Transcript N1 Place value, ordering and rounding

KS3 Mathematics
N1 Place value, ordering
and rounding
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Contents
N1 Place value, ordering and rounding
NN1.1 Place value
1
NN1.2 Powers of ten
1
NN1.3 Ordering decimals
1
NN1.4 Rounding
1
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Blank cheques
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Place value
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Multiplying by 10, 100 and 1000
What is 6.2 × 10?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
6
Units
tenths
6
2
2
hundredths
thousandths
When we multiply by ten the digits move one place to
the left.
6.2 × 10 = 62
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Multiplying by 10, 100 and 1000
What is 3.1 × 100?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
3
Tens
Units
tenths
1
1
3
0
hundredths
thousandths
When we multiply by one hundred the digits move two
places to the left.
We then add a zero place holder.
3.1 × 100 = 310
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Multiplying by 10, 100 and 1000
What is 0.7 × 1000?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
7
Tens
Units
tenths
7
0
0
0
hundredths
thousandths
When we multiply by one thousand the digits move three
places to the left.
We then add zero place holders.
0.7 × 1000 = 700
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Dividing by 10, 100 and 1000
What is 4.5 ÷ 10?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
Units
tenths
hundredths
4
0
5
4
5
thousandths
When we divide by ten the digits move one place to the
right.
When we write decimals it is usual to write a zero in the
units column when there are no whole numbers.
4.5 ÷ 10 = 0.45
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Dividing by 10, 100 and 1000
What is 9.4 ÷ 100?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
Units
tenths
hundredths
thousandths
9
0
4
0
9
4
When we divide by one hundred the digits move two
places to the right.
We need to add zero place holders.
9.4 ÷ 100 = 0.094
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Dividing by 10, 100 and 1000
What is 510 ÷ 1000?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
Units
tenths
hundredths
5
1
0
0
5
1
thousandths
When we divide by one thousand the digits move three
places to the right.
We add a zero before the decimal point.
510 ÷ 1000 = 0.51
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Spider diagram
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Multiplying and dividing by 10, 100 and 1000
Complete the following:
3.4 × 10 =
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34
73.8 ÷ 10 = 7.38
64.34 ÷ 100 = 0.6434
8310 ÷ 1000 = 8.31
1000 × 45.8 = 45 800
0.64 × 1000 = 640
43.7 × 100 = 4370
0.021 × 100 = 2.1
92.1 ÷ 10 = 9.21
250 ÷ 100 = 2.5
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Multiplying by 0.1 and 0.01
What is 4 × 0.1?
We can think of this as 4 lots of 0.1 or 0.1 + 0.1 + 0.1 + 0.1.
We can also think of this as 4 ×
1
10.
1
4 × 10 is equivalent to 4 ÷ 10.
Therefore:
4 × 0.1 = 0.4
Multiplying by 0.1
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is the same as
Dividing by 10
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Multiplying by 0.1 and 0.01
What is 3 × 0.01?
We can think of this as 3 lots of 0.01 or 0.01 + 0.01 + 0.01.
We can also think of this as 3 ×
3×
1
100
1
100 .
is equivalent to 3 ÷ 100.
Therefore:
3 × 0.01 = 0.03
Multiplying by 0.01
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is the same as
Dividing by 100
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Dividing by 0.1 and 0.01
What is 7 ÷ 0.1?
We can think of this as “How many 0.1s (tenths) are there
in 7?”.
There are ten 0.1s (tenths) in each whole one.
So, in 7 there are 7 × 10 tenths.
Therefore:
7 ÷ 0.1 = 70
Dividing by 0.1
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is the same as Multiplying by 10
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Dividing by 0.1 and 0.01
What is 12 ÷ 0.01?
We can think of this as “How many 0.01s (hundredths) are
there in 12?” .
There are a hundred 0.01s (hundredths) in each whole one.
So, in 12 there are 12 × 100 hundredths.
Therefore:
12 ÷ 0.01 = 1200
Dividing by 0.01
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is the same as Multiplying by 100
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Multiplying and dividing by 0.1 and 0.01
Complete the following:
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24 × 0.1 = 2.4
92.8 ÷ 0.01 = 9280
52 ÷ 0.01 = 5200
0.674 ÷ 0.001 = 674
0.01 × 950 = 9.5
470 × 0.001 = 0.47
31.2 × 0.1 = 3.12
830 × 0.01 = 8.3
6.51 ÷ 0.1 = 65.1
0.54 ÷ 0.1 = 5.4
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Multiplying by small multiples of 0.1
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Contents
N1 Place value, ordering and rounding
NN1.1 Place value
1
NN1.2 Powers of ten
1
NN1.3 Ordering decimals
1
NN1.4 Rounding
1
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Powers of ten
Our decimal number system is based on powers of ten.
We can write powers of ten using index notation.
10 = 101
100 = 10 × 10 = 102
1000 = 10 × 10 × 10 = 103
10 000 = 10 × 10 × 10 × 10 = 104
100 000 = 10 × 10 × 10 × 10 × 10 = 105
1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106
10 000 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 107 …
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Negative powers of ten
Any number raised to the power of 0 is 1, so
1 = 100
We use negative powers of ten to give us decimals.
0.1 =
1
10
1
101
=
1
0.01 = 100 =
1
102
1
0.001 = 1000 =
0.0001 =
1
10000
0.00001 =
= 10−2
1
103
=
1
100000
0.000001 =
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=10−1
= 10−3
1
104
=
1
1000000
= 10−4
1
105
=
= 10−5
1
106
= 10−6
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Standard form – writing large numbers
We can write very large numbers using standard form.
To write a number in standard form we write it as a number
between 1 and 10 multiplied by a power of ten.
For example, the average
distance from the earth to the
sun is about 150 000 000 km.
We can write this
number as 1.5 × 108 km.
A number
between 1 and 10
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A power of ten
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Standard form – writing large numbers
How can we write these numbers in standard form?
80 000 000 =
8 × 107
230 000 000 =
2.3 × 108
7.24 × 105
724 000 =
6 003 000 000 =
371.45 =
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6.003 × 109
3.7145 × 102
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Standard form – writing large numbers
These numbers are written in standard form. How can they
be written as ordinary numbers?
5 × 1010 =
50 000 000 000
7.1 × 106 =
7 100 000
4.208 × 1011 =
420 800 000 000
2.168 × 107 =
21 680 000
6.7645 × 103 =
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6764.5
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Standard form – writing small numbers
We can also write very small numbers using standard form.
To write a small number in standard form we write it as a
number between 1 and 10 multiplied by a negative power
of ten.
The actual width of this
shelled amoeba is
0.00013 m.
We can write this
number as 1.3 × 10−4 m.
A number
between 1 and 10
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A negative
power of 10
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Standard form – writing small numbers
How can we write these numbers in standard form?
0.0006 =
0.00000072 =
0.0000502 =
0.0000000329 =
0.001008 =
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6 × 10−4
7.2 × 10−7
5.02 × 10−5
3.29 × 10−8
1.008 × 10−3
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Standard form – writing small numbers
These numbers are written in standard form. How can they
be written as ordinary numbers?
8 × 10−4 =
0.0008
2.6 × 10−6 =
0.0000026
9.108 × 10−8 =
0.00000009108
7.329 × 10−5 =
0.00007329
8.4542 × 10−2 =
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0.084542
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Contents
N1 Place value, ordering and rounding
NN1.1 Place value
1
NN1.2 Powers of ten
1
NN1.3 Ordering decimals
1
NN1.4 Rounding
1
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Zooming in on a number line
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Decimal sequences
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Decimals on a number line
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Mid-points
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Comparing decimals
Which number is bigger:
1.72 or 1.702?
To compare two decimal numbers, look at each digit in order
from left to right:
1.72
1.702
These
The
2 is
digits
bigger
arethan
the same.
the 0 so:
1.72 > 1.702
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Comparing decimals
Which measurement is
bigger:
5.36 kg or 5371 g?
To compare two measurements, first write both
measurements using the same units.
We can convert the grams to kilograms by dividing
by 1000:
5371 g = 5.371 kg
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Comparing decimals
Which measurement is
bigger:
5.36 kg or 5.371 kg?
Next, compare the two decimal numbers by looking at each
digit in order from left to right:
5.36
5.371
These
The
7 is
digits
bigger
arethan
the same.
the 6 so:
5.36 < 5.371
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Comparing decimals
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Ordering decimals
Write these decimals in order from smallest to largest:
4.67
4.717
4.77
4.73
4.7
4.70
4.07
To order these decimals we must compare the digits in
Thesame
correct
order is:
the
position,
starting from the left.
The digits in the unit positions are the same, so this does
4.07
4.7
4.717
4.73
4.77
not
help. 4.67
Looking at the first decimal place tells us that 4.07 is the
smallest followed by 4.67.
Looking at the second decimal place of the remaining
numbers tells us that 4.7 is the smallest followed by 4.717,
4.73 and 4.77.
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Ordering decimals
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Dewey Decimal Classification System
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Contents
N1 Place value, ordering and rounding
NN1.1 Place value
1
NN1.2 Powers of ten
1
NN1.3 Ordering decimals
1
NN1.4 Rounding
1
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Rounding
We do not always need to know the exact value of a number.
For example:
There are 1432
pupils at Eastpark
Secondary School.
There are about one
and a half thousand
pupils at Eastpark
Secondary School.
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Rounding readings from scales
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Rounding whole numbers
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Rounding whole numbers
Round 34 871 to the nearest 100.
Look at the digit in the hundreds position.
We need to write down every digit up to this.
Look at the digit in the tens position.
If this digit is 5 or more then we need to round up the digit
in the hundreds position.
Solution:
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34871 = 34900 (to the nearest 100)
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Rounding whole numbers
Complete this table:
to the nearest
1000
to the nearest
100
to the nearest
10
37521
38000
37500
37520
274503
275000
274500
274500
7630918
7631000
7630900
7630920
9875
10000
9900
9880
452
0
500
450
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Rounding decimals
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Rounding decimals
Round 2.75241302 to one decimal place.
Look at the digit in the first decimal place.
We need to write down every digit up to this.
Look at the digit in the second decimal place.
If this digit is 5 or more then we need to round up the digit
in the first decimal place.
2.75241302 to 1 decimal place is 2.8.
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Rounding to a given number of decimal places
Complete this table:
to the nearest
whole number
to 1 d.p.
to 2 d.p.
to 3 d.p.
63.4721
63
63.5
63.47
63.472
87.6564
88
87.7
87.66
87.656
149.9875
150
150.0
149.99
149.988
3.54029
4
3.5
3.54
3.540
0.59999
1
0.6
0.60
0.600
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Rounding to significant figures
Numbers can also be rounded to a given number of
significant figures.
The first significant figure of a number is the first digit
which is not a zero.
For example:
4 890 351
This is the first significant figure
and
0.0007506
This is the first significant figure
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Rounding to significant figures
The second, third and fourth significant figures are the
digits immediately following the first significant figure,
including zeros.
For example:
4 890 351
This
is
the
first
significant
figurefigure
This
This
This
is is
the
isthe
the
second
third
fourth
significant
significant
significant
figure
figure
and
0.0007506
This
is
first
significant
figure
This
This
This
isthe
is
the
isthe
the
second
third
fourth
significant
significant
significant
figure
figure
figure
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Rounding to significant figures
Complete this table:
to 3 s. f.
to 2 s. f.
to 1 s. f.
6.3528
6.35
6.4
6
34.026
34.0
34
30
0.005708
0.00571
0.0057
0.006
150.932
151
150
200
0.0000784
0.000078
0.00008
0.00007835
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