DECIMALS
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DECIMALS

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Some topics in order to understand decimals, powers of ten etc.

Some topics in order to understand decimals, powers of ten etc.

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  • 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 36-39.
  • 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 6,304.97. 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 ( no tens). 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 0.1 from this number we need to take 1 from the tenths column, to give 6304.87. Compare this with subtracting 10 from tens column. The place value grid can also be used to demonstrate multiplying and dividing numbers by 10, 100, 1000, 0.1 and 0.01. Groups of digits can be dragged to the left and right to show this.
  • This is the first of three slides illustrating the effect of multiplying a decimal number by 10, 100 and 1000. The emphasis is when multiplying by 10, 100, 1000 numbers get bigger and when dividing by 10,100,1000 numbers get smaller. 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. Students may have prior experience of moving point – ask the question should it get bigger or smaller. Links: N9 Mental methods – multiplication and division S7 Measures – converting units
  • Again, stress 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.
  • The next three slides illustrate the effect of dividing a decimal number by 10, 100 and 1000. Link: N9 Mental methods – multiplication and division
  • Link: N9 Mental methods – multiplication and division
  • Link: N9 Mental methods – multiplication and division
  • 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.
  • 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.
  • 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.
  • Extend the discussion to multiplication by 0.01.
  • 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 10 3 , 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 10 9 (one thousand million). In the past, a British billion was one million million, or 10 12 . Links: N4 Powers and roots – powers
  • Discuss the convention of using negative integers to express tenths, hundredths, thousandths, and other negative powers of ten. Link: N4 Powers and roots – powers
  • Discuss the use of standard form to write large numbers. Discuss whether 10 ×10 5 and 1 ×10 4 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.
  • 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 in the number.
  • Discuss how each number written in standard form should be written in full.
  • The image of a shelled amoeba has been reproduced with the kind permission of Wim van Egmond © Microscopy UK http://www.microscopy-uk.org.uk/index.html
  • Notice that the power of ten is always ‘–’ and the number of zeros before the first significant figure including the one before the decimal point..
  • Notice that the power of ten tells us the number of zeros before the first significant figure including the one before the decimal point..
  • 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.
  • 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 0.001 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.
  • 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.
  • 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
  • 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 1.702. Some pupils may need reminding of the meaning of the ‘greater than’ symbol.
  • Stress 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.
  • 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.
  • Start by emphasising 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.
  • 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.
  • Practice 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 – 599.999; and within that maths books are usually filed beginning 510. Books about pets have numbers beginning 636. Books about dogs start with 636.7 and books about cats start with 636.8.
  • 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, 0.01 0.001 etc. depending on the level of accuracy required.
  • 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 10 o C including negative amounts. Introduce the convention of rounding up half-way values. Link: S7 Reading scales
  • This activity illustrates the rounding of large whole numbers to the nearest 10, 100, 1000, 10 000, 100 000 or 1 000 000 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.
  • Talk through the example. Note that 34871 is not equal to 34900. 34900 is an approximation. We must include ‘to the nearest 100’ to ensure that the equals sign is not used incorrectly.
  • Talk through each answer in the table.
  • 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, 0.001 or 0.0001. 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.
  • Talk through the example. Emphasize that 2.75241302 is not equal to 2.8. 2.8 is an approximation.
  • Talk through each answer in the table. Stress 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.
  • Discuss each example, including the use of zero place holders and zeros that are significant figures.

DECIMALS DECIMALS Presentation Transcript

  • © Boardworks Ltd 2004of 51 N1 Place value, ordering and rounding KS3 Mathematics
  • © Boardworks Ltd 2004of 51 N 1 N 1 N 1 N 1 N1.1 Place value Contents N1 Place value, ordering and rounding N1.3 Ordering decimals N1.4 Rounding N1.2 Powers of ten
  • © Boardworks Ltd 2004of 51 Place value
  • © Boardworks Ltd 2004of 51 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 When we multiply by ten the digits move one place to the left. 6 2 6.2 × 10 = 62 Multiplying by 10, 100 and 1000
  • © Boardworks Ltd 2004of 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 Thousands Hundreds Tens Units tenths hundredths thousandths 3 1 3 1 0 Multiplying by 10, 100 and 1000
  • © Boardworks Ltd 2004of 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 Thousands Hundreds Tens Units tenths hundredths thousandths 0 7 7 0 0 Multiplying by 10, 100 and 1000
  • © Boardworks Ltd 2004of 51 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 When we divide by ten the digits move one place to the right. 4 5 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
  • © Boardworks Ltd 2004of 51 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 When we divide by one hundred the digits move two places to the right. 9 4 We need to add zero place holders. 0 0 9.4 ÷ 100 = 0.094
  • © Boardworks Ltd 2004of 51 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 0 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
  • © Boardworks Ltd 20040 of 51 Spider diagram
  • © Boardworks Ltd 20041 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 ÷ = 7.3810 ÷ 1000 = 8.318310 0.64 × = 6401000 0.021 × 100 = 2.1 250 ÷ = 2.5100
  • © Boardworks Ltd 20042 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 which is equivalent to 4 ÷ 10 1 10 Therefore, 4 × 0.1 = 0.4 Multiplying by 0.1 Dividing by 10is the same as
  • © Boardworks Ltd 20043 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 100We can also think of this as 3 × which is equivalent to 3 ÷ 100 Therefore, 3 × 0.01 = 0.03 Multiplying by 0.01 Dividing by 100is the same as
  • © Boardworks Ltd 20044 of 51 N 1 N 1 N 1 N 1 N1.2 Powers of ten Contents N1.1 Place value N1 Place value, ordering and rounding N1.3 Ordering decimals N1.4 Rounding
  • © Boardworks Ltd 20045 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 = 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 …
  • © Boardworks Ltd 20046 of 51 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. 0.01 = = = 10-21 102 1 100 0.001 = = = 10-31 103 1 1000 0.0001 = = = 10-41 10000 1 104 0.00001 = = = 10-51 100000 1 105 0.000001 = = = 10-61 1000000 1 106 0.1 = = =10-11 10 1 101
  • © Boardworks Ltd 20047 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 × 108 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
  • © Boardworks Ltd 20048 of 51 How can we write these numbers in standard form? 80 000 000 = 8 × 107 230 000 000 = 2.3 × 108 724 000 = 7.24 × 105 6 003 000 000 = 6.003 × 109 371.45 = 3.7145 × 102 Standard form – writing large numbers
  • © Boardworks Ltd 20049 of 51 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 2.168 × 107 = 21 680 000 6.7645 × 103 = 6764.5 Standard form – writing large numbers
  • © Boardworks Ltd 20040 of 51 We can also write very small numbers using standard form. For example, the 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
  • © Boardworks Ltd 20041 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
  • © Boardworks Ltd 20042 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
  • © Boardworks Ltd 20043 of 51 N 1 N 1 N 1 N 1 N1.3 Ordering decimals Contents N1.4 Rounding N1.1 Place value N1 Place value, ordering and rounding N1.2 Powers of ten
  • © Boardworks Ltd 20044 of 51 Zooming in on a number line
  • © Boardworks Ltd 20045 of 51 Decimal sequences
  • © Boardworks Ltd 20046 of 51 Decimals on a number line
  • © Boardworks Ltd 20047 of 51 Mid-points
  • © Boardworks Ltd 20048 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
  • © Boardworks Ltd 20049 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
  • © Boardworks Ltd 20040 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
  • © Boardworks Ltd 20041 of 51 Comparing decimals
  • © Boardworks Ltd 20042 of 51 4.67 4.74.717 4.77 4.73 4.074.67 4.717 4.734.77 4.074.74.67 4.717 4.77 4.074.73 4.74.67 4.717 4.77 4.73 4.70 4.074.717 4.77 4.73 4.7 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: 4.07 4.67 4.7 4.717 4.73 4.77 Ordering decimals
  • © Boardworks Ltd 20043 of 51 Ordering decimals
  • © Boardworks Ltd 20044 of 51 Dewey Decimal Classification System
  • © Boardworks Ltd 20045 of 51 N 1 N 1 N 1 N 1 N1.4 Rounding Contents N1.3 Ordering decimals N1.1 Place value N1 Place value, ordering and rounding N1.2 Powers of ten
  • © Boardworks Ltd 20046 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.
  • © Boardworks Ltd 20047 of 51 Rounding readings from scales
  • © Boardworks Ltd 20048 of 51 Rounding whole numbers
  • © Boardworks Ltd 20049 of 51 Example Round 34 871 to the nearest 100.Round 34 871Round 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
  • © Boardworks Ltd 20040 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 38000 37500 37520 275000 274500 274500 7631000 7630900 7630920 10000 9900 9880 0 500 450
  • © Boardworks Ltd 20041 of 51 Rounding decimals
  • © Boardworks Ltd 20042 of 51 Round 2.75241302 to one decimal place.Round 2.75241302Round 2.75241302 Example 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
  • © Boardworks Ltd 20043 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. 63 63.5 63.47 63.472 88 87.7 87.66 87.656 150 150.0 149.99 149.988 4 3.5 3.54 3.540 1 0.6 0.60 0.600
  • © Boardworks Ltd 20044 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
  • © Boardworks Ltd 20045 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
  • © Boardworks Ltd 20046 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. 6.35 6.4 6 34.0 34 30 0.00571 0.0057 0.006 151 150 200 0.0000784 0.000078 0.00008 Rounding to significant figures