Download § 7-1 Introduction to Decimals

§ 7-1 Introduction to Decimals
Place Value
12.345678
Place Value
12.345678
Place Value
12.345678
tens
Place Value
12.345678
units
tens
Place Value
decimal point
units
tens
12.345678
Place Value
tenths
decimal point
units
tens
12.345678
Place Value
hundredths
tenths
decimal point
units
tens
12.345678
Place Value
thousandths
hundredths
tenths
decimal point
units
tens
12.345678
Place Value
ten-thousandths
thousandths
hundredths
tenths
decimal point
units
tens
12.345678
Place Value
hundred-thousandths
ten-thousandths
thousandths
hundredths
tenths
decimal point
units
tens
12.345678
Place Value
millionths
hundred-thousandths
ten-thousandths
thousandths
hundredths
tenths
decimal point
units
tens
12.345678
Numbers in Words
Example
How can we say the number .401?
Numbers in Words
Example
How can we say the number .401?
four hundred one thousandths
Numbers in Words
Example
How can we say the number .401?
four hundred one thousandths
Example
How do we write the number ‘two-hundred thirty-seven
ten-thousandths’?
Numbers in Words
Example
How can we say the number .401?
four hundred one thousandths
Example
How do we write the number ‘two-hundred thirty-seven
ten-thousandths’?
.0237
Expanded Form
When working with children, we want to represent these decimal
numbers as many ways as we can. One way is using expanded form.
What is expanded form?
Expanded Form
When working with children, we want to represent these decimal
numbers as many ways as we can. One way is using expanded form.
What is expanded form?
Example
Write .324 in expanded form.
Expanded Form
When working with children, we want to represent these decimal
numbers as many ways as we can. One way is using expanded form.
What is expanded form?
Example
Write .324 in expanded form.
.324 = 3 · 10−1 + 2 · 10−2 + 4 · 10−3
Expanded Form
When working with children, we want to represent these decimal
numbers as many ways as we can. One way is using expanded form.
What is expanded form?
Example
Write .324 in expanded form.
.324 = 3 · 10−1 + 2 · 10−2 + 4 · 10−3
Example
Write .0201 in expanded form.
Expanded Form
When working with children, we want to represent these decimal
numbers as many ways as we can. One way is using expanded form.
What is expanded form?
Example
Write .324 in expanded form.
.324 = 3 · 10−1 + 2 · 10−2 + 4 · 10−3
Example
Write .0201 in expanded form.
.0201 = 2 · 10−2 + 1 · 10−4
Expanded Fraction Form
Example
Write 13.243 using fraction expanded form.
Expanded Fraction Form
Example
Write 13.243 using fraction expanded form.
13.243 = 1(10) + 3(1) +
2
4
3
+
+
10 100 1000
Visually Representing Decimals
We have a couple of different ways we can represent decimals.
Visually Representing Decimals
We have a couple of different ways we can represent decimals.
1
the number line
Visually Representing Decimals
We have a couple of different ways we can represent decimals.
1
the number line
2
grids
Decimals on the Number Line
0
.2
.4
.6
.8
1
1.2
1.4
1.6
Decimals on the Number Line
0
.2
.4
.6
.8
1
1.2
1.4
1.6
Decimals on the Number Line
0
.2
.4
.6
.8
1
1.2
1.4
1.6
Decimals on the Number Line
0
.2
.4
.6
.8
1
1.2
1.4
1.6
Tenths Grid
Tenths Grid
Tenths Grid
Decimal Squares
Decimal Squares
Decimal Squares
Converting
Example
Convert .31 to a fraction in simplest form.
Converting
Example
Convert .31 to a fraction in simplest form.
Example
Convert .051 to a fraction in simplest form.
Converting
Example
Convert .31 to a fraction in simplest form.
Example
Convert .051 to a fraction in simplest form.
Example
Convert .0204 to a fraction in simplest form.
Converting
Example
Convert
37
100
to a decimal.
Converting
Example
Convert
37
100
to a decimal.
Example
Convert
201
10000
to a decimal.
Converting
Example
Convert
3
8
to a decimal.
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
8
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
3
= 3
8
2
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
=
8
3
= 3·
2
3
23
53
53
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
3
= 3
8
2
3 53
= 3· 3
2 5
3 · 53
=
103
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
3
= 3
8
2
3 53
= 3· 3
2 5
3 · 53
=
103
375
=
1000
Converting
Example
Convert
3
8
to a decimal.
Since we don’t have a denominator that is a power of 10, we need to
get there somehow. Suggestions?
3
3
= 3
8
2
3 53
= 3· 3
2 5
3 · 53
=
103
375
=
1000
= .375
Terminating Decimals
The question is, when can we writing a given rational number as a
terminating decimal?
Terminating Decimals
The question is, when can we writing a given rational number as a
terminating decimal?
Definition
A terminating decimal can be written with a finite number of non-zero
places to the right of the decimal point.
Terminating Decimals
The question is, when can we writing a given rational number as a
terminating decimal?
Definition
A terminating decimal can be written with a finite number of non-zero
places to the right of the decimal point.
Example
Which of the following can be written as terminating decimal?
10
21
6
30
5
30
7
125
2
18
3
32
3
60
11
121
Terminating Decimals
Definition
A rational number ab , with gcf(a, b) = 1, can be written as a
terminating decimal if and only if the prime factorization of the
denominator contains only powers of 2 and 5.
Ordering Decimals
Example
Which is larger?
.29 or .3?
Ordering Decimals
Example
Which is larger?
.29 or .3?
.003 or .00093?
Ordering Decimals
Example
Which is larger?
.29 or .3?
.003 or .00093?
.001 or .000999?
Ordering Decimals
Example
Which is larger?
.29 or .3?
.003 or .00093?
.001 or .000999?
Can we come up with a rule as to how to decide which is larger when
given two terminating decimals?