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**N1 Place value, ordering and rounding**

KS3 Mathematics The aim of this unit is to teach pupils to: Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 10. Compare and order decimals. Round numbers, including to a given number of decimal places or significant figures. Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp N1 Place value, ordering and rounding

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**N1 Place value, ordering and rounding**

Contents N1 Place value, ordering and rounding N1 N1.1 Place value N1 N1.2 Powers of ten N1 N1.3 Ordering decimals N1 N1.4 Rounding

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Blank cheques The aim of the activity is to ensure that pupils are able to read large numbers in the context of money. Ask pupils why they think it is necessary to write amounts of money in words on cheques. Start by revealing the amount of money on the cheque in figures and asking pupils how it would be written in words. Repeat the activity with the amount of money shown in words and ask pupils how it would be written in figures. Point out to pupils that in the context of money is read as ‘three hundred and two pounds, seventy-six’. If this number were not in the context of money it would be read as ‘three hundred and two point seven six’. When we say ‘seventy-six’ in the context of money we are referring to seventy-six pence. When we say ‘point seven six’ in the context of decimal numbers we are referring to 7 tenths and 6 hundredths. Also point out that the decimal 4.50 is usually written as 4.5. However, we cannot write £4.5 in the context of money. This must be written as £4.50. When writing amounts of money we cannot combine units, for example £6.73p is not allowed. This must be written either as £6.73 or as 673p.

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Place value Before starting the activity make sure that pupils can verbally describe the column headings and understand that the place value grid can be extended in both directions. Describe the decimal point as a means of separating the whole number part of a number from the fractional part. Start by dragging an example onto the grid, for example Ask what the 6 is worth in this position (6 thousand), what the 3 is worth in this position (3 hundred), and so on. Discuss the importance of the zero place holder. Ask pupils to tell you the most important digit in this number. Establish that this is the 6 because it is worth 6 thousand. Although 9 is a bigger digit, it is only worth 9 tenths. We call 6 the most significant digit. Ask pupils to add and subtract 0.001, 0.01 then 0.1 from the number. Note that to subtract 0.01 from this number we need to take 1 from the tenths column, then swap the 0 for a 9 to give Compare this with subtracting 10 from 907. Also, since there is a 9 in the tenths column, the effect of adding 0.1 is to add one to the digit in the units column to give Compare this with adding 100 to 4907. The place value grid can also be used to demonstrate multiplying and dividing numbers by 10, 100, 1000, 0.1 and Groups of digits can be dragged to the left and right to show this.

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**6 2 6 2 Multiplying by 10, 100 and 1000 What is 6.2 × 10?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 6 2 6 2 This is the first of three slides illustrating the effect of multiplying a decimal number by 10, 100 and 1000. Start by checking that pupils can multiply whole numbers by 10, 100 and 1000 and divide whole numbers ending in 0, 00 and 000 by 10, 100 and 1000. Tell pupils that we often need to be able to multiply and divide numbers by 10, 100 and 1000 in every day life, for example when converting between metric units. Give examples as necessary. Explain that we are now going to look at the effect of multiplying decimals by 10, 100 and 1000. Look at the question on the slide. Most pupils will be able to solve this without reference to the place value grid. Reveal each step on the slide and encourage the idea that the digits are moved one place to the left to make the number ten times bigger (rather than the decimal point being moved one place to the right). Links: N9 Mental methods – multiplication and division S7 Measures – converting units When we multiply by ten the digits move one place to the left. 6.2 × 10 = 62

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**3 1 3 1 Multiplying by 10, 100 and 1000 What is 3.1 × 100?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 3 1 3 1 When we multiply by one hundred the digits move two places to the left. Again, emphasize that it is the digits that are being moved two places to the left and not the decimal point that is being moved two places to the right. We then add a zero place holder. 3.1 × 100 = 310

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**7 7 Multiplying by 10, 100 and 1000 What is 0.7 × 1000?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 7 7 When we multiply by one thousand the digits move three places to the left. We then add zero place holders. 0.7 × 1000 = 700

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**4 5 4 5 Dividing by 10, 100 and 1000 What is 4.5 ÷ 10?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 4 5 4 5 When we divide by ten the digits move one place to the right. The next three slides illustrate the effect of dividing a decimal number by 10, 100 and 1000. Link: N9 Mental methods – multiplication and division 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

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**9 4 9 4 Dividing by 10, 100 and 1000 What is 9.4 ÷ 100?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 9 4 9 4 When we divide by one hundred the digits move two places to the right. Link: N9 Mental methods – multiplication and division We need to add zero place holders. 9.4 ÷ 100 = 0.094

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**5 1 5 1 Dividing by 10, 100 and 1000 What is 510 ÷ 1000?**

Let’s look at what happens on the place value grid. Thousands Hundreds Tens Units tenths hundredths thousandths 5 1 5 1 When we divide by one thousand the digits move three places to the right. Link: N9 Mental methods – multiplication and division We add a zero before the decimal point. 510 ÷ 1000 = 0.51

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Spider diagram This activity illustrates the effect of multiplying and dividing decimals by 10, 100 and 1000. Click on the green ovals to reveal the number inside. Reset the activity to change the central number.

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**Multiplying and dividing by 10, 100 and 1000**

Complete the following: 3.4 × 10 = 34 73.8 ÷ = 7.38 10 64.34 ÷ = 100 8310 ÷ 1000 = 8.31 1000 × 45.8 = 0.64 × = 640 1000 Complete the exercise as a class. An alternative would be to erase the answers from the slide (or change the animation order) and ask pupils to complete the exercise individually; you could also give pupils a worksheet with similar calculations. 43.7 × = 4370 100 0.021 × 100 = 2.1 92.1 ÷ 10 = 9.21 250 ÷ = 2.5 100

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**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 × . 1 10 4 × is equivalent to 4 ÷ 10. 1 10 Therefore: There are many ways in which to visualize the multiplication of a whole number by 0.1. Discuss as a group. Ask pupils what they think 2.7 × 0.1 is, or 32.8 × 0.1. 4 × 0.1 = 0.4 Multiplying by 0.1 Dividing by 10 is the same as

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**Multiplying by 0.1 and 0.01 What is 3 × 0.01?**

We can think of this as 3 lots of 0.01 or 1 100 We can also think of this as 3 × 3 × is equivalent to 3 ÷ 100. 1 100 Therefore: Extend the discussion to multiplication by 0.01. 3 × 0.01 = 0.03 Multiplying by 0.01 Dividing by 100 is the same as

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**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: Discuss ways to visualize dividing whole numbers by 0.1. Extend to dividing decimals by 0.1. 7 ÷ 0.1 = 70 Dividing by 0.1 Multiplying by 10 is the same as

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**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: Extend the discussion to dividing by 0.01. 12 ÷ 0.01 = 1200 Dividing by 0.01 Multiplying by 100 is the same as

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**Multiplying and dividing by 0.1 and 0.01**

Complete the following: 24 × 0.1 = 2.4 92.8 ÷ = 9280 0.01 52 ÷ = 5200 0.01 0.674 ÷ = 674 × 950 = 9.5 0.01 470 × = 0.47 0.001 Complete the exercise as a class. An alternative would be to erase the answers from the slide (or change the animation order) and ask pupils to complete the exercise individually. You could also give pupils a worksheet with similar calculations. 31.2 × = 3.12 0.1 830 × 0.01 = 8.3 6.51 ÷ 0.1 = 65.1 0.54 ÷ = 5.4 0.1

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**Multiplying by small multiples of 0.1**

Use this grid to demonstrate the effect of multiplying by 0.1 and small multiples of 0.1. Start by explaining to pupils that the large square represents one whole. It is one unit across and one unit down. The square is divided into ten equal parts across the top and so each division across the length must be equal to 0.1. The same is true down the side. Establish that the unit square is divided into 100 equal parts and so each small part is one hundredth of the large square or 0.01. Compare this with the fact that there are 100 mm2 in 1 cm2. Demonstrate various calculations by changing the size of the shaded area. For example, start by showing that 0.1 × 0.1 = One tenth of the width multiplied by one tenth of the length equals one hundredth of the whole unit. Ask pupils what 0.2 × 0.4 is equal to. It is a common error to give the answer as 0.8. Show, using the grid that the answer is 0.08. Change the number of unit squares to extend to more difficult problems. For example, 0.4 × 3.7 is equal to 0.4 × ( ) = = 1.48 Click on the yellow rectangles to hide the answers and ask pupils to calculate the solutions mentally using the diagram.

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**N1 Place value, ordering and rounding**

Contents N1 Place value, ordering and rounding N1 N1.1 Place value N1 N1.2 Powers of ten N1 N1.3 Ordering decimals N1 N1.4 Rounding

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**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 × 10 × 10 × 10 = 104 = 10 × 10 × 10 × 10 × 10 = 105 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). You may wish to mention that the standard meaning of a billion is 109 (one thousand million). In the past, a British billion was one million million, or 1012. Links: N4 Powers and roots – powers = 10 × 10 × 10 × 10 × 10 × 10 = 106 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 107 …

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**Negative powers of ten Any number raised to the power of 0 is 1, so**

1 = 100 We use negative powers of ten to give us decimals. 0.1 = = =10−1 1 10 101 0.01 = = = 10−2 1 102 100 0.001 = = = 10−3 1 103 1000 = = = 10−4 1 10000 104 Discuss the convention of using negative integers to express tenths, hundredths, thousandths, and other negative powers of ten. Link: N4 Powers and roots – powers = = = 10−5 1 100000 105 = = = 10−6 1 106

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**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 km. Discuss the use of standard form to write large numbers. Discuss whether 10×105 and 1×104 are correct examples of standard form. The first is not whereas the second is; if the discussion is interesting, you might want to discuss how to use inequality notation to show exactly what is meant by ‘between 1 and 10’ in the definition of standard form. We can write this number as 1.5 × 108 km. A number between 1 and 10 A power of ten

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**Standard form – writing large numbers**

How can we write these numbers in standard form? = 8 × 107 = 2.3 × 108 = 7.24 × 105 Discuss how each number should be written in standard form. Notice that the power of ten will always be one less than the number of digits before the decimal point in the number. = 6.003 × 109 = × 102

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

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**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 m. We can write this number as 1.3 × 10−4 m. The image of a shelled amoeba has been reproduced with the kind permission of Wim van Egmond © Microscopy UK A number between 1 and 10 A negative power of 10

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**Standard form – writing small numbers**

How can we write these numbers in standard form? = 6 × 10−4 = 7.2 × 10−7 = 5.02 × 10−5 Notice that the power of ten is always –1 times the number of zeros before the first significant figure, including the one before the decimal point. = 3.29 × 10−8 = 1.008 × 10−3

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**Standard form – writing small numbers**

These numbers are written in standard form. How can they be written as ordinary numbers? 8 × 10−4 = 0.0008 2.6 × 10−6 = 9.108 × 10−8 = 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 =

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**N1 Place value, ordering and rounding**

Contents N1 Place value, ordering and rounding N1 N1.1 Place value N1 N1.2 Powers of ten N1 N1.3 Ordering decimals N1 N1.4 Rounding

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**Zooming in on a number line**

Spend some time zooming in and out of the number line to demonstrate decimals on a number line. Start by using the zoom in tool to select the area between two whole numbers, for example 1 and 2. Ask pupils to tell you the value of a decimal that would lie between any two given divisions, and zoom in to check. Establish that by allowing decimals to any number of decimal places, an infinite number of decimals lie between any two given divisions; however many we have found, we can always find at least one more. Extend the activity by zooming in to show numbers to more decimal places.

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Decimal sequences The aim of this activity is to ensure that the pupils understand the positions of decimal numbers on a number line and that they are able to add (and subtract) small multiples of 0.1, 0.01 and from decimals. Start the activity by first establishing the size of the steps between each number. Ask pupils to tell you the values of the missing numbers (it is not necessary to do these in order). Press on an empty cell to reveal the number inside it. Press on the number of decimal places required will generate a new number line. Link: A4 Sequences – continuing sequences.

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**Decimals on a number line**

The aim of this activity is to help pupils to visualize the positions of decimal numbers on a number line and to estimate the size of decimals between two others by looking at the next decimal place. Click on an empty box to reveal the number inside it. Differentiate the activity by including positive and negative decimals.

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Mid-points This activity focuses on finding the mid-point between two numbers by looking at the next decimal place. For more difficult examples encourage pupils to deduce that a mid-point can be found by adding the two end points and dividing by two. Discuss why this works, referring to the mean. Extend the activity by using negative numbers; and by finding the mid-point between a negative and a positive number. Link: D3 Representing and interpreting data – calculating the mean

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**Which number is bigger:**

Comparing decimals 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: Explain that to compare two decimals we must look at digits in the same position starting from the left. Explain that the digit furthest to the left is the most significant digit. Talk through the example on the board: Both numbers have 1 unit, so that doesn’t help. Both numbers have 7 tenths, so that doesn’t help either. The first number has 2 hundredths and the second number has no hundredths. 1.72 must therefore be bigger than Some pupils may need reminding of the meaning of the ‘greater than’ symbol. 1.72 > 1.702

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**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: Emphasize that to compare measurements we must convert them to the same units. Ask pupils how to convert grams to kilograms before revealing this on the board. Point out that we also have chosen to convert both units to grams to compare the measurements. 5371 g = kg

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**Comparing decimals 5.36 < 5.371 Which measurement is bigger:**

5.36 kg or kg? Next, compare the two decimal numbers by looking at each digit in order from left to right: The 7 is bigger than the 6 so: These digits are the same. These digits are the same. 5.36 < 5.371

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Comparing decimals Drag and drop the correct sign into place. Remind pupils that for the ‘greater than’ and ‘less than’ symbols the open end faces towards the larger number. You may need to remind pupils of the unit conversions: There are 1000 grams in a kilogram. There are 1000 millilitres in a litre. There are 100 centimetres in a metre.

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Ordering decimals Write these decimals in order from smallest to largest: 4.67 4.7 4.717 4.77 4.73 4.07 4.67 4.717 4.77 4.07 4.73 4.7 4.67 4.717 4.73 4.77 4.07 4.7 4.67 4.717 4.77 4.73 4.70 4.07 4.717 4.77 4.73 4.7 To order these decimals we must compare the digits in the same position, starting from the left. The correct order is: 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. Start by emphasizing that, unlike whole numbers, we cannot order decimal numbers by looking at the number of digits in the number. We need to look at the digits in the same position starting from the left. In this example all of the numbers start with a 4 in the unit position, so that does not help us. We need to look at the next digit in each number. Talk through each point of the slide. We can write a zero in the second decimal place of 4.7 to help compare them. Point out that this zero does not change the number’s value. 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.

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Ordering decimals Drag and drop each card underneath in order according to suggestions from pupils. Compare units, tenths, hundredths and thousandths in turn. Ensure that the concept of working from the left until you find the highest digit is understood.

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**Dewey Decimal Classification System**

Practise the method for ordering decimals by completing the following activity: Claire has been doing a project on cats and returns her books to the library. Help Mrs Hooper the librarian to put the books in the correct order using the Dewey Decimal Classification System. In the 1870s Melvil Dewey invented a system of classifying library books on different subjects to make them easy to find. Each code has a three-digit whole number followed by a decimal part. Books are stored in DDC numerical order in libraries all over the world. You can find science books in the range 500 – ; and within that maths books are usually filed beginning Books about pets have numbers beginning 636. Books about dogs start with and books about cats start with

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**N1 Place value, ordering and rounding**

Contents N1 Place value, ordering and rounding N1 N1.1 Place value N1 N1.2 Powers of ten N1 N1.3 Ordering decimals N1 N1.4 Rounding

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**There are 1432 pupils at Eastpark Secondary School.**

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. Talk about rounding in real life contexts, for example the number of people at a football match. Ask pupils to give examples. State that we can also use rounding in maths to give approximate answers to calculations. We can round numbers to the nearest 1000, 100, 10, whole number, 0.1, etc. depending on the level of accuracy required.

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**Rounding readings from scales**

Use the slider to show a variety of readings from each scale. Use the measuring cylinder to round reading to the nearest 10ml, the scales to round numbers to the nearest 10g or 100g and the thermometer to round numbers to the nearest 10oC including negative amounts. Introduce the convention of rounding up half-way values. Link: S7 Reading scales

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**Rounding whole numbers**

This activity illustrates the rounding of large whole numbers to the nearest 10, 100, 1000, , or using a number line. Extend the activity by asking pupils to determine the values of the two end-points before revealing them and then predict where the given value will be on the number line.

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**Rounding whole numbers**

Round to the nearest 100. Round 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. Talk through the example. Note that is not equal to is an approximation. We must include ‘to the nearest 100’ to ensure that the equals sign is not used incorrectly. Solution: 34871 = (to the nearest 100)

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**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 Talk through each answer in the table. 9875 10000 9900 9880 452 500 450

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Rounding decimals This activity illustrates the rounding of decimal numbers on a number line to the nearest given power of 10, for example to the nearest 0.1, or Extend the activity by asking pupils to determine the values of the two end-points before revealing them and then predict where the given value will be on the number line.

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**Rounding decimals Round 2.75241302 to one decimal place.**

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. Talk through the example. Emphasize that is not equal to is an approximation. to 1 decimal place is 2.8.

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**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. 63 63.5 63.47 63.472 88 87.7 87.66 87.656 150 150.0 149.99 Talk through each answer in the table. Point out that the number of decimal places tells you the number of digits that must be written after the decimal point, even if the last digit (or digits) are zero. 4 3.5 3.54 3.540 1 0.6 0.60 0.600

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**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: This is the first significant figure and This is the first significant figure

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**Rounding to significant figures**

The second, third and fourth significant figures are the digits immediately following the first significant figure, including zeros. For example: This is the first significant figure This is the third significant figure This is the second significant figure This is the fourth significant figure and This is the first significant figure

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**Rounding to significant figures**

Complete this table: to 3 s. f. to 2 s. f. to 1 s. f. 6.3528 6.35 6.4 6 34.026 34.0 34 30 0.0057 0.006 Discuss each example, including the use of zero place holders and zeros that are significant figures. In addition ask pupils to round to 3 s. f., 2 s. f. and then to 1 s. f. to make sure pupils are always using the original number to do the rounding from. 151 150 200

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N4 Decimals and rounding

N4 Decimals and rounding

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