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

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© Boardworks Ltd 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 of 51 Blank cheques

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© Boardworks Ltd of 51 Place value

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© Boardworks Ltd 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 × 10 = 62 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 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 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 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 Multiplying by 10, 100 and 1000

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© Boardworks Ltd 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 ÷ 10 = 0.45

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© Boardworks Ltd 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 ÷ 100 = 0.094

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© Boardworks Ltd 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 ÷ 1000 =

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© Boardworks Ltd of 51 Spider diagram

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© Boardworks Ltd of 51 Multiplying and dividing by 10, 100 and 1000 Complete the following: 3.4 × 10 = ÷ = × 45.8 = × = ÷ 10 = ÷ = ÷ 1000 = × = × 100 = ÷ =

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© Boardworks Ltd 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 We can also think of this as 4 × × is equivalent to 4 ÷ Therefore: 4 × 0.1 = 0.4 Multiplying by 0.1Dividing by 10 is the same as

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

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© Boardworks Ltd 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 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 of 51 Complete the following: 24 × 0.1 = ÷ = × 950 = × = ÷ 0.1 = ÷ = ÷ = × = × 0.01 = ÷ = Multiplying and dividing by 0.1 and 0.01

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© Boardworks Ltd of 51 Multiplying by small multiples of 0.1

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© Boardworks Ltd 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 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 × 10 = = 10 × 10 × 10 = = 10 × 10 × 10 × 10 = = 10 × 10 × 10 × 10 × 10 = = 10 × 10 × 10 × 10 × 10 × 10 = = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10 7 …

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© Boardworks Ltd 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 = = = 10 − = = = 10 − = = = 10 − = = = 10 − = = = 10 − = = =10 −

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© Boardworks Ltd 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 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 of 51 How can we write these numbers in standard form? = 8 × = 2.3 × = 7.24 × = × = × 10 2 Standard form – writing large numbers

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© Boardworks Ltd of 51 These numbers are written in standard form. How can they be written as ordinary numbers? 5 × = × 10 6 = × = × 10 7 = × 10 3 = Standard form – writing large numbers

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© Boardworks Ltd of 51 We can also write very small numbers using standard form. The actual width of this shelled amoeba is 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 of 51 How can we write these numbers in standard form? = 6 × 10 − = 7.2 × 10 − = 5.02 × 10 − = 3.29 × 10 − = × 10 −3 Standard form – writing small numbers

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© Boardworks Ltd of 51 These numbers are written in standard form. How can they be written as ordinary numbers? 8 × 10 −4 = × 10 −6 = × 10 −8 = × 10 −5 = × 10 −2 = Standard form – writing small numbers

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© Boardworks Ltd 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 of 51 Zooming in on a number line

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© Boardworks Ltd of 51 Decimal sequences

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© Boardworks Ltd of 51 Decimals on a number line

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© Boardworks Ltd of 51 Mid-points

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© Boardworks Ltd 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 These digits are the same The 2 is bigger than the 0 so: > Comparing decimals

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© Boardworks Ltd 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 = kg Comparing decimals

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© Boardworks Ltd of 51 These digits are the same These digits are the same The 7 is bigger than the 6 so: Next, compare the two decimal numbers by looking at each digit in order from left to right: 5.36 < Which measurement is bigger: 5.36 kg or kg? Comparing decimals

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© Boardworks Ltd of 51 Comparing decimals

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© Boardworks Ltd of 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 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 The correct order is: Ordering decimals

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© Boardworks Ltd of 51 Ordering decimals

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© Boardworks Ltd of 51 Dewey Decimal Classification System

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© Boardworks Ltd 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 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 of 51 Rounding readings from scales

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© Boardworks Ltd of 51 Rounding whole numbers

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© Boardworks Ltd of 51 Round to the nearest 100.Round 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 = (to the nearest 100) Rounding whole numbers

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© Boardworks Ltd of 51 Rounding whole numbers Complete this table: to the nearest to the nearest 100 to the nearest

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© Boardworks Ltd of 51 Rounding decimals

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© Boardworks Ltd of 51 Round to one decimal place.Round 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 to 1 decimal place is 2.8. Rounding decimals

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© Boardworks Ltd of 51 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

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© Boardworks Ltd 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: and This is the first significant figure This is the first significant figure

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© Boardworks Ltd of 51 Rounding to significant figures For example: and This is the first significant figure This is the first significant figure The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros This is the second significant figure This is the second significant figure This is the third significant figure This is the third significant figure This is the fourth significant figure This is the fourth significant figure

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© Boardworks Ltd of 51 Complete this table: to 3 s. f to 2 s. f.to 1 s. f Rounding to significant figures

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