Large numbers Read these numbers then put them into your calculator. There are approximately 250 000 000 footballers worldwide 83 569 426 supporters watched European football teams in the season 2001–2002. Photo credit: © Lario Tus 2010, Shutterstock.com Football information from: http://web.archive.org/web/20060915133001/http://access.fifa.com/infoplus/IP-199_01E_big-count.pdf) http://www.guardian.co.uk/football/2003/oct/09/theknowledge.sport) The total attendance in the Premier League in 2007–8 was 13 500 000 spectators

Large and small numbers Try to read each of these numbers. The number of atoms in a ‘mole’ is: 602 000 000 000 000 000 000 000 The diameter of an iron atom is: 0.000000000074 m The wavelength of red light is: 0.00000065 m Photo credit: © Sashkin 2010, Shutterstock.com The distance to the nearest star (Proxima Centauri) is: 25 284 000 000 000 miles

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). 10 000 = 10 × 10 × 10 × 10 = 104 100 000 = 10 × 10 × 10 × 10 × 10 = 105 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106…

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 0.0001 = = = 10–4 1 10000 104 0.00001 = = = 10–5 1 100000 105 0.000001 = = = 10–6… 1 1000000 106

Very large numbers Use your calculator to work out the answer to 40 000 000 × 50 000 000. 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 40 000 000 and 7 zeros in 50 000 000. 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 = 2 000 000 000 000 000

Very small numbers Use you calculator to work out the answer to 0.000045 ÷ 30 000 000. 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 = 0.0000000000015

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.

Standard form – writing large numbers For example, the mass of the planet earth is about 5 970 000 000 000 000 000 000 000 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

Standard form – writing large numbers Write these numbers in standard form. 80 000 000 = 8 × 107 230 000 000 = 2.3 × 108 724 000 = 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. 6 003 000 000 = 6.003 × 109 371.45 = 3.7145 × 102

Standard form – writing large numbers 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 Teacher notes Discuss how each number written in standard form should be written in full. 2.168 × 107 = 21 680 000 6.7645 × 103 = 6764.5

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 0.00013 m. This is written in standard form as: Image credit: © Wim van Egmond, Microscopy UK http://www.microscopy-uk.org.uk/index.html 1.3 × 10–4 m. number between 1 and 10 power of ten

Standard form – writing small numbers Write these numbers in standard form. 0.0006 = 6 × 10–4 0.00000072 = 7.2 × 10–7 0.0000502 = 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.. 0.0000000329 = 3.29 × 10–8 0.001008 = 1.008 × 10–3

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 = 0.0000026 9.108 × 10–8 = 0.00000009108 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 = 0.00007329 8.4542 × 10–2 = 0.084542

Standard form matching

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

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

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

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

Converting and ordering Change the numbers to standard form and put them in order from lowest to highest. 0.53 × 107 46 × 106 67 000 000 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

Calculations involving standard form

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

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?

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

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.

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 – 1.285 × 10-6 kg so the water weighs more Small sand – 0.00175 kg Large sand – 0.0022 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?