
– validation of the answer
CRITICAL THINKING QUESTIONS
1. What is the result when a number is multiplied by zero? by one?
2. What shortcut can you use to multiply by 10, 100, 1000 etc. (powers of 10)?
3. When multiplying whole numbers, what are some notation techniques that can be used to improve accuracy?
80
Section 1.4 — Multiplying Whole Numbers
81
4. What is the relationship between multiplication and addition?
5. Why is there always a zero at the end of the second partial product?
6. What does the Distributive Property of Multiplication over Addition have to do with calculating the sum
of the partial products to get the final answer in a multiplication problem? Give an example.
7. Even though the Commutative Property of Multiplication allows you to divide the quotient by either
factor to get the other, why is it a good idea to divide by the multiplier (bottom number) to validate the
quotient?
Chapter 1 — Whole Numbers
82
TIPS
FOR
SUCCESS
•
Know confidently all single digit multiplication facts. Work to improve your proficiency, speed, and accuracy.
•
Show all of your work neatly and legibly, with proper notation and vertical alignment.
•
Use graph paper or lined paper turned sideways to help align place value columns accurately.
•
Use ending zeros to align partial products correctly.
•
Always validate!
DEMONSTRATE YOUR UNDERSTANDING
1. Estimate the answer by rounding each factor to its largest place value before you multiply.
a) 5,320 × 879
b) 923 × 79
c) 56,789 × 3725
2. Perform the indicated operation in each of the following and validate your answers.
Problem
a)
95 × 28
Worked Solution
Validation
83
Section 1.4 — Multiplying Whole Numbers
Problem
b)
57 × 82
c) Find the product of 389
and 17.
d) 712 × 108
Worked Solution
Validation
Chapter 1 — Whole Numbers
84
Problem
e) Multiply 603 by 184.
f) 2709 × 417
g) 48 × 13 × 76
Worked Solution
Validation
85
Section 1.4 — Multiplying Whole Numbers
IDENTIFY
AND
CORRECT
THE
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
Correct Process
1) Multiply: 349 × 37
Did not align
the product of
multiplication by
30 correctly. The
zero (for the ones
placeholder) is
missing.
Should be 10470
rather than 1047.
2) Multiply:
34,569 × 307
3) 47 × 506 =
349
× 37
2443
10470
12,913
Answer: 12,913
Validation
349 9
37 12913
–111
181
–148
333
–333
0
Chapter 1 — Whole Numbers
86
Worked Solution
Identify Errors
or Validate
Correct Process
4) Find the product of
509 and 93.
5) 548 × 18 =
ADDITIONAL EXERCISES
Perform the indicated operation in each of the following. Validate your answers.
1. 59 × 83
2. 123 × 456
3. Find the product of 478 and 15.
4. 612 × 209
5. Multiply 805 by 178.
6. 4307 × 265
7. 82 × 17 × 53
Validation