KS3 Mathematics

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N1 Place value, ordering and rounding

KS3 Mathematics


N1

N1

N1

N1

N1.1 Place value

Contents


N1.3 Ordering decimals

N1.4 Rounding

N1.2 Powers of ten


Blank cheques


Place value


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


What is 3.1 × 100?


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


3 13 1 0



What is 0.7 × 1000?


When we multiply by one thousand the digits move three places to the left.

We then add zero place holders.

0.7 × 1000 = 700


0 77 0 0



Dividing by 10, 100 and 1000

What is 4.5 ÷ 10?



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



What is 9.4 ÷ 100?



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



What is 510 ÷ 1000?



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


Spider diagram


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


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


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


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


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


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


Multiplying by small multiples of 0.1


N1

N1

N1

N1

N1.2 Powers of ten

Contents

N1.1 Place value



N1.4 Rounding


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 …


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


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


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



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



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


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



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



N1

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N1


Contents

N1.4 Rounding

N1.1 Place value


N1.2 Powers of ten


Zooming in on a number line


Decimal sequences


Decimals on a number line


Mid-points


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


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


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



5 . 3 6

5 . 3 7 1


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


Comparing decimals


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


Ordering decimals


Dewey Decimal Classification System


N1

N1

N1

N1

N1.4 Rounding

Contents


N1.1 Place value


N1.2 Powers of ten


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.


Rounding readings from scales


Rounding whole numbers


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)




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


Rounding decimals


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


Rounding to a given number of decimal places


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


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


4 890 351



For example:

4 890 351

and0.0007506


0.0007506


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



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


KS3 Mathematics

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