1. N1 Place value, ordering and rounding
KS3 Mathematics
The aim of this unit is to teach pupils to:
Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 10.
Compare and order decimals.
Round numbers, including to a given number of decimal places or significant figures.
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
N1 Place value, ordering and rounding

2. N1 Place value, ordering and rounding
Contents
N1 Place value, ordering and rounding N1
N1.1 Place value N1
N1.2 Powers of ten N1
N1.3 Ordering decimals N1
N1.4 Rounding

3. 6 2 6 2 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
Units
tenths
hundredths
thousandths 6 2 6 2
This is the first of three slides illustrating the effect of multiplying a decimal number by 10, 100 and 1000.
Start by checking that pupils can multiply whole numbers by 10, 100 and 1000 and divide whole numbers ending in 0, 00 and 000 by 10, 100 and 1000.
Tell pupils that we often need to be able to multiply and divide numbers by 10, 100 and 1000 in every day life, for example when converting between metric units. Give examples as necessary.
Explain that we are now going to look at the effect of multiplying decimals by 10, 100 and 1000.
Look at the question on the slide. Most pupils will be able to solve this without reference to the place value grid.
Reveal each step on the slide and encourage the idea that the digits are moved one place to the left to make the number ten times bigger (rather than the decimal point being moved one place to the right).
Links:
N9 Mental methods – multiplication and division
S7 Measures – converting units
When we multiply by ten the digits move one place to the left.
6.2 × 10 = 62

4. 3 1 3 1 Multiplying by 10, 100 and 1000 What is 3.1 × 100?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
Units
tenths
hundredths
thousandths 3 1 3 1
When we multiply by one hundred the digits move two places to the left.
Again, stress that it is the digits that are being moved two places to the left and not the decimal point that is being moved two places to the right.
We then add a zero place holder.
3.1 × 100 = 310

5. 7 7 Multiplying by 10, 100 and 1000 What is 0.7 × 1000?
Let’s look at what happens on the place value grid.
Thousands
Hundreds
Tens
Units
tenths
hundredths
thousandths 7 7
When we multiply by one thousand the digits move three places to the left.
We then add zero place holders.
0.7 × 1000 = 700

6. 4 5 4 5 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 4 5
When we divide by ten the digits move one place to the right.
The next three slides illustrate the effect of dividing a decimal number by 10, 100 and 1000.
Link:
N9 Mental methods – multiplication and division
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

7. 9 4 9 4 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 9 4
When we divide by one hundred the digits move two places to the right.
Link:
N9 Mental methods – multiplication and division
We need to add zero place holders.
9.4 ÷ 100 = 0.094

8. 5 1 5 1 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 5 1
When we divide by one thousand the digits move three places to the right.
Link:
N9 Mental methods – multiplication and division
We add a zero before the decimal point.
510 ÷ 1000 = 0.51

9. Multiplying and dividing by 10, 100 and 1000
Complete the following:
3.4 × 10 = 34
73.8 ÷ = 7.38 10
64.34 ÷ =
100
8310
÷ 1000 = 8.31
1000
× 45.8 =
0.64 × = 640
1000
Complete the exercise as a class.
An alternative would be to erase the answers from the slide (or change the animation order) and ask pupils to complete the exercise individually; you could also give pupils a worksheet with similar calculations.
43.7 × = 4370
100
0.021 × 100 =
2.1
92.1 ÷ 10 =
9.21
250 ÷ = 2.5
100

10. Multiplying by 0.1 and 0.01 What is 4 × 0.1?
We can think of this as 4 lots of 0.1 or
We can also think of this as 4 × 1 10
4 × is equivalent to 4 ÷ 10 1 10
Therefore,
There are many ways in which to visualize the multiplication of a whole number by 0.1. Discuss as a group.
Ask pupils what they think 2.7 × 0.1 is, or 32.8 × 0.1.
4 × 0.1 = 0.4
Multiplying by 0.1
Dividing by 10
is the same as

11. Multiplying by 0.1 and 0.01 What is 3 × 0.01?
We can think of this as 3 lots of 0.01 or 1
100
We can also think of this as 3 ×
3 × is equivalent to 3 ÷ 100 1
100
Therefore,
Extend the discussion to multiplication by 0.01.
3 × 0.01 = 0.03
Multiplying by 0.01
Dividing by 100
is the same as

12. 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,
Discuss ways to visualize dividing whole numbers by 0.1. Extend to dividing decimals by 0.1.
7 ÷ 0.1 = 70
Dividing by 0.1
Multiplying by 10
is the same as

13. 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,
Extend the discussion to dividing by 0.01.
12 ÷ 0.01 = 1200
Dividing by 0.01
Multiplying by 100
is the same as

14. Multiplying and dividing by 0.1 and 0.01
Complete the following:
24 × 0.1 =
2.4
92.8 ÷ = 9280
0.01
52 ÷ = 5200
0.01
0.674
÷ = 674
× 950 = 9.5
0.01
470 × = 0.47
0.001
Complete the exercise as a class.
An alternative would be to erase the answers from the slide (or change the animation order) and ask pupils to complete the exercise individually; you could also give pupils a worksheet with similar calculations.
31.2 × = 3.12
0.1
830 × 0.01 =
8.3
6.51 ÷ 0.1 =
65.1
0.54 ÷ = 5.4
0.1

15. N1 Place value, ordering and rounding
Contents
N1 Place value, ordering and rounding N1
N1.1 Place value N1
N1.2 Powers of ten N1
N1.3 Ordering decimals N1
N1.4 Rounding

16. 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 × 10 × 10 × 10 = 104
= 10 × 10 × 10 × 10 × 10 = 105
Discuss the use of index notation to describe numbers like 10, 100 and 1000 as powers of 10. Be aware that pupils often confuse powers with multiples and reinforce the idea of a power as a number, in this case 10, repeatedly multiplied by itself.
Make sure that pupils know that 103, for example, is said as “ten to the power of three”.
Explain that the index tells us how many 0s will follow the 1 (this is only true for positive integer powers of ten).
You may wish to mention that the standard meaning of a billion is 109 (one thousand million). In the past, a British billion was one million million, or 1012.
Links:
N4 Powers and roots – powers
= 10 × 10 × 10 × 10 × 10 × 10 = 106
= 10 × 10 × 10 × 10 × 10 × 10 × 10 = 107 …

17. 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 = = =10-1 1 10
101
0.01 = = = 10-2 1
102
100
0.001 = = = 10-3 1
103
1000
= = = 10-4 1
10000
104
Discuss the convention of using negative integers to express tenths, hundredths, thousandths, and other negative powers of ten.
Link:
N4 Powers and roots – powers
= = = 10-5 1
100000
105
= = = 10-6 1
106

18. 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 km.
Discuss the use of standard form to write large numbers.
Discuss whether 10×105 and 1×104 are correct examples of standard form. The first is not whereas the second is; if the discussion is interesting, you might want to discuss how to use inequality notation to show exactly what is meant by ‘between 1 and 10’ in the definition of standard form.
We can write this number as 1.5 × 108 km.
A number between 1 and 10
A power of ten

19. Standard form – writing large numbers
How can we write these numbers in standard form? =
8 × 107 =
2.3 × 108 =
7.24 × 105
Discuss how each number should be written in standard form.
Notice that the power of ten will always be one less than the number of digits in the number. =
6.003 × 109 =
× 102

20. Standard form – writing large numbers
These numbers are written in standard form. How can they be written as ordinary numbers?
5 × 1010 =
7.1 × 106 =
4.208 × 1011 =
Discuss how each number written in standard form should be written in full.
2.168 × 107 =
× 103 =
6764.5

21. 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.
For example, the width of this shelled amoeba is m.
We can write this number as 1.3 × 10-4 m.
The image of a shelled amoeba has been reproduced with the kind permission of Wim van Egmond © Microscopy UK
A number between 1 and 10
A negative power of 10

22. Standard form – writing small numbers
How can we write these numbers in standard form? =
6 × 10-4 =
7.2 × 10-7 =
5.02 × 10-5
Notice that the power of ten is always ‘–’ and the number of zeros before the first significant figure including the one before the decimal point.. =
3.29 × 10-8 =
1.008 × 10-3

23. 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 =
9.108 × 10-8 =
Notice that the power of ten tells us the number of zeros before the first significant figure including the one before the decimal point..
7.329 × 10-5 =
× 10-2 =

24. N1 Place value, ordering and rounding
Contents
N1 Place value, ordering and rounding N1
N1.1 Place value N1
N1.2 Powers of ten N1
N1.3 Ordering decimals N1
N1.4 Rounding

25. Which number is bigger:
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:
These digits are the same.
These digits are the same.
The 2 is bigger than the 0 so:
Explain that to compare two decimals we must look at digits in the same position starting from the left. Explain that the digit furthest to the left is the most significant digit.
Talk through the example on the board:
Both numbers have 1 unit, so that doesn’t help.
Both numbers have 7 tenths, so that doesn’t help either.
The first number has 2 hundredths and the second number has no hundredths.
1.72 must therefore be bigger than
Some pupils may need reminding of the meaning of the ‘greater than’ symbol.
1.72 > 1.702

26. 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:
Stress that to compare measurements we must convert them to the same units.
Ask pupils how to convert grams to kilograms before revealing this on the board.
Point out that we also have chosen to convert both units to grams to compare the measurements.
5371 g = kg

27. Comparing decimals 5.36 < 5.371 Which measurement is bigger:
5.36 kg or kg?
Next, compare the two decimal numbers by looking at each digit in order from left to right:
The 7 is bigger than the 6 so:
These digits are the same.
These digits are the same.
5.36 < 5.371

28. Ordering decimals
Write these decimals in order from smallest to largest:
4.67
4.7
4.717
4.77
4.73
4.07
4.67
4.717
4.77
4.07
4.73
4.7
4.67
4.717
4.73
4.77
4.07
4.7
4.67
4.717
4.77
4.73
4.70
4.07
4.717
4.77
4.73
4.7
To order these decimals we must compare the digits in the same position, starting from the left.
The correct order is:
The digits in the unit positions are the same, so this does not help.
4.07
4.67
4.7
4.717
4.73
4.77
Looking at the first decimal place tells us that 4.07 is the smallest followed by 4.67
Start by emphasising that, unlike whole numbers, we cannot order decimal numbers by looking at the number of digits in the number. We need to look at the digits in the same position starting from the left.
In this example all of the numbers start with a 4 in the unit position, so that does not help us. We need to look at the next digit in each number.
Talk through each point of the slide. We can write a zero in the second decimal place of 4.7 to help compare them. Point out that this zero does not change the number’s value.
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.

29. N1 Place value, ordering and rounding
Contents
N1 Place value, ordering and rounding N1
N1.1 Place value N1
N1.2 Powers of ten N1
N1.3 Ordering decimals N1
N1.4 Rounding

30. There are 1432 pupils at Eastpark Secondary School.
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.
Talk about rounding in real life contexts, for example the number of people at a football match. Ask pupils to give examples.
State that we can also use rounding in maths to give approximate answers to calculations.
We can round numbers to the nearest 1000, 100, 10, whole number, 0.1, etc. depending on the level of accuracy required.

31. Rounding whole numbers
Example
Round
Round
Round 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.
Talk through the example.
Note that is not equal to is an approximation. We must include ‘to the nearest 100’ to ensure that the equals sign is not used incorrectly.
If this digit is 5 or more then we need to round up the digit in the hundreds position.
Solution:
34871 = (to the nearest 100)

32. 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
Talk through each answer in the table.
9875
10000
9900
9880
452
500
450

33. Rounding decimals Example 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.
Talk through the example.
Emphasize that is not equal to is an approximation.
If this digit is 5 or more then we need to round up the digit in the first decimal place.
to 1 decimal place is 2.8.

34. 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
63.5
63.47
63.472 88
87.7
87.66
87.656
150
150.0
149.99
Talk through each answer in the table.
Stress that the number of decimal places tells you the number of digits that must be written after the decimal point, even if the last digit (or digits) are zero. 4
3.5
3.54
3.540 1
0.6
0.60
0.600