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## KS3 Mathematics

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**KS3 Mathematics**N1 Place value, ordering and rounding**N1 Place value, ordering and rounding**Contents • N1 N1.2 Powers of ten • N1 N1.1 Place value N1.3 Ordering decimals • N1 N1.4 Rounding • N1**Multiplying by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 6 2 6 2 What is 6.2 × 10? Let’s look at what happens on the place value grid. When we multiply by ten the digits move one place to the left. 6.2 × 10 = 62**Multiplying by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 3 1 3 1 What is 3.1 × 100? Let’s look at what happens on the place value grid. 0 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**Multiplying by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 0 7 7 What is 0.7 × 1000? Let’s look at what happens on the place value grid. 0 0 When we multiply by one thousand the digits move three places to the left. We then add zero place holders. 0.7 × 1000 = 700**Dividing by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 4 5 4 5 What is 4.5 ÷ 10? Let’s look at what happens on the place value grid. 0 When we divide by ten the digits move one place to the right. 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**Dividing by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 9 4 9 4 What is 9.4 ÷ 100? Let’s look at what happens on the place value grid. 0 0 When we divide by one hundred the digits move two places to the right. We need to add zero place holders. 9.4 ÷ 100 = 0.094**Dividing by 10, 100 and 1000**Thousands Hundreds Tens Units tenths hundredths thousandths 5 1 0 5 1 What is 510 ÷ 1000? Let’s look at what happens on the place value grid. 0 When we divide by one thousand the digits move three places to the right. We add a zero before the decimal point. 510 ÷ 1000 = 0.51**Multiplying and dividing by 10, 100 and 1000**3.4 × 10 = 73.8 ÷ = 7.38 64.34 ÷ = 0.6434 ÷ 1000 = 8.31 × 45.8 = 45 800 0.64 × = 640 43.7 × = 4370 0.021 × 100 = 92.1 ÷ 10 = 250 ÷ = 2.5 Complete the following: 34 10 100 8310 1000 1000 100 2.1 100 9.21**Multiplying by 0.1 and 0.01**1 We can also think of this as 4 × . 10 1 4 × is equivalent to 4 ÷ 10. 10 Multiplying by 0.1 Dividing by 10 is the same as 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. Therefore: 4 × 0.1 = 0.4**Multiplying by 0.1 and 0.01**1 We can also think of this as 3 × . 100 1 3 × is equivalent to 3 ÷ 100. 100 Multiplying by 0.01 Dividing by 100 is the same as What is 3 × 0.01? We can think of this as 3 lots of 0.01 or 0.01 + 0.01 + 0.01. Therefore: 3 × 0.01 = 0.03**Dividing by 0.1 and 0.01**Dividing by 0.1 Multiplying by 10 is the same as 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 and 0.01**Dividing by 0.01 Multiplying by 100 is the same as 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**Multiplying and dividing by 0.1 and 0.01**24 × 0.1 = 92.8 ÷ = 9280 52 ÷ = 5200 × 950 = 9.5 ÷ 0.001 = 674 470 × = 0.47 31.2 × = 3.12 830 × 0.01 = 6.51 ÷ 0.1 = 0.54 ÷ = 5.4 Complete the following: 2.4 0.01 0.01 0.674 0.01 0.001 8.3 0.1 65.1 0.1**N1 Place value, ordering and rounding**Contents N1.1 Place value • N1 • N1 N1.2 Powers of ten N1.3 Ordering decimals • N1 N1.4 Rounding • N1**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**1 0.1 = = =10−1 1 10 101 1 0.01 = = = 10−2 102 1 1 1 1000000 1 0.001 = = = 10−3 1 10000 100000 103 100 1 1000 1 0.0001 = = = 10−4 104 1 0.00001 = = = 10−5 105 1 0.000001 = = = 10−6 106 Any number raised to the power of 0 is 1, so 1 = 100 We use negative powers of ten to give us decimals.**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 150 000 000 km. We can write this number as 1.5 × 108 km. A number between 1 and 10 A power of ten**Standard form – writing large numbers**80 000 000 = 230 000 000 = 724 000 = 6 003 000 000 = 371.45 = How can we write these numbers in standard form? 8 × 107 2.3 × 108 7.24 × 105 6.003 × 109 3.7145 × 102**Standard form – writing large numbers**5 × 1010 = 7.1 × 106 = 4.208 × 1011 = 2.168 × 107 = 6.7645 × 103 = These numbers are written in standard form. How can they be written as ordinary numbers? 50 000 000 000 7 100 000 420 800 000 000 21 680 000 6764.5**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. The actual width of this shelled amoeba is 0.00013 m. We can write this number as 1.3 × 10−4 m. A number between 1 and 10 A negative power of 10**Standard form – writing small numbers**0.0006 = 0.00000072 = 0.0000502 = 0.0000000329 = 0.001008 = How can we write these numbers in standard form? 6 × 10−4 7.2 × 10−7 5.02 × 10−5 3.29 × 10−8 1.008 × 10−3**Standard form – writing small numbers**8 × 10−4 = 2.6 × 10−6 = 9.108 × 10−8 = 7.329 × 10−5 = 8.4542 × 10−2 = These numbers are written in standard form. How can they be written as ordinary numbers? 0.0008 0.0000026 0.00000009108 0.00007329 0.084542**N1 Place value, ordering and rounding**Contents N1.1 Place value • N1 N1.2 Powers of ten • N1 N1.3 Ordering decimals • N1 N1.4 Rounding • N1**Comparing decimals**Which number is bigger: 1.72 or 1.702? 1 . 7 2 1 . 7 2 1 . 7 0 2 1 . 7 0 2 These digits are the same. The 2 is bigger than the 0 so: These digits are the same. To compare two decimal numbers, look at each digit in order from left to right: 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 6 5 . 3 7 1 5 . 3 7 1 The 7 is bigger than the 6 so: These digits are the same. These digits are the same. Which measurement is bigger: 5.36 kg or 5.371 kg? Next, compare the two decimal numbers by looking at each digit in order from left to right: 5 . 3 6 5 . 3 7 1 5.36 < 5.371**Ordering decimals**4.67 4.67 4.67 4.67 4.717 4.717 4.717 4.717 4.717 4.77 4.77 4.77 4.77 4.77 4.73 4.73 4.73 4.73 4.73 4.7 4.7 4.7 4.70 4.7 4.07 4.07 4.07 4.07 Write these decimals in order from smallest to largest: The correct order is: 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. 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. 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.**N1 Place value, ordering and rounding**Contents N1.1 Place value • N1 N1.2 Powers of ten • N1 N1.4 Rounding N1.3 Ordering decimals • N1 • N1**Rounding**There are 1432 pupils at Eastpark Secondary School. There are about one and a half thousand pupils at Eastpark Secondary School. We do not always need to know the exact value of a number. For example:**Rounding whole numbers**Round 34 871 to the nearest 100. Round 34871 Round 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**Complete this table: to the nearest 1000 to the nearest 100 to the nearest 10 37521 38000 37500 37520 274503 275000 274500 274500 7630918 7631000 7630900 7630920 9875 10000 9900 9880 452 0 500 450**Rounding decimals**Round 2.75241302 to one decimal place. Round 2.75241302 Round 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.**Roundingtoagivennumberofdecimalplaces**Complete this table: to the nearest whole number to 1 d.p. to 2 d.p. to 3 d.p. 63.4721 63 63.5 63.47 63.472 87.6564 88 87.7 87.66 87.656 149.9875 150 150.0 149.99 149.988 3.54029 4 3.5 3.54 3.540 0.59999 1 0.6 0.60 0.600**Rounding to significant figures**For example: 4 890 351 4 890 351 This is the first significant figure and 0.0007506 0.0007506 This is the first significant figure 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.**Rounding to significant figures**4890351 0.0007506 4890351 This is the fourth significant figure This is the fourth significant figure This is the third significant figure For example: 0.0007506 4 890 351 4890 351 4 890 351 This is the third significant figure This is the first significant figure This is the second significant figure and 0.0007506 0.0007506 0.0007506 This is the first significant figure This is the second significant figure The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros.