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6 Year Helping your child with Maths Book 1 2015 Please look after this booklet Book1 1 Theme Pictograms and Grouped Data Your child will already have been introduced to bar graphs when they were younger but in Years 5 and 6 they will work with more complex graphs including line graphs and pie charts, demanding a more sophisticated understanding of data and how it can be represented. A Pictogram is simply a way of representing a number of items using pictures. For example: In the picture on the left we have a pictorial representation of the number of pets counted in a survey. The person has decided that each individual picture of an animal = 2 pets. Therefore the results of the survey are as follows: Dogs = 7 Rabbits = 4 Cats = 9 Other = 4 Can you work out the results if each picture stood for 10 animals? Grouped data, quite simply, is data that has been sorted into ranges of values rather than recording each single value. Example: Let’s say we decided to carry out a survey of the ages of people at a football match. As the people left we would ask their age and record their answers in a tally chart like this: AGE FREQUENCY Below 10 llll llll llll 14 10-19 llll llll llll llll llll llll lll 33 20-29 llll llll llll llll llll llll llll llll llll llll llll 54 30+ llll llll llll llll 19 1 This data would then be plotted onto a bar graph or pie chart. As we have grouped the data there would only be four bars in the bar graph or four slices in the pie. Why not get a piece of paper and draw the graph? How could you accurately draw a pie chart to represent this data? Clue 1: Protractor! Clue 2: 120 people watched the football match and a circular pie is equivalent to an angle of 360°. Clue 3: 3 x 12 =36. Still stuck? Ask your teacher! That’s all there is to it really. Of course the main skill expected of Year 6 pupils is to be able to interpret information from graphs including line graphs and pie charts. In Year 4 and 5 children have become confident at carrying out simple surveys and representing the information in a graph. In Year 6 children need to be a little more ‘flexible’ in the way they tackle graph work. Certainly in recent years KS2 SATs questions have featured a wide variety of graphs, demanding a good understanding of different scales and the ability to interpret the data presented. For a ‘different’ type of data handling question, access the electronic version of this booklet on the school website and click here. NOTES: 2 2 Theme Understanding Metric and Imperial Units of Length and Weight This section looks at different units of length and weight. The emphasis is to gain a good understanding of how to convert between different metric units of measurement. LENGTH Facts to LEARN: 1m = 100cm (and so 100 is the ‘key’ number when converting between the two) 1m = 1000mm (and so 1000 is the ‘key’ number when converting between the two) 1cm = 10mm (and so 10 is the ‘key’ number when converting between the two) 1km = 1000m (and so 1000 is the ‘key’ number when converting between the two) Once these facts have been ‘internalised’ then converting from one unit to another is straightforward – as long as you remember the ‘key’ number! Converting between Metres and Centimetres (key number = 100) To change m to cm simply MULTIPLY by 100. For example: 1.23m = 1.23 x 100 = 123cm To change cm to m simply DIVIDE by 100. For example: 1056cm = 1056 ÷ 100 = 10.56m Converting between Metres and Millimetres (key number = 1000) To change m to mm simply MULTIPLY by 1000. For example: 9.75m = 9.75 x 1000 = 9750mm To change mm to m simply DIVIDE by 1000. For example: 3 234mm = 234 ÷ 1000 = 0.234m Converting between Centimetres and Millimetres (key number = 10) To change cm to mm simply MULTIPLY by 10. For example: 1.5cm = 1.5 x 10 = 15mm To change mm to cm simply DIVIDE by 10. For example: 246mm = 246 ÷ 10 = 24.6cm Converting between Kilometres and Metres (key number = 1000) To change km to m simply MULTIPLY by 1000. For example: 3.06km = 3.06 x 1000 = 3060m To change m to km simply DIVIDE by 1000. For example: 1029m = 1029 ÷ 1000 = 1.029km Imperial Units of Length Some familiarity with imperial measures is expected. Children should be able to understand and use approximate equivalences between metric units and imperial units such as inches, pounds and pints. They should also be able to convert between miles and kilometres. If there is a question in SATs it is usually something like: “If 4km is equivalent to 2.5 miles, how many km are there in 17.5 miles?” or 0_______________________________________________________________50 miles 0_______________________________________________________________80km “Approximately how many kilometres are equivalent to 35 miles?” WEIGHT Facts to LEARN: 1kg = 1000g (and so 1000 is the ‘key’ number when converting between the two) 1 Tonne = 1000kg (and so 1000 is the ‘key’ number when converting between the two) 4 We are helped here by the fact that the ‘key’ number is always 1000, and therefore there is less chance for confusion! Converting between Kilograms and grams (key number = 1000) To change kg into g simply MULTIPLY by 1000. For example: 1.15kg = 1.15 x 1000 = 1150g To change g into kg simply DIVIDE by 1000. For example: 874g = 874 ÷ 1000 = 0.874kg Converting between Tonnes and Kilograms (key number = 1000) To change tonnes into kg simply MULTIPLY by 1000. For example: 1.03 Tonnes = 1.03 x 1000 = 1030kg To change kg into tones simply DIVIDE by 1000. For example: 1607kg = 1607 ÷ 1000 = 1.607tonnes Imperial Units of Weight If there is a question in SATs it is usually something like: “If 2.2lb(pounds) = 1kg, how many pounds are equivalent to 12kg?” or “If 1oz(ounce) = 30g and there are 16oz in 1lb, what weight in lbs is approximately equivalent to 5kg?” Clearly if children are to convert comfortably between different METRIC units of measure then the ability to multiply/divide by 10/100/1000 is crucial. When your child was in Year 5 they learnt how to do this by moving the digits to the LEFT (multiplying) or to the RIGHT (dividing). It works like this: Multiplying by 10 – move the digits ONE place left like this: TH H T 1 U . t 1 . 2 2 . 0 So - 1.2 x 10 = 12 5 Multiplying by 100 – move the digits TWO places left like this: TH 1 H 3 T U . t h 1 3 . 2 5 2 5 . So - 13.25 x 100 = 1325 Multiplying by 1000 – move the digits THREE places left like this: TTH TH 1 6 H 0 T U . t h 1 6 . 0 5 5 0 . So - 16.05 x 1000 = 16050 Notice here we need to put a zero in the units column to make the answer correct Dividing by 10 – move the digits ONE place right like this: TH H T U . 2 5 6 . 2 5 . t So - 256 ÷ 10 = 25.6 6 6 Dividing by 100 – move the digits TWO places right like this: TH H T U . 1 6 . 0 . t h So - 16 ÷ 100 = 0.16 1 6 Dividing by 1000 – move the digits THREE places right like this: TH 2 H T 0 5 U . 0 . 2 . t h th So - 2050 ÷ 1000 = 2.05 0 5 0 Notice that we do not need to put the final zero in the answer. Need more practice? Use the grid below to do just that! TTH TH H T U . . . . . . . . . . 7 t h th 3 Theme Angles Measuring and Estimating Angles Angles are measured in degrees. Children have already used protractors in Year 5, but firstly we need to make sure they can remember how to use them! It is also important that children are able to estimate the size of any given angle. In order to help them ‘estimate’ intelligently, they need to first of all recognize angles as ‘acute’, ‘obtuse’ or ‘reflex’. Acute: angles that are LESS than 90˚ are called acute angles. 8 An angle of exactly 90˚ is called a ‘right-angle’ and looks like this: Obtuse: angles that are GREATER than 90˚ but LESS than 180˚ are called obtuse angles. An angle of exactly 180˚ is a straight line! I don’t think we need to draw one of those for you! Reflex: angles that are GREATER than 180˚ but less than 360˚ are called reflex. 9 An angle of 360˚ is a full turn or a circle – I don’t think we need to draw you one of those either! Using a Protractor Protractors can be 180˚ or 360˚. Your child will be taught how to use both. Some children prefer one to the other. In their tests they can choose to use the one they are most comfortable with. Protractors can be used to measure angles and draw angles. Your child will be shown how to do this. For a good ‘on-line’ tutorial on how to use a protractor, find this page on the electronic version of this booklet on the school website and click here . When the tutorial loads choose option 3 – ‘Measuring Angles’. 10 Calculating Angles Children in Year 6 also need to be able to calculate angles using the rules they know about the number of degrees in a triangle, on a straight line etc. Again it would be really useful if your child could learn by heart these few important rules: (i) Angles that meet at a point total 360˚ The simplest way to think about this is to look at a cross. Here we have 4 right angles meeting at a point, and 4 x 90˚ = 360˚ Here is an example where none of the angles are right angles: 140˚ + 87˚ + 80˚ + 53˚ = 360˚ So in the following example we can calculate the missing angle by applying the rule: 110˚ + 75˚ + 63˚ + 50˚ = 298˚ 298˚ + Angle C = 360˚ Angle C = 360˚ - 298˚ = 62˚ 11 Looking at angles in relation to a clock face is quite a common way of testing a Year 6 pupil’s understanding of this topic, so it’s probably worth having a quick look at this now. Here is a clock face showing 4 o’clock. Can you calculate the obtuse angle that is made by the two hands? The key to this is knowing that each section of the clock face must equal 30˚. (12 x 30 = 360˚) So the answer to the question is 4 x 30˚ = 120˚. Now try this one: Imagine the time has advanced to 4.45 – How many degrees will the minute hand have moved through? Answer: The minute hand will turn through 9 ‘sections’ of the clock face to reach 4.45, and therefore the angle = 9 x 30˚ = 270˚ (ii) The three angles of a triangle total 180˚ All triangles contain three angles that total 180˚. So if we know two of the angles we can easily calculate the third: In this example we can see that Angle A = 40˚ and Angle B = 60˚. So Angle C must = 80˚ because 40˚ + 60˚ + 80˚ = 180˚ Equilateral Triangles Equilateral triangles are triangles where all the sides (and therefore the angles) are all the same size. So each angle of an equilateral triangle will equal 60˚. (60 + 60 + 60 = 180) Isosceles Triangles Isosceles triangles are triangles where two of the sides (and therefore two of the angles) are the same size. Look at this example of an isosceles triangle: 12 Notice that the two equal angles are at the end of the equal sides. 72˚ + 72˚ + 36˚ = 180˚. Notice the = sign on the two equal sides. If you know that a triangle is ‘isosceles’ then you only need to know one of the angles to be able to work out the other two. Look at the next example. Right angled IsoscelesTriangle C 42˚ A B In this isosceles triangle sides A and B are of equal length and one of the angles is 42˚. Can you work out the other two angles? Solution: If A and B are equal in size then the angle between sides C and B must be equal to the angle between A and C (ie 42˚). So if we add these two angles together we get 42 + 42 = 84. Therefore the third angle must be 180 – 84 = 96˚ Some space for your own jottings/notes: 13 Scalene Triangles A scalene triangle is one where all the sides (and therefore all the angles) are different in size. For example: (iii) Angles on a straight line total 180˚ 150˚ x˚ In this example we can see that angle x must equal 30˚ Because 150 + 30 = 180˚. Now try this one. Can you calculate the value of angle K? A 41˚ K B 20˚ We can see that AB is a straight line and that it has three angles sat on it. We are told the size of two of these angles (41˚ + 20˚ = 61˚). Therefore angle K must be 180˚ - 61˚ = 119˚. If you log on to the school website and access the electronic version of this booklet you can click here and you will be able to view a powerpoint presentation entitled ‘Missing Angles’. This should clarify things if you are still a bit unsure! 14 4 Theme The Four Rules (Again!) Yes folks – it’s here again! For the next couple of weeks your child will be reminded about various written and mental strategies that they could use in adding, subtracting, multiplying and dividing. Your child will probably already have a favourite method for each of these operations, but will their method be the most appropriate for the job? Furthermore, can they apply their calculation strategies effectively to solve written problems? Adding and Subtracting During the course of KS2 your child has learnt various methods for adding and subtracting numbers, including formal written methods (column addition/subtraction). However we cannot stress enough the importance of mental strategies, and indeed the vast majority of KS2 test questions can be answered through a combination of mental calculations and written jottings. Example: Mr Smith is a poor teacher and so he has been trying to earn some extra money in the school holidays by washing his friends’ cars. He earned £45 in week 1, £38 in week 2, £27 in week 3 and £35 in week 4. He now has enough money to enjoy himself during the last two weeks! How much did Mr Smith earn altogether? Now you may be sat at home reading this thinking the best way to add these up is to write them out in a column addition. Certainly if you were taught in a primary school in the 1960’s or 70’s, then this ‘formal’ way of adding numbers would have been the obvious method. Like this: 45 38 27 +35 145 2 This method is fine if you are able to write the sum down neatly and remember to add on the number you have ‘carried’. Some children are not good at this. Therefore they may prefer a more ‘informal’ way of adding: 15 45 + 38 + 27 + 35 We begin by adding the ‘tens’, looking for number bonds as we go: 40 + 30 + 30 = 100 + 20 = 120 This leaves: 5+8+7+5 5 + 5 = 10 8 + 7 = 15 10 + 15 = 25 Then 120 + 25 = 145 So Mr Smith earns £145 by washing cars. Compensatory addition and subtraction This is an important mental strategy, and one that many children fail to use. Example: 5246 + 2997 Again many ‘older’ people would simply write one number on top of the other and perform a column addition without giving it a moment’s thought. Others might ‘partition’ the numbers into thousands, hundreds, tens etc. and add up separately before combining all the totals at the end. The clever mathematicians, however, can see that 2997 is just three less than 3000. So: 5246 + 3000 = 8246 Now by adding 3000 we have actually added 3 too many, and therefore we now need to subtract 3 to make the answer correct: 8246 – 3 = 8243 and therefore: 5246 + 2997 = 8243 Here is another example, this time in a ‘take-away’ sum: 8506 – 398 In this example the smart way is to initially subtract 400. So 16 8506 – 400 = 8106 But by taking away 400, we have actually subtracted 2 extra, so we must now adjust our calculation accordingly by adding the 2 back on: 8106 + 2 = 8108 Therefore 8506 – 398 = 8108. Simple isn’t it! See if you can use this ‘compensatory’ method to solve this next problem: Mrs Smith went to Blackpool for the day. She had £100 in her purse. She bought a lilo for £3.99 and a beach ball for £2.99. For her lunch she enjoyed fish and chips (£2.50) and a ‘99’ ice cream which actually cost £0.99! In the afternoon she bought a pass for the Pleasure Beach costing £15.95. Having been sick on the ‘Pepsi Max’ she decided to call it a day and go home – but not before having 3 rides on Denis the Donkey (99p per ride – ‘buy two get one free!) How much money did Mrs Smith have left out of her original £100? Use this box for your jottings: £ 17 Adding and Subtracting Decimals It’s only very small but that decimal point can create total chaos in the minds of our young mathematicians! And yet there is no need to panic – we can use our knowledge of whole number calculations to help us. Here is an example: What is 11.38 - 7.69? Step 1: make a rough estimate = 11 – 7 = 4 Step 2: ignore the decimal point so that the sum reads 1138 – 769 Step 3: subtract 769 from 1138 using any method you like. Answer = 369 Step 4: put the decimal point into the answer so that it confirms your estimate = 3.69 Another way of looking at it is to say that because there are 2 digits after the decimal point in the numbers in the question, then there will be 2 digits after the decimal point in the answer. BANANA SKIN!! What if we are faced with a question like this? What is 17.45 – 13.2? Here you can see that there are 2 digits after the dp in the first number but only 1 digit after the dp in the second. This is what you have to do: Step 1: still make an estimate = 17 – 13 = 4 Step 2: make the 13.2 into 13.20 (this doesn’t change the value of the number but it does make both numbers in the question have 2 digits after the dp) Step 3: ignore the decimal point so that the sum now reads 1745 – 1320 Step 4: subtract 1320 from 1745 using any method you like. Answer = 425 Step 5: put the decimal point into the answer so that it confirms your estimate = 4.25 Counting On Your child has been taught this method of subtraction in Year 4 and 5. We think it is really quite a safe method. Many Year 6 pupils make errors in subtraction calculations because they have forgotten how useful this method is! Here is a reminder of how it works: Example: Mrs Asady has just bought a new pair of pom-poms for her cheerleading club. They cost £7.65. She paid for them with a £20 note. How much change did she receive? +5p +30p +£12 £7.65_____£7.70____________£8.00________________________________£20.00 So you can see that we have counted on from £7.65 to £20 in three stages: 18 £7.65 up to £7.70 £7.70 up to £8 £8 up to £20 = a jump of 5p = a jump of 30p = a jump of £12 So altogether we have jumped £12.35. So this must be Mrs Asady’s change. Now let’s try another one: Calculate 7.01 – 1.82 Estimate first: 7 – 2 = 5 Ignore the decimal point to give us 701 – 182 Now count on from 182 to 701 +8 +10 +500 +1 182______190_________200__________________________700____701 So altogether we have jumped on 519. Now put the decimal point back in so that it confirms your estimate = 5.19 Multiplying and Dividing This section of work begins again with a ‘refresher’ of ideas and methods covered in Year 5. Again we must reiterate the importance of ‘Times Tables’ knowledge. If your child is able to learn these multiplication/division facts well, then they will have an important tool in their mathematical kitbags! DOUBLING AND HALVING This is an important mathematical skill and one which will help your child in many areas of multiplication and division. Here are some top tips: X2 means you double the number X4 means you double and double again X8 means you double, double and double again. ÷2 means halve the number ÷4 means halve and halve again ÷8 means halve, halve and halve again. The x16 table is the x8 table doubled The x 18 table is the x9 table doubled and so on… The starting point for multiplying by 25 or 50 is x 100. For example: 19 What is 13 x 25? 13 x 100 13 x 50 13 x 25 = 1300 = 650 = 325 We can also use this method to calculate multiples close to 50 or 25. For example: There are 24 rows of seats in a theatre and each row contains 17 seats. How many seats are there altogether in the theatre? If there were 25 rows of seats, we could work it out like this: 100 x 17 50 x 17 25 x 17 = 1700 = 850 = 425 But there were 24 rows not 25, so that is one row less (ie 17 seats less) So 24 x 17 = 408 Another useful tip to know is that in any multiplication sum, if you double one side and halve the other the answer will be the same. For example: 14 x 35 is the same as 7(half of 14) x 70(double 35) 7 x 70 = 7 x 7 x 10 = 490 Here is another example: 18 x 40 = 9 x 80 = 9 x 8 x 10 = 720 It is worth remembering this little ‘trick’ as it can turn quite difficult and scary looking sums into little pussycats. Miaow! 20 5 Theme Mixed Numbers and Improper Fractions This mini theme is a reprise of a topic covered in Year 5, so your child should know something already – famous last words! Last year we did lots of visual representations of Mixed Numbers and Improper Fractions; this year we’ll try a quicker way of moving between the two! First of all let’s make sure we all understand what the two terms mean. A MIXED NUMBER is where you have a whole number and a fraction together. For example: 3½ is a mixed number An IMPROPER FRACTION is where the numerator (top number) is greater than the denominator (bottom number). For example: 7 2 is an improper fraction Year 6 children will be taught how to convert from mixed numbers to improper fractions and viceversa. Here are the two rules in a nutshell: (i) “To turn a mixed number into an improper fraction multiply the whole number by the denominator, add the numerator and then put the total over the denominator” For example 3½ Start by taking the whole number (3) and multiply it by the denominator (2) So 3 x 2 = 6 Next add the numerator (1), so that makes 7 Finally put this total on top of the denominator (2), so this gives us 7 21 2 (ii) “To change an improper fraction into a mixed number simply divide the numerator by the denominator. Any remainder then becomes the numerator over the original denominator” For example: 7 2 Start by dividing the numerator (7) by the denominator (2) 7 ÷ 2 = 3 remainder 1 So our answer of 3 becomes the whole number and the remainder becomes So 7 2 =3 ½ Why not try a few examples of your own in this box! 22 ½ 6 Theme Equivalent Fractions Here we go with another aspect of fractions work. It is really important that Year 6 children are able to recognise the common fraction equivalents (including decimals and percentages) and that they have some strategies to use if they are not sure. All fractions where the numerator is numerator) are equivalent to 1 of the denominator (the denominator is double the 2 1 : 2 1 2 3 4 = = = etc. 2 4 6 8 All fractions where the numerator is 1 of the denominator (the denominator is four times the 4 1 numerator) are equivalent to : 4 1 2 3 4 = = = etc. 4 8 12 16 All fractions where the numerator is numerator) are equivalent to 1 of the denominator (the denominator is three times the 3 1 : 3 1 2 3 4 = = = etc 3 6 9 12 You can work out equivalents of 1 1 , etc in exactly the same way. 5 6 23 This fraction wall will help you spot common equivalents: Here is another way of working out equivalent fractions. If you take any fraction and multiply or divide the numerator and denominator by the same number you will create an equivalent fraction. For example: 28 40 If we multiply top and bottom by 2 we get 56 28 . This fraction is therefore equivalent to . 80 40 As long as we multiply top and bottom by same number we will create an equivalent fraction. If we 84 28 multiply by 3 we get , and so this fraction is equivalent to . 120 40 We can also create equivalent fractions by dividing top and bottom by the same number. Let us 28 14 stick with as our example. Let us divide by 2: 28 ÷ 2 = 14, and 40 ÷ 2 = 20. Therefore is 40 20 7 equivalent. We could divide by 2 again: 14 ÷ 2 = 7, and 20 ÷ 2 = 10. This gives us . Some of you 10 bright sparks may have realised that we could have divided by 4 at the start instead: 28 ÷ 4 = 7, and 40 ÷ 4 = 10. 24 7 we can see that we are unable to divide any further. When this 10 happens we say that the fraction is in its simplest form. If we now look at this fraction Look at the table below. Can you change the fractions into their simplest forms? Remember you will have to divide numerator and denominator by the same number. Can you spot the fraction that is already in its simplest form? 2 4 4 16 3 9 200 400 150 600 6 12 8 32 3 12 4 5 3 24 5 10 6 18 4 20 5 20 3 15 4 12 2 8 5 20 10 40 11 22 7 21 60 180 7 28 10 30 6 24 9 18 12 36 50 100 25 75 80 240 Now have some fun using this interactive resource on equivalent fractions: (You will need to access the electronic version of this booklet via the Learning Platform) http://www.learningplanet.com/sam/ff/index.asp 25 7 Theme Ordering Fractions The next strand in our ‘Fractions’ work is all to do with ordering fractions from smallest to largest or vice versa. The skills we have gained through our work on equivalence will be put to good use here! Ordering fractions that have the same denominator is simple. It is when fractions have different denominators that problems might be encountered. Rule number 1: Don’t panic! You already have the skills and the knowledge to tackle any question to do with ordering fractions. Let’s look at a typical SATs question: Put these fractions in order starting with the smallest: 7 10 2 3 7 15 1 2 4 5 Now, as you can see, each of these fractions has a different denominator. So what we need to do is to change them so that they all have the same denominator. But how do we do that? Simple. Start by looking at the denominators. We need to find a number that is a common multiple of 10, 3, 15, 2 and 5. In other words we need a number that all these denominators will divide into exactly. Have you worked it out yet? That’s right – it’s 30. The next step is to take each of the fractions in turn and change them into thirtieths – fractions out of 30. Here’s how to do it: 7 ? = So the denominator of 10 has become a denominator of 30. In other words we have 10 30 multiplied the denominator by 3. Therefore we must multiply the numerator by 3 as well to give an equivalent fraction out of 30. Well 7 x 3 = 21, so 7 21 = 10 30 Now let’s move onto the next one: 26 2 ? = This time the denominator of 3 has been multiplied by 10 to make 30, so we must multiply 3 30 the numerator by 10 also: 2 20 = 3 30 7 ? = multiply by 2 to give: 15 30 7 14 = 15 30 1 15 = I hope you spotted this one without too much trouble! And finally – 2 30 4 ? = This time we need to multiply by 6 to give: 5 30 4 24 = 5 30 So our results are: Original Fraction 7 10 2 3 7 15 1 2 4 5 Fraction out of 30 21 30 20 30 14 30 15 30 24 30 So the original fractions arranged from smallest to largest are: 7 15 1 2 2 3 27 7 10 4 5