Download - KS3 Mathematics
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N1
N1
N1
N1
N1.1 Place value
Contents
N1 Place value, ordering and rounding
N1.3 Ordering decimals
N1.4 Rounding
N1.2 Powers of ten
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What is 6.2 × 10?
Let’s look at what happens on the place value grid.
Thousands Hundreds Tens Units tenths hundredths thousandths
6 2
When we multiply by ten the digits move one place to the left.
6 2
6.2 × 10 = 62
Multiplying by 10, 100 and 1000
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What is 3.1 × 100?
Let’s look at what happens on the place value grid.
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
Thousands Hundreds Tens Units tenths hundredths thousandths
3 13 1 0
Multiplying by 10, 100 and 1000
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What is 0.7 × 1000?
Let’s look at what happens on the place value grid.
When we multiply by one thousand the digits move three places to the left.
We then add zero place holders.
0.7 × 1000 = 700
Thousands Hundreds Tens Units tenths hundredths thousandths
0 77 0 0
Multiplying by 10, 100 and 1000
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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 thousandths
4 5
When we divide by ten the digits move one place to the right.
4 5
When we write decimals it is usual to write a zero in the units column when there are no whole numbers.
0
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 4
When we divide by one hundred the digits move two places to the right.
9 4
We need to add zero place holders.
0 0
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 thousandths
5 1 0
When we divide by one thousand the digits move three places to the right.
We add a zero before the decimal point.
0
510 ÷ 1000 = 0.51
5 1
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Multiplying and dividing by 10, 100 and 1000
Complete the following:
3.4 × 10 = 34
64.34 ÷ = 0.6434100
× 45.8 = 45 8001000
43.7 × = 4370100
92.1 ÷ 10 = 9.21
73.8 ÷ = 7.3810
÷ 1000 = 8.318310
0.64 × = 6401000
0.021 × 100 = 2.1
250 ÷ = 2.5100
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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 × .110
4 × is equivalent to 4 ÷ 10.110
Therefore:
4 × 0.1 = 0.4
Multiplying by 0.1 Dividing by 10is the same as
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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.
1100We can also think of this as 3 × .
3 × is equivalent to 3 ÷ 100.1100
Therefore:
3 × 0.01 = 0.03
Multiplying by 0.01 Dividing by 100is the same as
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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 Multiplying by 10is the same as
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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 Multiplying by 100is the same as
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Complete the following:
24 × 0.1 = 2.4
52 ÷ = 52000.01
× 950 = 9.50.01
31.2 × = 3.120.1
6.51 ÷ 0.1 = 65.1
92.8 ÷ = 92800.01
÷ 0.001 = 6740.674
470 × = 0.470.001
830 × 0.01 = 8.3
0.54 ÷ = 5.40.1
Multiplying and dividing by 0.1 and 0.01
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N1
N1
N1
N1
N1.2 Powers of ten
Contents
N1.1 Place value
N1 Place value, ordering and rounding
N1.3 Ordering decimals
N1.4 Rounding
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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.
0.01 = = = 10−21102
1100
0.001 = = = 10−31103
11000
0.0001 = = = 10−4110000
1104
0.00001 = = = 10−51100000
1105
0.000001 = = = 10−611000000
1106
0.1 = = =10−1110
1101
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Standard form – writing large numbers
We can write very large numbers using standard form.
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.
To write a number in standard form we write it as a number between 1 and 10 multiplied by a power of ten.
A number between 1 and 10
A power of ten
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How can we write these numbers in standard form?
80 000 000 = 8 × 107
230 000 000 = 2.3 × 108
724 000 = 7.24 × 105
6 003 000 000 = 6.003 × 109
371.45 = 3.7145 × 102
Standard form – writing large numbers
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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 = 6764.5
Standard form – writing large numbers
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We can also write very small numbers using standard form.
The actual width of this shelled amoeba is 0.00013 m.
We can write this number as 1.3 × 10−4 m.
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.
A number between 1 and 10
A negative power of 10
Standard form – writing small numbers
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How can we write these numbers in standard form?
0.0006 = 6 × 10−4
0.00000072 = 7.2 × 10−7
0.0000502 = 5.02 × 10−5
0.0000000329 = 3.29 × 10−8
0.001008 = 1.008 × 10−3
Standard form – writing small numbers
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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 = 0.084542
Standard form – writing small numbers
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N1.3 Ordering decimals
Contents
N1.4 Rounding
N1.1 Place value
N1 Place value, ordering and rounding
N1.2 Powers of ten
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Which number is bigger:
1.72 or 1.702?
To compare two decimal numbers, look at each digit in order from left to right:
These digits are the same.
1 . 7 2
1 . 7 0 2
These digits are the same.
1 . 7 2
1 . 7 0 2
The 2 is bigger than the 0 so:
1 . 7 2
1 . 7 0 2
1.72 > 1.702
Comparing decimals
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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
Comparing decimals
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These digits are the same.
5 . 3 6
5 . 3 7 1
These digits are the same.
5 . 3 6
5 . 3 7 1
The 7 is bigger than the 6 so:
5 . 3 6
5 . 3 7 1
Next, compare the two decimal numbers by looking at each digit in order from left to right:
5.36 < 5.371
Which measurement is bigger:
5.36 kg or 5.371 kg?
Comparing decimals
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4.67 4.74.717 4.77 4.73 4.074.67 4.717 4.734.77 4.074.74.67 4.717 4.77 4.074.73 4.74.67 4.717 4.77 4.73 4.70 4.074.717 4.77 4.73 4.7
Write these decimals in order from smallest to largest:
To order these decimals we must compare the digits in the same position, starting from the left.
The digits in the unit positions are the same, so this does not help.
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.
The correct order is:
4.07 4.67 4.7 4.717 4.73 4.77
Ordering decimals
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N1
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N1
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N1.4 Rounding
Contents
N1.3 Ordering decimals
N1.1 Place value
N1 Place value, ordering and rounding
N1.2 Powers of ten
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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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Round 34 871 to the nearest 100.Round 34 871Round 34 871
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: 34871 = 34900 (to the nearest 100)
Rounding whole numbers
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Rounding whole numbers
Complete this table:
37521
274503
7630918
9875
to the nearest 1000
452
to the nearest 100
to the nearest
10
38000 37500 37520
275000 274500 274500
7631000 7630900 7630920
10000 9900 9880
0 500 450
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Round 2.75241302 to one decimal place.Round 2.75241302Round 2.75241302
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.
Rounding decimals
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Rounding to a given number of decimal places
Complete this table:
63.4721
87.6564
149.9875
3.54029
0.59999
to the nearest whole number to 1 d.p. to 2 d.p. to 3 d.p.
63 63.5 63.47 63.472
88 87.7 87.66 87.656
150 150.0 149.99 149.988
4 3.5 3.54 3.540
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
and0.0007506
This is the first significant figure
0.0007506
This is the first significant figure
4 890 351
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Rounding to significant figures
For example:
4 890 351
and0.0007506
This is the first significant figure
0.0007506
This is the first significant figure
4 890 351
The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros.
0.0007506
This is the second significant figure
4 890 351
This is the second significant figure
4 890 351
This is the third significant figure
0.0007506
This is the third significant figure
4 890 351
This is the fourth significant figure
0.0007506
This is the fourth significant figure