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Fractions guide(pdf)

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  • 1. Fractions GuideAbout this program About the author Why fractions? Begin Copyright notice
  • 2. About this programThis is a co-learning program. If you have bought this or borrowed this orsimply copied this with the intention of passing it to your son or daughteror student so that they can teach themselves, you have wasted your time.This is a framework from which a good tutor or parent can devise a seriesof lessons for young people. The best plan for the parent or tutor (that‟syou) is to preview the material in these lessons and familiarise yourselfwith the menu system, and then go through it again with your child orstudent. Have a large pile of paper nearby so that you can practise thematerial and (if necessary) learn the material together.This is not a game. I have not installed anything in here other than the barebones of a long and dry introduction to fractions. In my experience, youdon‟t need flying saucers and shotguns to make a good lesson in maths.You just need a really good grown-up to accompany the child on his or herjourney. This program is your roadmap. Return
  • 3. About the authorMy name is David Coulson. At the time of writing, I am 47 years old andhave been teaching or tutoring maths for 20 out of the last 25 years. I didnot make this program for profit or out of any naïve notions of furtheringmy career. This is intended to be a freebie so that kids who can learnfractions and have the desire to learn fractions will actually have thechance to learn fractions. In this respect I am thinking most of thehundreds of youngsters I have tutored from my home, who surprised mewith their desire to learn maths simply because it was easy and fun, andthereby re-convinced me that maths is not everything it‟s cracked down tobe. I can be contacted at dtcoulson@gmail.com Return
  • 4. Why fractions?Let‟s not regurgitate any party political manifestos here. We all knowthat ninety percent of grown-ups never use fractions, and of those thatdo, ninety percent of them never need fractions more severe than ¾ and½ and ¼ . That means that only one percent of the users of this programwill actually use the material beyond passing the next maths test atschool.The trouble is that none of us know who that one percent is. While itwould be naïve of me to believe that every parent wants their child to bepart of that one percent, very few of us are prepared to close the door onour children and say that they are definitely not in that one percent.So we shakily encourage our kids to learn fractions, whether they needto or not. For the ninety-nine percent of students who won‟t use fractionsmuch after leaving school, rest assured that a child who can learn to dofractions can learn to do just about anything. Fractions, therefore, like allof maths and all the other unworldly things taught at school, are toolsthat sharpen a child‟s mind and prepare him or her for…. Well, whoreally knows? Return
  • 5. CopyrightThis program of course is covered by copyright laws. Thisprogram has been released free to the public domain on faiththat it will be used as it was designed, i.e., as a tool foreducation. You may copy the program and circulate it as youplease as long as the price of the program remains zero. Inparticular, removing my name from this program and on-selling it as your own material is strictly forbidden. As well asbeing illegal, it‟s clearly a dirty thing to do. This appliesregardless of whether the offender is an individual, a privateorganisation or a government department or any othercategory not mentioned.
  • 6. Begin To say you are good at fractions means you know how to… Add fractions Subtract fractions Multiply fractions Divide fractions Convert fractions Memorise fractionsManipulate fractions Estimate fractions Choose a topicEnd Show or test yourself
  • 7. Add – deciderAddition : Choose a level Main Index Addition SubtractionMultiplication Easiest Division Conversion Memorise Manipulate Estimate Definitions Hardest Self-test Previous Last viewed
  • 8. Addition : Basics add – L1 EG1 Main Index Addition Subtraction Fractions can only be added if theMultiplication denominators are the same. Division Conversion Therefore turn the quarter Memorise into a twelfth by scaling up. Manipulate Estimate Definitions You can turn 4/ into 1/ by Self-test 12 3 scaling down. Previous Last viewed More examples
  • 9. Addition : Basics add – L1 EG2 Main Index Addition Subtraction A third can‟t be added to a ninth.Multiplication Division Conversion Turn the third into a ninth by Memorise tripling the numbers. Manipulate Estimate Definitions Self-test 14/ be converted into 15/9 9 can Previous Last viewed Solved problems
  • 10. Addition : Basics Add – L1 SP Main Index Addition Try these yourself SubtractionMultiplication 2/ + 1 /6 = 5 /6 ½ + 1/10 = 6/10 3 Division Conversion ¼ + 3 /8 = 5 /8 1/ 9 + 1/27 = 4/27 Memorise Manipulate 1/ + 1/3 = 6/15 5/ + 11/ = 61/ Estimate 15 6 60 60 Definitions 5/ + 1/39 = 16/ 3/ + 1/14 = 5/28 Self-test 13 39 28 Previous Last viewed Next level
  • 11. Addition : Level 2add – L2 EG1 Main Index What do you do when the denominator of one fraction Addition can‟t be turned into the denominator of the other fraction? Subtraction Multiplication Division Conversion Memorise Both fractions have to be scaled up to Manipulate something else. Estimate Definitions Self-test Previous Last viewed More examples
  • 12. Addition : Level 2 Add – L2 EG2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 13. Addition : Level 2 add – L2 SP Main Index Addition Subtraction Try these yourselfMultiplication Division 1/ + 1/8 = 5/24 1/ + 1/15 = 5/30 = 1/6 12 10 Conversion Memorise 3/ + 1/24 = 11/ 1/ + 1/9 = 8/45 16 48 15 Manipulate Estimate 5/ + 7 /9 = 64/ 3/ + 3/10 = 21/ Definitions 21 63 25 50 Self-test Previous Last viewed Next level
  • 14. Addition : Picnic table method add – L3 EG1 Main Index Sometimes the denominators are so different that Addition there is no obvious way to make them the same. SubtractionMultiplication Division Conversion Memorise In that case, draw this thing that Manipulate looks like an upturned picnic table. Estimate Definitions Self-test Then cross-multiply along the red lines Previous Last viewed More examples
  • 15. Addition : Picnic table method add – L3 EG2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 16. add – L3 SPAddition : Picnic table method Main Index Addition SubtractionMultiplication Try these yourself Division 9/ + 9/11 = 189/ ¾ + 2/9 = 35/ Conversion 10 110 36 Memorise Manipulate 1/ 100 + 1/63 = 163/ 6300 ½ + 4/9 = 17/ 18 Estimate Definitions 7/ + 1/5 = 43/ 2/ + 2/7 = 40/ 8 40 13 91 Self-test Previous Last viewed Next level
  • 17. Addition : Mixed fractions add – L4 EG1 Main Index Addition Separate the fractional parts Subtraction from the integer parts and thenMultiplication add them separately Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed More examples
  • 18. Addition : Mixed fractions add – L4 EG2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 19. add – L4 SPAddition : Mixed fractions Main Index Addition SubtractionMultiplication Try these yourself Division 11/2 + 9/10 = 22/5 11/3 + 14/ 15 = 24/15 Conversion Memorise Manipulate 11/4 + 13/4 = 3 11/9 + 11/3 = 24/9 Estimate Definitions 21/8 + 19/10 = 41/40 35/9 + 44/7 = 88/63 Self-test Previous End of topic Last viewed Choose another from the sidebar
  • 20. Subtraction : Choose a level Sub - decider Main Index Addition SubtractionMultiplication Easiest Basics Division Conversion Memorise Level 2 Manipulate Estimate Picnic table method Definitions Self-test Mixed Hardest fractions Previous Last viewed
  • 21. sub – L1 EG1Subtraction : Basics Main Index Subtraction works exactly the same way as addition. Addition If you don‟t recognise what I‟m doing, see the lesson Subtraction on additionMultiplication Division Conversion Memorise Manipulate The 2/3 has to be scaled up to 4/6 Estimate Any fraction which can be Definitions scaled down (such as 3/6) Self-test should be scaled down, at the end of a problem. Previous Last viewed More examples
  • 22. sub – L1 EG2Subtraction : Basics Main Index Addition SubtractionMultiplication A scale factor of 3 will turn the fifth into a fifteenth. Division Conversion Memorise Manipulate Estimate Definitions When the denominators are the Self-test same, the fractions can be combined into one. Previous Last viewed Solved problems
  • 23. Subtraction : Basics Sub – L1 SP Main Index Addition Subtraction Try these yourselfMultiplication Division 2/ - 1 /9 = 5 /9 ½ - 1/20 = 9/20 3 Conversion Memorise ¼ - 3/16 = 1/16 1/ 9 - 1/27 = 2/27 Manipulate Estimate 14/ - 1/9 = 9/45 5/ - 1/60 = 49/ 45 6 60 Definitions Self-test 5/ - 1/39 = 14/ 21/ - 1/14 = 19/ 13 39 28 28 Previous Last viewed Next level
  • 24. Sub – L2 EG1Subtraction : Level 2 Main Index Addition SubtractionMultiplication Division Conversion Scale up both fractions Memorise to twelfths Manipulate Estimate Definitions Self-test ( If you don‟t know what I‟m doing, see the lesson on addition. ) Previous Last viewed More examples
  • 25. Sub – L2 EG2Subtraction : Level 2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Scale factor 3 Scale factor 4 Estimate Definitions Self-test Previous Last viewed Solved problems
  • 26. Sub – L2 SPSubtraction : Level 2 Main Index Addition Subtraction Try these yourselfMultiplication Division 1/ - 1/12 = 1/24 1/ - 1/15 = 1/30 8 10 Conversion Memorise 3/ - 1/24 = 7/48 1/ - 1/15 = 2/45 16 9 Manipulate Estimate 5/ - 2/9 = 1/63 9/ - 3/10 = 3/50 Definitions 21 25 Self-test Previous Last viewed Next level
  • 27. Subtraction : Picnic table method Sub – L3 EG1 Main Index Addition Use the picnic table method when the denominators cannot be combined in any easy way. SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed More examples
  • 28. Subtraction : Picnic table method Sub – L3 SP Main Index Addition Subtraction Try these yourselfMultiplication Division 9/ - 9/ = 9/110 ¾ - 2/9 = 19/36 10 11 Conversion Memorise 1/ 63 - 1/100 = 37/6300 ½ - 2/9 = 5/18 Manipulate Estimate 7/ - 1/5 = 27/40 10/ - 2/ = 44/91 Definitions 8 13 7 Self-test Previous Last viewed Next level
  • 29. Subtraction : Mixed Fractions Sub – L4 EG1 Main Index Addition Sometimes the integer and fractional parts of a pair of Subtraction mixed fractions can be subtracted separately.Multiplication Division Conversion Memorise Manipulate Estimate For this to work, the fraction on the left has to be bigger than the fraction on the right This won‟t Definitions always be the case. Self-test Previous Last viewed Different technique
  • 30. Subtraction : Mixed Fractions Sub – L4 EG2 Main Index In some cases it will be necessary to turn the Addition mixed fractions into top-heavy fractions. SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 31. Subtraction : Mixed Fractions Sub – L4 SP Main Index Addition Subtraction Try these yourselfMultiplication Division 11/2 - 9/10 = 3/5 11/3 - 14/ 15 = 6/15 Conversion Memorise 13/4 - 11/4 = 1/2 11/3 - 11/9 = 2/9 Manipulate Estimate 21/8 - 19/10 = 9/40 85/9 - 24/7 = 562/63 Definitions Self-test Previous End of topic Last viewed Choose another from the sidebar
  • 32. Multi - deciderMultiplication : Choose a level Main Index Addition SubtractionMultiplication Division Easiest Basics Conversion Memorise Cross cancellation Manipulate Estimate Integers Definitions Self-test Hardest Mixed fractions Previous Last viewed
  • 33. Multi – L1Multiplication : Basics Main Index Addition Subtraction Simply multiply the tops andMultiplication multiply the bottoms Division Conversion Memorise Manipulate Estimate Definitions Fractions should always be scaled Self-test down if possible. Previous Last viewed Solved problems
  • 34. Multiplication : Basics Main Index Multi – L1 SP Addition SubtractionMultiplication Division Conversion Try these yourself Memorise 3/ x 2/5 = 6/35 Manipulate 7 Estimate 2/ x 8/9 = 16/81 Definitions 9 Self-test 3/ x 3/10 = 9/100 10 Previous Last viewed Next level
  • 35. Multiplication : Cross-cancellation Multi – L2 cross Main Index It‟s possible to scale Addition down across different Subtraction fractions.Multiplication Division But only for multiplication! Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Why does this work?
  • 36. Multiplication : Cross-cancellation Multi – L2 cross - why Main Index Cross-cancellation works because Addition Subtraction produces the same result asMultiplication Division The numerators can be swapped Conversion without upsetting the calculation. Memorise Manipulate Estimate In that case, you can scale down any numerator with any denominator wherever fractions are being Definitions multiplied. Self-test Previous Last viewed Solved problems
  • 37. Multiplication : Cross-cancellation Multi – L2 SP Main Index Addition SubtractionMultiplication Try these yourself Division 13/ x 7 /5 = 13/ 8/ x 45/ = 6 /5 Conversion 7 5 25 12 Memorise 20/ x 18/ = 40/ 17/ x 30/ = 6 /5 Manipulate 9 7 7 25 17 Estimate Definitions 5/ x 3/10 = 1/4 19/ x 22/ = 2 6 11 19 Self-test Previous Last viewed Next level
  • 38. Multiplication : Fraction of an integer Multi - L3 integers Main Index Addition The word „of‟ always means multiply Subtraction Turn the integer into a fraction by writing a „1‟ under it.Multiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 39. Multiplication : Fraction of an integer Multi – L3 SP Main Index Addition SubtractionMultiplication Try these yourself Division 2/ of $105 = $70 7/ of $600 = $140 Conversion 3 30 Memorise Manipulate 3/ of 105 m = 63 m 9/ of $15 = $13.50 5 10 Estimate Definitions 1/ 3/ 8 of $176 = $22 8 of 40% = 15% Self-test Previous Last viewed Next level
  • 40. Multiplication : Mixed fractions Multi – L4 mixed Main Index Addition SubtractionMultiplication Mixed factions have to be converted to top- Division heavy fractions before they can be multiplied. Conversion Memorise Manipulate Estimate And at the end of the calculation, the fraction Definitions should be turned back into mixed form. Self-test Previous Last viewed Solved problems
  • 41. Multiplication : Mixed fractions Multi – L4 SP Main Index Addition Subtraction Try these yourselfMultiplication Division 13/ 7 x 7/5 = 13/5 8/ 25 x 45/12 = 6/5 Conversion Memorise 20/ x 18/7 = 40/7 17/ 30/ = 6 /5 Manipulate 9 25 x 17 Estimate 5/ x 3/10 = 1/4 19/ x 22/19 = 2 Definitions 6 11 Self-test Previous End of topic Last viewed Choose another from the sidebar
  • 42. Division : Choose a level Div - decider Main Index Addition SubtractionMultiplication Division Easiest Basics Conversion Memorise Mixed fractions Manipulate Estimate Hardest Integers as Definitions fractions Self-test Previous Last viewed
  • 43. Division : Basics Div – L1 EG1 Main Index Addition Tip the back fraction upside Subtraction down and multiply instead ofMultiplication divide. Division Conversion Memorise Manipulate Estimate Definitions Convert the final answer into Self-test a mixed fraction if possible. Previous Last viewed More examples
  • 44. Division : Basics Div – L1 EG2 Main Index Addition SubtractionMultiplication Division Cross cancel Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 45. Division : Basics Div – L1 SP Main Index Addition Subtraction Try these yourselfMultiplication 2/ 2/ = 12/5 1/ 3/ = 1 /3 Division 5 7 4 4 Conversion Memorise 4/ 2/ 9 15 = 12/3 9/ 10 3/ 20 = 11/2 Manipulate Estimate 45/ 3/ = 3 4/ 2/ = 2/ Definitions 100 20 15 5 3 Self-test Previous Last viewed Next level
  • 46. Div – L2 – MixedDivision : Mixed fractions fractions Main Index Addition Mixed fractions have to be Subtraction converted to top-heavy form firstMultiplication Division Conversion Memorise Manipulate Turn the division into a multiplication Estimate Definitions Self-test Previous Last viewed Solved problems
  • 47. Division : Mixed Fractions Div – L2 SP Main Index Addition SubtractionMultiplication Try these yourself Division Conversion 11/2 2/ 3 = 21/4 1/ 5 12/3 = 3/25 Memorise Manipulate 1 3/ 4 1/ = 7 41/2 3/ = 6 4 4 Estimate Definitions 11/6 2/ 3 = 13/4 1/ 2 22/3 = 3/16 Self-test Previous Last viewed Next level
  • 48. Division : Integers Div – L3 – integers Main Index Addition Awkward divisions can Subtraction often be solved easily byMultiplication using fraction techniques. Division Conversion Memorise Scale down Manipulate Estimate Definitions Convert to mixed fraction Self-test Convert to decimal Previous Last viewed More examples
  • 49. Division : Integers Div – L3 - 2 Main Index Set the division Addition up as a fraction SubtractionMultiplication Division Conversion scale down Memorise Manipulate Estimate Convert to a Definitions mixed fraction Self-test Convert to decimal Previous Last viewed Solved problems
  • 50. Division : Integers Div – L3 - SP Main Index Addition SubtractionMultiplication Try these yourself Division Conversion 150 100 = 1.5 125 75 = 1.67 Memorise Manipulate 150 90 = 1.67 200 15 = 13.33 Estimate 150 120 = 1.25 256 48 = 5.33 Definitions Self-test Previous End of topic Last viewed Choose another topic from the sidebar
  • 51. Memorise : Why memorise? Mem – why Main Index Asking kids to memorise isn‟t as fashionable as it used to Addition be. Grown-ups act as if asking kids to memorise basic Subtraction mathematical facts is a form of torture, even though kidsMultiplication memorise the alphabet, telephone numbers, the route to school and even the names of their parents and siblings! Division Kids memorise whether you want them to or not, so why Conversion not let that include a handful of basic mathematical Memorise relationships?A very small set of remembered facts can Manipulate boost a child‟s mathematical skill tenfold, in my opinion. That increase in skill will manifest not just in terms of Estimate what a child can do but in how quickly and how easily Definitions the child can do it. The ultimate objective in maths is to Self-test make the job easy. With the same investment of time and energy that it takes to learn to use a calculator, a child can learn to calculate in his head and never need a Previous calculator again. Last viewed Begin tour
  • 52. Memorise : Beginners level Mem – set 1 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next level up
  • 53. Memorise : Junior level Mem – set 2 - thirds Main Index These numbers aren‟t Addition so commonly used. Subtraction But see how easyMultiplication they are to remember! Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Intermediate level
  • 54. Memorise : Intermediate level Mem – set 3 - eights Main Index Addition SubtractionMultiplication Can you see the Division pattern in the numbers? Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Expert level
  • 55. Memorise : Expert level Mem – sevenths Main Index Addition Subtraction 14 Except for the last digit, the stringMultiplication is made up of multiples of 7. Division 28 Conversion 56 Memorise Manipulate Estimate 7= 0.285714… 2/ All other sevenths are made from Definitions the same repeating string, starting 7= 0.428571… 3/ Self-test from a different part of the string. 7= 0.571428… 4/ 7= 0.714285… 5/ Previous 7= 0.857142… 6/ Last viewed Snob level
  • 56. Memorise : Snob level Mem - elevenths Main Index 1/ 11 = 0.090909… This set is hardly Addition important but the 2/ 11 = 0.181818… Subtraction numbers are easyMultiplication 3/ 11 = 0.272727… (if you know the nine times table) Division 4/ 11 = 0.363636… and they look Conversion 5/ 11 = 0.454545… impressive. Memorise 6/ 11 = 0.545454… Manipulate 7/ 11 = 0.636363… Estimate 8/ 11 = 0.727272… Definitions Self-test 9/ 11 = 0.818181… 10/ 11 = 0.909090… Previous Last viewed Summary
  • 57. Memorise : Summary Mem – summary Main Index Nearly all other decimals can be calculated or Addition estimated from the few that you have memorised. Subtraction For example….Multiplication 1/ 6 = half of 1/3 = 0.1666… Division Conversion 1/ 12 = half of 1/6 = 0.08333… Memorise Manipulate 7/ 12 = ½ + 1/12 = 0.58333… Estimate 5/ = 10/ = 10 x 1/12 = 0.8333 Definitions 6 12 Self-test 1/ 15 = 2/ 30 = 0.06666…. Previous End of topic Last viewed Choose another topic from the sidebar
  • 58. Manipulate fractions : Intro Man - intro Main Index By „manipulating‟ a fraction, I mean breaking it apart Addition and reforming it in such a way that hard calculations Subtraction become easy, easy enough to do in your head.Multiplication Division Before commencing, be sure that you have Conversion memorised most or all of the transformations in Memorise the „memorising fractions‟ lesson. If you haven‟t, Manipulate go there now. Otherwise none of this will make any sense to you. Estimate Definitions Self-test Previous Last viewed Begin
  • 59. Manipulating numbers with halves Man - halves Main Index Addition SubtractionMultiplication Think: 2.5 is Alternatively, it may be Division half of five. easier simply to add the numbers two and a half Conversion times. Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next
  • 60. Manipulating numbers with halves Man - halves - 2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next
  • 61. Manipulating numbers with thirds Man - thirds Main Index Any percentage between about 30% and 35% can be Addition approximated by the fraction 1/3 without introducing much error. SubtractionMultiplication Division Conversion We can improve on the estimate by subtracting 1% of the original number. Memorise Manipulate Estimate Definitions This is still wrong by one third of Self-test one percent (i.e., one third of $18). Previous Last viewed Next
  • 62. Manipulating numbers with quarters Man - quarters Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next
  • 63. Manipulating numbers with fifths Man - fifths Main Index Addition This means that any number divided by five is the number doubled and Subtraction divided by ten.Multiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Using the same logic, multiplying a number by 5 means halving it and putting a zero on the back. Previous Last viewed Next
  • 64. Manipulating numbers with eighths Man - eighths Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next
  • 65. Manipulating other numbers Man - other numbers Main Index Addition 35% of $1200 Subtraction = (25% + 10%) of $1200Multiplication = $300 + $120 Division = $420 Conversion Memorise Manipulate Estimate End of topic Definitions Self-test Choose another topic from the Previous sidebar Last viewed
  • 66. Estimation : Intro est - intro Main Index In this section you‟ll learn how to replace an Addition awkward fraction (or decimal) with an easy Subtraction fraction that is more or less the same size. ByMultiplication doing this you will introduce a tiny amount of Division error (which may or may not be a problem) but you will dramatically reduce the complexity of Conversion a problem. Memorise Make sure you have memorised most or all of Manipulate the transforms listed in the „memorise‟ section Estimate before coming here. This section will make no Definitions sense to you otherwise. Self-test Previous Last viewed Begin
  • 67. Estimation by rounding est 1 – from frac Main Index Addition Subtraction One strategy is to remove theMultiplication last digit of both the numerator and the denominator. Division Conversion Memorise Manipulate Estimate (The exact answer is $2038) Definitions Self-test Previous Last viewed Next
  • 68. Estimation of percentages Est 2 – from perc Main Index Addition Another strategy – one that works for percentages and decimals – is to find a fraction that closely matches the Subtraction percentage or decimal.Multiplication Division 2/ (67%) Conversion 3 Memorise Manipulate 7/ 68% 10 (70%) Estimate Definitions Self-test 5/ (71%) 7 Previous Last viewed Which one do you think is the best? Answer
  • 69. Estimation of percentages Est 3 – examples Main Index The truth is, there is no „best‟ replacement. Addition Subtraction Which replacement you choose depends on which number you need to to multiply the fraction against.Multiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Next
  • 70. Estimation : Challenge Est challenge Main Index Match each percentage with a fraction Addition that is roughly the same size SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test End of topic Previous Choose another topic from the sidebar Last viewed
  • 71. Conv – deciderConversions : Choose a topic Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed
  • 72. Conversions : Fractions to decimals Conv – frac to dec Main Index Decimals are fractions where the only allowable Addition denominators are 10, 100, 1000, 10000, . SubtractionMultiplication Division Therefore to convert a Conversion fraction into a decimal, you must scale it up until Memorise the denominator becomes Manipulate one of those numbers Estimate Definitions Self-test Previous Last viewed Another technique
  • 73. Conv – frac to dec – 2Conversions : Fractions to Decimals by dividing Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test (Most people „round off‟after two or three digits) Previous Last viewed Solved problems
  • 74. Conversions : Fractions to Decimals Conv – frac to dec - SP Main Index Addition Subtraction Try these yourselfMultiplication ¼ = 0.25 1/ 9 = 0.111… Division Conversion ¾ = 0.75 1/ 6 = 0.166… Memorise Manipulate 1/ 3 = 0.333… 1/ 11 = 0.0909… Estimate Definitions 7/ 15 = 0.466… 8/ 13 = 0.615… Self-test Previous End of topic Last viewed Choose a new topic from the sidebar
  • 75. Conversions : Decimals to Fractions Conv – dec to frac Main Index Addition Scale up by 10 or 100 Subtraction or 1000 until theMultiplication decimal point vanishes. Division Conversion Memorise Manipulate Estimate Definitions In short, count the number of digits in the decimal, and Self-test write the same number of zeroes in the denominator. Previous Last viewed More examples
  • 76. Conversions : Decimals to Fractions Conv – dec to frac - 2 Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 77. Conversions : Decimals to Fractions Conv – dec to frac - SP Main Index Addition Subtraction Try these yourselfMultiplication Division 0.375 = 3/8 0.15 = 3/20 Conversion Memorise 0.05 = 1/20 1.12 = 13/25 Manipulate Estimate 0.555 = 111/200 2.65 = 213/20 Definitions Self-test End of topic Previous Choose another topic from the sidebar Last viewed
  • 78. Conversions : Fractions to Percentages Conv – frac to perc Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Round off Previous Last viewed Next example
  • 79. Conversions : Fractions to Percentages Conv – frac to perc – divide Main Index Addition SubtractionMultiplication Division Conversion Convert to decimal Memorise Manipulate Estimate Definitions Self-test Round off Previous Last viewed Solved problems
  • 80. Conversions : Fractions to Percentages Conv – frac to perc - SP Main Index Addition Subtraction Try these yourselfMultiplication Division 3/ = 38% 2/ = 28% 8 7 Conversion Memorise 4/ = 44% 5/ = 83% 9 6 Manipulate Estimate 2/ 4/ 11 = 9% 5 = 80% Definitions Self-test End of topic Previous Choose another topic from the sidebar Last viewed
  • 81. Conversions : Percentages to Fractions Conv – perc to frac Main Index Addition The denominator will always be 100 Subtraction (until you scale down).Multiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Previous Last viewed Solved problems
  • 82. Conv – perc to frac -Conversions : Percentages to Fractions SP Main Index Addition Subtraction Try these yourselfMultiplication Division 90% = 9/10 5% = 1/20 Conversion Memorise 36% = 9/25 12% = 3/25 Manipulate Estimate Definitions 35% = 7/20 12.5% = 1/8 Self-test End of topic Previous Choose another topic from the sidebar Last viewed
  • 83. Conversions : Top-heavy to Mixed fractions Conv – TH to MX Main Index Addition SubtractionMultiplication Division Conversion 5 means “22 divided by 5.” 22/ Memorise Manipulate Estimate Definitions Self-test Previous Last viewed More examples
  • 84. Conversions : Top-heavy to Mixed fractions Conv – TH to MX – 2 Main Index Addition Subtraction When the numbers in theMultiplication fraction are similar in size, Division you know that the integer Conversion part will be a 1. Memorise Manipulate Therefore, to get the new numerator, simply subtract: Estimate 13 – 9 = 4 Definitions Self-test Previous Last viewed Solved problems
  • 85. Conversions : Top-heavy to Mixed fractions Conv – TH to MX - SP Main Index Addition SubtractionMultiplication Try these yourself Division 22/ = 31/7 20/ = 62/3 Conversion 7 3 Memorise 42/ = 42/3 45/ = 5 Manipulate 9 9 Estimate 13/ = 16/7 25/ = 21/2 7 10 Definitions Self-test End of topic Previous Choose a new topic from the sidebar Last viewed
  • 86. Conversions : Mixed to Top-heavy fractions Conv – MX to TH Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Begin with the denominator. (3) Estimate Multiply by the integer. (3x2 = 6) Definitions Then add on the numerator. (6+ 2 = 8) Self-test This creates the new numerator The denominator stays the same. Previous Last viewed Different explanation
  • 87. Conversions : Mixed to Top-heavy fractions Conv – MX to TH - pic Main Index Addition SubtractionMultiplication Division Conversion = 8/3 Memorise Manipulate Estimate How many thirds can you see? Definitions Self-test Previous Last viewed Solved problems
  • 88. Conversions : Mixed to Top-heavy fractions Conv – MX to TH - SP Main Index Addition Try these yourself SubtractionMultiplication 31/8 = 25/ 8 21/10 = 21/ 10 Division Conversion Memorise 18/9 = 17/ 9 91/11 = 100/ 11 Manipulate Estimate 51/2 = 11/ 2 11/5 = 6/5 Definitions Self-test Previous End of topic. Last viewed Choose another topic from the sidebar
  • 89. Conversions : Scaling up Conv - Sc up Main Index Addition Scaling up means multiplying Subtraction the numbers in a fraction byMultiplication something to make them larger. Division 6/ 9 is the same size as 2/3. Conversion Memorise Manipulate Estimate Scaling up is like cutting the Definitions fraction into smaller pieces Self-test Why scale up? Previous Last viewed How do I know what to multiply by?
  • 90. Conversions : why scale up? Conv - Sc up - why Main Index Addition Scaling up is usually necessary when Subtraction fractions are to be added or subtracted.Multiplication Division Conversion For example, ½ has to be rewritten Memorise as 5/10 for it to be added to 1/10. Manipulate Estimate Definitions Self-test The fraction itself has not grown any bigger or smaller, even though the numbers have changed. Previous Last viewed
  • 91. Conversions : Knowing what to multiply by Conv – sc up - Knowing what to „scale up‟ by Main Index Addition Example: 1/ + 1/18 Subtraction 3Multiplication Division A „3‟ has to be turned into an ‟18‟ Conversion Think: “3 times what makes 18?” Memorise The scale factor has to be a 6. Manipulate Estimate Definitions Self-test Scaling up usually occurs in fraction addition and subtraction. Previous Last viewed
  • 92. Conversions : Scaling down Conv – sc dn Main Index Addition This means making the numbers in a Subtraction fraction smaller. You divide top andMultiplication bottom by some convenient number, Division called a scale factor. Conversion Memorise Manipulate Scaling down is like gluing the Estimate parts of a fraction together so that there are fewer of them. Definitions Self-test Previous Last viewed Why do this?
  • 93. Conversions : Why scale down? Conv – sc dn - why Main Index Addition Scaling down keeps the numbers in a calculation Subtraction small so that the calculation is easier to do.Multiplication Division Conversion Divide top and bottom Memorise numbers by „4‟ Manipulate Estimate Definitions Self-test Previous Last viewed How do I know what to divide by?
  • 94. Conversions : Finding a scale factor Conv – sc dn - Knowing what to „scale down‟ by Main Index Addition Subtraction Look for a number that dividesMultiplication both the top and bottom numbers. Division Conversion Memorise „3‟ will do. So will „2‟. Manipulate So will „6‟ and ‟12‟. Estimate Definitions Self-test Divide by any one of these numbers. Then divide again until the numbers can go no further. Previous Last viewed Scale factors
  • 95. Conversions : Fraction to picture Conv – frac to pic Main Index Draw a rectangle, Addition Then divide it into 5 parts Subtraction (ie, according to theMultiplication denominator). Division Then shade in (from the Conversion bottom) 3 parts (I.e, according to the numerator) Memorise Manipulate Being able to „draw‟ a fraction means knowing the Estimate size of a fraction. If you know roughly how big the Definitions fraction is, you can estimate how big an answer Self-test should be even before you do the formal calculation. For example, from this drawing you can see Previous that 3/5 is bigger than a half Last viewed Next example
  • 96. Conversions : Fraction to picture Conv – frac to pic – 2 Main Index Addition Subtraction Which fraction is bigger: ¾ or 5/7?Multiplication A quick sketch can tell you. Division Conversion Memorise Manipulate The fractions are close enough that you can Estimate use ¾ in place of 5/7 (to make the calculation Definitions easier) without introducing too much error. Self-test But then again, maybe 2/3 is closer to 5/7, and is easier still to work with. (Try it out). Previous Last viewed Next example
  • 97. Conversions : Fraction to picture Conv – frac to pic - 3 Main Index Addition SubtractionMultiplication Division Conversion Memorise It‟s not practical to „cut‟ the rectangle into 100 pieces. Manipulate However, we can approximate the fraction by ignoring the second digit of each number in the fraction. Estimate Definitions Self-test Therefore 63% of some number should be slightly larger than half of that number. Previous Last viewed End of topic
  • 98. Conversions : Picture to fraction Conv – pic to frac Main Index Addition What fraction is this? SubtractionMultiplication Division Conversion (Get a ruler and measure the lines yourself. Your measurements will 14 cm Memorise probably be different from mine, Manipulate but your fraction will end up being Estimate almost exactly the same as mine.) 8.5 cm Definitions Self-test Previous Last viewed Next example
  • 99. Conversions : Picture to fraction Conv – pic to frac – 2 Main Index Addition Even without a ruler you can estimate the Subtraction size of a fraction, as long as you have something to compare it against. The stackMultiplication of hollow boxes, for example.The fraction Division is approximately 1.5 boxes out of 3.75 Conversion boxes. Memorise Manipulate Estimate Definitions Self-test End of topic Previous Choose a new topic from the sidebar Last viewed
  • 100. Definitions : Index Defn - decider Main Index Addition Scale factors Subtraction NumeratorsMultiplication Division Denominators Conversion Integers Memorise Top-heavy fractions Manipulate Mixed fractions Estimate Decimals Definitions Percentages Self-test Rounding off Last viewed
  • 101. Definitions : Scale factors Defn - Scale factors Main Index Scale factors are numbers which turn one fraction into Addition another without changing the size of the fraction. SubtractionMultiplication Division Conversion For example ½ can be turned into 2/4 by Memorise doubling the top and bottom numbers. Manipulate The scale factor in that case is „2‟. Estimate Definitions ½ can be turned into 3/6 using a scale factor of „3‟. Self-test Scale factors multiply to make the numbers larger Previous or divide to make the numbers smaller. Last viewed
  • 102. Definitions : Numerators Defn - Numerators Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Numerators tell Estimate you how many pieces you have Definitions Self-test Previous Last viewed
  • 103. Definitions : Denominators Defn - Denominators Main Index Addition SubtractionMultiplication Division Conversion Memorise Manipulate Estimate Definitions Self-test Denominators tell you Large denominators Previous how small the pieces are mean small pieces Last viewed
  • 104. Definitions : Integers Defn – integers Main Index Addition Integers are the ordinary numbers we use for Subtraction counting.Multiplication Division Integers can be turned into fractions very easily, Conversion simply by writing a „1‟ underneath them Memorise Why would I want to do this? Manipulate Estimate Definitions Self-test Some top-heavy fractions can be written as a Previous mixture of integers and ordinary fractions. Last viewed
  • 105. Definitions : Top-heavy fractions Defn – TH fractions Main Index Addition Top heavy fractions are fractions in which the number on top is bigger than the number Subtraction underneath.Multiplication 21/ and 17/4 are both top-heavy fractions. They 5 Division can also be written in mixed form as 4 1/5 and Conversion 41/4. However, they are much easier to work with Memorise if they are in top-heavy form, especially in Manipulate multiplications and divisions. Estimate Top-heavy fractions are usually called „improper‟ fractions in the textbooks. In my opinion, this is a Definitions thoroughly unhelpful name because it suggests Self-test that there is something wrong with them. Previous Last viewed
  • 106. Definitions : Mixed fractions Defn – MX fractions Main Index Mixed fractions are fractions like 2 1/2 and 41/5, Addition that is fractions which have an integer in front Subtraction as well as a purely fractional part.Multiplication Fractions in mixed form are easier to visualise Division than the same fractions in top-heavy form but Conversion they are not so easy to do calculations with. Memorise Therefore fractions are not usually turned into Manipulate mixed form until the very end of the problem. Estimate Definitions Self-test Previous Last viewed
  • 107. Definitions : Decimals Defn – decimals Main Index Addition Decimals are numbers like 0.25, 1.125, 12.556 and 3.14159. They are identical to fractions which Subtraction have either 10, 100, 1000 (etc) written underneath.Multiplication 0.25 therefore is exactly the same as 25/100. Division Conversion Decimal notation is just another way of writing down Memorise fractions. Just as ideas can be expressed in English. Manipulate German and Japanese, numbers can be written as Estimate fractions, decimals and percentages. Each „language‟ has its advantages and disadvantages. For example, it Definitions is difficult to add and subtract fractions but easy to Self-test multiply and divide them. At the other extreme, adding and subtracting decimals is a piece of cake, but multiplying and dividing decimals usually Previous requires a piece of paper. Last viewed More
  • 108. Definitions : Decimals Defn – decimals 2 Main Index Addition Not surprisingly, there are ways to convert fractions Subtraction into decimals and decimals into fractions, and so on. You should master these so that you can takeMultiplication advantage of the strengths of each „language‟. Division For example, if you need to add two fractions, try Conversion converting both to decimals. If you want to multiply Memorise two decimals, try converting to fractions first. Manipulate Estimate Definitions Self-test Previous Last viewed
  • 109. Definitions : Percentages Defn – percentages Main Index A percentage is a decimal rounded (normally) to Addition two decimal places and treated as if it were an Subtraction integer. Doing so simply makes the number easierMultiplication to say and to visualise. Division Conversion For example, Memorise 1/ 3 = 0.333… = 33 1/3% 33%. Manipulate Estimate While it is easy to add and subtract percentages, Definitions most of the time when you see a percentage, it is being multiplied against an integer. The best way Self-test to do this is to turn the percentage into a fraction, because percentages are just plain lousy for multiplication. Previous Last viewed
  • 110. Definitions : Rounding off Defn – rounding off Main Index Addition Rounding a decimal number off means shortening it Subtraction by a few decimal places to make it easier to handle.Multiplication Doing so reduces the accuracy, which may or may Division not cause problems. Therefore you need to exercise a Conversion bit of common-world wisdom in deciding how many Memorise digits to remove. Manipulate Estimate For example 25.275 cm should be rounded to 25.3 cm Definitions because no-one except a lab technician deals with Self-test measurements more accurate than plus-or-minus one millimetre. Previous Last viewed
  • 111. Self test : Intro Self test start Main Index There are no answers to the questions on this self-test. Addition You either know how to get the answer or you don‟t. If Subtraction you have even the slightest doubt about your ability toMultiplication get the answer then you should do the lessons provided Division on the sidebar. Conversion Have a look at the questions. They cover just about Memorise every topic covered in this program. Manipulate Estimate Definitions Self-test Previous Begin test Last viewed
  • 112. Self test : Page 1 of 3 ST 1 – add sub mult Main Index 2/ + 2/7 = ?? 35/9 + 44/7 = ?? Addition 13 Subtraction 3/ 3/ 19/ x 22/ = ??Multiplication 25 + 10 = ?? 11 19 Division 3/ 10/ 2/ Conversion 28 + 1/14 = ?? 13 - 7 = ?? Memorise Manipulate 21/ 28 - 1/14 = ?? 85/9 - 24/7 = ?? Estimate 9/ 3/ 3/ 3/ 25 - 10 = ?? 10 x 10 = ?? Definitions Self-test 3/ of 40% = ?? 19/ x 22/ = ?? 8 11 19 Previous More Last viewed
  • 113. Self test : Page 2 of 3 ST 2 – frac dec perc Fill in the blanks Main Index Addition Fraction Decimal Percentage 2/ Subtraction 3Multiplication 0.2 Division 75% Conversion 1/ Memorise 9 Manipulate 0.1666… Estimate 85% 1/ Definitions 6 Self-test 0.0625 12.5% Previous More Last viewed
  • 114. Self test : Page 3 of 3 ST 3 – MX TH Main Index Fill in the blanks Addition Mixed Top heavy Subtraction 12/3Multiplication 101/ Division 4 Conversion 16/15 Memorise 20/ 5 Manipulate Estimate 67/8 100/ Definitions 9 Self-test 33/20 22/ 7 Previous Last viewed