Instructional Manual Math Book 2

Transcription

Instruction Manual
EDUSS NCC Mathematics Instruction
Multiplication
Division
Decimals
Volume 2 of 4, K-7, Mathematics
Eduss Broadcast & Media, Inc., a Colorado corporation
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EDUSS Broadcast & Media Inc.
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Published by EDUSS Broadcast & Media Inc.
Copyright by EDUSS Broadcast & Media Inc.
Second Edition 2004, All rights reserved
ISBN 0-620-21809-6
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2
Content
Multiplication
Introduction
Multiplication Tables
Vertical Multiplication
42 - 51
43 - 44
45 - 47
48 - 51
Division
Introduction
52 - 62
53 - 57
DivisionTables
Recording Division - short
Division by Inspection
Decimals
Introduction
Decimal Fractions
Adding Decimals
Multiplying Decimals
Copyright by EDUSS Broadcast & Media Inc. 2004, All rights reserved
3
56 - 56
58 - 61
62 - 62
63 - 80
64 - 69
70 - 72
73 - 74
75 - 80
Multiplication
EDUSS NCC: Student Instruction Manual
VolumeVII: NCTM K-7
Mathematics
Multiplication
2
1
Multiplication
Multiplication
When we do multiplication problems or equations we use the times
symbol to represent the multiplying of numbers. We also use
the equal sign to indicate the answer.
Symbol
Meaning
X
Used in:
When we add five(5), four (4) times we get twenty (20);
Ed
+
Multiply 1 X 1 = 1
If we have to write or say this
equation we will do it like this.
1
Multiplication is a special way of doing addition.
1 = 1
or
1 times 1 equals 1
=
+
5 + 5 + 5 + 5 = 20
4 x 5 = 20
equals
times
X
+
3
4
Multiplication
Multiplication
Multiplication is a special way of doing addition.
When we add 1(one), 6 (six) times the answer is 6 (six);
Now let’s look at it in another way.
1
4
= 5
rows of
4
1
or
4
rows
4
of 5 = 20
1
x
1
2
3
4
5
6
+
2
2
1
1
1
=
1
2
3
4
5
6
or
1x6=6
Multiplication
Let’s look at another example:
2+2+2=6
1
1
6
1
1
We can also say the answer to a multiplication problem is the product.
Multiplication
2
1
1+1+1+1+1+1=6
5
1
1
Multiplication is associated with repeated addition. We
can write the above exercise as follows:
5
5
+ + + + + =
1
1
1
+
2
1
=
1
1
1
2
2
2
5 x 4 = 5 + 5 + 5 + 5 = 20
3
1
2
3
4
2x3=6
5
+
1
2
3
4
5
+
1
2
3
5
4
+
1
2
3
4
5
=
1
2
3
2 times 3 = 6
4
Multiplication is associated with repeated addition:
5
x
1
2
3
4
=
1
5
4
2
1
3
4
1
2
1
3
3
4
2
5
2
3
5
4
Multiplication Introduction
43
5
VolumeVII: NCTM K-7
Mathematics
Multiplication
8
7
Multiplication
See if you can solve the following problem:
Ann has 2 red jellybeans, 2 yellow jellybeans
2 blue jellybeans, 2 green jellybeans and
2 orange jellybeans.
Multiplication
See if you can solve the following problem:
Ed
2
We can see there are 5 groups of 2.
5 x 2 = 10
How many jellybeans has Ann got altogether?
9
10
Multiplication
Multiplication
See if you can work out the following answer:
5x2=
4
We can also write or say it in the following manner:
2
5
2 times 5
or
2 multiplied by 5
or
2 by 5
or
5 two’s
1
2
x5
2
3
4
2
53
4
5
11
4 times 5
or
4 multiplied by 5
or
4 by 5
or
5 fours
4
x5
20
12
Multiplication
Multiplication
Which of the following problems can be solved by using
multiplication?
We can solve problem a) and problem c) with multiplication. Let’s see why.
a) 4 rugby teams played in the competition with 15 players
on each team. How many rugby players took part in the
competition?
With problem a) there are 4 teams with 15 players each. The
equation will look as follow:
a) 4 rugby teams played in the competition with 15 players
on each team. How many rugby players took part in the
competition?
15
x4
?
b) 4 rugby teams were going to play in the competition with
each team having 15 players. One of the teams withdrew.
How many teams are left in the competition?
b) 4 rugby teams were going to play in the competition with
each team having 15 players. One of the teams withdrew.
How many teams are left in the competition?
With problem b) there are 4 teams with and one team withdraws.
This is a subtraction problem. It does not matter how many players
are on each team:
c) Next year there will be 10 teams with 15 players
on each team. How many players will be in next years
competition?
c) Next year there will be 10 teams with 15 players
on each team. How many players will be in next years
competition?
15
x10
With problem c) there are 10 teams with 15 players each. The
?
equation will look as follow:
15
x4
60
4
- 1
3
15
x10
150
Multiplication Introduction
44
VolumeVII: NCTM K-7
Mathematics
Multiplication
14
13
Multiplication
Multiplication
The 1 times table or 1x table
To make multiplication easier we can learn and use
multiplication tables: If you practice them, you can
recall them when needed.
We will start with:
the 1x table,
the 2x table,
the 5x table and
the 10x table.
1
1
1
1
1
1
1
1
1
1
1
1
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
5
9
10
3
4
8
1
7
11
12
2
6
1
1
1
1
1
1
Ed
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
1
2
3
4
5
6
15
1
1
1
1
1
1
x
x
x
x
x
x
7=7
8=8
9=9
10 = 10
11 = 11
12 = 12
16
Multiplication
Multiplication
2
2
2
2
2
2
2
2
2
2
2
2
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
20
6
8 10 12 14 16 18
2
4
22
24
5
5
5
5
5
5
5
5
5
5
5
5
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
50
55
60
5 10 15 20 25 30 35 40 45
2
2
2
2
2
2
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
2
4
6
8
10
12
2
2
2
2
2
2
x
x
x
x
x
x
7 = 14
8 = 16
9 = 18
10 = 20
11 = 22
12 = 24
5
5
5
5
5
5
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
5
10
15
20
25
30
5
5
5
5
5
5
x
x
x
x
x
x
7 = 35
8 = 40
9 = 45
10 = 50
11 = 55
12 = 60
17
Multiplication
10
10 10 10 10 10 10 10 10 10
10
10
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
10 20 30 40 50 60 70 80 90 100 110 120
10
10
10
10
10
10
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
10
20
30
40
50
60
10
10
10
10
10
10
x
x
x
x
x
x
7 = 70
8 = 80
9 = 90
10 = 100
11 = 110
12 = 120
Vertical Addition Continues...
45
VolumeVII: NCTM K-7
Mathematics
Multiplication
20
19
Multiplication
Multiplication
Now once you know your 1x, 2x, 5x and
10x tables it will be easy to learn the
rest.
3
3
3
3
3
3
3
3
3
3
3
3
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
9 12 15 18 21 24 27
30
33
36
3
6
Remember you must
practice them so that you
can recall them when
you need them.
Ed
3
3
3
3
3
3
Now the 3x, 4x, 6x, 7x, 8x, 9x, 11x and 12x tables.
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
3
6
9
12
15
18
21
3
3
3
3
3
3
x
x
x
x
x
x
7 = 21
8 = 24
9 = 27
10 = 30
11 = 33
12 = 36
22
Multiplication
Multiplication
4
4
4
4
4
4
4
4
4
4
4
4
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
40
4
48
8 12 16 20 24 28 32 36
44
6
6
6
6
6
6
6
6
6
6
6
6
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
60
6 12 16 24 30 36 42 48 54
66
72
4
4
4
4
4
4
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
4
8
12
16
20
24
4
4
4
4
4
4
x
x
x
x
x
x
7 = 28
8 = 32
9 = 36
10 = 40
11 = 44
12 = 48
6
6
6
6
6
6
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
6
12
18
24
30
36
23
6
6
6
6
6
6
x
x
x
x
x
x
7 = 42
8 = 48
9 = 54
10 = 60
11 = 66
12 = 72
24
Multiplication
Multiplication
7
7
7
7
7
7
7
7
7
7
7
7
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
70
7 14 21 28 35 42 49 56 63
77
84
8
8
8
8
8
8
8
8
8
8
8
8
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
80
8 16 24 32 40 48 56 64 72
88
96
7
7
7
7
7
7
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
7
14
21
28
35
42
7
7
7
7
7
7
x
x
x
x
x
x
7 = 49
8 = 56
9 = 63
10 = 70
11 = 77
12 = 84
8
8
8
8
8
8
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
8
16
24
32
40
48
8
8
8
8
8
8
x
x
x
x
x
x
7 = 56
8 = 64
9 = 72
10 = 80
11 = 88
12 = 96
46
VolumeVII: NCTM K-7
Mathematics
Multiplication
26
25
Multiplication
Multiplication
9
9
9
9
9
9
9
9
9
9
9
9
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
90
9 18 27 36 45 54 63 72 81
99 108
11
11 11 11 11 11 11 11 11 11
11
11
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
11 22 33 44 55 66 77 88 99 110 121 132
9
9
9
9
9
9
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
9
18
27
36
45
54
9
9
9
9
9
9
x
x
x
x
x
x
7 = 63
8 = 72
9 = 81
10 = 90
11 = 99
12 = 108
11
11
11
11
11
11
27
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
11
22
33
44
55
66
11
11
11
11
11
11
x
x
x
x
x
x
28
Multiplication
12 12 12 12 12 12 12 12 12 12
12
12
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12
12 24 36 48 60 72 84 96 108 120 132 144
12
12
12
12
12
12
x
x
x
x
x
x
1
2
3
4
5
6
=
=
=
=
=
=
12
24
36
48
60
72
12
12
12
12
12
12
x
x
x
x
x
x
7 = 84
8 = 96
9 = 108
10 = 120
11 = 132
12 = 144
29
30
47
7 = 77
8 = 88
9 = 99
10 = 110
11 = 121
12 = 132
VolumeVII: NCTM K-7
Mathematics
Multiplication
2
1
Multiplication
Units:
Now let’s look at how we multiply numbers
with units of tens and ones.
1
1 2 3 4 5 6 7 8 9 10
We are going to
use the Vertical Multiplication
method:
Ed
Remember when we did units we saw that a set of
numbers is made up of units. Let’s look again at ones,
tens and hundreds.
= 1 one unit
= 10 ten units
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Hundreds
4
Multiplication
Multiplication
Let’s do this example: 25 x 2 = ?
We have learned how to multiply lower case units of one.
Now let’s look at how we multiply with unit patterns
of tens and ones:
1
and
2
3
4
x 2 =
3
4
25
x2
1
5
2
tens
ones
5
2
3
4
6
5
7
8
9
10
2
3
4
5
1
25
x2
0
5
In the first column (ones) we
multiply the 2 and the 5: The
answer is 10. As we did in addition
we write the 0 (ones) and we carry
the 1 tens
6
Multiplication
Multiplication
Let’s look at this step using the blocks:
1
25
x2
0
Tens
= 100 one hundred units
3
4
x2
Tens
= 40 forty units
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Before we look at
multiplication with multiples
we will review units:
Ones
1
Tens
25
x2
0
5
Ed
Two times 5 is equal to 10. As we have
seen when we did units that, 10
represents 1(tens) and 0 (ones).
x
5
5
48
Ones
5
5
VolumeVII: NCTM K-7
Mathematics
Multiplication
8
7
Multiplication
Multiplication
Now looking at the blocks we can see
what our example looks like after
the first step.
Tens
Now the units of ten:
Ones
Tens
1
1
25
x2
0
We multiply the 2 with
the 2(tens) in the tens column.
25
x2
0
5
5
Ed
Ones
5
5
The answer is 4 (tens).
9
10
Multiplication
Tens
We add the 1(tens) that we
carried over from the
ones column.
Multiplication
Ones
Tens
5
1
5
1
We add the 1(tens) to the 4 (tens)
11
2
3
4
1
x 4 =
5
2
3
4
5
6
7
8
Now the units of ten:
9
10
11
12
2
2
15
x4
60
3
4
5
2
15
x4
0
5(tens)
or
50
Multiplication
Let’s do another example: 15 x 4 = ?
3
Ed
12
Multiplication
2
5
25
x2
50
25
x2
0
1
Ones
5
multiply the 4 and the 5: The
answer is 20. As we did in addition
we write the 0 (ones) and we carry
the 2 (tens).
Now we multiply the 4 with
the 1 in the tens column.
The answer is 4, but we still
need to add the 2 (tens) that
we carried over from the
(ones) column.
This will give us an answer of 6.
The answer for this problem is 60
49
VolumeVII: NCTM K-7
Mathematics
Multiplication
14
13
Multiplication
Multiplication
Now what happens when we multiply 4 by 15 : 4 x 15 = ?
1
4 x
2
3
1
2
=
3
2
3
4
6
5
8
7
9
10
11
4
x15
20
12
2
3
4
4
5
5
Ed
4
x15
?
In the first column
(ones) we multiply
the 5 (ones) and
the 4 (ones):
The answer is 20.
?
15
16
Multiplication
4
x15
20
40
?
Multiplication
Now we need to multiply the
second column (tens).
We multiply the 1 with the 4.
Because the 1 is in the tens column
it actually represents 10.
The answer is 40.
Because we are multiplying
with the tens, we will write the
answer under the 20 in the tens
column.
All we need to do now
is add the 20 and the
40 together.
This gives us an
answer of 60.
Next step: we add the 20 and the 40 together.
17
18
Multiplication
Multiplication(Long Multiplication)
2
3
4
2
1
3
H
15 x 12 = ?
Let’s do an example of 2, two-digit numbers:
1
2
3
4
5
6
7
8
9
10
11
12
1
13
14
15
16
17
3
4
5
6
7
8
5
9
10
Ed
1
15
x12
0
T
Ones
15
x12
30
18
2
x 12 =
4
x15
20
40
60
multiply the 2 and the 5.
The answer is 10.
As we did in addition we write
the 0 (ones) and we place
We multiply the 2(ones)
and the 1(tens):
The answer is 2. Now we
must remember to add
the 1(tens) that was carried.
The answer is 3 (tens).
Write answer in tens column.
?
50
VolumeVII: NCTM K-7
Mathematics
Multiplication
20
19
Multiplication
H
T
Multiplication
Ones
15
x12
30
50
?
H
Now we need to multiply
the second column (tens). We
multiply the 1(tens) with the
5(ones). Because the 1 is
in the tens column it actually
represents 10. The answer is 50.
Because we are multiplying
with the tens, we will write the
answer under the 30 in the tens
column.
T
Ones
15
x12
30
150
?
21
Multiplication
H
T
Ones
15
x12
30
150
180
We now add the 150 and the 30
together. The answer is 180.
Write the answer in the correct
columns below the 150.
Multiplication
H
T
Ones
15
x10
150
Multiplication
T
22
There are also other methods to use
when multiplying larger numbers.
If we take the same example,
15 x 12 = ?
H
Ones
15
x10
?
Ed
+
T
Ones
15
x 2
?
23
H
Step1:
15 x 10 = 150. We
know when we
multiply with 10
we add a 0 to the
number.
H
Next we multiply 1(tens) with
1(tens) and write the answer
next to the 50.
Remember because we are
multiplying in the tens
columns we are multiplying
tens with tens therefore the
answer will be in the
hundreds column.
T
Ones
15
x 2
30
Step 2:
Now 15 x 2 = 30. The
2 is the remainder
after we used the 10.
Step 3:All we have to do now is add the two together.
150 + 30 = 180
51
We can
do it
this way.
Division
VolumeVII: NCTM K-7
Mathematics
Division
1
2
When we do division problems or equations we use the division symbol to
represent the dividing of numbers.
Let’s start with division in relation to multiplication.
Division can be described as sharing and grouping.
Symbol
Meaning
Used in:
Division 6
If we have to say this
equation we will do it like this.
Ed
In multiplication we said that:
4 rows
4 x 3 = 12
X
or
3=2
3 rows
3 x 4 = 12
divide equals
Ed
=
12
6 divided by 3 equals 2
3
4
Let’s answer these two questions.
We can also say that:
1. How many rows of 4 go into 12?
4 rows
12 = 3 rows of 4
3 rows
?
or
Ed
12 = 4 rows of 3
=
12
12
5
6
3 rows
?
Ed
12
12
We can see that there are
3 rows of 4 in 12.
Division Introduction
53
VolumeVII: NCTM K-7
Mathematics
Division
8
7
Division
Division
So we can see that 12 is made up of 3 rows of 4 or 4 rows of 3.
There are 3 rows of 4 in 12.
3 rows
12 =
4 rows
There are 4 rows of 3 in 12.
4 rows
12 =
12
We can see that there are
4 rows of 3 in 12.
9
10
Division
Division
We can now make the following deductions:
We can also see that:
If 4 rows of 3 add up to 12, then we can say:
If 3 rows of 4 add up to 12, then we can say:
4
3 will divide into 12 , 4 times:
4 will divide into 12 , 3 times:
times
3 divide into
3
4 divide into
We will write it this way:
12
Ed
We will write it this way:
3=4
12
4=3
11
Division
If I have 8 jellybeans and 2 children, how many jellybeans will each child get.
8jellybeans
2children = ?
4
times
equal to or (=)
2 divide into 8
8jellybeans
2children = 4
Division continues...
54
times
VolumeVII: NCTM K-7
Mathematics
Division
13
Division
Division can be divided into two groups,
Let’s first look
at sharing.
14
Division
sharing and grouping:
We know the total number of plates but we do not know how many donuts go into each plate.
We know the total number of baskets but we do not know how many
donuts go into each plate:
Example: I have to put 12 donuts into 3 plates.
Example: I have to put 12 donuts into 3 plates.
How many donuts go into each of the 3 plates so that each plate has an equal amount of donuts.
How many donuts go into each of the 3 plates so that each plate has an equal amount of donuts.
If we share the 12 donuts equally between
the 3 plates each plate will have 4 donuts.
Let’s share the donuts one by one between the plates.
12
Division
Sharing is when:
Sharing is when:
15
16
Division
Sharing is when:
3=4
Sharing is when:
apples go into each basket:
apples go into each basket:
Let’s look at another sharing example:
Example: There are 15 apples on the tree. I have 3 children with one basket each.
We have to divide the 15 apples on the tree equally between the 3 baskets.
How many apples will go into each basket?
Example: There are 15 apples on the tree. I have 3 children with one basket each.
We have to divide the 15 apples on the tree equally between the 3 baskets.
How many apples will go into each basket?
Let’s share the apples between the 3 baskets.
15 apples divided into 3 baskets will give us 5 apples in each basket.
17
18
Division
Division
Example: There are 4 cars to transport 20 people. How many people will
go into each car if we want to divide the 20 people equally?
Grouping is when:
Now we can look
at grouping
We know the total number of apples in each basket but we
do not know how many baskets.
Example: I have 12 apples to put into baskets holding 3 each.
We need to know how many people will go equally into each car?
20
people
We need to know how many baskets we will need?
We know that each basket must hold 3 apples so we
must first divide the 12 apples in groups of 3.
4
cars
?
We can see that if we divide 20 people with into 4 cars we will fit 5 people in each car.
20
4=5
?
?
?
55
?
?
VolumeVII: NCTM K-7
Mathematics
Division
19
Division
Now we can look
at grouping
20
Division
Let’s look at another
grouping examples:
Grouping is when:
We know the total number of apples in each basket but we
do not know how many baskets.
Example: I have 12 apples to put into baskets holding 3 each.
Grouping is when:
We know the total number of people in each bus but we
do not know how many buses.
Example: There are 24 people to transport. Each bus will take 6 people.
How many buses will we need to transport all the people?
We need to know how many buses we will need to transport 24 people?
We need to know how many baskets we will need?
24
people
We now have 4 groups of 3 so we know we need 4 baskets.
6
6
6
6
4
busses
We can see that if we divide 24 people with 6 we will need 4 buses to transport the people.
24
21
Division
6=4
22
Division
The 2x table compared with division
We can see that division and multiplication go hand in hand.
In multiplication we suggested that you learn your tables.
If you know your tables you will find division much easier.
2
2
2
2
2
2
2
2
2
2
2
2
Compare times tables with division.
Multiplication
Division
2x2=4
as to
4
2 x 5 = 10
as to
10
2=2
2=5
x
x
x
x
x
x
x
x
x
x
x
x
1=2
2=4
3=6
4=8
5 = 10
6 = 12
7 = 14
8 = 16
9 = 18
10 = 20
11 = 22
12 = 24
23
2
4
6
8
10
12
14
16
18
20
22
24
2=1
2=2
2=3
2=4
2=5
2=6
2=7
2=8
2=9
2 = 10
2 = 11
2 = 12
24
Division
Division
5
5
5
5
5
5
5
5
5
5
5
5
x
x
x
x
x
x
x
x
x
x
x
x
1=5
2 = 10
3 = 15
4 = 20
5 = 25
6 = 30
7 = 35
8 = 40
9 = 45
10 = 50
11 = 55
12 = 60
5
10
15
20
25
30
35
40
45
50
55
60
5
5
5
5
5
5
5
5
5
5
5
5
=
=
=
=
=
=
=
=
=
=
=
=
1
2
3
4
5
6
7
8
9
10
11
12
10
10
10
10
10
10
10
10
10
10
10
10
x
x
x
x
x
x
x
x
x
x
x
x
1 = 10
2 = 20
3 = 30
4 = 40
5 = 50
6 = 60
7 = 70
8 = 80
9 = 90
10 = 100
11 = 110
12 = 120
10
20
30
40
50
60
70
80
90
100
110
120
56
10
10
10
10
10
10
10
10
10
10
10
10
=
=
=
=
=
=
=
=
=
=
=
=
1
2
3
4
5
6
7
8
9
10
11
12
VolumeVII: NCTM K-7
Mathematics
Division
26
25
Division
Division
Let’s look at an example:
Recording of division
We write division in the following manner:
T O
T O
?
3 18
6
3=2
or
2
6
3
18
3=?
or
Ed
How many times does 3 go into 18 ?
27
27
Division
Let’s do this example of division by inspection:
Division
18
3=?
or
?
3 18
How many times does 3 go into 18 ?
3
=?
Ed
?
6 times
Ed
3=7
or
3
3 goes into ?, 7 times.
6
3 18
29
Division
?
3=7
or
3
7
?
3 goes into ?, 7 times.
Because we know there is a relation between multiplication
and division we can say the following:
3 times 7 = 21 = 3 x 7 = 21
that means
7
3 21 = 21
3=7
57
7
?
VolumeVII: NCTM K-7
Mathematics
Division
2
1
Short Division
Short Division
Recording of division
2
We write division in the following manner:
3=2
or
3
6 divided by 2 is equal to 3
The reason we record division in the following manner, is when we
get more complex problems it will be easier to solve:
Example:
T O
6
3
6
2
6
In this equation we want to know how many times 2 will
divide into 248:
To make it easier we divide 2 into 248, step by step:
HT O
Ed
?
2 248
Let’s look at short division.
3
4
Short Division
Short Division
HTO
HTO
?
?
We will divide 2 into 248, step by step starting with the
hundreds then the tens and last we will do the ones.
2 248
2 248
H TO
Step 1
H TO
Step 2
1
2 248
2 will divide into 2,
1 time and we will
record the answer
on the top.
2 times and we will
record the answer
on the top.
12
2 248
5
6
Short Division
Short Division with Regrouping
We will now look at short division with regrouping, in more detail.
HTO
?
2 248
Step 3
4 times and we will
record the answer
on the top.
Example:
We can see that 3 will
not divide into 2.
H TO
124
2 248
225
We
can see
that 2 will divide
into 248,
124 times.
3=?
or
?
3 225
?
3 225
H T O
?
3 225
Compare with
long division
method.
Recording Division - Short
58
VolumeVII: NCTM K-7
Mathematics
Division
8
7
not divide into 2.
We will therefore divide
3 into 22. We can see
that 3 divides into 22, 7
times.
7
3 225
H T O
We will therefore divide
3 into 22. We can see that
3 divides into 22, 7 times.
7
3 225
21
Because 3 does not
divide into 22 without a
remainder we need to
write the remainder down.
7
1
3 225
9
7
1
3 225
H T O
divide into 7, 1 time.
75
3 225
21
15
We write down the
5 on top of the bar.
We can see that 3 will divide
into 225, 75 times.
We know that 4 x 1 = 4,
we subtract 4 from 7
and get 3.
We write the
remainder next to the 4
in the tens column.
4=?
?
4 744
1
4 744
12
Because 4 does not divide
into 7 without a remainder
we need to write the
remainder down.
744
or
Compare with
long division
method.
11
744
Compare with
long division
method.
10
7
3 225
21
1
and get 1. We write the
remainder next to the 5.
Compare with
long division
method.
Because we know that 3
will not divide into 1
remainder we now bring
the remaining 5 ones into
the equation.
This will give us 15 and
we know that 3 divides
into 15, 5 times.
H T O
4=?
or
?
4 744
8 times. We write the 8
on top of the bar.
1
3
4 744
744
4=?
or
?
4 744
18
3
4 744
59
VolumeVII: NCTM K-7
Mathematics
Division
14
13
744
and get a remainder of 2.
4=?
or
?
4 744
We now divide 4 into
24. We know that 4
divides into 24, 6 times.
18
3 2
4 744
We write the remainder
next to the 4 in the
ones column.
744
4=?
or
?
4 744
1 86
3 2
4 744
We can see that 4 will divide into 744, 186 times.
15
16
We will now look at short division with a remainder.
tens
hundreds
ones
H
T
tens
hundreds
ones
rem.
O
H
1
1
1
1
1
We will start by dividing 2. Two will divide into 2, 1 time.
Write the answer on top in the hundreds column.
17
18
tens
hundreds
ones
H
T
13
H
T
rem.
O
1 331
1
1
1
1
1
1
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
2
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
1
1 3 rem.
2 267
1 2 3 4 5 6 7 8 9 10
1
1
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
2
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
tens
hundreds
ones
rem.
O
1 rem.
2 267
1
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
Let’s use the M.A.B. Blocks to help us do this example. In this example we
are going to divide 267 by 2. We will see that 2 will divide into 267 but there
will be a remainder left. Let’s do it step by step.
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
1
2
2 267
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1
1
1
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
2
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
rem.
O
1
rem.
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
T
1 33rem.1
2 267
Next we will divide 7 by 2. Two will divide into 7, 3 times but there is still
remainder because we know that 2 x 3 is equal to 6.
We can see that 7 ones minus 6 ones will give 1 remainder.
Write the 3 on top in the ones column and the remainder on the right.
Next we will divide 6 by 2. Two will divide into 6, 3 times.
Write the answer on top in the tens column.
60
VolumeVII: NCTM K-7
Mathematics
Division
20
19
3145
divide into 3, 1 time.
2=?
Because 2 does not divide
into 3 without a remainder
we need to write the
remainder down.
Th H T O
1
2 3145
rem.
and get 1.
We write the
remainder next to the 1
in the tens column.
21
3145
5 times. We write the 5
on top of the bar.
2=?
We write the remainder
next to the 4.
rem.
23
Write the 7 in the tens
column on top.
Now 2 x 7 = 14 so
there is no remainder.
rem.
3145
2=?
Th H T O
15
1 1
2 3145
rem.
24
14. We know that 2
1
1
2 3145
and get a remainder of 1.
Th H T O
3145
Th H T O
15
1
2 3145
2=?
22
3145
2=?
5. We know that 2
Th H T O
15 7
1 1
2 3145
Write the answer on
top in the ones column.
rem.
We know that 2 x 2 = 4.
We subtract the 4 from the
5 and write the
1 remainder on top.
3145
2=?
Th H T O
1 5 72 1
1 1
2 3145
rem.
We can see that 2 will divide into 3145, 1572 times with 1 remainder.
61
VolumeVII: NCTM K-7
Mathematics
Division
2
1
Division by Inspection:
Let’s do this example of division by inspection:
Which number, divided by 3, produces a result of 7?
?
Ed
?
3=7
3
3 goes into ? , 7 times.
4
3=7
or
Ed
3
?
3=7
7
?
or
3
7
?
that means
7
3 21 = 21
Because we know that division is the inverse operation
of multiplication we can say the following:
3=7
because
3 times 7 = 21 = 3 x 7 = 21
Ed
7 x 3 = 21
5
6
Which number, divided by 3, produces a result of 7?
Because we know that division is the inverse operation
of multiplication we can say the following:
Ed
?
4=6
that means
6
4 24 = 24
4=6
because
Division by Inspection
62
Decimals
VolumeVII: NCTM K-7
Mathematics
Decimals
2
1
Decimal Numbers
Decimal Numbers
What do we do when we compare decimal numbers with
hundredths, to decimal numbers with thousandths?
When we compare decimal numbers, we use
the following relationship signs.
>=<
bigger than
Ed
equals
1.05 ? 1.723
smaller than
We can compare all decimal number by filling in the
zeros, to make both decimal numbers, thousandths?
3
4
Decimal Numbers
Decimal Numbers
It will then look like this:
Which number is bigger?
1.050 < 1.723
We can see clearly that 50 thousandths,
is smaller than 723 thousandths.
2.233 < 2.238
Ed
Ed
1.050 < 1.723
5
6
Decimal Numbers
Decimal Numbers
We start by looking at the ones to the
left of the decimal point.
The two whole numbers are equal. Therefore we
know look to the digits right of the decimal point.
Ed
Ones
Same
2.233 ? 2.238
2.233 ? 2.238
Decimals
Decimals
64
VolumeVII: NCTM K-7
Mathematics
Decimals
8
7
Decimal Numbers
Decimal Numbers
We now compare the tenths in the two numbers.
Which of the tenth are bigger?
We now compare the hundredths in the two numbers.
Ed
Ed
Which of the hundredth are bigger?
Same
Same
2.233 ? 2.238
2.233 ? 2.238
The tenths in these two numbers are the same, namely 2 tenths.
The hundredths in these two numbers are the
same, namely 3 hundredths.
9
10
Decimal Numbers
Ed
Decimal Numbers
Here are a few examples to study:
We now compare the thousandths in the two numbers.
Ed
Which of the thousandths are bigger?
bigger
2.233 < 2.238
The thousandths in the numbers on the right is bigger than
the thousandths in the number on the left.
1.0 ? 0.1
1.0 > 0.1
0.05 ? 0.5
0.05 < 0.5
Therefore the numbers on the right is bigger than the number on the left.
11
12
Decimal Numbers
Decimal Numbers
When we multiply the number by 10 we move each number
in the digit, one position to the left, because the number then
gets 10 times bigger.
Here are a few examples to study:
2.5 ? 3.2
2.5 > 3.2
23.754 x 10 = ?
x 10
1.28 ? 1.5
1.28 < 1.5
23.754 x 10 = 237.54
Decimals
65
VolumeVII: NCTM K-7
Mathematics
Decimals
14
13
Decimal Numbers
Decimal Numbers
When we divide the number by 10 we move each number
in the digit, one position to the right, because the number
then gets 10 times smaller.
23.754
When we multiply the number by 100 we move each number
in the digit, two positions to the left, because the number then
gets 100 times bigger.
10 = ?
23.754 x 100 = ?
10
x 100
23.754
10 = 2.3754
23.754 x 100 = 2375.4
15
16
Decimal Numbers
Decimal Numbers
When we divide the number by 100 we move each number
in the digit, two positions to the right, because the number
then gets 100 times smaller.
23.754
Study a few examples:
100 = ?
100
23.754
100 = 0.23754
Ed
2.5 x 1 = 2.5
2.5
1 = 2.5
2.5 x 10 = 25
2.5
10 = 0.25
2.5 x 100 = 250.0
2.5
100 = 0.025
2.5 x 1000 = 2500.0
2.5
1000 = 0.0025
17
18
Decimal Numbers
Decimal Numbers
The rounding off of decimal numbers.
We follow the same rule as rounding off of normal numbers.
23.751
Ed
23.751
23.75
Rounded off
Ed
Ed
Rounding off of this decimal number to two places after
the decimal point to the nearest hundredths.
Decimals
66
VolumeVII: NCTM K-7
Mathematics
Decimals
20
19
Decimal Numbers
Decimal Numbers
Round this number off to two places after the
decimal point or to the nearest hundredths.
We start by identifying the hundredths.
Ed
23.254
23.254
We now have to identify the thousandths that follow the hundredths,
to determine whether we round the number up or it stays the same.
Ed
When the digit is less than 5, the preceding digit
stays the same. If the digit is higher than 5 the preceding
digit is rounded up to the next number.
21
22
Decimal Numbers
Decimal Numbers
If the thousandths in this number was 6 what will we round the number off to?
Ed
23.254
Ed
23.256
The thousandths in this number is smaller than 5 so the
hundredths number remains the same and 4 falls away.
The thousandths in this number is bigger than 5 so the hundredths
number will be rounded up to the next value which is 6.
The answer is 23.25
The answer is 23.26
23
24
Decimal Numbers
Decimal Numbers
Now we are going to investigate converting common
fractions to decimal fractions.
2
10
=
?
4
10
=
?
3 =
5
4 =
210
?
?
4
110
1
4
=
Remember decimal fractions consist of tenth, hundredths and thousandths.
Before we can convert any common fraction we first have to change it to
tenths, hundredths and thousandths.
1
10
?
=
1
100
?
1
1000
Let’s convert these common fractions to decimal fractions:
Decimals
5
= 0.1
= 0.01
= 0.001
Ed
VolumeVII: NCTM K-7
Mathematics
Decimals
26
25
Decimal Numbers
2
10
=
?
4
10
=
?
3 =
5
4 =
210
?
?
1
10 =
4 =
110
?
1
4
=
Decimal Numbers
0.1
2
10
2 =
10 0.2
?
?
4
10
=
?
3 =
5
4 =
210
We know that tenths are
one digit after the decimal
point so two-tenths is equal
to zero point two.
Remember before we can convert
any common fraction we first have
to change it to tenths, hundredths
and thousandths.
=
?
?
1
10 =
4 =
110
?
1
4
=
4 =
10 0.4
?
We know that tenths are
one digit after the decimal
point so four-tenths is equal
to zero point four.
Remember before we can convert any
common fraction we first have to
change it to tenths, hundredths
and thousandths.
27
Decimal Numbers
0.1
28
1
10 =
Decimal Numbers
0.1
1
10 =
0.1
We first
have to convert
the fraction to tenths.
2
10
=
?
4
10
=
?
3 =
5
4 =
210
?
?
4 =
110
?
1
4
=
?
2
10
4 =
1.4
110
?
4
10
=
?
3 =
5
4 =
210
We know that the 2 wholes stay the
same when converted to decimal
fractions. We know that tenths
are one digit after the decimal
point so four-tenths is equal
to zero point four.
and thousandths.
=
?
?
1
4
=
?
=
?
4
10
=
?
1
10 =
3 =
5
4 =
210
?
?
4 =
110
?
1
4
=
?
and thousandths.
3x2 = 6
5 x 2 10
30
0.1
Decimal Numbers
1
100=
Now we can
convert the common
fraction to a decimal fraction
2
10
?
10
To change the fraction to tenths, we
have to multiply the denominator with
2 and then the numerator with 2.
and thousandths.
29
Decimal Numbers
3
5
4 =
110
?
We first have
to convert the fraction to
tenths, hundredths or thousandths.
3x2 = 6
5 x 2 10
2
10
=
?
4
10
=
?
3 =
5
6 =
10 0.6
4 =
210
?
?
4 =
110
?
1
4
=
?
and thousandths.
We know that tenths are one
digit after the decimal point
so six-tenths is equal
to zero point six.
Decimals
0.01
68
1
4
In this fraction we can not convert this
fraction to tenths because the denominator
4 can not to be converted to tenths,
therefore we try hundredths.
1 x 25 = 25
4 x 25
100
Remember what we multiply the
denominator with must also
be multiplied with
the numerator.
VolumeVII: NCTM K-7
Mathematics
Decimals
32
31
Decimal Numbers
1
100=
0.01
Now we are going to investigate converting
decimal fractions to common fractions.
Now we can
convert the common
fraction to a decimal fraction
2
10
=
?
4
10
=
3 =
5
4 =
210
?
?
1
4
=
?
and thousandths.
6 = 3
0.6 = 10
5
25 =
100 0.25
We know that hundredths are two
digit after the decimal point
so 25 hundredths is equal
to zero point two five.
Let’s look at an example:
33
34
Decimal Numbers
Decimal Numbers
Remember decimal fractions consist of tenth, hundredths and thousandths.
1
10
1
100
1
1000
= 0.1
Convert this decimal fraction to a common fraction.
0.7
Ed
We know that tenths are one digit after the decimal point so we know that the common
fraction will be tenths.
= 0.01
7
0.7 = 10
= 0.001
Therefore the common fraction of a decimal fraction is a
tenth, hundredth or thousandth.
Let’s do another example:
Convert this decimal fraction to a common fraction.
0.04
We know that hundredths are two digit after the decimal point so we know that the
common fraction will be hundredths.
4
0.04 = 100
Ed
We can see that zero point zero four as a common fraction is four-hundredths.
Decimals
Ed
We can see that zero point seven as a common fraction is seven-tenths.
35
Decimal Numbers
Ed
1 x 25 = 25
4 x 25
100
4 =
110
?
?
Decimal Numbers
69
VolumeVII: NCTM K-7
Mathematics
Decimals
2
1
Decimal Fractions
Decimal Fractions
Can you still remember the place value table?
We use a decimal sign, the decimal point, to indicate that the
digits to the left of the decimal point are wholes and the digits
to the right of the decimal point are parts of the wholes.
The place value of each digit is ten times smaller than
the digit on the left.
thousands
hundreds
tens
ones
Th
H
T
O
10
10
1000
tens
ones
H
T
O
5
7
4 3
1
Ed
Wholes
Let’s have a look at the place value
of numbers smaller than 1.
tens
ones
H
T
O
?
Parts of the
wholes
4
Decimal Fractions
Let’s investigate the first place value to the right
of the ones and the decimal point.
We use a decimal sign, the decimal point, to indicate that the
digits to the left of the decimal point are wholes and the digits
to the right of the decimal point are parts of the wholes.
hundreds
?
.
decimal
point
3
Th
?
With decimal numbers, we have tenths, hundredths and thousandths.
Decimal Fractions
thousands
decimal
point
1
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
9 10
9 10
9 10
9 10
9 10
9 10
9 10
9 10
9 10
1 2 3 4 5 6 7 8 9 10
8
8
8
8
8
8
8
8
8
hundreds
Th
10
10
1 2 3 4 5 6 7 8 9 10
7
7
7
7
7
7
7
7
7
1 2 3 4 5 6 7 8 9 10
6
6
6
6
6
6
6
6
6
1 2 3 4 5 6 7 8 9 10
5
5
5
5
5
5
5
5
5
1 2 3 4 5 6 7 8 9 10
4
4
4
4
4
4
4
4
4
1 2 3 4 5 6 7 8 9 10
3
3
3
3
3
3
3
3
3
1 2 3 4 5 6 7 8 9 10
2
2
2
2
2
2
2
2
2
1 2 3 4 5 6 7 8 9 10
theo
1 2 3 4 5 6 7 8 9 10
1
11
111
1111
1
1111 1
11111 1
1
11111111
11111 111
1
11111 1111
11111 111
11111 11
11111 1
11111
11111
1111
111
11
1
100
thousands
decimal
point
tenths
We know that the place value is 10 times smaller than 1-ones.
10
Ones
hundredths thousandths
Tenths
Ed
5
7
4 3
Wholes
.
decimal
point
.
Ed
Parts of the
wholes
With decimal numbers, we have tenths, hundredths and thousandths.
We also know that the place value is part of
.
the Ones (whole) . . a fraction.
5
6
Decimal Fractions
Decimal Fractions
Let’s look at the following illustration using blocks.
The large block
represents
one-whole.
Therefore the place value to the right of Ones and
the decimal point is tenths.
10
10
Ed
Ed
10
10
Ones
Tenths
0 . 1
1 whole
1
- We write it as a fraction: 10
When we divide one Ones (1 whole) into ten equal
1 ).
parts, one part of the whole equals one tenth ( 10
- As a decimal fraction: 0.1
1 whole
Decimals Fractions
70
- We say: zero point one
VolumeVII: NCTM K-7
Mathematics
Decimals
8
7
Decimal Fractions
Decimal Fractions
Let’s look at a few more examples:
Let’s look at a few more examples:
Diagram
Common Fraction
Decimal Fraction
3
10
7
10
2
10
0.3
0.7
0.2
12
10
1.2
Diagram
Common Fraction
Decimal Fraction
12
10
1.2
Ed
Remember, the first place after the
decimal point is tenths.
9
10
Decimal Fractions
Decimal Fractions
Let’s now investigate the second place value to the
right of the ones and the decimal point.
The blocks
represents
tenths.
We know that the place value is 10 times smaller than 1 tenths.
10
10
Ones
Tenths
10
Hundredths
Ed
Ed
0 . 0
1
10
0.1
.
When we divide one Tenths (1 tenths) into ten equal parts,
1
one part of the tenths equals one hundredths ( 100
).
11
12
Decimal Fractions
Decimal Fractions
Therefore the second place value to the right of Ones
and the decimal point is hundredths.
10
Ones
0 .
Diagram
Common Fraction
Decimal Fraction
10
Tenths
0
Hundredths
1
Ed
1
10
7 0.07
100
1
- We write it as a fraction: 100
1
10
Remember, the second place after the decimal point is hundredths.
- We say: zero point zero one
1 whole
Decimals Fractions
71
VolumeVII: NCTM K-7
Mathematics
Decimals
14
13
Decimal Fractions
Decimal Fractions
Let’s now look at the third place value to the
right of the ones and the decimal point.
The blocks
represents
one-hundredths.
We know that the place value is 10 times smaller than 1 hundredths.
10
Ones
10
Tenths
10
10
Hundredths Thousandths
Ed
Ed
0 . 0
0
1
10
0.01
.
When we divide one Hundredths (1 hundredths) into ten equal
1
parts, one part of the hundredths equals one thousandths( 1000
).
15
16
Decimal Fractions
Decimal Fractions
Therefore the third place value to the right of Ones
and the decimal point is thousandths.
10
Ed
Ones
10
0
- We write it as a fraction:
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Hundredths Thousandths
Tenths
0 .
Common Fraction
Diagram
0
1
Decimal Fraction
8 0.008
1000
Ed
1
1000
Remember, the third place after the decimal point is thousandths.
- We say: zero point zero zero one
17
18
Decimal Fractions
0.001 =
0.01 =
0.1
=
1
=
10
=
100
=
1000
=
thousandths
hundredths
tenths
Ones
Tens
Hundreds
Thousands
Decimal Fractions
This represents the place value table with thousands,
hundreds, tens, ones, tenths, hundredths and thousandths.
1.0
1.0 0.1
0.1
The decimal point
identifies the
position of the
ones and tenths
places and does
not represent a
place itself.
thousands
hundreds
tens
ones
Th
H
T
O
5
7
4 3
decimal
point
tenths
.
3
2
7
Ed
5743.327
Decimals Fractions
hundredths thousandths
72
VolumeVII: NCTM K-7
Mathematics
Decimals
2
1
Decimal Fractions - Addition:
The same principles that apply to addition of counting
numbers also apply to decimal numbers.
H
T
T O
Ones
234
+ 322
556
We start by adding tenths.
Ensure you write a zero in the
place of the ones when you
write a decimal fraction without
ones, like this one.
t h
4 3 .1 2
+ 2 4 .3 3
6 7.4 5
O
t
0 .3
+ 0 .2
?
Ed
With decimal numbers we have to place the decimal
point in right position to obtain the correct answer.
3
4
We start by adding tenths.
All we need to do is add the tenths.
O
3 tenths + 2 tenths = 5 tenths.
We can also write and add 0.3 + 0.2 as a common fraction.
t
0 .3
+ 0 .2
0 .5
Type the 5 in the tenths column.
Remember the decimal point.
There are no ones so
Ed
remember to type the zero
in the ones column.
... 0.3 + 0.2 = 0.5
Ed
3
2
5
+
=
10
10
10
5
We know that
10 = 0.5
... 0.3 + 0.2 = 0.5
5
6
We can now add numbers consisting of ones and tens.
First add the tenths.
6 tenths + 2 tenths = 8 tenths.
Ed
O
We can now add numbers consisting of ones and tens.
Now add the ones.
t
O
4 ones + 3 ones = 7 ones.
4 .6
+ 3 .2
8
Type the 7 in the ones column.
Ed
4 .6
+ 3 .2
7 .8
... 4.6 + 3.2 = 7.8
Adding Decimals
t
73
VolumeVII: NCTM K-7
Mathematics
Decimals
8
7
Let’s add tens, ones and tenths that require carrying over.
T O
First add the tenths.
6 tenths + 6 tenths. This
equals 12 tenths.
12 tenths = 1 ones + 2 tenths
t
4 2 .6
+ 3 8 .6
?
Ed
Ed
9
10
T O
1 1
11 ones = 1 tens + 1 ones.
Type the 1 ones in the ones
column and carry over
the 1 tens to the
tens column.
Ed
t
4 2 .6
+ 3 8 .6
2
Type the 2 tenths in the tenths
column and carry over the
1 whole to the ones column.
Now we add the ones.
2 ones + 8 ones. This
equals 10 ones. Add the
1 ones that was carried
over to the 10. This equal 11.
T 1O
Now we add the tens.
4 tens + 3 tens. This
equals 7 tens. Add the
1 tens that was carried
over to the 7. This equal 8.
t
4 2 .6
+ 3 8 .6
1 .2
Type the 8 tens
in the tens
column.
T 1O
1
Ed
4 2 .6
+ 3 8 .6
8 1 .2
... 42.6 + 38.6 = 81.2
Adding Decimals
t
74
VolumeVII: NCTM K-7
Mathematics
Decimals
2
1
Decimal Fractions - Multiplications:
The same principles that apply to multiplication of counting
numbers also apply to decimal numbers.
T O
H T Ones
15
x12
30
150
180
We start by multiplying tenths with one digit number.
What is 0.3 multiplied by 2?
O
t h
4 3 .1 2
x 1 2 .3 3
5 3 1.6 7
t
0 .3
x 2
?
Ed
With decimal numbers we have to place the decimal
point in right position to obtain the correct answer.
3
4
O
We start by multiplying 3 tenths
with 2. This is equal to 6 tenths.
t
0 .3
x 2
0 .8
Ed
O
We start by multiplying 3 tenths
with 2. This is equal to 6 tenths.
0 .3
x 2
0 .8
Remember the decimal point!
Now multiply the 0 Ones with
the 2, this equals 0 Ones.
Type the 0 in the ones column.
5
The answer is zero point eight.
We will again multiply tenths with a one digit number.
Ed
Ed
6
t
O
Multiply 6 tents with 3. This
equals 18 tenths.
t
0 .6
x 3
?
1
1 ones to the ones column.
Ed
O
0 .6
x 3
8
75
t
VolumeVII: NCTM K-7
Mathematics
Decimals
8
7
equals 18 tenths.
1
O
1 ones to the ones column.
Now multiply the 0 ones with 3,
this equals 0. Add the 1 ones
that was carried over and write
the answer in the ones column.
t
0 .6
x 3
1. 8
H O
t
5 .6
x
3
?
Ed
Ed
The answer is one point eight.
9
10
equals 18 tenths.
Now multiply 3 by 5 ones, this
equals 15 ones. Add the 1 ones
that was carried over to the 15.
This equal 16.
H O
1
t
5 .6
x
3
.8
column and carry
over the 1 ones to
the ones
Ed
column.
16 ones = 6 ones + 1 tens
Type the 6 ones in the
ones column and
Ed
carry over the
1 tens to the
tens column.
T O
1
1
t
5 .6
x
3
6.8
11
There are no ones to multiply.
The 1 tens which were carried
over must be added. Now type
Ed
T O
1
1
t
5 .6
x
3
1 6.8
The answer is sixteen point eight.
Multiplying Decimals Continue...
76
VolumeVII: NCTM K-7
Mathematics
Decimals
14
13
Let’s study an example where we multiply a decimal number with a two-digit number.
O
t
2 .6
x 12
?
Ed
O
equals 12 tenths.
t
2 .6
x 12
1 .2
column and the 1 ones in the
ones column.
(0.6 x 2)
Ed
15
16
O
Next we multiply 2 ones with 2.
This equals 4 ones.
4 ones = 4 ones + 0 tenths
column. Remember the
decimal point and type 0
in the tenths column
Ed
as a place holder.
Next we multiply 10 with 6 tenths.
This equals 60 tenths
t
2 .6
x 12
1 .2
4 .0
Ed
(0.6 x 2)
(2.0 x 2)
(0.6 x 10)
18
Type the 2 tens in the tens
column. Type the 0 in
the ones column
Remember the decimal
point and type 0 in
the tenths column
as a place holder.
t
2 .6
x 12
1 .2
4 .0
6 .0
column. Remember the
decimal point and type 0
in the tenths column
Ed
as a place holder.
(0.6 x 2)
(2.0 x 2)
17
Next we multiply 10 with 2 ones.
This equals 20 tenths
20 ones = 2 tens + 0 ones
O
O
Now we can add.
t
2 .6
x 12
1 .2
4 .0
6 .0
2 0 .0
O
The answer is 31.2
thirty-one point two
(0.6 x 2)
(2.0 x 2)
(0.6 x 10)
(2.0 x 10)
Ed
2 .6
x 12
1 .2
4 .0
6 .0
2 0 .0
3 1 .2
77
t
(0.6 x 2)
(2.0 x 2)
(0.6 x 10)
(2.0 x 10)
VolumeVII: NCTM K-7
Mathematics
Decimals
20
19
Let’s study an example where we multiply a
decimal number with a multiple of 10.
We know that 30 equals 10 x 3.
Now we can proceed a follows:
6.8 x 30 = ?
6.8 x 30 = 6.8 x (10 x 3)
Ed
Ed
21
22
Ed
We simplify the calculation,
by first multiplying 6.8 by 10.
Now we multiply 68 by 3.
The answer is 204.
6.8 x 30 = 6.8 x (10 x 3)
= (6.8 x 10) x 3
= 68 x 3
Ed
6.8 x 30 = 6.8 x (10 x 3)
= (6.8 x 10) x 3
= 68 x 3
= 204
23
24
The number of decimal places in the answer, is the sum of the number
of decimal places in the numbers that are being multiplied.
The number of decimal places in the answer, is the sum of the number
of decimal places in the numbers that are being multiplied.
In this example we are multiplying 2 decimal places.
In this example we are multiplying 5 decimal places.
This means that the answer will have two
decimal places after the decimal point.
This means that the answer will have five
decimal places after the decimal point.
0.1 x 0.2 = 0.02
1
2
Ed
Ed
0.01 x 0.002 = 0.00002
1 2
3 4 5
1 2
Let’s look t another example:
78
1 2 3 4 5
VolumeVII: NCTM K-7
Mathematics
Decimals
26
25
Let’s study another example where a decimal
fraction is multiplied by a decimal fraction.
0.4 is the same as;
What does 0.4 x 0.2 mean?
- four-tenths
- 4
10
Ed
27
28
0.2 is the same as;
- two-tenths
- 2
10
Ed
0.4 x 0.2 is the same as;
4 x 2
8
=
10
10
100
Ed
29
Ed
30
Yes and
8
again,
100
is 0.08
Ed
... 0.4 x 0.2 = 0.08
Multiplying Decimals Continue...
79
VolumeVII: NCTM K-7
Mathematics
Decimals
32
31
What is 2.6 multiplied by 0.6?
2 .6
x 0 .6
?
Multiply 6 by 6, this equals 36.
Type the 6 and carry over the 3.
3
2 .6
x 0 .6
6
Ed
33
Ed
34
Multiply 6 by 2, this equals 12.
Add the 3 that was carried over, this equals 15.
The decimal point is placed in such a way in the answer, 156, that the number
of digits after the decimal point, equals the sum of the number of digits after the
decimal point in 2.6 and 0.6. Therefore there will be 2 decimal points.
Type the 15. Where does the decimal point go?
Start on the right hand side of the answer and count two places or digits to the left.
The answer is 1 point 56.
3
Ed
3
2 .6
x 0 .6
156
Ed
2 .6
x 0 .6
1.5 6
80
EDUCATIONAL SOFTWARE SOLUTIONS
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ISBN 118768831-2
®
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Copyright by EDUSS Broadcast & Media Inc. 2004, All rights reserved. ISBN 0-620-21809-6

Instructional Manual Math Book 2

Transcription

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