3. Large numbers Read these numbers then put them into your calculator.
There are approximately footballers worldwide
supporters watched European football teams in the season 2001–2002.
Photo credit: © Lario Tus 2010, Shutterstock.com
Football information from:
The total attendance in the Premier League in 2007–8 was spectators

4. Large and small numbers
Try to read each of these numbers.
The number of atoms in a ‘mole’ is:
The diameter of an iron atom is: m
The wavelength of red light is: m
Photo credit: © Sashkin 2010, Shutterstock.com
The distance to the nearest star (Proxima Centauri) is:
miles

5. Powers of ten The decimal number system is based on powers of ten.
Powers of ten are written using index notation.
10 = 101
100 = 10 × 10 = 102
1000 = 10 × 10 × 10 = 103
Teacher notes
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).
= 10 × 10 × 10 × 10 = 104
= 10 × 10 × 10 × 10 × 10 = 105
= 10 × 10 × 10 × 10 × 10 × 10 = 106…

6. Negative powers of ten Any number raised to the power of 0 is 1, so
100 = 1
Decimals can be written using negative powers of ten.
0.1 = = =10–1 1 10
101
0.01 = = = 10–2 1
102
100
0.001 = = = 10–3 1
103
1000
Teacher notes
Talk through the use of negative integers to represent decimals. This is discussed in the context of the place value system in Unit 2: Decimals
= = = 10–4 1
10000
104
= = = 10–5 1
100000
105
= = = 10–6… 1
106

7. Very large numbers Use your calculator to work out the answer to×
Your calculator may display the answer as: 2
×10 15 2 E 15 2 15
What does the 15 mean?
Teacher notes
Different models of calculator may show the answer in different ways. Many will leave out the ×10 and will have EXP before the power or nothing at all.
Discuss how many zeros there will be in the answer. 4 × 5 is 20. There are 7 zeros in and 7 zeros in That means that the answer will have 14 zeros plus the zero from the 20, making 15 zeros altogether.
The 15 means that the answer is 2 multiplied by 10, 15 times in a row:
2 × 1015 =

8. Very small numbers Use you calculator to work out the answer to÷
Your calculator may display the answer as:
1.5
×10
–12
1.5 E
–12
1.5
–12
What does the –12 mean?
Teacher notes
Point out that if we include the 0 before the decimal point the answer has 12 zeros altogether.
The –12 means that the 1.5 is divided by 10, 12 times in row.
1.5 × 10–12 =

10. Standard form
2 × 1015 and 1.5 × 10–12 are examples of numbers written in standard form.
Numbers written in standard form have two parts:
number
between
1 and 10
power of 10 ×
This way of writing a number is also called standard index form or scientific notation.
Teacher notes
Point out that the numbers between 1 and 10 do not include the number 10.
Any number can be written using standard form, however it is usually used to write very large or very small numbers.

11. Standard form – writing large numbers
For example, the mass of the planet earth is about
kg.
We can write this in standard form as a number between 1 and 10 multiplied by a power of 10.
Photo credit: © Max FX 2010, Shutterstock.com
5.97 × 1024 kg
number between 1 and 10
power of ten

12. Standard form – writing large numbers
Write these numbers in standard form. =
8 × 107 =
2.3 × 108 =
7.24 × 105
Teacher notes
Discuss how each number should be written in standard form.
Notice that for large numbers the power of ten will always be one less than the number of digits in the whole part of the number. =
6.003 × 109 =
× 102

13. 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 =
Teacher notes
Discuss how each number written in standard form should be written in full.
2.168 × 107 =
× 103 =
6764.5

14. Standard form – writing small numbers
We can write very small numbers using negative powers of ten.
For example, the width of this shelled amoeba is m.
This is written in standard form as:
Image credit: © Wim van Egmond, Microscopy UK
1.3 × 10–4 m.
number between 1 and 10
power of ten

15. Standard form – writing small numbers
Write these numbers in standard form. =
6 × 10–4 =
7.2 × 10–7 =
5.02 × 10–5
Teacher notes
Notice that the power of ten is always minus the number of zeros before the first significant figure including the one before the decimal point.. =
3.29 × 10–8 =
1.008 × 10–3

16. Standard form – writing small numbers
These numbers are written in standard form. How would they be written as ordinary numbers?
8 × 10–4 =
0.0008
2.6 × 10–6 =
9.108 × 10–8 =
Teacher notes
Again, 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 =

17. Standard form matching

18. Which number is incorrect?
Teacher notes
Ask pupils how the number that is incorrectly written can be expressed correctly in standard form before revealing the answer.
Remind pupils of the special case of 100 = 1

20. Ordering numbers in standard form
Write these numbers in order from smallest to largest:
5.3 × 10–4, 6.8 × 10–5, 4.7 × 10–3, 1.5 × 10–4.
First compare the powers of 10.
10–5 is smaller than 10–4. That means that 6.8 × 10 –5 is the smallest number in the list.
When two or more numbers have the same power of ten, the number parts are compared. 5.3 × 10–4 is larger than
1.5 × 10–4 so the correct order is:
6.8 × 10–5, 1.5 × 10–4, 5.3 × 10–4, 4.7 × 10–3

21. Ordering planet sizes Teacher notes
The diameter of each planet is given in standard form. Ask a volunteer to come to the board and put the in the correct order from smallest to biggest.
Image credit: © Jurgen Ziewe 2010, Shutterstock.com

22. Ordering elements Teacher notes Correct order: Titanium 2 × 10–10
Gold 1.79 × 10–10
Silver 1.75 × 10–10
Copper 1.57 × 10–10
Carbon 9.1 × 10–11
Nitrogen 7.5 × 10–11
Oxygen 6.5 × 10–11
Helium 4.9 × 10–11
Photo credit: © caesart 2010, Shutterstock.com

23. Converting and ordering
Change the numbers to standard form and put them in order from lowest to highest.
0.53 × 107
46 × 106
Teacher notes
This is a written activity.
The order is
5.3 × 106
8.32 × 106
4.6 × 107
6.7 × 107
7 × 107
7.1 × 107
8 × 108
0.08 × 1010
832 × 104
0.71 × 108
7000 × 104

25. Calculations involving standard form

26. Standard form calculations
Teacher notes
Answers are:
6.9  1026
3  105
8.7  106
5  1011
6.3  1012
5.6  10-16

27. Travelling to Mars
Teacher notes
The answer is worked out on the next slide.
How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?

28. Calculations involving standard form
How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
distance
distance
Rearrange speed =
to give time =
time
speed
8.32 × 107
Teacher notes
Remind pupils that 107 ÷ 103 = 104 because the indices are subtracted when dividing.
Photo credit: © Stephen Girimont 2010, Shutterstock.com
Time to reach Mars =
2.6 × 103
= (8.32 ÷ 2.6) × (107 ÷ 103)
= 3.2 × 104 hours

29. Calculations involving standard form
Use your calculator to work out how long 3.2 × 104 hours is in years.
Enter 3.2 × 104 into your calculator.
Divide by 24 to give the equivalent number of days.
Teacher notes
Make sure that pupils are able to enter numbers given in standard form into their calculators.
Photo credit: © Provasilich 2010, Shutterstock.com
Divide by 365 to give the equivalent number of years.
3.2 × 104 hours is over 3½ years.

30. In the lab The table shows some weight of very small items.
Which weighs more...
3.56 × 1020 Hydrogen atoms or 4.3 × 1019 water molecules?
Hydrogen atom
1.67×10-27 kg
Water molecule
2.99×10-26 kg
Silver atom
1.79×10-25 kg
Lead atom
3.45×10-25 kg
Small grain of sand
3.5×10-10 kg
Large grain of sand
1.1×10-5 kg
1 Euro coin
0.008 kg
5 million small grains of sand or 200 large grains?
How many atoms of silver are needed to have the same weight as a 1 Euro coin?
Teacher notes
This slide tests how well students can work with numbers in standard form, as well as testing the understanding of what type of calculation is needed to answer the questions. Students will need to be encouraged to show their calculations.
Answers:
Hydrogen – 5.95 × 10-7 kg
Water – × 10-6 kg so the water weighs more
Small sand – kg
Large sand – kg so the large sand weighs more
4.47 × 1022 atoms of silver
Moles: Hydrogen – 0.001kg (or 1g)
Water – 0.018kg
Silver – 0.108kg
Lead – 0.208kg
55.6 moles make up 1kg of water
Ask How many moles of water make 1kg?
A mole of a substance is 6.02 × 1023 atoms or molecules of the substance. What does a mole of Hydrogen, Water, Silver and Lead weigh?