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© Boardworks Ltd 2006 1 of 51 N1 Place value, ordering and rounding KS3 Mathematics

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© Boardworks Ltd 2006 2 of 51 N1N1 N1N1 N1N1 N1N1 N1.1 Place value Contents N1 Place value, ordering and rounding N1.3 Ordering decimals N1.4 Rounding N1.2 Powers of ten

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© Boardworks Ltd 2006 3 of 51 Blank cheques

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© Boardworks Ltd 2006 4 of 51 Place value

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© Boardworks Ltd 2006 5 of 51 What is 6.2 × 10? Let’s look at what happens on the place value grid. ThousandsHundredsTensUnitstenthshundredthsthousandths 62 When we multiply by ten the digits move one place to the left. 62 6.2 × 10 = 62 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 2006 6 of 51 What is 3.1 × 100? Let’s look at what happens on the place value grid. 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 ThousandsHundredsTensUnitstenthshundredthsthousandths 31 31 0 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 2006 7 of 51 What is 0.7 × 1000? Let’s look at what happens on the place value grid. When we multiply by one thousand the digits move three places to the left. We then add zero place holders. 0.7 × 1000 = 700 ThousandsHundredsTensUnitstenthshundredthsthousandths 07 7 00 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 2006 8 of 51 Dividing by 10, 100 and 1000 What is 4.5 ÷ 10? Let’s look at what happens on the place value grid. ThousandsHundredsTensUnitstenthshundredthsthousandths 45 When we divide by ten the digits move one place to the right. 45 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

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© Boardworks Ltd 2006 9 of 51 Dividing by 10, 100 and 1000 What is 9.4 ÷ 100? Let’s look at what happens on the place value grid. ThousandsHundredsTensUnitstenthshundredthsthousandths 94 When we divide by one hundred the digits move two places to the right. 94 We need to add zero place holders. 0 0 9.4 ÷ 100 = 0.094

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© Boardworks Ltd 2006 10 of 51 Dividing by 10, 100 and 1000 What is 510 ÷ 1000? Let’s look at what happens on the place value grid. ThousandsHundredsTensUnitstenthshundredthsthousandths 510 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

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© Boardworks Ltd 2006 11 of 51 Spider diagram

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© Boardworks Ltd 2006 12 of 51 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 ÷ = 7.38 10 ÷ 1000 = 8.318310 0.64 × = 640 1000 0.021 × 100 =2.1 250 ÷ = 2.5 100

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© Boardworks Ltd 2006 13 of 51 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 ×. 1 10 4 × is equivalent to 4 ÷ 10. 1 10 Therefore: 4 × 0.1 = 0.4 Multiplying by 0.1Dividing by 10 is the same as

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© Boardworks Ltd 2006 14 of 51 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. 1 100 We can also think of this as 3 ×.3 × is equivalent to 3 ÷ 100. 1 100 Therefore: 3 × 0.01 = 0.03 Multiplying by 0.01Dividing by 100 is the same as

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© Boardworks Ltd 2006 15 of 51 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 0.1Multiplying by 10 is the same as

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© Boardworks Ltd 2006 16 of 51 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 0.01Multiplying by 100 is the same as

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© Boardworks Ltd 2006 17 of 51 Complete the following: 24 × 0.1 =2.4 52 ÷ = 5200 0.01 × 950 = 9.50.01 31.2 × = 3.12 0.1 6.51 ÷ 0.1 =65.1 92.8 ÷ = 9280 0.01 ÷ 0.001 = 6740.674 470 × = 0.47 0.001 830 × 0.01 = 8.3 0.54 ÷ = 5.4 0.1 Multiplying and dividing by 0.1 and 0.01

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© Boardworks Ltd 2006 18 of 51 Multiplying by small multiples of 0.1

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© Boardworks Ltd 2006 19 of 51 N1N1 N1N1 N1N1 N1N1 N1.2 Powers of ten Contents N1.1 Place value N1 Place value, ordering and rounding N1.3 Ordering decimals N1.4 Rounding

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© Boardworks Ltd 2006 20 of 51 Powers of ten Our decimal number system is based on powers of ten. We can write powers of ten using index notation. 10 = 10 1 100 = 10 × 10 = 10 2 1000 = 10 × 10 × 10 = 10 3 10 000 = 10 × 10 × 10 × 10 = 10 4 100 000 = 10 × 10 × 10 × 10 × 10 = 10 5 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 10 6 10 000 000 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10 7 …

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© Boardworks Ltd 2006 21 of 51 Negative powers of ten Any number raised to the power of 0 is 1, so 1 = 10 0 We use negative powers of ten to give us decimals. 0.01 = = = 10 −2 1 10 2 1 100 0.001 = = = 10 −3 1 10 3 1 1000 0.0001 = = = 10 −4 1 10000 1 10 4 0.00001 = = = 10 −5 1 100000 1 10 5 0.000001 = = = 10 −6 1 1000000 1 10 6 0.1 = = =10 −1 1 10 1 10 1

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© Boardworks Ltd 2006 22 of 51 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 × 10 8 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

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© Boardworks Ltd 2006 23 of 51 How can we write these numbers in standard form? 80 000 000 = 8 × 10 7 230 000 000 = 2.3 × 10 8 724 000 = 7.24 × 10 5 6 003 000 000 = 6.003 × 10 9 371.45 = 3.7145 × 10 2 Standard form – writing large numbers

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© Boardworks Ltd 2006 24 of 51 These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 10 10 = 50 000 000 000 7.1 × 10 6 = 7 100 000 4.208 × 10 11 = 420 800 000 000 2.168 × 10 7 = 21 680 000 6.7645 × 10 3 = 6764.5 Standard form – writing large numbers

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© Boardworks Ltd 2006 25 of 51 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

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© Boardworks Ltd 2006 26 of 51 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 Standard form – writing small numbers

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© Boardworks Ltd 2006 27 of 51 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 Standard form – writing small numbers

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© Boardworks Ltd 2006 28 of 51 N1N1 N1N1 N1N1 N1N1 N1.3 Ordering decimals Contents N1.4 Rounding N1.1 Place value N1 Place value, ordering and rounding N1.2 Powers of ten

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© Boardworks Ltd 2006 29 of 51 Zooming in on a number line

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© Boardworks Ltd 2006 30 of 51 Decimal sequences

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© Boardworks Ltd 2006 31 of 51 Decimals on a number line

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© Boardworks Ltd 2006 32 of 51 Mid-points

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© Boardworks Ltd 2006 33 of 51 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 These digits are the same. 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

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© Boardworks Ltd 2006 34 of 51 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

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© Boardworks Ltd 2006 35 of 51 These digits are the same. 5. 3 6 5. 3 7 1 These digits are the same. 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

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© Boardworks Ltd 2006 36 of 51 Comparing decimals

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© Boardworks Ltd 2006 37 of 51 4.674.74.7174.774.734.074.674.7174.734.774.074.74.674.7174.774.074.734.74.674.7174.774.734.704.074.7174.774.734.7 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: 4.074.674.74.7174.734.77 Ordering decimals

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© Boardworks Ltd 2006 38 of 51 Ordering decimals

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© Boardworks Ltd 2006 39 of 51 Dewey Decimal Classification System

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© Boardworks Ltd 2006 40 of 51 N1N1 N1N1 N1N1 N1N1 N1.4 Rounding Contents N1.3 Ordering decimals N1.1 Place value N1 Place value, ordering and rounding N1.2 Powers of ten

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© Boardworks Ltd 2006 41 of 51 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.

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© Boardworks Ltd 2006 42 of 51 Rounding readings from scales

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© Boardworks Ltd 2006 43 of 51 Rounding whole numbers

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© Boardworks Ltd 2006 44 of 51 Round 34 871 to the nearest 100.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

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© Boardworks Ltd 2006 45 of 51 Rounding whole numbers Complete this table: 37521 274503 7630918 9875 to the nearest 1000 452 to the nearest 100 to the nearest 10 380003750037520 275000274500 763100076309007630920 1000099009880 0500450

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© Boardworks Ltd 2006 46 of 51 Rounding decimals

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© Boardworks Ltd 2006 47 of 51 Round 2.75241302 to one decimal place.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. Rounding decimals

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© Boardworks Ltd 2006 48 of 51 Rounding to a given number of decimal places Complete this table: 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. 6363.563.4763.472 8887.787.6687.656 150150.0149.99149.988 43.53.543.540 10.60.600.600

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© Boardworks Ltd 2006 49 of 51 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 and 0.0007506 This is the first significant figure 0.0007506 This is the first significant figure 4 890 351

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© Boardworks Ltd 2006 50 of 51 Rounding to significant figures For example: 4 890 351 and 0.0007506 This is the first significant figure 0.0007506 This is the first significant figure 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

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© Boardworks Ltd 2006 51 of 51 Complete this table: 6.3528 34.026 0.005708 150.932 to 3 s. f. 0.00007835 to 2 s. f.to 1 s. f. 6.356.46 34.03430 0.005710.00570.006 151150200 0.00007840.0000780.00008 Rounding to significant figures

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