California Mathematics Content Standards
REFRESHER
LESSON
22
• Writing Decimal Numbers in
Expanded Notation
5.MR 1.0, 1.1 Analyze problems by identifying
relationships, distinguishing relevant from irrelevant
information, sequencing and prioritizing information,
and observing patterns.
5.MR 1.0, 1.2 Determine when and how to break a
problem into simpler parts.
5.NS 2.0, 2.1 Add, subtract, multiply, and divide
with decimals; add with negative integers; subtract
positive integers from negative integers; and verify
the reasonableness of the results.
• Mentally Multiplying Decimal
Numbers by 10 and by 100
Refresher Concept
We may use expanded notation to write decimal numbers just as we
have used expanded notation to write whole numbers. The values of
some decimal places are shown in this table:
hundredths
thousandths
1
100
1
1000
tenths
decimal point
ones
Decimal Place Values
1
10
1
.
We write 4.025 in expanded notation this way:
(4 ⫻ 1) ⫹ a2 ⫻
1
1
b ⫹ a5 ⫻
b
100
1000
The zero that serves as a placeholder is usually not included in
expanded notation.
Example 1
Write 5.06 in expanded notation.
The 5 is in the ones place, and the 6 is in the hundredths place.
(5 ⴛ 1) ⴙ a6 ⴛ
1
b
100
We say the word and when we see a decimal point. Read 5.06 as
“five and six hundreths.”
Saxon Math Intermediate 6
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45
Example 2
1
1
Write (4 × 10
) ∙ (5 × 1000 ) as a decimal number.
We write the decimal number with a 4 in the tenths place and a 5 in the
thousandths place. No digits in the ones place or the hundredths place
are indicated, so we write zeros in those places.
0.405
When we multiply whole numbers by 10 or by 100, we can find the
product mentally by attaching zeros to the whole number we are
multiplying.
24 × 10 = 240
24 × 100 = 2400
It may seem that we are just attaching zeros, but we are actually
shifting the digits to the left. When we multiply 24 by 10, the digits shift
one place to the left. When we multiply 24 by 100, the digits shift two
places to the left. In each product zeros hold the 2 and the 4 in their
proper places.
1000s
100s
2
10s
1s
2
4
24
2
4
0
24 × 10 (one-place shift)
4
0
0
24 × 100 (two-place shift)
When we multiply a decimal number by 10, the digits shift one place
to the left. When we multiply a decimal number by 100, the digits
shift two places to the left. Here we show the products when 0.24 is
multiplied by 10 and by 100.
10s
2
1s
1
s
10
1
s
100
0
2
4
2
4
4
0.24
0.24 × 10 (one-place shift)
0.24 × 100 (two-place shift)
Although it is the digits that are shifting one or two places to the left,
we get the same effect by shifting the decimal point one or two places
to the right.
46
0.24 × 10 = 2.4
0.24 × 100 = 24. = 24
one-place shift
two-place shift
© Harcourt Achieve Inc. and Stephen Hake. All rights reserved.
Saxon Math Intermediate 6
Example 3
Multiply: 3.75 × 10
Since we are multiplying by 10, the product will have the same digits as
3.75, but the digits will be shifted one place. The product will be ten times
as large, so we mentally shift the decimal point one place to the right.
3.75 × 10 = 37.5 (one-place shift)
We do not need to attach any zeros, because the decimal point serves to
hold the digits in their proper places.
Example 4
Multiply: 3.75 × 100
When multiplying by 100, we mentally shift the decimal point two places
to the right.
3.75 × 100 = 375. = 375 (two-place shift)
We do not need to attach zeros. Since there are no decimal places, we
may leave off the decimal point.
Example 5
Multiply:
1.2 10
ⴛ
0.4 10
Multiplying both 1.2 and 0.4 by 10 shifts each decimal point one place.
1.2 10 12
⫻
⫽
0.4 10
4
The expression
12
4
means “12 divided by 4.”
12
⫽3
4
Refresher Practice
Write these numbers in expanded notation:
1
a. 2.05 (2 × 1) + 5 × ____
100
1
b. 20.5 (2 × 10) + 5 × ___
10
1
1 ) + 5 × _____
c. 0.205 (2 × ___
10
1000
(
)
)
(
)
(
Write these numbers in decimal form:
d. (7 ⫻ 10) ⫹ a8 ⫻
e. a6 ⫻
Saxon Math Intermediate 6
1
b
10
70.8
1
1
b
b ⫹ a4 ⫻
100
10
0.64
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47
Mentally calculate each product:
f. 0.35 × 10 3.5
h. 2.5 × 10 25
j. 0.125 × 10 1.25
Conclude
g. 0.35 × 100 35
i. 2.5 × 100 250
k. 0.125 × 100 12.5
For the following statements, answer “true” or “false”:
l. If 0.04 is multiplied by 10, the product is a whole number.
false
m. If 0.04 is multiplied by 100, the product is a whole number. true
Multiply as shown. Then complete the division.
n.
48
1.5 10
⫻
0.5 10
3
o.
100
2.5
⫻
0.05 100
50
© Harcourt Achieve Inc. and Stephen Hake. All rights reserved.
Saxon Math Intermediate 6