© Boardworks Ltd 2008 1 of 51 N1 Place value, ordering and rounding Maths Age 11-14.
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Transcript of © Boardworks Ltd 2008 1 of 51 N1 Place value, ordering and rounding Maths Age 11-14.
© Boardworks Ltd 2008 1 of 51
N1 Place value, ordering and rounding
Maths Age 11-14
© Boardworks Ltd 2008 2 of 51
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
© Boardworks Ltd 2008 3 of 51
Blank cheques
© Boardworks Ltd 2008 4 of 51
Place value
© Boardworks Ltd 2008 5 of 51
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
© Boardworks Ltd 2008 6 of 51
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
© Boardworks Ltd 2008 7 of 51
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
© Boardworks Ltd 2008 8 of 51
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
© Boardworks Ltd 2008 9 of 51
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
© Boardworks Ltd 2008 10 of 51
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
© Boardworks Ltd 2008 11 of 51
Spider diagram
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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
© Boardworks Ltd 2008 13 of 51
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
© Boardworks Ltd 2008 14 of 51
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
© Boardworks Ltd 2008 15 of 51
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
© Boardworks Ltd 2008 16 of 51
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
© Boardworks Ltd 2008 17 of 51
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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Multiplying by small multiples of 0.1
© Boardworks Ltd 2008 19 of 51
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
© Boardworks Ltd 2008 20 of 51
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 …
© Boardworks Ltd 2008 21 of 51
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
© Boardworks Ltd 2008 22 of 51
N1
N1
N1
N1
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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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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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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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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Ordering decimals
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Dewey Decimal Classification System
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N1
N1
N1
N1
N1.4 Rounding
Contents
N1.3 Ordering decimals
N1.1 Place value
N1 Place value, ordering and rounding
N1.2 Powers of ten
© Boardworks Ltd 2008 35 of 51
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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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
© Boardworks Ltd 2008 40 of 51
Rounding decimals
© Boardworks Ltd 2008 41 of 51
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
© Boardworks Ltd 2008 43 of 51
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
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Complete this table:
6.3528
34.026
0.005708
150.932
to 3 s. f.
0.00007835
to 2 s. f. to 1 s. f.
6.35 6.4 6
34.0 34 30
0.00571 0.0057 0.006
151 150 200
0.0000784 0.000078 0.00008
Rounding to significant figures